From 500c5319f462f623cce4c6077aa2586206ac3bcf Mon Sep 17 00:00:00 2001 From: "Documenter.jl" Date: Thu, 3 Oct 2024 20:29:35 +0000 Subject: [PATCH] build based on dc667f2 --- previews/PR114/.documenter-siteinfo.json | 1 + previews/PR114/assets/cstr.svg | 101 ++ previews/PR114/assets/documenter.js | 1064 +++++++++++++++++ previews/PR114/assets/imc.svg | 216 ++++ previews/PR114/assets/logo-dark.svg | 58 + previews/PR114/assets/logo.svg | 58 + previews/PR114/assets/pendulum.svg | 72 ++ previews/PR114/assets/plot_controller.svg | 114 ++ previews/PR114/assets/plot_estimator.svg | 106 ++ previews/PR114/assets/plot_model.svg | 42 + previews/PR114/assets/readme_result.svg | 108 ++ .../PR114/assets/themes/catppuccin-frappe.css | 1 + .../PR114/assets/themes/catppuccin-latte.css | 1 + .../assets/themes/catppuccin-macchiato.css | 1 + .../PR114/assets/themes/catppuccin-mocha.css | 1 + .../PR114/assets/themes/documenter-dark.css | 7 + .../PR114/assets/themes/documenter-light.css | 9 + previews/PR114/assets/themeswap.js | 84 ++ previews/PR114/assets/warner.js | 52 + previews/PR114/func_index/index.html | 2 + previews/PR114/index.html | 2 + .../internals/predictive_control/index.html | 112 ++ previews/PR114/internals/sim_model/index.html | 2 + .../PR114/internals/state_estim/index.html | 215 ++++ previews/PR114/manual/installation/index.html | 2 + previews/PR114/manual/linmpc/index.html | 134 +++ previews/PR114/manual/mtk/index.html | 102 ++ previews/PR114/manual/nonlinmpc/index.html | 148 +++ previews/PR114/manual/plot10_NonLinMPC.svg | 82 ++ previews/PR114/manual/plot11_NonLinMPC.svg | 82 ++ previews/PR114/manual/plot1_LinMPC.svg | 118 ++ previews/PR114/manual/plot1_MTK.svg | 82 ++ previews/PR114/manual/plot1_NonLinMPC.svg | 39 + previews/PR114/manual/plot2_LinMPC.svg | 126 ++ previews/PR114/manual/plot2_MTK.svg | 82 ++ previews/PR114/manual/plot2_NonLinMPC.svg | 143 +++ previews/PR114/manual/plot3_LinMPC.svg | 124 ++ previews/PR114/manual/plot3_NonLinMPC.svg | 82 ++ previews/PR114/manual/plot4_LinMPC.svg | 128 ++ previews/PR114/manual/plot4_NonLinMPC.svg | 82 ++ previews/PR114/manual/plot5_NonLinMPC.svg | 82 ++ previews/PR114/manual/plot6_NonLinMPC.svg | 84 ++ previews/PR114/manual/plot7_NonLinMPC.svg | 84 ++ previews/PR114/manual/plot8_NonLinMPC.svg | 82 ++ previews/PR114/manual/plot9_NonLinMPC.svg | 82 ++ previews/PR114/objects.inv | Bin 0 -> 2394 bytes previews/PR114/public/generic_func/index.html | 133 +++ previews/PR114/public/plot_sim/index.html | 47 + .../public/predictive_control/index.html | 105 ++ previews/PR114/public/sim_model/index.html | 116 ++ previews/PR114/public/state_estim/index.html | 119 ++ previews/PR114/search_index.js | 3 + previews/PR114/siteinfo.js | 1 + 53 files changed, 4923 insertions(+) create mode 100644 previews/PR114/.documenter-siteinfo.json create mode 100644 previews/PR114/assets/cstr.svg create mode 100644 previews/PR114/assets/documenter.js create mode 100644 previews/PR114/assets/imc.svg create mode 100644 previews/PR114/assets/logo-dark.svg create mode 100644 previews/PR114/assets/logo.svg create mode 100644 previews/PR114/assets/pendulum.svg create mode 100644 previews/PR114/assets/plot_controller.svg create mode 100644 previews/PR114/assets/plot_estimator.svg create mode 100644 previews/PR114/assets/plot_model.svg create mode 100644 previews/PR114/assets/readme_result.svg create mode 100644 previews/PR114/assets/themes/catppuccin-frappe.css create mode 100644 previews/PR114/assets/themes/catppuccin-latte.css create mode 100644 previews/PR114/assets/themes/catppuccin-macchiato.css create mode 100644 previews/PR114/assets/themes/catppuccin-mocha.css create mode 100644 previews/PR114/assets/themes/documenter-dark.css create mode 100644 previews/PR114/assets/themes/documenter-light.css create mode 100644 previews/PR114/assets/themeswap.js create mode 100644 previews/PR114/assets/warner.js create mode 100644 previews/PR114/func_index/index.html create mode 100644 previews/PR114/index.html create mode 100644 previews/PR114/internals/predictive_control/index.html create mode 100644 previews/PR114/internals/sim_model/index.html create mode 100644 previews/PR114/internals/state_estim/index.html create mode 100644 previews/PR114/manual/installation/index.html create mode 100644 previews/PR114/manual/linmpc/index.html create mode 100644 previews/PR114/manual/mtk/index.html create mode 100644 previews/PR114/manual/nonlinmpc/index.html create mode 100644 previews/PR114/manual/plot10_NonLinMPC.svg create mode 100644 previews/PR114/manual/plot11_NonLinMPC.svg create mode 100644 previews/PR114/manual/plot1_LinMPC.svg create mode 100644 previews/PR114/manual/plot1_MTK.svg create mode 100644 previews/PR114/manual/plot1_NonLinMPC.svg create mode 100644 previews/PR114/manual/plot2_LinMPC.svg create mode 100644 previews/PR114/manual/plot2_MTK.svg create mode 100644 previews/PR114/manual/plot2_NonLinMPC.svg create mode 100644 previews/PR114/manual/plot3_LinMPC.svg create mode 100644 previews/PR114/manual/plot3_NonLinMPC.svg create mode 100644 previews/PR114/manual/plot4_LinMPC.svg create mode 100644 previews/PR114/manual/plot4_NonLinMPC.svg create mode 100644 previews/PR114/manual/plot5_NonLinMPC.svg create mode 100644 previews/PR114/manual/plot6_NonLinMPC.svg create mode 100644 previews/PR114/manual/plot7_NonLinMPC.svg create mode 100644 previews/PR114/manual/plot8_NonLinMPC.svg create mode 100644 previews/PR114/manual/plot9_NonLinMPC.svg create mode 100644 previews/PR114/objects.inv create mode 100644 previews/PR114/public/generic_func/index.html create mode 100644 previews/PR114/public/plot_sim/index.html create mode 100644 previews/PR114/public/predictive_control/index.html create mode 100644 previews/PR114/public/sim_model/index.html create mode 100644 previews/PR114/public/state_estim/index.html create mode 100644 previews/PR114/search_index.js create mode 100644 previews/PR114/siteinfo.js diff --git a/previews/PR114/.documenter-siteinfo.json b/previews/PR114/.documenter-siteinfo.json new file mode 100644 index 00000000..251fdea5 --- /dev/null +++ b/previews/PR114/.documenter-siteinfo.json @@ -0,0 +1 @@ +{"documenter":{"julia_version":"1.10.5","generation_timestamp":"2024-10-03T20:29:30","documenter_version":"1.7.0"}} \ No newline at end of file diff --git a/previews/PR114/assets/cstr.svg b/previews/PR114/assets/cstr.svg new file mode 100644 index 00000000..f3f24964 --- /dev/null +++ b/previews/PR114/assets/cstr.svg @@ -0,0 +1,101 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/previews/PR114/assets/documenter.js b/previews/PR114/assets/documenter.js new file mode 100644 index 00000000..82252a11 --- /dev/null +++ b/previews/PR114/assets/documenter.js @@ -0,0 +1,1064 @@ +// Generated by Documenter.jl +requirejs.config({ + paths: { + 'highlight-julia': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia.min', + 'headroom': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/headroom.min', + 'jqueryui': 'https://cdnjs.cloudflare.com/ajax/libs/jqueryui/1.13.2/jquery-ui.min', + 'katex-auto-render': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/contrib/auto-render.min', + 'jquery': 'https://cdnjs.cloudflare.com/ajax/libs/jquery/3.7.0/jquery.min', + 'headroom-jquery': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/jQuery.headroom.min', + 'katex': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min', + 'highlight': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/highlight.min', + 'highlight-julia-repl': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia-repl.min', + }, + shim: { + "highlight-julia": { + "deps": [ + "highlight" + ] + }, + "katex-auto-render": { + "deps": [ + "katex" + ] + }, + "headroom-jquery": { + "deps": [ + "jquery", + "headroom" + ] + }, + "highlight-julia-repl": { + "deps": [ + "highlight" + ] + } +} +}); +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'katex', 'katex-auto-render'], function($, katex, renderMathInElement) { +$(document).ready(function() { + renderMathInElement( + document.body, + { + "delimiters": [ + { + "left": "$", + "right": "$", + "display": false + }, + { + "left": "$$", + "right": "$$", + "display": true + }, + { + "left": "\\[", + "right": "\\]", + "display": true + } + ] +} + + ); +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'highlight', 'highlight-julia', 'highlight-julia-repl'], function($) { +$(document).ready(function() { + hljs.highlightAll(); +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +let timer = 0; +var isExpanded = true; + +$(document).on( + "click", + ".docstring .docstring-article-toggle-button", + function () { + let articleToggleTitle = "Expand docstring"; + const parent = $(this).parent(); + + debounce(() => { + if (parent.siblings("section").is(":visible")) { + parent + .find("a.docstring-article-toggle-button") + .removeClass("fa-chevron-down") + .addClass("fa-chevron-right"); + } else { + parent + .find("a.docstring-article-toggle-button") + .removeClass("fa-chevron-right") + .addClass("fa-chevron-down"); + + articleToggleTitle = "Collapse docstring"; + } + + parent + .children(".docstring-article-toggle-button") + .prop("title", articleToggleTitle); + parent.siblings("section").slideToggle(); + }); + } +); + +$(document).on("click", ".docs-article-toggle-button", function (event) { + let articleToggleTitle = "Expand docstring"; + let navArticleToggleTitle = "Expand all docstrings"; + let animationSpeed = event.noToggleAnimation ? 0 : 400; + + debounce(() => { + if (isExpanded) { + $(this).removeClass("fa-chevron-up").addClass("fa-chevron-down"); + $("a.docstring-article-toggle-button") + .removeClass("fa-chevron-down") + .addClass("fa-chevron-right"); + + isExpanded = false; + + $(".docstring section").slideUp(animationSpeed); + } else { + $(this).removeClass("fa-chevron-down").addClass("fa-chevron-up"); + $("a.docstring-article-toggle-button") + .removeClass("fa-chevron-right") + .addClass("fa-chevron-down"); + + isExpanded = true; + articleToggleTitle = "Collapse docstring"; + navArticleToggleTitle = "Collapse all docstrings"; + + $(".docstring section").slideDown(animationSpeed); + } + + $(this).prop("title", navArticleToggleTitle); + $(".docstring-article-toggle-button").prop("title", articleToggleTitle); + }); +}); + +function debounce(callback, timeout = 300) { + if (Date.now() - timer > timeout) { + callback(); + } + + clearTimeout(timer); + + timer = Date.now(); +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require([], function() { +function addCopyButtonCallbacks() { + for (const el of document.getElementsByTagName("pre")) { + const button = document.createElement("button"); + button.classList.add("copy-button", "fa-solid", "fa-copy"); + button.setAttribute("aria-label", "Copy this code block"); + button.setAttribute("title", "Copy"); + + el.appendChild(button); + + const success = function () { + button.classList.add("success", "fa-check"); + button.classList.remove("fa-copy"); + }; + + const failure = function () { + button.classList.add("error", "fa-xmark"); + button.classList.remove("fa-copy"); + }; + + button.addEventListener("click", function () { + copyToClipboard(el.innerText).then(success, failure); + + setTimeout(function () { + button.classList.add("fa-copy"); + button.classList.remove("success", "fa-check", "fa-xmark"); + }, 5000); + }); + } +} + +function copyToClipboard(text) { + // clipboard API is only available in secure contexts + if (window.navigator && window.navigator.clipboard) { + return window.navigator.clipboard.writeText(text); + } else { + return new Promise(function (resolve, reject) { + try { + const el = document.createElement("textarea"); + el.textContent = text; + el.style.position = "fixed"; + el.style.opacity = 0; + document.body.appendChild(el); + el.select(); + document.execCommand("copy"); + + resolve(); + } catch (err) { + reject(err); + } finally { + document.body.removeChild(el); + } + }); + } +} + +if (document.readyState === "loading") { + document.addEventListener("DOMContentLoaded", addCopyButtonCallbacks); +} else { + addCopyButtonCallbacks(); +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'headroom', 'headroom-jquery'], function($, Headroom) { + +// Manages the top navigation bar (hides it when the user starts scrolling down on the +// mobile). +window.Headroom = Headroom; // work around buggy module loading? +$(document).ready(function () { + $("#documenter .docs-navbar").headroom({ + tolerance: { up: 10, down: 10 }, + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +$(document).ready(function () { + let meta = $("div[data-docstringscollapsed]").data(); + + if (meta?.docstringscollapsed) { + $("#documenter-article-toggle-button").trigger({ + type: "click", + noToggleAnimation: true, + }); + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +/* +To get an in-depth about the thought process you can refer: https://hetarth02.hashnode.dev/series/gsoc + +PSEUDOCODE: + +Searching happens automatically as the user types or adjusts the selected filters. +To preserve responsiveness, as much as possible of the slow parts of the search are done +in a web worker. Searching and result generation are done in the worker, and filtering and +DOM updates are done in the main thread. The filters are in the main thread as they should +be very quick to apply. This lets filters be changed without re-searching with minisearch +(which is possible even if filtering is on the worker thread) and also lets filters be +changed _while_ the worker is searching and without message passing (neither of which are +possible if filtering is on the worker thread) + +SEARCH WORKER: + +Import minisearch + +Build index + +On message from main thread + run search + find the first 200 unique results from each category, and compute their divs for display + note that this is necessary and sufficient information for the main thread to find the + first 200 unique results from any given filter set + post results to main thread + +MAIN: + +Launch worker + +Declare nonconstant globals (worker_is_running, last_search_text, unfiltered_results) + +On text update + if worker is not running, launch_search() + +launch_search + set worker_is_running to true, set last_search_text to the search text + post the search query to worker + +on message from worker + if last_search_text is not the same as the text in the search field, + the latest search result is not reflective of the latest search query, so update again + launch_search() + otherwise + set worker_is_running to false + + regardless, display the new search results to the user + save the unfiltered_results as a global + update_search() + +on filter click + adjust the filter selection + update_search() + +update_search + apply search filters by looping through the unfiltered_results and finding the first 200 + unique results that match the filters + + Update the DOM +*/ + +/////// SEARCH WORKER /////// + +function worker_function(documenterSearchIndex, documenterBaseURL, filters) { + importScripts( + "https://cdn.jsdelivr.net/npm/minisearch@6.1.0/dist/umd/index.min.js" + ); + + let data = documenterSearchIndex.map((x, key) => { + x["id"] = key; // minisearch requires a unique for each object + return x; + }); + + // list below is the lunr 2.1.3 list minus the intersect with names(Base) + // (all, any, get, in, is, only, which) and (do, else, for, let, where, while, with) + // ideally we'd just filter the original list but it's not available as a variable + const stopWords = new Set([ + "a", + "able", + "about", + "across", + "after", + "almost", + "also", + "am", + "among", + "an", + "and", + "are", + "as", + "at", + "be", + "because", + "been", + "but", + "by", + "can", + "cannot", + "could", + "dear", + "did", + "does", + "either", + "ever", + "every", + "from", + "got", + "had", + "has", + "have", + "he", + "her", + "hers", + "him", + "his", + "how", + "however", + "i", + "if", + "into", + "it", + "its", + "just", + "least", + "like", + "likely", + "may", + "me", + "might", + "most", + "must", + "my", + "neither", + "no", + "nor", + "not", + "of", + "off", + "often", + "on", + "or", + "other", + "our", + "own", + "rather", + "said", + "say", + "says", + "she", + "should", + "since", + "so", + "some", + "than", + "that", + "the", + "their", + "them", + "then", + "there", + "these", + "they", + "this", + "tis", + "to", + "too", + "twas", + "us", + "wants", + "was", + "we", + "were", + "what", + "when", + "who", + "whom", + "why", + "will", + "would", + "yet", + "you", + "your", + ]); + + let index = new MiniSearch({ + fields: ["title", "text"], // fields to index for full-text search + storeFields: ["location", "title", "text", "category", "page"], // fields to return with results + processTerm: (term) => { + let word = stopWords.has(term) ? null : term; + if (word) { + // custom trimmer that doesn't strip @ and !, which are used in julia macro and function names + word = word + .replace(/^[^a-zA-Z0-9@!]+/, "") + .replace(/[^a-zA-Z0-9@!]+$/, ""); + + word = word.toLowerCase(); + } + + return word ?? null; + }, + // add . as a separator, because otherwise "title": "Documenter.Anchors.add!", would not + // find anything if searching for "add!", only for the entire qualification + tokenize: (string) => string.split(/[\s\-\.]+/), + // options which will be applied during the search + searchOptions: { + prefix: true, + boost: { title: 100 }, + fuzzy: 2, + }, + }); + + index.addAll(data); + + /** + * Used to map characters to HTML entities. + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + const htmlEscapes = { + "&": "&", + "<": "<", + ">": ">", + '"': """, + "'": "'", + }; + + /** + * Used to match HTML entities and HTML characters. + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + const reUnescapedHtml = /[&<>"']/g; + const reHasUnescapedHtml = RegExp(reUnescapedHtml.source); + + /** + * Escape function from lodash + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + function escape(string) { + return string && reHasUnescapedHtml.test(string) + ? string.replace(reUnescapedHtml, (chr) => htmlEscapes[chr]) + : string || ""; + } + + /** + * RegX escape function from MDN + * Refer: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Guide/Regular_Expressions#escaping + */ + function escapeRegExp(string) { + return string.replace(/[.*+?^${}()|[\]\\]/g, "\\$&"); // $& means the whole matched string + } + + /** + * Make the result component given a minisearch result data object and the value + * of the search input as queryString. To view the result object structure, refer: + * https://lucaong.github.io/minisearch/modules/_minisearch_.html#searchresult + * + * @param {object} result + * @param {string} querystring + * @returns string + */ + function make_search_result(result, querystring) { + let search_divider = `
`; + let display_link = + result.location.slice(Math.max(0), Math.min(50, result.location.length)) + + (result.location.length > 30 ? "..." : ""); // To cut-off the link because it messes with the overflow of the whole div + + if (result.page !== "") { + display_link += ` (${result.page})`; + } + searchstring = escapeRegExp(querystring); + let textindex = new RegExp(`${searchstring}`, "i").exec(result.text); + let text = + textindex !== null + ? result.text.slice( + Math.max(textindex.index - 100, 0), + Math.min( + textindex.index + querystring.length + 100, + result.text.length + ) + ) + : ""; // cut-off text before and after from the match + + text = text.length ? escape(text) : ""; + + let display_result = text.length + ? "..." + + text.replace( + new RegExp(`${escape(searchstring)}`, "i"), // For first occurrence + '$&' + ) + + "..." + : ""; // highlights the match + + let in_code = false; + if (!["page", "section"].includes(result.category.toLowerCase())) { + in_code = true; + } + + // We encode the full url to escape some special characters which can lead to broken links + let result_div = ` + +
+
${escape(result.title)}
+
${result.category}
+
+

+ ${display_result} +

+
+ ${display_link} +
+ + ${search_divider} + `; + + return result_div; + } + + self.onmessage = function (e) { + let query = e.data; + let results = index.search(query, { + filter: (result) => { + // Only return relevant results + return result.score >= 1; + }, + combineWith: "AND", + }); + + // Pre-filter to deduplicate and limit to 200 per category to the extent + // possible without knowing what the filters are. + let filtered_results = []; + let counts = {}; + for (let filter of filters) { + counts[filter] = 0; + } + let present = {}; + + for (let result of results) { + cat = result.category; + cnt = counts[cat]; + if (cnt < 200) { + id = cat + "---" + result.location; + if (present[id]) { + continue; + } + present[id] = true; + filtered_results.push({ + location: result.location, + category: cat, + div: make_search_result(result, query), + }); + } + } + + postMessage(filtered_results); + }; +} + +// `worker = Threads.@spawn worker_function(documenterSearchIndex)`, but in JavaScript! +const filters = [ + ...new Set(documenterSearchIndex["docs"].map((x) => x.category)), +]; +const worker_str = + "(" + + worker_function.toString() + + ")(" + + JSON.stringify(documenterSearchIndex["docs"]) + + "," + + JSON.stringify(documenterBaseURL) + + "," + + JSON.stringify(filters) + + ")"; +const worker_blob = new Blob([worker_str], { type: "text/javascript" }); +const worker = new Worker(URL.createObjectURL(worker_blob)); + +/////// SEARCH MAIN /////// + +// Whether the worker is currently handling a search. This is a boolean +// as the worker only ever handles 1 or 0 searches at a time. +var worker_is_running = false; + +// The last search text that was sent to the worker. This is used to determine +// if the worker should be launched again when it reports back results. +var last_search_text = ""; + +// The results of the last search. This, in combination with the state of the filters +// in the DOM, is used compute the results to display on calls to update_search. +var unfiltered_results = []; + +// Which filter is currently selected +var selected_filter = ""; + +$(document).on("input", ".documenter-search-input", function (event) { + if (!worker_is_running) { + launch_search(); + } +}); + +function launch_search() { + worker_is_running = true; + last_search_text = $(".documenter-search-input").val(); + worker.postMessage(last_search_text); +} + +worker.onmessage = function (e) { + if (last_search_text !== $(".documenter-search-input").val()) { + launch_search(); + } else { + worker_is_running = false; + } + + unfiltered_results = e.data; + update_search(); +}; + +$(document).on("click", ".search-filter", function () { + if ($(this).hasClass("search-filter-selected")) { + selected_filter = ""; + } else { + selected_filter = $(this).text().toLowerCase(); + } + + // This updates search results and toggles classes for UI: + update_search(); +}); + +/** + * Make/Update the search component + */ +function update_search() { + let querystring = $(".documenter-search-input").val(); + + if (querystring.trim()) { + if (selected_filter == "") { + results = unfiltered_results; + } else { + results = unfiltered_results.filter((result) => { + return selected_filter == result.category.toLowerCase(); + }); + } + + let search_result_container = ``; + let modal_filters = make_modal_body_filters(); + let search_divider = `
`; + + if (results.length) { + let links = []; + let count = 0; + let search_results = ""; + + for (var i = 0, n = results.length; i < n && count < 200; ++i) { + let result = results[i]; + if (result.location && !links.includes(result.location)) { + search_results += result.div; + count++; + links.push(result.location); + } + } + + if (count == 1) { + count_str = "1 result"; + } else if (count == 200) { + count_str = "200+ results"; + } else { + count_str = count + " results"; + } + let result_count = `
${count_str}
`; + + search_result_container = ` +
+ ${modal_filters} + ${search_divider} + ${result_count} +
+ ${search_results} +
+
+ `; + } else { + search_result_container = ` +
+ ${modal_filters} + ${search_divider} +
0 result(s)
+
+
No result found!
+ `; + } + + if ($(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").removeClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(search_result_container); + } else { + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(` +
Type something to get started!
+ `); + } +} + +/** + * Make the modal filter html + * + * @returns string + */ +function make_modal_body_filters() { + let str = filters + .map((val) => { + if (selected_filter == val.toLowerCase()) { + return `${val}`; + } else { + return `${val}`; + } + }) + .join(""); + + return ` +
+ Filters: + ${str} +
`; +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Modal settings dialog +$(document).ready(function () { + var settings = $("#documenter-settings"); + $("#documenter-settings-button").click(function () { + settings.toggleClass("is-active"); + }); + // Close the dialog if X is clicked + $("#documenter-settings button.delete").click(function () { + settings.removeClass("is-active"); + }); + // Close dialog if ESC is pressed + $(document).keyup(function (e) { + if (e.keyCode == 27) settings.removeClass("is-active"); + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +$(document).ready(function () { + let search_modal_header = ` +
+
+

+ + + + +

+
+
+ +
+
+ `; + + let initial_search_body = ` +
Type something to get started!
+ `; + + let search_modal_footer = ` + + `; + + $(document.body).append( + ` + + ` + ); + + document.querySelector(".docs-search-query").addEventListener("click", () => { + openModal(); + }); + + document + .querySelector(".close-search-modal") + .addEventListener("click", () => { + closeModal(); + }); + + $(document).on("click", ".search-result-link", function () { + closeModal(); + }); + + document.addEventListener("keydown", (event) => { + if ((event.ctrlKey || event.metaKey) && event.key === "/") { + openModal(); + } else if (event.key === "Escape") { + closeModal(); + } + + return false; + }); + + // Functions to open and close a modal + function openModal() { + let searchModal = document.querySelector("#search-modal"); + + searchModal.classList.add("is-active"); + document.querySelector(".documenter-search-input").focus(); + } + + function closeModal() { + let searchModal = document.querySelector("#search-modal"); + let initial_search_body = ` +
Type something to get started!
+ `; + + searchModal.classList.remove("is-active"); + document.querySelector(".documenter-search-input").blur(); + + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".documenter-search-input").val(""); + $(".search-modal-card-body").html(initial_search_body); + } + + document + .querySelector("#search-modal .modal-background") + .addEventListener("click", () => { + closeModal(); + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Manages the showing and hiding of the sidebar. +$(document).ready(function () { + var sidebar = $("#documenter > .docs-sidebar"); + var sidebar_button = $("#documenter-sidebar-button"); + sidebar_button.click(function (ev) { + ev.preventDefault(); + sidebar.toggleClass("visible"); + if (sidebar.hasClass("visible")) { + // Makes sure that the current menu item is visible in the sidebar. + $("#documenter .docs-menu a.is-active").focus(); + } + }); + $("#documenter > .docs-main").bind("click", function (ev) { + if ($(ev.target).is(sidebar_button)) { + return; + } + if (sidebar.hasClass("visible")) { + sidebar.removeClass("visible"); + } + }); +}); + +// Resizes the package name / sitename in the sidebar if it is too wide. +// Inspired by: https://github.com/davatron5000/FitText.js +$(document).ready(function () { + e = $("#documenter .docs-autofit"); + function resize() { + var L = parseInt(e.css("max-width"), 10); + var L0 = e.width(); + if (L0 > L) { + var h0 = parseInt(e.css("font-size"), 10); + e.css("font-size", (L * h0) / L0); + // TODO: make sure it survives resizes? + } + } + // call once and then register events + resize(); + $(window).resize(resize); + $(window).on("orientationchange", resize); +}); + +// Scroll the navigation bar to the currently selected menu item +$(document).ready(function () { + var sidebar = $("#documenter .docs-menu").get(0); + var active = $("#documenter .docs-menu .is-active").get(0); + if (typeof active !== "undefined") { + sidebar.scrollTop = active.offsetTop - sidebar.offsetTop - 15; + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Theme picker setup +$(document).ready(function () { + // onchange callback + $("#documenter-themepicker").change(function themepick_callback(ev) { + var themename = $("#documenter-themepicker option:selected").attr("value"); + if (themename === "auto") { + // set_theme(window.matchMedia('(prefers-color-scheme: dark)').matches ? 'dark' : 'light'); + window.localStorage.removeItem("documenter-theme"); + } else { + // set_theme(themename); + window.localStorage.setItem("documenter-theme", themename); + } + // We re-use the global function from themeswap.js to actually do the swapping. + set_theme_from_local_storage(); + }); + + // Make sure that the themepicker displays the correct theme when the theme is retrieved + // from localStorage + if (typeof window.localStorage !== "undefined") { + var theme = window.localStorage.getItem("documenter-theme"); + if (theme !== null) { + $("#documenter-themepicker option").each(function (i, e) { + e.selected = e.value === theme; + }); + } + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// update the version selector with info from the siteinfo.js and ../versions.js files +$(document).ready(function () { + // If the version selector is disabled with DOCUMENTER_VERSION_SELECTOR_DISABLED in the + // siteinfo.js file, we just return immediately and not display the version selector. + if ( + typeof DOCUMENTER_VERSION_SELECTOR_DISABLED === "boolean" && + DOCUMENTER_VERSION_SELECTOR_DISABLED + ) { + return; + } + + var version_selector = $("#documenter .docs-version-selector"); + var version_selector_select = $("#documenter .docs-version-selector select"); + + version_selector_select.change(function (x) { + target_href = version_selector_select + .children("option:selected") + .get(0).value; + window.location.href = target_href; + }); + + // add the current version to the selector based on siteinfo.js, but only if the selector is empty + if ( + typeof DOCUMENTER_CURRENT_VERSION !== "undefined" && + $("#version-selector > option").length == 0 + ) { + var option = $( + "" + ); + version_selector_select.append(option); + } + + if (typeof DOC_VERSIONS !== "undefined") { + var existing_versions = version_selector_select.children("option"); + var existing_versions_texts = existing_versions.map(function (i, x) { + return x.text; + }); + DOC_VERSIONS.forEach(function (each) { + var version_url = documenterBaseURL + "/../" + each + "/"; + var existing_id = $.inArray(each, existing_versions_texts); + // if not already in the version selector, add it as a new option, + // otherwise update the old option with the URL and enable it + if (existing_id == -1) { + var option = $( + "" + ); + version_selector_select.append(option); + } else { + var option = existing_versions[existing_id]; + option.value = version_url; + option.disabled = false; + } + }); + } + + // only show the version selector if the selector has been populated + if (version_selector_select.children("option").length > 0) { + version_selector.toggleClass("visible"); + } +}); + +}) diff --git a/previews/PR114/assets/imc.svg b/previews/PR114/assets/imc.svg new file mode 100644 index 00000000..765bfc82 --- /dev/null +++ b/previews/PR114/assets/imc.svg @@ -0,0 +1,216 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/previews/PR114/assets/logo-dark.svg b/previews/PR114/assets/logo-dark.svg new file mode 100644 index 00000000..03b9cfa9 --- /dev/null +++ b/previews/PR114/assets/logo-dark.svg @@ -0,0 +1,58 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/previews/PR114/assets/logo.svg b/previews/PR114/assets/logo.svg new file mode 100644 index 00000000..651b92d1 --- /dev/null +++ b/previews/PR114/assets/logo.svg @@ -0,0 +1,58 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/previews/PR114/assets/pendulum.svg b/previews/PR114/assets/pendulum.svg new file mode 100644 index 00000000..dbc874eb --- /dev/null +++ b/previews/PR114/assets/pendulum.svg @@ -0,0 +1,72 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/previews/PR114/assets/plot_controller.svg b/previews/PR114/assets/plot_controller.svg new file mode 100644 index 00000000..5d488e7b --- /dev/null +++ b/previews/PR114/assets/plot_controller.svg @@ -0,0 +1,114 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + \ No newline at end of file diff --git a/previews/PR114/assets/plot_estimator.svg b/previews/PR114/assets/plot_estimator.svg new file mode 100644 index 00000000..61489639 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.button.is-black.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-black.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-black.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-frappe .button.is-black.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;box-shadow:none;color:#0a0a0a}html.theme--catppuccin-frappe .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-frappe .button.is-black.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-black.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-black.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-black.is-inverted.is-outlined.is-focused{background-color:#fff;color:#0a0a0a}html.theme--catppuccin-frappe .button.is-black.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-black.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-black.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-black.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--catppuccin-frappe .button.is-black.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-frappe .button.is-light{background-color:#f5f5f5;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-light:hover,html.theme--catppuccin-frappe .button.is-light.is-hovered{background-color:#eee;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-light:focus,html.theme--catppuccin-frappe .button.is-light.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-light:focus:not(:active),html.theme--catppuccin-frappe .button.is-light.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(245,245,245,0.25)}html.theme--catppuccin-frappe .button.is-light:active,html.theme--catppuccin-frappe .button.is-light.is-active{background-color:#e8e8e8;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-light[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-light{background-color:#f5f5f5;border-color:#f5f5f5;box-shadow:none}html.theme--catppuccin-frappe .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);color:#f5f5f5}html.theme--catppuccin-frappe 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.button.is-light.is-outlined.is-focused{background-color:#f5f5f5;border-color:#f5f5f5;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-light.is-outlined.is-loading::after{border-color:transparent transparent #f5f5f5 #f5f5f5 !important}html.theme--catppuccin-frappe .button.is-light.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-light.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-light.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-light.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-light.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-light.is-outlined{background-color:transparent;border-color:#f5f5f5;box-shadow:none;color:#f5f5f5}html.theme--catppuccin-frappe .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-light.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-light.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-light.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-light.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#f5f5f5}html.theme--catppuccin-frappe .button.is-light.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-light.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-light.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-light.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #f5f5f5 #f5f5f5 !important}html.theme--catppuccin-frappe 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kbd.button.is-inverted{background-color:#fff;color:#414559}html.theme--catppuccin-frappe .button.is-dark.is-inverted:hover,html.theme--catppuccin-frappe .content kbd.button.is-inverted:hover,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-hovered,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-frappe .button.is-dark.is-inverted[disabled],html.theme--catppuccin-frappe .content kbd.button.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-dark.is-inverted,fieldset[disabled] html.theme--catppuccin-frappe .content kbd.button.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#414559}html.theme--catppuccin-frappe .button.is-dark.is-loading::after,html.theme--catppuccin-frappe .content kbd.button.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-frappe 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!important}html.theme--catppuccin-frappe .button.is-dark.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .content kbd.button.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-dark.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .content kbd.button.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-dark.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .content kbd.button.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-dark.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-frappe .content kbd.button.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-frappe .button.is-dark.is-outlined[disabled],html.theme--catppuccin-frappe .content kbd.button.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-dark.is-outlined,fieldset[disabled] html.theme--catppuccin-frappe .content kbd.button.is-outlined{background-color:transparent;border-color:#414559;box-shadow:none;color:#414559}html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined.is-focused{background-color:#fff;color:#414559}html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #414559 #414559 !important}html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined[disabled],html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined,fieldset[disabled] html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-frappe .button.is-primary,html.theme--catppuccin-frappe .docstring>section>a.button.docs-sourcelink{background-color:#8caaee;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-primary:hover,html.theme--catppuccin-frappe .docstring>section>a.button.docs-sourcelink:hover,html.theme--catppuccin-frappe .button.is-primary.is-hovered,html.theme--catppuccin-frappe .docstring>section>a.button.is-hovered.docs-sourcelink{background-color:#81a2ec;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-primary:focus,html.theme--catppuccin-frappe 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.button.is-success{background-color:#a6d189;border-color:#a6d189;box-shadow:none}html.theme--catppuccin-frappe .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);color:#a6d189}html.theme--catppuccin-frappe .button.is-success.is-inverted:hover,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#a6d189}html.theme--catppuccin-frappe .button.is-success.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-success.is-outlined{background-color:transparent;border-color:#a6d189;color:#a6d189}html.theme--catppuccin-frappe .button.is-success.is-outlined:hover,html.theme--catppuccin-frappe .button.is-success.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-success.is-outlined:focus,html.theme--catppuccin-frappe .button.is-success.is-outlined.is-focused{background-color:#a6d189;border-color:#a6d189;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success.is-outlined.is-loading::after{border-color:transparent transparent #a6d189 #a6d189 !important}html.theme--catppuccin-frappe .button.is-success.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-success.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-success.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-success.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-success.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-success.is-outlined{background-color:transparent;border-color:#a6d189;box-shadow:none;color:#a6d189}html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#a6d189}html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #a6d189 #a6d189 !important}html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success.is-light{background-color:#f4f9f0;color:#446a29}html.theme--catppuccin-frappe .button.is-success.is-light:hover,html.theme--catppuccin-frappe .button.is-success.is-light.is-hovered{background-color:#edf6e7;border-color:transparent;color:#446a29}html.theme--catppuccin-frappe .button.is-success.is-light:active,html.theme--catppuccin-frappe .button.is-success.is-light.is-active{background-color:#e6f2de;border-color:transparent;color:#446a29}html.theme--catppuccin-frappe .button.is-warning{background-color:#e5c890;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning:hover,html.theme--catppuccin-frappe .button.is-warning.is-hovered{background-color:#e3c386;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning:focus,html.theme--catppuccin-frappe .button.is-warning.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning:focus:not(:active),html.theme--catppuccin-frappe .button.is-warning.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(229,200,144,0.25)}html.theme--catppuccin-frappe .button.is-warning:active,html.theme--catppuccin-frappe .button.is-warning.is-active{background-color:#e0be7b;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-warning{background-color:#e5c890;border-color:#e5c890;box-shadow:none}html.theme--catppuccin-frappe .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-inverted:hover,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-warning.is-outlined{background-color:transparent;border-color:#e5c890;color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-outlined:hover,html.theme--catppuccin-frappe .button.is-warning.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-warning.is-outlined:focus,html.theme--catppuccin-frappe .button.is-warning.is-outlined.is-focused{background-color:#e5c890;border-color:#e5c890;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning.is-outlined.is-loading::after{border-color:transparent transparent #e5c890 #e5c890 !important}html.theme--catppuccin-frappe .button.is-warning.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-warning.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-warning.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-warning.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-warning.is-outlined{background-color:transparent;border-color:#e5c890;box-shadow:none;color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #e5c890 #e5c890 !important}html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning.is-light{background-color:#fbf7ee;color:#78591c}html.theme--catppuccin-frappe .button.is-warning.is-light:hover,html.theme--catppuccin-frappe .button.is-warning.is-light.is-hovered{background-color:#f9f2e4;border-color:transparent;color:#78591c}html.theme--catppuccin-frappe .button.is-warning.is-light:active,html.theme--catppuccin-frappe .button.is-warning.is-light.is-active{background-color:#f6edda;border-color:transparent;color:#78591c}html.theme--catppuccin-frappe .button.is-danger{background-color:#e78284;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-danger:hover,html.theme--catppuccin-frappe .button.is-danger.is-hovered{background-color:#e57779;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-danger:focus,html.theme--catppuccin-frappe .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-danger:focus:not(:active),html.theme--catppuccin-frappe .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(231,130,132,0.25)}html.theme--catppuccin-frappe .button.is-danger:active,html.theme--catppuccin-frappe .button.is-danger.is-active{background-color:#e36d6f;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-danger[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-danger{background-color:#e78284;border-color:#e78284;box-shadow:none}html.theme--catppuccin-frappe .button.is-danger.is-inverted{background-color:#fff;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-inverted:hover,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-frappe .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-frappe .button.is-danger.is-outlined{background-color:transparent;border-color:#e78284;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-outlined:hover,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-danger.is-outlined:focus,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-focused{background-color:#e78284;border-color:#e78284;color:#fff}html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #e78284 #e78284 !important}html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-frappe .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-danger.is-outlined{background-color:transparent;border-color:#e78284;box-shadow:none;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #e78284 #e78284 !important}html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-frappe .button.is-danger.is-light{background-color:#fceeee;color:#9a1e20}html.theme--catppuccin-frappe .button.is-danger.is-light:hover,html.theme--catppuccin-frappe .button.is-danger.is-light.is-hovered{background-color:#fae3e4;border-color:transparent;color:#9a1e20}html.theme--catppuccin-frappe .button.is-danger.is-light:active,html.theme--catppuccin-frappe .button.is-danger.is-light.is-active{background-color:#f8d8d9;border-color:transparent;color:#9a1e20}html.theme--catppuccin-frappe .button.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--catppuccin-frappe .button.is-small:not(.is-rounded),html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--catppuccin-frappe .button.is-normal{font-size:1rem}html.theme--catppuccin-frappe .button.is-medium{font-size:1.25rem}html.theme--catppuccin-frappe .button.is-large{font-size:1.5rem}html.theme--catppuccin-frappe .button[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button{background-color:#737994;border-color:#626880;box-shadow:none;opacity:.5}html.theme--catppuccin-frappe .button.is-fullwidth{display:flex;width:100%}html.theme--catppuccin-frappe .button.is-loading{color:transparent !important;pointer-events:none}html.theme--catppuccin-frappe .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--catppuccin-frappe .button.is-static{background-color:#292c3c;border-color:#626880;color:#838ba7;box-shadow:none;pointer-events:none}html.theme--catppuccin-frappe .button.is-rounded,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--catppuccin-frappe .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--catppuccin-frappe .buttons .button{margin-bottom:0.5rem}html.theme--catppuccin-frappe .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--catppuccin-frappe .buttons:last-child{margin-bottom:-0.5rem}html.theme--catppuccin-frappe .buttons:not(:last-child){margin-bottom:1rem}html.theme--catppuccin-frappe .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--catppuccin-frappe .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--catppuccin-frappe .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--catppuccin-frappe .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--catppuccin-frappe .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-frappe .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--catppuccin-frappe .buttons.has-addons .button:last-child{margin-right:0}html.theme--catppuccin-frappe .buttons.has-addons .button:hover,html.theme--catppuccin-frappe .buttons.has-addons .button.is-hovered{z-index:2}html.theme--catppuccin-frappe .buttons.has-addons .button:focus,html.theme--catppuccin-frappe .buttons.has-addons .button.is-focused,html.theme--catppuccin-frappe .buttons.has-addons .button:active,html.theme--catppuccin-frappe .buttons.has-addons .button.is-active,html.theme--catppuccin-frappe .buttons.has-addons .button.is-selected{z-index:3}html.theme--catppuccin-frappe .buttons.has-addons .button:focus:hover,html.theme--catppuccin-frappe .buttons.has-addons .button.is-focused:hover,html.theme--catppuccin-frappe .buttons.has-addons .button:active:hover,html.theme--catppuccin-frappe .buttons.has-addons .button.is-active:hover,html.theme--catppuccin-frappe .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--catppuccin-frappe .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-frappe .buttons.is-centered{justify-content:center}html.theme--catppuccin-frappe .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--catppuccin-frappe .buttons.is-right{justify-content:flex-end}html.theme--catppuccin-frappe .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--catppuccin-frappe .button.is-responsive.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--catppuccin-frappe .button.is-responsive,html.theme--catppuccin-frappe .button.is-responsive.is-normal{font-size:.65625rem}html.theme--catppuccin-frappe .button.is-responsive.is-medium{font-size:.75rem}html.theme--catppuccin-frappe .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--catppuccin-frappe .button.is-responsive.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--catppuccin-frappe .button.is-responsive,html.theme--catppuccin-frappe .button.is-responsive.is-normal{font-size:.75rem}html.theme--catppuccin-frappe .button.is-responsive.is-medium{font-size:1rem}html.theme--catppuccin-frappe .button.is-responsive.is-large{font-size:1.25rem}}html.theme--catppuccin-frappe .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--catppuccin-frappe .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--catppuccin-frappe .container{max-width:992px}}@media screen and (max-width: 1215px){html.theme--catppuccin-frappe .container.is-widescreen:not(.is-max-desktop){max-width:1152px}}@media screen and (max-width: 1407px){html.theme--catppuccin-frappe .container.is-fullhd:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}@media screen and (min-width: 1216px){html.theme--catppuccin-frappe .container:not(.is-max-desktop){max-width:1152px}}@media screen and (min-width: 1408px){html.theme--catppuccin-frappe .container:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}html.theme--catppuccin-frappe .content li+li{margin-top:0.25em}html.theme--catppuccin-frappe .content p:not(:last-child),html.theme--catppuccin-frappe .content dl:not(:last-child),html.theme--catppuccin-frappe .content ol:not(:last-child),html.theme--catppuccin-frappe .content ul:not(:last-child),html.theme--catppuccin-frappe .content blockquote:not(:last-child),html.theme--catppuccin-frappe .content pre:not(:last-child),html.theme--catppuccin-frappe .content table:not(:last-child){margin-bottom:1em}html.theme--catppuccin-frappe .content h1,html.theme--catppuccin-frappe .content h2,html.theme--catppuccin-frappe .content h3,html.theme--catppuccin-frappe .content h4,html.theme--catppuccin-frappe .content h5,html.theme--catppuccin-frappe .content h6{color:#c6d0f5;font-weight:600;line-height:1.125}html.theme--catppuccin-frappe .content h1{font-size:2em;margin-bottom:0.5em}html.theme--catppuccin-frappe .content h1:not(:first-child){margin-top:1em}html.theme--catppuccin-frappe .content h2{font-size:1.75em;margin-bottom:0.5714em}html.theme--catppuccin-frappe .content h2:not(:first-child){margin-top:1.1428em}html.theme--catppuccin-frappe .content 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body.has-spaced-navbar-fixed-bottom{padding-bottom:6rem}html.theme--catppuccin-frappe a.navbar-item.is-active,html.theme--catppuccin-frappe .navbar-link.is-active{color:#8caaee}html.theme--catppuccin-frappe a.navbar-item.is-active:not(:focus):not(:hover),html.theme--catppuccin-frappe .navbar-link.is-active:not(:focus):not(:hover){background-color:rgba(0,0,0,0)}html.theme--catppuccin-frappe .navbar-item.has-dropdown:focus .navbar-link,html.theme--catppuccin-frappe .navbar-item.has-dropdown:hover .navbar-link,html.theme--catppuccin-frappe .navbar-item.has-dropdown.is-active .navbar-link{background-color:rgba(0,0,0,0)}}html.theme--catppuccin-frappe .hero.is-fullheight-with-navbar{min-height:calc(100vh - 4rem)}html.theme--catppuccin-frappe .pagination{font-size:1rem;margin:-.25rem}html.theme--catppuccin-frappe .pagination.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.pagination{font-size:.75rem}html.theme--catppuccin-frappe 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.button.is-white:focus:not(:active),html.theme--catppuccin-latte .button.is-white.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(255,255,255,0.25)}html.theme--catppuccin-latte .button.is-white:active,html.theme--catppuccin-latte .button.is-white.is-active{background-color:#f2f2f2;border-color:transparent;color:#0a0a0a}html.theme--catppuccin-latte .button.is-white[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-white{background-color:#fff;border-color:#fff;box-shadow:none}html.theme--catppuccin-latte .button.is-white.is-inverted{background-color:#0a0a0a;color:#fff}html.theme--catppuccin-latte .button.is-white.is-inverted:hover,html.theme--catppuccin-latte .button.is-white.is-inverted.is-hovered{background-color:#000}html.theme--catppuccin-latte .button.is-white.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-white.is-inverted{background-color:#0a0a0a;border-color:transparent;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-white.is-loading::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--catppuccin-latte .button.is-white.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-white.is-outlined:hover,html.theme--catppuccin-latte .button.is-white.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-white.is-outlined:focus,html.theme--catppuccin-latte .button.is-white.is-outlined.is-focused{background-color:#fff;border-color:#fff;color:#0a0a0a}html.theme--catppuccin-latte .button.is-white.is-outlined.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-white.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-white.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-white.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-white.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--catppuccin-latte .button.is-white.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-white.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-white.is-inverted.is-outlined{background-color:transparent;border-color:#0a0a0a;color:#0a0a0a}html.theme--catppuccin-latte .button.is-white.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-white.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-white.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-white.is-inverted.is-outlined.is-focused{background-color:#0a0a0a;color:#fff}html.theme--catppuccin-latte .button.is-white.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-white.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-white.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-white.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-white.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-white.is-inverted.is-outlined{background-color:transparent;border-color:#0a0a0a;box-shadow:none;color:#0a0a0a}html.theme--catppuccin-latte .button.is-black{background-color:#0a0a0a;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-black:hover,html.theme--catppuccin-latte .button.is-black.is-hovered{background-color:#040404;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-black:focus,html.theme--catppuccin-latte .button.is-black.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-black:focus:not(:active),html.theme--catppuccin-latte .button.is-black.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(10,10,10,0.25)}html.theme--catppuccin-latte .button.is-black:active,html.theme--catppuccin-latte .button.is-black.is-active{background-color:#000;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-black[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-black{background-color:#0a0a0a;border-color:#0a0a0a;box-shadow:none}html.theme--catppuccin-latte .button.is-black.is-inverted{background-color:#fff;color:#0a0a0a}html.theme--catppuccin-latte .button.is-black.is-inverted:hover,html.theme--catppuccin-latte .button.is-black.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-latte .button.is-black.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-black.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#0a0a0a}html.theme--catppuccin-latte .button.is-black.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;color:#0a0a0a}html.theme--catppuccin-latte .button.is-black.is-outlined:hover,html.theme--catppuccin-latte .button.is-black.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-black.is-outlined:focus,html.theme--catppuccin-latte .button.is-black.is-outlined.is-focused{background-color:#0a0a0a;border-color:#0a0a0a;color:#fff}html.theme--catppuccin-latte .button.is-black.is-outlined.is-loading::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--catppuccin-latte .button.is-black.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-black.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-black.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-black.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-black.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;box-shadow:none;color:#0a0a0a}html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined.is-focused{background-color:#fff;color:#0a0a0a}html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-light{background-color:#f5f5f5;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-light:hover,html.theme--catppuccin-latte .button.is-light.is-hovered{background-color:#eee;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-light:focus,html.theme--catppuccin-latte .button.is-light.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-light:focus:not(:active),html.theme--catppuccin-latte .button.is-light.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(245,245,245,0.25)}html.theme--catppuccin-latte .button.is-light:active,html.theme--catppuccin-latte .button.is-light.is-active{background-color:#e8e8e8;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-light[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-light{background-color:#f5f5f5;border-color:#f5f5f5;box-shadow:none}html.theme--catppuccin-latte .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);color:#f5f5f5}html.theme--catppuccin-latte .button.is-light.is-inverted:hover,html.theme--catppuccin-latte .button.is-light.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-light.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#f5f5f5}html.theme--catppuccin-latte .button.is-light.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-latte .button.is-light.is-outlined{background-color:transparent;border-color:#f5f5f5;color:#f5f5f5}html.theme--catppuccin-latte .button.is-light.is-outlined:hover,html.theme--catppuccin-latte .button.is-light.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-light.is-outlined:focus,html.theme--catppuccin-latte .button.is-light.is-outlined.is-focused{background-color:#f5f5f5;border-color:#f5f5f5;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-light.is-outlined.is-loading::after{border-color:transparent transparent #f5f5f5 #f5f5f5 !important}html.theme--catppuccin-latte .button.is-light.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-light.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-light.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-light.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-latte .button.is-light.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-light.is-outlined{background-color:transparent;border-color:#f5f5f5;box-shadow:none;color:#f5f5f5}html.theme--catppuccin-latte .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-light.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-light.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-light.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-light.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#f5f5f5}html.theme--catppuccin-latte .button.is-light.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-light.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-light.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-light.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #f5f5f5 #f5f5f5 !important}html.theme--catppuccin-latte .button.is-light.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-dark,html.theme--catppuccin-latte .content kbd.button{background-color:#ccd0da;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-dark:hover,html.theme--catppuccin-latte .content kbd.button:hover,html.theme--catppuccin-latte .button.is-dark.is-hovered,html.theme--catppuccin-latte .content kbd.button.is-hovered{background-color:#c5c9d5;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-dark:focus,html.theme--catppuccin-latte .content kbd.button:focus,html.theme--catppuccin-latte .button.is-dark.is-focused,html.theme--catppuccin-latte .content kbd.button.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-dark:focus:not(:active),html.theme--catppuccin-latte .content kbd.button:focus:not(:active),html.theme--catppuccin-latte .button.is-dark.is-focused:not(:active),html.theme--catppuccin-latte .content kbd.button.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(204,208,218,0.25)}html.theme--catppuccin-latte .button.is-dark:active,html.theme--catppuccin-latte .content kbd.button:active,html.theme--catppuccin-latte .button.is-dark.is-active,html.theme--catppuccin-latte .content kbd.button.is-active{background-color:#bdc2cf;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-dark[disabled],html.theme--catppuccin-latte .content kbd.button[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-dark,fieldset[disabled] html.theme--catppuccin-latte .content kbd.button{background-color:#ccd0da;border-color:#ccd0da;box-shadow:none}html.theme--catppuccin-latte .button.is-dark.is-inverted,html.theme--catppuccin-latte .content kbd.button.is-inverted{background-color:rgba(0,0,0,0.7);color:#ccd0da}html.theme--catppuccin-latte .button.is-dark.is-inverted:hover,html.theme--catppuccin-latte .content kbd.button.is-inverted:hover,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-hovered,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-dark.is-inverted[disabled],html.theme--catppuccin-latte .content kbd.button.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-dark.is-inverted,fieldset[disabled] html.theme--catppuccin-latte .content kbd.button.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#ccd0da}html.theme--catppuccin-latte .button.is-dark.is-loading::after,html.theme--catppuccin-latte .content kbd.button.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-latte .button.is-dark.is-outlined,html.theme--catppuccin-latte .content kbd.button.is-outlined{background-color:transparent;border-color:#ccd0da;color:#ccd0da}html.theme--catppuccin-latte .button.is-dark.is-outlined:hover,html.theme--catppuccin-latte .content kbd.button.is-outlined:hover,html.theme--catppuccin-latte .button.is-dark.is-outlined.is-hovered,html.theme--catppuccin-latte .content kbd.button.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-dark.is-outlined:focus,html.theme--catppuccin-latte .content kbd.button.is-outlined:focus,html.theme--catppuccin-latte .button.is-dark.is-outlined.is-focused,html.theme--catppuccin-latte .content kbd.button.is-outlined.is-focused{background-color:#ccd0da;border-color:#ccd0da;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-dark.is-outlined.is-loading::after,html.theme--catppuccin-latte .content kbd.button.is-outlined.is-loading::after{border-color:transparent transparent #ccd0da #ccd0da !important}html.theme--catppuccin-latte .button.is-dark.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .content kbd.button.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-dark.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .content kbd.button.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-dark.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .content kbd.button.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-dark.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-latte .content kbd.button.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-latte .button.is-dark.is-outlined[disabled],html.theme--catppuccin-latte .content kbd.button.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-dark.is-outlined,fieldset[disabled] html.theme--catppuccin-latte .content kbd.button.is-outlined{background-color:transparent;border-color:#ccd0da;box-shadow:none;color:#ccd0da}html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#ccd0da}html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ccd0da #ccd0da !important}html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined[disabled],html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined,fieldset[disabled] html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-primary,html.theme--catppuccin-latte .docstring>section>a.button.docs-sourcelink{background-color:#1e66f5;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-primary:hover,html.theme--catppuccin-latte .docstring>section>a.button.docs-sourcelink:hover,html.theme--catppuccin-latte .button.is-primary.is-hovered,html.theme--catppuccin-latte .docstring>section>a.button.is-hovered.docs-sourcelink{background-color:#125ef4;border-color:transparent;color:#fff}html.theme--catppuccin-latte 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.button.is-success.is-outlined.is-loading::after{border-color:transparent transparent #40a02b #40a02b !important}html.theme--catppuccin-latte .button.is-success.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-success.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-success.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-success.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-success.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-success.is-outlined{background-color:transparent;border-color:#40a02b;box-shadow:none;color:#40a02b}html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-focused{background-color:#fff;color:#40a02b}html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #40a02b #40a02b !important}html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-success.is-light{background-color:#f1fbef;color:#40a12b}html.theme--catppuccin-latte .button.is-success.is-light:hover,html.theme--catppuccin-latte .button.is-success.is-light.is-hovered{background-color:#e8f8e5;border-color:transparent;color:#40a12b}html.theme--catppuccin-latte .button.is-success.is-light:active,html.theme--catppuccin-latte .button.is-success.is-light.is-active{background-color:#e0f5db;border-color:transparent;color:#40a12b}html.theme--catppuccin-latte .button.is-warning{background-color:#df8e1d;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-warning:hover,html.theme--catppuccin-latte .button.is-warning.is-hovered{background-color:#d4871c;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-warning:focus,html.theme--catppuccin-latte .button.is-warning.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-warning:focus:not(:active),html.theme--catppuccin-latte .button.is-warning.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(223,142,29,0.25)}html.theme--catppuccin-latte .button.is-warning:active,html.theme--catppuccin-latte .button.is-warning.is-active{background-color:#c8801a;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-warning[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-warning{background-color:#df8e1d;border-color:#df8e1d;box-shadow:none}html.theme--catppuccin-latte .button.is-warning.is-inverted{background-color:#fff;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-inverted:hover,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-latte .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-warning.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-warning.is-outlined{background-color:transparent;border-color:#df8e1d;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-outlined:hover,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-warning.is-outlined:focus,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-focused{background-color:#df8e1d;border-color:#df8e1d;color:#fff}html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading::after{border-color:transparent transparent #df8e1d #df8e1d !important}html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-warning.is-outlined{background-color:transparent;border-color:#df8e1d;box-shadow:none;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-focused{background-color:#fff;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #df8e1d #df8e1d !important}html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-warning.is-light{background-color:#fdf6ed;color:#9e6515}html.theme--catppuccin-latte .button.is-warning.is-light:hover,html.theme--catppuccin-latte .button.is-warning.is-light.is-hovered{background-color:#fbf1e2;border-color:transparent;color:#9e6515}html.theme--catppuccin-latte .button.is-warning.is-light:active,html.theme--catppuccin-latte .button.is-warning.is-light.is-active{background-color:#faebd6;border-color:transparent;color:#9e6515}html.theme--catppuccin-latte .button.is-danger{background-color:#d20f39;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-danger:hover,html.theme--catppuccin-latte .button.is-danger.is-hovered{background-color:#c60e36;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-danger:focus,html.theme--catppuccin-latte .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-danger:focus:not(:active),html.theme--catppuccin-latte .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(210,15,57,0.25)}html.theme--catppuccin-latte .button.is-danger:active,html.theme--catppuccin-latte .button.is-danger.is-active{background-color:#ba0d33;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-danger[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-danger{background-color:#d20f39;border-color:#d20f39;box-shadow:none}html.theme--catppuccin-latte .button.is-danger.is-inverted{background-color:#fff;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-inverted:hover,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-latte .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-danger.is-outlined{background-color:transparent;border-color:#d20f39;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-outlined:hover,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-danger.is-outlined:focus,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-focused{background-color:#d20f39;border-color:#d20f39;color:#fff}html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #d20f39 #d20f39 !important}html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-danger.is-outlined{background-color:transparent;border-color:#d20f39;box-shadow:none;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #d20f39 #d20f39 !important}html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-danger.is-light{background-color:#feecf0;color:#e9113f}html.theme--catppuccin-latte .button.is-danger.is-light:hover,html.theme--catppuccin-latte .button.is-danger.is-light.is-hovered{background-color:#fde0e6;border-color:transparent;color:#e9113f}html.theme--catppuccin-latte .button.is-danger.is-light:active,html.theme--catppuccin-latte .button.is-danger.is-light.is-active{background-color:#fcd4dd;border-color:transparent;color:#e9113f}html.theme--catppuccin-latte .button.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--catppuccin-latte .button.is-small:not(.is-rounded),html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--catppuccin-latte .button.is-normal{font-size:1rem}html.theme--catppuccin-latte .button.is-medium{font-size:1.25rem}html.theme--catppuccin-latte .button.is-large{font-size:1.5rem}html.theme--catppuccin-latte .button[disabled],fieldset[disabled] html.theme--catppuccin-latte .button{background-color:#9ca0b0;border-color:#acb0be;box-shadow:none;opacity:.5}html.theme--catppuccin-latte .button.is-fullwidth{display:flex;width:100%}html.theme--catppuccin-latte .button.is-loading{color:transparent !important;pointer-events:none}html.theme--catppuccin-latte .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--catppuccin-latte .button.is-static{background-color:#e6e9ef;border-color:#acb0be;color:#8c8fa1;box-shadow:none;pointer-events:none}html.theme--catppuccin-latte .button.is-rounded,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--catppuccin-latte .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--catppuccin-latte .buttons .button{margin-bottom:0.5rem}html.theme--catppuccin-latte .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--catppuccin-latte .buttons:last-child{margin-bottom:-0.5rem}html.theme--catppuccin-latte .buttons:not(:last-child){margin-bottom:1rem}html.theme--catppuccin-latte .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--catppuccin-latte .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--catppuccin-latte .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--catppuccin-latte .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--catppuccin-latte .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-latte .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--catppuccin-latte .buttons.has-addons .button:last-child{margin-right:0}html.theme--catppuccin-latte .buttons.has-addons .button:hover,html.theme--catppuccin-latte .buttons.has-addons .button.is-hovered{z-index:2}html.theme--catppuccin-latte .buttons.has-addons .button:focus,html.theme--catppuccin-latte .buttons.has-addons .button.is-focused,html.theme--catppuccin-latte .buttons.has-addons .button:active,html.theme--catppuccin-latte .buttons.has-addons .button.is-active,html.theme--catppuccin-latte .buttons.has-addons .button.is-selected{z-index:3}html.theme--catppuccin-latte .buttons.has-addons .button:focus:hover,html.theme--catppuccin-latte .buttons.has-addons .button.is-focused:hover,html.theme--catppuccin-latte .buttons.has-addons .button:active:hover,html.theme--catppuccin-latte .buttons.has-addons .button.is-active:hover,html.theme--catppuccin-latte .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--catppuccin-latte .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-latte .buttons.is-centered{justify-content:center}html.theme--catppuccin-latte .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--catppuccin-latte .buttons.is-right{justify-content:flex-end}html.theme--catppuccin-latte .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--catppuccin-latte .button.is-responsive.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--catppuccin-latte .button.is-responsive,html.theme--catppuccin-latte .button.is-responsive.is-normal{font-size:.65625rem}html.theme--catppuccin-latte .button.is-responsive.is-medium{font-size:.75rem}html.theme--catppuccin-latte .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--catppuccin-latte .button.is-responsive.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--catppuccin-latte .button.is-responsive,html.theme--catppuccin-latte .button.is-responsive.is-normal{font-size:.75rem}html.theme--catppuccin-latte .button.is-responsive.is-medium{font-size:1rem}html.theme--catppuccin-latte .button.is-responsive.is-large{font-size:1.25rem}}html.theme--catppuccin-latte .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--catppuccin-latte .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--catppuccin-latte .container{max-width:992px}}@media screen and (max-width: 1215px){html.theme--catppuccin-latte .container.is-widescreen:not(.is-max-desktop){max-width:1152px}}@media screen and (max-width: 1407px){html.theme--catppuccin-latte .container.is-fullhd:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}@media screen and (min-width: 1216px){html.theme--catppuccin-latte .container:not(.is-max-desktop){max-width:1152px}}@media screen and (min-width: 1408px){html.theme--catppuccin-latte .container:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}html.theme--catppuccin-latte .content li+li{margin-top:0.25em}html.theme--catppuccin-latte .content p:not(:last-child),html.theme--catppuccin-latte .content dl:not(:last-child),html.theme--catppuccin-latte .content ol:not(:last-child),html.theme--catppuccin-latte 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h3:not(:first-child){margin-top:1.3333em}html.theme--catppuccin-latte .content h4{font-size:1.25em;margin-bottom:0.8em}html.theme--catppuccin-latte .content h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--catppuccin-latte .content h6{font-size:1em;margin-bottom:1em}html.theme--catppuccin-latte .content blockquote{background-color:#e6e9ef;border-left:5px solid #acb0be;padding:1.25em 1.5em}html.theme--catppuccin-latte .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-latte .content ol:not([type]){list-style-type:decimal}html.theme--catppuccin-latte .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--catppuccin-latte .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--catppuccin-latte .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--catppuccin-latte .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--catppuccin-latte .content 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.file.is-link.is-active .file-cta{background-color:#0b57ef;border-color:transparent;color:#fff}html.theme--catppuccin-latte .file.is-info .file-cta{background-color:#179299;border-color:transparent;color:#fff}html.theme--catppuccin-latte .file.is-info:hover .file-cta,html.theme--catppuccin-latte .file.is-info.is-hovered .file-cta{background-color:#15878e;border-color:transparent;color:#fff}html.theme--catppuccin-latte .file.is-info:focus .file-cta,html.theme--catppuccin-latte .file.is-info.is-focused .file-cta{border-color:transparent;box-shadow:0 0 0.5em rgba(23,146,153,0.25);color:#fff}html.theme--catppuccin-latte .file.is-info:active .file-cta,html.theme--catppuccin-latte .file.is-info.is-active .file-cta{background-color:#147d83;border-color:transparent;color:#fff}html.theme--catppuccin-latte .file.is-success .file-cta{background-color:#40a02b;border-color:transparent;color:#fff}html.theme--catppuccin-latte .file.is-success:hover .file-cta,html.theme--catppuccin-latte .file.is-success.is-hovered .file-cta{background-color:#3c9628;border-color:transparent;color:#fff}html.theme--catppuccin-latte .file.is-success:focus .file-cta,html.theme--catppuccin-latte .file.is-success.is-focused .file-cta{border-color:transparent;box-shadow:0 0 0.5em rgba(64,160,43,0.25);color:#fff}html.theme--catppuccin-latte .file.is-success:active .file-cta,html.theme--catppuccin-latte .file.is-success.is-active .file-cta{background-color:#388c26;border-color:transparent;color:#fff}html.theme--catppuccin-latte .file.is-warning .file-cta{background-color:#df8e1d;border-color:transparent;color:#fff}html.theme--catppuccin-latte .file.is-warning:hover .file-cta,html.theme--catppuccin-latte .file.is-warning.is-hovered .file-cta{background-color:#d4871c;border-color:transparent;color:#fff}html.theme--catppuccin-latte .file.is-warning:focus .file-cta,html.theme--catppuccin-latte .file.is-warning.is-focused .file-cta{border-color:transparent;box-shadow:0 0 0.5em rgba(223,142,29,0.25);color:#fff}html.theme--catppuccin-latte .file.is-warning:active .file-cta,html.theme--catppuccin-latte .file.is-warning.is-active .file-cta{background-color:#c8801a;border-color:transparent;color:#fff}html.theme--catppuccin-latte .file.is-danger .file-cta{background-color:#d20f39;border-color:transparent;color:#fff}html.theme--catppuccin-latte .file.is-danger:hover .file-cta,html.theme--catppuccin-latte .file.is-danger.is-hovered .file-cta{background-color:#c60e36;border-color:transparent;color:#fff}html.theme--catppuccin-latte .file.is-danger:focus .file-cta,html.theme--catppuccin-latte .file.is-danger.is-focused .file-cta{border-color:transparent;box-shadow:0 0 0.5em rgba(210,15,57,0.25);color:#fff}html.theme--catppuccin-latte .file.is-danger:active .file-cta,html.theme--catppuccin-latte .file.is-danger.is-active .file-cta{background-color:#ba0d33;border-color:transparent;color:#fff}html.theme--catppuccin-latte .file.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.file{font-size:.75rem}html.theme--catppuccin-latte .file.is-normal{font-size:1rem}html.theme--catppuccin-latte .file.is-medium{font-size:1.25rem}html.theme--catppuccin-latte .file.is-medium .file-icon .fa{font-size:21px}html.theme--catppuccin-latte .file.is-large{font-size:1.5rem}html.theme--catppuccin-latte .file.is-large .file-icon .fa{font-size:28px}html.theme--catppuccin-latte .file.has-name .file-cta{border-bottom-right-radius:0;border-top-right-radius:0}html.theme--catppuccin-latte .file.has-name .file-name{border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-latte .file.has-name.is-empty .file-cta{border-radius:.4em}html.theme--catppuccin-latte .file.has-name.is-empty .file-name{display:none}html.theme--catppuccin-latte .file.is-boxed .file-label{flex-direction:column}html.theme--catppuccin-latte .file.is-boxed .file-cta{flex-direction:column;height:auto;padding:1em 3em}html.theme--catppuccin-latte .file.is-boxed .file-name{border-width:0 1px 1px}html.theme--catppuccin-latte .file.is-boxed .file-icon{height:1.5em;width:1.5em}html.theme--catppuccin-latte .file.is-boxed .file-icon .fa{font-size:21px}html.theme--catppuccin-latte .file.is-boxed.is-small .file-icon .fa,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.is-boxed .file-icon .fa{font-size:14px}html.theme--catppuccin-latte .file.is-boxed.is-medium .file-icon .fa{font-size:28px}html.theme--catppuccin-latte .file.is-boxed.is-large .file-icon .fa{font-size:35px}html.theme--catppuccin-latte .file.is-boxed.has-name .file-cta{border-radius:.4em .4em 0 0}html.theme--catppuccin-latte .file.is-boxed.has-name .file-name{border-radius:0 0 .4em .4em;border-width:0 1px 1px}html.theme--catppuccin-latte .file.is-centered{justify-content:center}html.theme--catppuccin-latte .file.is-fullwidth .file-label{width:100%}html.theme--catppuccin-latte .file.is-fullwidth .file-name{flex-grow:1;max-width:none}html.theme--catppuccin-latte .file.is-right{justify-content:flex-end}html.theme--catppuccin-latte .file.is-right .file-cta{border-radius:0 .4em .4em 0}html.theme--catppuccin-latte .file.is-right .file-name{border-radius:.4em 0 0 .4em;border-width:1px 0 1px 1px;order:-1}html.theme--catppuccin-latte .file-label{align-items:stretch;display:flex;cursor:pointer;justify-content:flex-start;overflow:hidden;position:relative}html.theme--catppuccin-latte .file-label:hover .file-cta{background-color:#c5c9d5;color:#41445a}html.theme--catppuccin-latte .file-label:hover .file-name{border-color:#a5a9b8}html.theme--catppuccin-latte .file-label:active .file-cta{background-color:#bdc2cf;color:#41445a}html.theme--catppuccin-latte .file-label:active .file-name{border-color:#9ea2b3}html.theme--catppuccin-latte .file-input{height:100%;left:0;opacity:0;outline:none;position:absolute;top:0;width:100%}html.theme--catppuccin-latte .file-cta,html.theme--catppuccin-latte .file-name{border-color:#acb0be;border-radius:.4em;font-size:1em;padding-left:1em;padding-right:1em;white-space:nowrap}html.theme--catppuccin-latte .file-cta{background-color:#ccd0da;color:#4c4f69}html.theme--catppuccin-latte .file-name{border-color:#acb0be;border-style:solid;border-width:1px 1px 1px 0;display:block;max-width:16em;overflow:hidden;text-align:inherit;text-overflow:ellipsis}html.theme--catppuccin-latte .file-icon{align-items:center;display:flex;height:1em;justify-content:center;margin-right:.5em;width:1em}html.theme--catppuccin-latte .file-icon .fa{font-size:14px}html.theme--catppuccin-latte .label{color:#41445a;display:block;font-size:1rem;font-weight:700}html.theme--catppuccin-latte .label:not(:last-child){margin-bottom:0.5em}html.theme--catppuccin-latte .label.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.label{font-size:.75rem}html.theme--catppuccin-latte .label.is-medium{font-size:1.25rem}html.theme--catppuccin-latte .label.is-large{font-size:1.5rem}html.theme--catppuccin-latte .help{display:block;font-size:.75rem;margin-top:0.25rem}html.theme--catppuccin-latte .help.is-white{color:#fff}html.theme--catppuccin-latte .help.is-black{color:#0a0a0a}html.theme--catppuccin-latte .help.is-light{color:#f5f5f5}html.theme--catppuccin-latte .help.is-dark,html.theme--catppuccin-latte .content kbd.help{color:#ccd0da}html.theme--catppuccin-latte .help.is-primary,html.theme--catppuccin-latte .docstring>section>a.help.docs-sourcelink{color:#1e66f5}html.theme--catppuccin-latte .help.is-link{color:#1e66f5}html.theme--catppuccin-latte .help.is-info{color:#179299}html.theme--catppuccin-latte .help.is-success{color:#40a02b}html.theme--catppuccin-latte .help.is-warning{color:#df8e1d}html.theme--catppuccin-latte .help.is-danger{color:#d20f39}html.theme--catppuccin-latte .field:not(:last-child){margin-bottom:0.75rem}html.theme--catppuccin-latte .field.has-addons{display:flex;justify-content:flex-start}html.theme--catppuccin-latte .field.has-addons .control:not(:last-child){margin-right:-1px}html.theme--catppuccin-latte .field.has-addons .control:not(:first-child):not(:last-child) .button,html.theme--catppuccin-latte .field.has-addons .control:not(:first-child):not(:last-child) .input,html.theme--catppuccin-latte .field.has-addons .control:not(:first-child):not(:last-child) #documenter .docs-sidebar form.docs-search>input,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control:not(:first-child):not(:last-child) form.docs-search>input,html.theme--catppuccin-latte .field.has-addons .control:not(:first-child):not(:last-child) .select select{border-radius:0}html.theme--catppuccin-latte .field.has-addons .control:first-child:not(:only-child) .button,html.theme--catppuccin-latte .field.has-addons .control:first-child:not(:only-child) .input,html.theme--catppuccin-latte .field.has-addons .control:first-child:not(:only-child) #documenter .docs-sidebar form.docs-search>input,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control:first-child:not(:only-child) form.docs-search>input,html.theme--catppuccin-latte .field.has-addons .control:first-child:not(:only-child) .select select{border-bottom-right-radius:0;border-top-right-radius:0}html.theme--catppuccin-latte .field.has-addons .control:last-child:not(:only-child) .button,html.theme--catppuccin-latte .field.has-addons .control:last-child:not(:only-child) .input,html.theme--catppuccin-latte .field.has-addons .control:last-child:not(:only-child) #documenter .docs-sidebar form.docs-search>input,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control:last-child:not(:only-child) form.docs-search>input,html.theme--catppuccin-latte .field.has-addons .control:last-child:not(:only-child) .select select{border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-latte .field.has-addons .control .button:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control .button.is-hovered:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control .input:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input:not([disabled]):hover,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control .input.is-hovered:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input.is-hovered:not([disabled]),html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input.is-hovered:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control .select select:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control .select select.is-hovered:not([disabled]){z-index:2}html.theme--catppuccin-latte .field.has-addons .control .button:not([disabled]):focus,html.theme--catppuccin-latte .field.has-addons .control .button.is-focused:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control .button:not([disabled]):active,html.theme--catppuccin-latte .field.has-addons .control .button.is-active:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control .input:not([disabled]):focus,html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input:not([disabled]):focus,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input:not([disabled]):focus,html.theme--catppuccin-latte .field.has-addons .control .input.is-focused:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input.is-focused:not([disabled]),html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input.is-focused:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control .input:not([disabled]):active,html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input:not([disabled]):active,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input:not([disabled]):active,html.theme--catppuccin-latte .field.has-addons .control .input.is-active:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input.is-active:not([disabled]),html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input.is-active:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control .select select:not([disabled]):focus,html.theme--catppuccin-latte .field.has-addons .control .select select.is-focused:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control .select select:not([disabled]):active,html.theme--catppuccin-latte .field.has-addons .control .select select.is-active:not([disabled]){z-index:3}html.theme--catppuccin-latte .field.has-addons .control .button:not([disabled]):focus:hover,html.theme--catppuccin-latte .field.has-addons .control .button.is-focused:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control .button:not([disabled]):active:hover,html.theme--catppuccin-latte .field.has-addons .control .button.is-active:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control .input:not([disabled]):focus:hover,html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input:not([disabled]):focus:hover,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input:not([disabled]):focus:hover,html.theme--catppuccin-latte .field.has-addons .control .input.is-focused:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input.is-focused:not([disabled]):hover,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input.is-focused:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control .input:not([disabled]):active:hover,html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input:not([disabled]):active:hover,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input:not([disabled]):active:hover,html.theme--catppuccin-latte .field.has-addons .control .input.is-active:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input.is-active:not([disabled]):hover,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input.is-active:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control .select select:not([disabled]):focus:hover,html.theme--catppuccin-latte .field.has-addons .control .select select.is-focused:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control .select select:not([disabled]):active:hover,html.theme--catppuccin-latte .field.has-addons .control .select select.is-active:not([disabled]):hover{z-index:4}html.theme--catppuccin-latte .field.has-addons .control.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-latte .field.has-addons.has-addons-centered{justify-content:center}html.theme--catppuccin-latte .field.has-addons.has-addons-right{justify-content:flex-end}html.theme--catppuccin-latte .field.has-addons.has-addons-fullwidth .control{flex-grow:1;flex-shrink:0}html.theme--catppuccin-latte .field.is-grouped{display:flex;justify-content:flex-start}html.theme--catppuccin-latte .field.is-grouped>.control{flex-shrink:0}html.theme--catppuccin-latte .field.is-grouped>.control:not(:last-child){margin-bottom:0;margin-right:.75rem}html.theme--catppuccin-latte .field.is-grouped>.control.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-latte .field.is-grouped.is-grouped-centered{justify-content:center}html.theme--catppuccin-latte .field.is-grouped.is-grouped-right{justify-content:flex-end}html.theme--catppuccin-latte .field.is-grouped.is-grouped-multiline{flex-wrap:wrap}html.theme--catppuccin-latte .field.is-grouped.is-grouped-multiline>.control:last-child,html.theme--catppuccin-latte .field.is-grouped.is-grouped-multiline>.control:not(:last-child){margin-bottom:0.75rem}html.theme--catppuccin-latte .field.is-grouped.is-grouped-multiline:last-child{margin-bottom:-0.75rem}html.theme--catppuccin-latte 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.icon:only-child,html.theme--catppuccin-latte .navbar-link .icon:only-child{margin-left:-0.25rem;margin-right:-0.25rem}html.theme--catppuccin-latte a.navbar-item,html.theme--catppuccin-latte .navbar-link{cursor:pointer}html.theme--catppuccin-latte a.navbar-item:focus,html.theme--catppuccin-latte a.navbar-item:focus-within,html.theme--catppuccin-latte a.navbar-item:hover,html.theme--catppuccin-latte a.navbar-item.is-active,html.theme--catppuccin-latte .navbar-link:focus,html.theme--catppuccin-latte .navbar-link:focus-within,html.theme--catppuccin-latte .navbar-link:hover,html.theme--catppuccin-latte .navbar-link.is-active{background-color:rgba(0,0,0,0);color:#1e66f5}html.theme--catppuccin-latte .navbar-item{flex-grow:0;flex-shrink:0}html.theme--catppuccin-latte .navbar-item img{max-height:1.75rem}html.theme--catppuccin-latte .navbar-item.has-dropdown{padding:0}html.theme--catppuccin-latte .navbar-item.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-latte .navbar-item.is-tab{border-bottom:1px solid transparent;min-height:4rem;padding-bottom:calc(0.5rem - 1px)}html.theme--catppuccin-latte .navbar-item.is-tab:focus,html.theme--catppuccin-latte .navbar-item.is-tab:hover{background-color:rgba(0,0,0,0);border-bottom-color:#1e66f5}html.theme--catppuccin-latte .navbar-item.is-tab.is-active{background-color:rgba(0,0,0,0);border-bottom-color:#1e66f5;border-bottom-style:solid;border-bottom-width:3px;color:#1e66f5;padding-bottom:calc(0.5rem - 3px)}html.theme--catppuccin-latte .navbar-content{flex-grow:1;flex-shrink:1}html.theme--catppuccin-latte .navbar-link:not(.is-arrowless){padding-right:2.5em}html.theme--catppuccin-latte .navbar-link:not(.is-arrowless)::after{border-color:#fff;margin-top:-0.375em;right:1.125em}html.theme--catppuccin-latte .navbar-dropdown{font-size:0.875rem;padding-bottom:0.5rem;padding-top:0.5rem}html.theme--catppuccin-latte .navbar-dropdown .navbar-item{padding-left:1.5rem;padding-right:1.5rem}html.theme--catppuccin-latte .navbar-divider{background-color:rgba(0,0,0,0.2);border:none;display:none;height:2px;margin:0.5rem 0}@media screen and (max-width: 1055px){html.theme--catppuccin-latte .navbar>.container{display:block}html.theme--catppuccin-latte .navbar-brand .navbar-item,html.theme--catppuccin-latte .navbar-tabs .navbar-item{align-items:center;display:flex}html.theme--catppuccin-latte .navbar-link::after{display:none}html.theme--catppuccin-latte .navbar-menu{background-color:#1e66f5;box-shadow:0 8px 16px rgba(10,10,10,0.1);padding:0.5rem 0}html.theme--catppuccin-latte .navbar-menu.is-active{display:block}html.theme--catppuccin-latte .navbar.is-fixed-bottom-touch,html.theme--catppuccin-latte .navbar.is-fixed-top-touch{left:0;position:fixed;right:0;z-index:30}html.theme--catppuccin-latte .navbar.is-fixed-bottom-touch{bottom:0}html.theme--catppuccin-latte .navbar.is-fixed-bottom-touch.has-shadow{box-shadow:0 -2px 3px rgba(10,10,10,0.1)}html.theme--catppuccin-latte .navbar.is-fixed-top-touch{top:0}html.theme--catppuccin-latte .navbar.is-fixed-top .navbar-menu,html.theme--catppuccin-latte .navbar.is-fixed-top-touch .navbar-menu{-webkit-overflow-scrolling:touch;max-height:calc(100vh - 4rem);overflow:auto}html.theme--catppuccin-latte html.has-navbar-fixed-top-touch,html.theme--catppuccin-latte body.has-navbar-fixed-top-touch{padding-top:4rem}html.theme--catppuccin-latte html.has-navbar-fixed-bottom-touch,html.theme--catppuccin-latte body.has-navbar-fixed-bottom-touch{padding-bottom:4rem}}@media screen and (min-width: 1056px){html.theme--catppuccin-latte .navbar,html.theme--catppuccin-latte .navbar-menu,html.theme--catppuccin-latte .navbar-start,html.theme--catppuccin-latte .navbar-end{align-items:stretch;display:flex}html.theme--catppuccin-latte .navbar{min-height:4rem}html.theme--catppuccin-latte .navbar.is-spaced{padding:1rem 2rem}html.theme--catppuccin-latte .navbar.is-spaced .navbar-start,html.theme--catppuccin-latte .navbar.is-spaced .navbar-end{align-items:center}html.theme--catppuccin-latte .navbar.is-spaced a.navbar-item,html.theme--catppuccin-latte .navbar.is-spaced .navbar-link{border-radius:.4em}html.theme--catppuccin-latte .navbar.is-transparent a.navbar-item:focus,html.theme--catppuccin-latte .navbar.is-transparent a.navbar-item:hover,html.theme--catppuccin-latte .navbar.is-transparent a.navbar-item.is-active,html.theme--catppuccin-latte .navbar.is-transparent .navbar-link:focus,html.theme--catppuccin-latte .navbar.is-transparent .navbar-link:hover,html.theme--catppuccin-latte .navbar.is-transparent .navbar-link.is-active{background-color:transparent !important}html.theme--catppuccin-latte .navbar.is-transparent .navbar-item.has-dropdown.is-active .navbar-link,html.theme--catppuccin-latte .navbar.is-transparent .navbar-item.has-dropdown.is-hoverable:focus .navbar-link,html.theme--catppuccin-latte .navbar.is-transparent .navbar-item.has-dropdown.is-hoverable:focus-within 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.navbar-dropdown.is-boxed,.navbar.is-spaced html.theme--catppuccin-latte .navbar-item.is-hoverable:hover .navbar-dropdown,html.theme--catppuccin-latte .navbar-item.is-hoverable:hover .navbar-dropdown.is-boxed{opacity:1;pointer-events:auto;transform:translateY(0)}html.theme--catppuccin-latte .navbar-menu{flex-grow:1;flex-shrink:0}html.theme--catppuccin-latte .navbar-start{justify-content:flex-start;margin-right:auto}html.theme--catppuccin-latte .navbar-end{justify-content:flex-end;margin-left:auto}html.theme--catppuccin-latte .navbar-dropdown{background-color:#1e66f5;border-bottom-left-radius:8px;border-bottom-right-radius:8px;border-top:1px solid rgba(0,0,0,0.2);box-shadow:0 8px 8px rgba(10,10,10,0.1);display:none;font-size:0.875rem;left:0;min-width:100%;position:absolute;top:100%;z-index:20}html.theme--catppuccin-latte .navbar-dropdown .navbar-item{padding:0.375rem 1rem;white-space:nowrap}html.theme--catppuccin-latte .navbar-dropdown a.navbar-item{padding-right:3rem}html.theme--catppuccin-latte .navbar-dropdown a.navbar-item:focus,html.theme--catppuccin-latte .navbar-dropdown a.navbar-item:hover{background-color:rgba(0,0,0,0);color:#8c8fa1}html.theme--catppuccin-latte .navbar-dropdown a.navbar-item.is-active{background-color:rgba(0,0,0,0);color:#1e66f5}.navbar.is-spaced html.theme--catppuccin-latte .navbar-dropdown,html.theme--catppuccin-latte .navbar-dropdown.is-boxed{border-radius:8px;border-top:none;box-shadow:0 8px 8px rgba(10,10,10,0.1), 0 0 0 1px rgba(10,10,10,0.1);display:block;opacity:0;pointer-events:none;top:calc(100% + (-4px));transform:translateY(-5px);transition-duration:86ms;transition-property:opacity, transform}html.theme--catppuccin-latte .navbar-dropdown.is-right{left:auto;right:0}html.theme--catppuccin-latte .navbar-divider{display:block}html.theme--catppuccin-latte .navbar>.container .navbar-brand,html.theme--catppuccin-latte .container>.navbar .navbar-brand{margin-left:-.75rem}html.theme--catppuccin-latte .navbar>.container .navbar-menu,html.theme--catppuccin-latte .container>.navbar .navbar-menu{margin-right:-.75rem}html.theme--catppuccin-latte .navbar.is-fixed-bottom-desktop,html.theme--catppuccin-latte .navbar.is-fixed-top-desktop{left:0;position:fixed;right:0;z-index:30}html.theme--catppuccin-latte .navbar.is-fixed-bottom-desktop{bottom:0}html.theme--catppuccin-latte .navbar.is-fixed-bottom-desktop.has-shadow{box-shadow:0 -2px 3px rgba(10,10,10,0.1)}html.theme--catppuccin-latte .navbar.is-fixed-top-desktop{top:0}html.theme--catppuccin-latte html.has-navbar-fixed-top-desktop,html.theme--catppuccin-latte body.has-navbar-fixed-top-desktop{padding-top:4rem}html.theme--catppuccin-latte html.has-navbar-fixed-bottom-desktop,html.theme--catppuccin-latte body.has-navbar-fixed-bottom-desktop{padding-bottom:4rem}html.theme--catppuccin-latte html.has-spaced-navbar-fixed-top,html.theme--catppuccin-latte body.has-spaced-navbar-fixed-top{padding-top:6rem}html.theme--catppuccin-latte html.has-spaced-navbar-fixed-bottom,html.theme--catppuccin-latte body.has-spaced-navbar-fixed-bottom{padding-bottom:6rem}html.theme--catppuccin-latte a.navbar-item.is-active,html.theme--catppuccin-latte .navbar-link.is-active{color:#1e66f5}html.theme--catppuccin-latte a.navbar-item.is-active:not(:focus):not(:hover),html.theme--catppuccin-latte .navbar-link.is-active:not(:focus):not(:hover){background-color:rgba(0,0,0,0)}html.theme--catppuccin-latte .navbar-item.has-dropdown:focus .navbar-link,html.theme--catppuccin-latte .navbar-item.has-dropdown:hover .navbar-link,html.theme--catppuccin-latte .navbar-item.has-dropdown.is-active .navbar-link{background-color:rgba(0,0,0,0)}}html.theme--catppuccin-latte .hero.is-fullheight-with-navbar{min-height:calc(100vh - 4rem)}html.theme--catppuccin-latte .pagination{font-size:1rem;margin:-.25rem}html.theme--catppuccin-latte .pagination.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.pagination{font-size:.75rem}html.theme--catppuccin-latte .pagination.is-medium{font-size:1.25rem}html.theme--catppuccin-latte .pagination.is-large{font-size:1.5rem}html.theme--catppuccin-latte .pagination.is-rounded .pagination-previous,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.pagination .pagination-previous,html.theme--catppuccin-latte .pagination.is-rounded .pagination-next,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.pagination .pagination-next{padding-left:1em;padding-right:1em;border-radius:9999px}html.theme--catppuccin-latte .pagination.is-rounded .pagination-link,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.pagination .pagination-link{border-radius:9999px}html.theme--catppuccin-latte .pagination,html.theme--catppuccin-latte .pagination-list{align-items:center;display:flex;justify-content:center;text-align:center}html.theme--catppuccin-latte .pagination-previous,html.theme--catppuccin-latte .pagination-next,html.theme--catppuccin-latte .pagination-link,html.theme--catppuccin-latte .pagination-ellipsis{font-size:1em;justify-content:center;margin:.25rem;padding-left:.5em;padding-right:.5em;text-align:center}html.theme--catppuccin-latte .pagination-previous,html.theme--catppuccin-latte .pagination-next,html.theme--catppuccin-latte .pagination-link{border-color:#acb0be;color:#1e66f5;min-width:2.5em}html.theme--catppuccin-latte .pagination-previous:hover,html.theme--catppuccin-latte .pagination-next:hover,html.theme--catppuccin-latte .pagination-link:hover{border-color:#9ca0b0;color:#04a5e5}html.theme--catppuccin-latte .pagination-previous:focus,html.theme--catppuccin-latte .pagination-next:focus,html.theme--catppuccin-latte .pagination-link:focus{border-color:#9ca0b0}html.theme--catppuccin-latte .pagination-previous:active,html.theme--catppuccin-latte .pagination-next:active,html.theme--catppuccin-latte .pagination-link:active{box-shadow:inset 0 1px 2px rgba(10,10,10,0.2)}html.theme--catppuccin-latte .pagination-previous[disabled],html.theme--catppuccin-latte .pagination-previous.is-disabled,html.theme--catppuccin-latte .pagination-next[disabled],html.theme--catppuccin-latte .pagination-next.is-disabled,html.theme--catppuccin-latte .pagination-link[disabled],html.theme--catppuccin-latte .pagination-link.is-disabled{background-color:#acb0be;border-color:#acb0be;box-shadow:none;color:#616587;opacity:0.5}html.theme--catppuccin-latte .pagination-previous,html.theme--catppuccin-latte .pagination-next{padding-left:.75em;padding-right:.75em;white-space:nowrap}html.theme--catppuccin-latte .pagination-link.is-current{background-color:#1e66f5;border-color:#1e66f5;color:#fff}html.theme--catppuccin-latte .pagination-ellipsis{color:#9ca0b0;pointer-events:none}html.theme--catppuccin-latte .pagination-list{flex-wrap:wrap}html.theme--catppuccin-latte .pagination-list li{list-style:none}@media screen and (max-width: 768px){html.theme--catppuccin-latte .pagination{flex-wrap:wrap}html.theme--catppuccin-latte .pagination-previous,html.theme--catppuccin-latte .pagination-next{flex-grow:1;flex-shrink:1}html.theme--catppuccin-latte .pagination-list li{flex-grow:1;flex-shrink:1}}@media screen and (min-width: 769px),print{html.theme--catppuccin-latte .pagination-list{flex-grow:1;flex-shrink:1;justify-content:flex-start;order:1}html.theme--catppuccin-latte .pagination-previous,html.theme--catppuccin-latte .pagination-next,html.theme--catppuccin-latte .pagination-link,html.theme--catppuccin-latte .pagination-ellipsis{margin-bottom:0;margin-top:0}html.theme--catppuccin-latte .pagination-previous{order:2}html.theme--catppuccin-latte .pagination-next{order:3}html.theme--catppuccin-latte .pagination{justify-content:space-between;margin-bottom:0;margin-top:0}html.theme--catppuccin-latte .pagination.is-centered .pagination-previous{order:1}html.theme--catppuccin-latte .pagination.is-centered .pagination-list{justify-content:center;order:2}html.theme--catppuccin-latte .pagination.is-centered .pagination-next{order:3}html.theme--catppuccin-latte .pagination.is-right .pagination-previous{order:1}html.theme--catppuccin-latte .pagination.is-right .pagination-next{order:2}html.theme--catppuccin-latte .pagination.is-right .pagination-list{justify-content:flex-end;order:3}}html.theme--catppuccin-latte .panel{border-radius:8px;box-shadow:#171717;font-size:1rem}html.theme--catppuccin-latte .panel:not(:last-child){margin-bottom:1.5rem}html.theme--catppuccin-latte .panel.is-white .panel-heading{background-color:#fff;color:#0a0a0a}html.theme--catppuccin-latte .panel.is-white .panel-tabs a.is-active{border-bottom-color:#fff}html.theme--catppuccin-latte .panel.is-white .panel-block.is-active .panel-icon{color:#fff}html.theme--catppuccin-latte .panel.is-black .panel-heading{background-color:#0a0a0a;color:#fff}html.theme--catppuccin-latte .panel.is-black .panel-tabs a.is-active{border-bottom-color:#0a0a0a}html.theme--catppuccin-latte .panel.is-black .panel-block.is-active .panel-icon{color:#0a0a0a}html.theme--catppuccin-latte .panel.is-light .panel-heading{background-color:#f5f5f5;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .panel.is-light .panel-tabs a.is-active{border-bottom-color:#f5f5f5}html.theme--catppuccin-latte .panel.is-light .panel-block.is-active .panel-icon{color:#f5f5f5}html.theme--catppuccin-latte .panel.is-dark .panel-heading,html.theme--catppuccin-latte .content kbd.panel .panel-heading{background-color:#ccd0da;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .panel.is-dark .panel-tabs a.is-active,html.theme--catppuccin-latte .content kbd.panel .panel-tabs a.is-active{border-bottom-color:#ccd0da}html.theme--catppuccin-latte .panel.is-dark .panel-block.is-active .panel-icon,html.theme--catppuccin-latte .content kbd.panel .panel-block.is-active .panel-icon{color:#ccd0da}html.theme--catppuccin-latte .panel.is-primary .panel-heading,html.theme--catppuccin-latte .docstring>section>a.panel.docs-sourcelink .panel-heading{background-color:#1e66f5;color:#fff}html.theme--catppuccin-latte .panel.is-primary .panel-tabs a.is-active,html.theme--catppuccin-latte .docstring>section>a.panel.docs-sourcelink .panel-tabs a.is-active{border-bottom-color:#1e66f5}html.theme--catppuccin-latte .panel.is-primary .panel-block.is-active .panel-icon,html.theme--catppuccin-latte .docstring>section>a.panel.docs-sourcelink .panel-block.is-active .panel-icon{color:#1e66f5}html.theme--catppuccin-latte .panel.is-link .panel-heading{background-color:#1e66f5;color:#fff}html.theme--catppuccin-latte .panel.is-link .panel-tabs a.is-active{border-bottom-color:#1e66f5}html.theme--catppuccin-latte .panel.is-link .panel-block.is-active .panel-icon{color:#1e66f5}html.theme--catppuccin-latte .panel.is-info 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kbd.button.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-dark.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .content kbd.button.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-dark.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .content kbd.button.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-dark.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-macchiato .content kbd.button.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-dark.is-outlined[disabled],html.theme--catppuccin-macchiato .content kbd.button.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-dark.is-outlined,fieldset[disabled] html.theme--catppuccin-macchiato .content kbd.button.is-outlined{background-color:transparent;border-color:#363a4f;box-shadow:none;color:#363a4f}html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined.is-focused{background-color:#fff;color:#363a4f}html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #363a4f #363a4f !important}html.theme--catppuccin-macchiato 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.button.is-link.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-link.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-link.is-outlined.is-focused{background-color:#8aadf4;border-color:#8aadf4;color:#fff}html.theme--catppuccin-macchiato .button.is-link.is-outlined.is-loading::after{border-color:transparent transparent #8aadf4 #8aadf4 !important}html.theme--catppuccin-macchiato .button.is-link.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-link.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-link.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-link.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-link.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-link.is-outlined{background-color:transparent;border-color:#8aadf4;box-shadow:none;color:#8aadf4}html.theme--catppuccin-macchiato .button.is-link.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-macchiato .button.is-link.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-link.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-link.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-link.is-inverted.is-outlined.is-focused{background-color:#fff;color:#8aadf4}html.theme--catppuccin-macchiato .button.is-link.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-link.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-link.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-link.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #8aadf4 #8aadf4 !important}html.theme--catppuccin-macchiato .button.is-link.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-link.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-macchiato .button.is-link.is-light{background-color:#ecf2fd;color:#0e3b95}html.theme--catppuccin-macchiato .button.is-link.is-light:hover,html.theme--catppuccin-macchiato .button.is-link.is-light.is-hovered{background-color:#e1eafc;border-color:transparent;color:#0e3b95}html.theme--catppuccin-macchiato .button.is-link.is-light:active,html.theme--catppuccin-macchiato .button.is-link.is-light.is-active{background-color:#d5e2fb;border-color:transparent;color:#0e3b95}html.theme--catppuccin-macchiato 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.button.is-info{background-color:#8bd5ca;border-color:#8bd5ca;box-shadow:none}html.theme--catppuccin-macchiato .button.is-info.is-inverted{background-color:rgba(0,0,0,0.7);color:#8bd5ca}html.theme--catppuccin-macchiato .button.is-info.is-inverted:hover,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-info.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-info.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#8bd5ca}html.theme--catppuccin-macchiato .button.is-info.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-info.is-outlined{background-color:transparent;border-color:#8bd5ca;color:#8bd5ca}html.theme--catppuccin-macchiato .button.is-info.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-info.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-focused{background-color:#8bd5ca;border-color:#8bd5ca;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-loading::after{border-color:transparent transparent #8bd5ca #8bd5ca !important}html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-info.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-info.is-outlined{background-color:transparent;border-color:#8bd5ca;box-shadow:none;color:#8bd5ca}html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#8bd5ca}html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #8bd5ca #8bd5ca !important}html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-info.is-light{background-color:#f0faf8;color:#276d62}html.theme--catppuccin-macchiato .button.is-info.is-light:hover,html.theme--catppuccin-macchiato .button.is-info.is-light.is-hovered{background-color:#e7f6f4;border-color:transparent;color:#276d62}html.theme--catppuccin-macchiato .button.is-info.is-light:active,html.theme--catppuccin-macchiato .button.is-info.is-light.is-active{background-color:#ddf3f0;border-color:transparent;color:#276d62}html.theme--catppuccin-macchiato .button.is-success{background-color:#a6da95;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success:hover,html.theme--catppuccin-macchiato .button.is-success.is-hovered{background-color:#9ed78c;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success:focus,html.theme--catppuccin-macchiato .button.is-success.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success:focus:not(:active),html.theme--catppuccin-macchiato .button.is-success.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(166,218,149,0.25)}html.theme--catppuccin-macchiato .button.is-success:active,html.theme--catppuccin-macchiato .button.is-success.is-active{background-color:#96d382;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-success{background-color:#a6da95;border-color:#a6da95;box-shadow:none}html.theme--catppuccin-macchiato .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);color:#a6da95}html.theme--catppuccin-macchiato .button.is-success.is-inverted:hover,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#a6da95}html.theme--catppuccin-macchiato .button.is-success.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-success.is-outlined{background-color:transparent;border-color:#a6da95;color:#a6da95}html.theme--catppuccin-macchiato .button.is-success.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-success.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-focused{background-color:#a6da95;border-color:#a6da95;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading::after{border-color:transparent transparent #a6da95 #a6da95 !important}html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-success.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-success.is-outlined{background-color:transparent;border-color:#a6da95;box-shadow:none;color:#a6da95}html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#a6da95}html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #a6da95 #a6da95 !important}html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success.is-light{background-color:#f2faf0;color:#386e26}html.theme--catppuccin-macchiato .button.is-success.is-light:hover,html.theme--catppuccin-macchiato .button.is-success.is-light.is-hovered{background-color:#eaf6e6;border-color:transparent;color:#386e26}html.theme--catppuccin-macchiato .button.is-success.is-light:active,html.theme--catppuccin-macchiato .button.is-success.is-light.is-active{background-color:#e2f3dd;border-color:transparent;color:#386e26}html.theme--catppuccin-macchiato .button.is-warning{background-color:#eed49f;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning:hover,html.theme--catppuccin-macchiato .button.is-warning.is-hovered{background-color:#eccf94;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning:focus,html.theme--catppuccin-macchiato .button.is-warning.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning:focus:not(:active),html.theme--catppuccin-macchiato .button.is-warning.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(238,212,159,0.25)}html.theme--catppuccin-macchiato .button.is-warning:active,html.theme--catppuccin-macchiato .button.is-warning.is-active{background-color:#eaca89;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-warning{background-color:#eed49f;border-color:#eed49f;box-shadow:none}html.theme--catppuccin-macchiato .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-inverted:hover,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-warning.is-outlined{background-color:transparent;border-color:#eed49f;color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-warning.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-focused{background-color:#eed49f;border-color:#eed49f;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading::after{border-color:transparent transparent #eed49f #eed49f !important}html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-warning.is-outlined{background-color:transparent;border-color:#eed49f;box-shadow:none;color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #eed49f #eed49f !important}html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning.is-light{background-color:#fcf7ee;color:#7e5c16}html.theme--catppuccin-macchiato .button.is-warning.is-light:hover,html.theme--catppuccin-macchiato .button.is-warning.is-light.is-hovered{background-color:#faf2e3;border-color:transparent;color:#7e5c16}html.theme--catppuccin-macchiato .button.is-warning.is-light:active,html.theme--catppuccin-macchiato .button.is-warning.is-light.is-active{background-color:#f8eed8;border-color:transparent;color:#7e5c16}html.theme--catppuccin-macchiato .button.is-danger{background-color:#ed8796;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-danger:hover,html.theme--catppuccin-macchiato .button.is-danger.is-hovered{background-color:#eb7c8c;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-danger:focus,html.theme--catppuccin-macchiato .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-danger:focus:not(:active),html.theme--catppuccin-macchiato .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(237,135,150,0.25)}html.theme--catppuccin-macchiato .button.is-danger:active,html.theme--catppuccin-macchiato .button.is-danger.is-active{background-color:#ea7183;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-danger[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-danger{background-color:#ed8796;border-color:#ed8796;box-shadow:none}html.theme--catppuccin-macchiato .button.is-danger.is-inverted{background-color:#fff;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-inverted:hover,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-macchiato .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-danger.is-outlined{background-color:transparent;border-color:#ed8796;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-outlined:hover,html.theme--catppuccin-macchiato 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.button.is-danger.is-outlined{background-color:transparent;border-color:#ed8796;box-shadow:none;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ed8796 #ed8796 !important}html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-macchiato .button.is-danger.is-light{background-color:#fcedef;color:#971729}html.theme--catppuccin-macchiato .button.is-danger.is-light:hover,html.theme--catppuccin-macchiato .button.is-danger.is-light.is-hovered{background-color:#fbe2e6;border-color:transparent;color:#971729}html.theme--catppuccin-macchiato .button.is-danger.is-light:active,html.theme--catppuccin-macchiato .button.is-danger.is-light.is-active{background-color:#f9d7dc;border-color:transparent;color:#971729}html.theme--catppuccin-macchiato .button.is-small,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--catppuccin-macchiato .button.is-small:not(.is-rounded),html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--catppuccin-macchiato .button.is-normal{font-size:1rem}html.theme--catppuccin-macchiato .button.is-medium{font-size:1.25rem}html.theme--catppuccin-macchiato .button.is-large{font-size:1.5rem}html.theme--catppuccin-macchiato .button[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button{background-color:#6e738d;border-color:#5b6078;box-shadow:none;opacity:.5}html.theme--catppuccin-macchiato .button.is-fullwidth{display:flex;width:100%}html.theme--catppuccin-macchiato .button.is-loading{color:transparent !important;pointer-events:none}html.theme--catppuccin-macchiato .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--catppuccin-macchiato .button.is-static{background-color:#1e2030;border-color:#5b6078;color:#8087a2;box-shadow:none;pointer-events:none}html.theme--catppuccin-macchiato .button.is-rounded,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--catppuccin-macchiato .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--catppuccin-macchiato .buttons .button{margin-bottom:0.5rem}html.theme--catppuccin-macchiato .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--catppuccin-macchiato .buttons:last-child{margin-bottom:-0.5rem}html.theme--catppuccin-macchiato .buttons:not(:last-child){margin-bottom:1rem}html.theme--catppuccin-macchiato .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--catppuccin-macchiato .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--catppuccin-macchiato .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--catppuccin-macchiato .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--catppuccin-macchiato .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-macchiato .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--catppuccin-macchiato .buttons.has-addons .button:last-child{margin-right:0}html.theme--catppuccin-macchiato .buttons.has-addons .button:hover,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-hovered{z-index:2}html.theme--catppuccin-macchiato .buttons.has-addons .button:focus,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-focused,html.theme--catppuccin-macchiato .buttons.has-addons .button:active,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-active,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-selected{z-index:3}html.theme--catppuccin-macchiato .buttons.has-addons .button:focus:hover,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-focused:hover,html.theme--catppuccin-macchiato .buttons.has-addons .button:active:hover,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-active:hover,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--catppuccin-macchiato .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-macchiato .buttons.is-centered{justify-content:center}html.theme--catppuccin-macchiato .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--catppuccin-macchiato .buttons.is-right{justify-content:flex-end}html.theme--catppuccin-macchiato .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--catppuccin-macchiato .button.is-responsive.is-small,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--catppuccin-macchiato .button.is-responsive,html.theme--catppuccin-macchiato .button.is-responsive.is-normal{font-size:.65625rem}html.theme--catppuccin-macchiato .button.is-responsive.is-medium{font-size:.75rem}html.theme--catppuccin-macchiato .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--catppuccin-macchiato .button.is-responsive.is-small,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--catppuccin-macchiato .button.is-responsive,html.theme--catppuccin-macchiato .button.is-responsive.is-normal{font-size:.75rem}html.theme--catppuccin-macchiato .button.is-responsive.is-medium{font-size:1rem}html.theme--catppuccin-macchiato .button.is-responsive.is-large{font-size:1.25rem}}html.theme--catppuccin-macchiato .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--catppuccin-macchiato .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--catppuccin-macchiato .container{max-width:992px}}@media screen and (max-width: 1215px){html.theme--catppuccin-macchiato .container.is-widescreen:not(.is-max-desktop){max-width:1152px}}@media screen and (max-width: 1407px){html.theme--catppuccin-macchiato .container.is-fullhd:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}@media screen and (min-width: 1216px){html.theme--catppuccin-macchiato .container:not(.is-max-desktop){max-width:1152px}}@media screen and (min-width: 1408px){html.theme--catppuccin-macchiato .container:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}html.theme--catppuccin-macchiato .content li+li{margin-top:0.25em}html.theme--catppuccin-macchiato .content p:not(:last-child),html.theme--catppuccin-macchiato .content dl:not(:last-child),html.theme--catppuccin-macchiato .content ol:not(:last-child),html.theme--catppuccin-macchiato .content ul:not(:last-child),html.theme--catppuccin-macchiato .content blockquote:not(:last-child),html.theme--catppuccin-macchiato .content pre:not(:last-child),html.theme--catppuccin-macchiato .content table:not(:last-child){margin-bottom:1em}html.theme--catppuccin-macchiato .content h1,html.theme--catppuccin-macchiato .content h2,html.theme--catppuccin-macchiato .content h3,html.theme--catppuccin-macchiato .content h4,html.theme--catppuccin-macchiato .content h5,html.theme--catppuccin-macchiato .content h6{color:#cad3f5;font-weight:600;line-height:1.125}html.theme--catppuccin-macchiato .content h1{font-size:2em;margin-bottom:0.5em}html.theme--catppuccin-macchiato .content h1:not(:first-child){margin-top:1em}html.theme--catppuccin-macchiato .content h2{font-size:1.75em;margin-bottom:0.5714em}html.theme--catppuccin-macchiato .content h2:not(:first-child){margin-top:1.1428em}html.theme--catppuccin-macchiato .content h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--catppuccin-macchiato .content h3:not(:first-child){margin-top:1.3333em}html.theme--catppuccin-macchiato .content h4{font-size:1.25em;margin-bottom:0.8em}html.theme--catppuccin-macchiato .content h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--catppuccin-macchiato .content h6{font-size:1em;margin-bottom:1em}html.theme--catppuccin-macchiato .content blockquote{background-color:#1e2030;border-left:5px solid #5b6078;padding:1.25em 1.5em}html.theme--catppuccin-macchiato .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-macchiato .content ol:not([type]){list-style-type:decimal}html.theme--catppuccin-macchiato .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--catppuccin-macchiato .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--catppuccin-macchiato .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--catppuccin-macchiato .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--catppuccin-macchiato .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-macchiato .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--catppuccin-macchiato .content ul ul ul{list-style-type:square}html.theme--catppuccin-macchiato .content dd{margin-left:2em}html.theme--catppuccin-macchiato .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--catppuccin-macchiato .content figure:not(:first-child){margin-top:2em}html.theme--catppuccin-macchiato .content figure:not(:last-child){margin-bottom:2em}html.theme--catppuccin-macchiato .content figure img{display:inline-block}html.theme--catppuccin-macchiato .content figure figcaption{font-style:italic}html.theme--catppuccin-macchiato .content pre{-webkit-overflow-scrolling:touch;overflow-x:auto;padding:0;white-space:pre;word-wrap:normal}html.theme--catppuccin-macchiato .content sup,html.theme--catppuccin-macchiato .content sub{font-size:75%}html.theme--catppuccin-macchiato .content table{width:100%}html.theme--catppuccin-macchiato .content table td,html.theme--catppuccin-macchiato .content table th{border:1px solid #5b6078;border-width:0 0 1px;padding:0.5em 0.75em;vertical-align:top}html.theme--catppuccin-macchiato .content table th{color:#b5c1f1}html.theme--catppuccin-macchiato .content table th:not([align]){text-align:inherit}html.theme--catppuccin-macchiato .content table thead td,html.theme--catppuccin-macchiato .content table thead th{border-width:0 0 2px;color:#b5c1f1}html.theme--catppuccin-macchiato .content table tfoot td,html.theme--catppuccin-macchiato .content table tfoot th{border-width:2px 0 0;color:#b5c1f1}html.theme--catppuccin-macchiato .content table tbody tr:last-child td,html.theme--catppuccin-macchiato .content table tbody tr:last-child th{border-bottom-width:0}html.theme--catppuccin-macchiato .content .tabs li+li{margin-top:0}html.theme--catppuccin-macchiato .content.is-small,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.content{font-size:.75rem}html.theme--catppuccin-macchiato .content.is-normal{font-size:1rem}html.theme--catppuccin-macchiato .content.is-medium{font-size:1.25rem}html.theme--catppuccin-macchiato .content.is-large{font-size:1.5rem}html.theme--catppuccin-macchiato .icon{align-items:center;display:inline-flex;justify-content:center;height:1.5rem;width:1.5rem}html.theme--catppuccin-macchiato .icon.is-small,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.icon{height:1rem;width:1rem}html.theme--catppuccin-macchiato .icon.is-medium{height:2rem;width:2rem}html.theme--catppuccin-macchiato .icon.is-large{height:3rem;width:3rem}html.theme--catppuccin-macchiato .icon-text{align-items:flex-start;color:inherit;display:inline-flex;flex-wrap:wrap;line-height:1.5rem;vertical-align:top}html.theme--catppuccin-macchiato .icon-text .icon{flex-grow:0;flex-shrink:0}html.theme--catppuccin-macchiato .icon-text .icon:not(:last-child){margin-right:.25em}html.theme--catppuccin-macchiato .icon-text .icon:not(:first-child){margin-left:.25em}html.theme--catppuccin-macchiato div.icon-text{display:flex}html.theme--catppuccin-macchiato .image,html.theme--catppuccin-macchiato #documenter .docs-sidebar .docs-logo>img{display:block;position:relative}html.theme--catppuccin-macchiato .image img,html.theme--catppuccin-macchiato #documenter .docs-sidebar .docs-logo>img img{display:block;height:auto;width:100%}html.theme--catppuccin-macchiato .image img.is-rounded,html.theme--catppuccin-macchiato #documenter .docs-sidebar .docs-logo>img img.is-rounded{border-radius:9999px}html.theme--catppuccin-macchiato .image.is-fullwidth,html.theme--catppuccin-macchiato #documenter .docs-sidebar .docs-logo>img.is-fullwidth{width:100%}html.theme--catppuccin-macchiato .image.is-square img,html.theme--catppuccin-macchiato #documenter .docs-sidebar .docs-logo>img.is-square img,html.theme--catppuccin-macchiato .image.is-square .has-ratio,html.theme--catppuccin-macchiato #documenter .docs-sidebar .docs-logo>img.is-square .has-ratio,html.theme--catppuccin-macchiato .image.is-1by1 img,html.theme--catppuccin-macchiato #documenter .docs-sidebar .docs-logo>img.is-1by1 img,html.theme--catppuccin-macchiato .image.is-1by1 .has-ratio,html.theme--catppuccin-macchiato #documenter .docs-sidebar .docs-logo>img.is-1by1 .has-ratio,html.theme--catppuccin-macchiato .image.is-5by4 img,html.theme--catppuccin-macchiato #documenter .docs-sidebar .docs-logo>img.is-5by4 img,html.theme--catppuccin-macchiato .image.is-5by4 .has-ratio,html.theme--catppuccin-macchiato #documenter .docs-sidebar .docs-logo>img.is-5by4 .has-ratio,html.theme--catppuccin-macchiato .image.is-4by3 img,html.theme--catppuccin-macchiato #documenter .docs-sidebar .docs-logo>img.is-4by3 img,html.theme--catppuccin-macchiato .image.is-4by3 .has-ratio,html.theme--catppuccin-macchiato #documenter .docs-sidebar .docs-logo>img.is-4by3 .has-ratio,html.theme--catppuccin-macchiato .image.is-3by2 img,html.theme--catppuccin-macchiato #documenter .docs-sidebar .docs-logo>img.is-3by2 img,html.theme--catppuccin-macchiato .image.is-3by2 .has-ratio,html.theme--catppuccin-macchiato #documenter .docs-sidebar .docs-logo>img.is-3by2 .has-ratio,html.theme--catppuccin-macchiato .image.is-5by3 img,html.theme--catppuccin-macchiato #documenter .docs-sidebar .docs-logo>img.is-5by3 img,html.theme--catppuccin-macchiato .image.is-5by3 .has-ratio,html.theme--catppuccin-macchiato #documenter .docs-sidebar .docs-logo>img.is-5by3 .has-ratio,html.theme--catppuccin-macchiato .image.is-16by9 img,html.theme--catppuccin-macchiato #documenter .docs-sidebar .docs-logo>img.is-16by9 img,html.theme--catppuccin-macchiato .image.is-16by9 .has-ratio,html.theme--catppuccin-macchiato #documenter .docs-sidebar .docs-logo>img.is-16by9 .has-ratio,html.theme--catppuccin-macchiato .image.is-2by1 img,html.theme--catppuccin-macchiato #documenter .docs-sidebar .docs-logo>img.is-2by1 img,html.theme--catppuccin-macchiato .image.is-2by1 .has-ratio,html.theme--catppuccin-macchiato #documenter .docs-sidebar .docs-logo>img.is-2by1 .has-ratio,html.theme--catppuccin-macchiato .image.is-3by1 img,html.theme--catppuccin-macchiato #documenter .docs-sidebar .docs-logo>img.is-3by1 img,html.theme--catppuccin-macchiato .image.is-3by1 .has-ratio,html.theme--catppuccin-macchiato #documenter .docs-sidebar .docs-logo>img.is-3by1 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select[disabled]:-moz-placeholder,html.theme--catppuccin-macchiato .textarea[disabled]:-moz-placeholder,html.theme--catppuccin-macchiato .input[disabled]:-moz-placeholder,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input[disabled]:-moz-placeholder,fieldset[disabled] html.theme--catppuccin-macchiato .select select:-moz-placeholder,fieldset[disabled] html.theme--catppuccin-macchiato .textarea:-moz-placeholder,fieldset[disabled] html.theme--catppuccin-macchiato .input:-moz-placeholder,fieldset[disabled] html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input:-moz-placeholder{color:rgba(245,247,253,0.3)}html.theme--catppuccin-macchiato .select select[disabled]:-ms-input-placeholder,html.theme--catppuccin-macchiato .textarea[disabled]:-ms-input-placeholder,html.theme--catppuccin-macchiato .input[disabled]:-ms-input-placeholder,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input[disabled]:-ms-input-placeholder,fieldset[disabled] html.theme--catppuccin-macchiato .select select:-ms-input-placeholder,fieldset[disabled] html.theme--catppuccin-macchiato .textarea:-ms-input-placeholder,fieldset[disabled] html.theme--catppuccin-macchiato .input:-ms-input-placeholder,fieldset[disabled] html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input:-ms-input-placeholder{color:rgba(245,247,253,0.3)}html.theme--catppuccin-macchiato .textarea,html.theme--catppuccin-macchiato .input,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input{box-shadow:inset 0 0.0625em 0.125em rgba(10,10,10,0.05);max-width:100%;width:100%}html.theme--catppuccin-macchiato .textarea[readonly],html.theme--catppuccin-macchiato .input[readonly],html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input[readonly]{box-shadow:none}html.theme--catppuccin-macchiato 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form.docs-search>input.is-active{box-shadow:0 0 0 0.125em rgba(255,255,255,0.25)}html.theme--catppuccin-macchiato .is-black.textarea,html.theme--catppuccin-macchiato .is-black.input,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.is-black{border-color:#0a0a0a}html.theme--catppuccin-macchiato .is-black.textarea:focus,html.theme--catppuccin-macchiato .is-black.input:focus,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.is-black:focus,html.theme--catppuccin-macchiato .is-black.is-focused.textarea,html.theme--catppuccin-macchiato .is-black.is-focused.input,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.is-focused,html.theme--catppuccin-macchiato .is-black.textarea:active,html.theme--catppuccin-macchiato .is-black.input:active,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.is-black:active,html.theme--catppuccin-macchiato 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kbd.button.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined.is-focused{background-color:#fff;color:#313244}html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .content 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html.theme--catppuccin-mocha .button.is-link.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-mocha .button.is-link.is-light{background-color:#ebf3fe;color:#063c93}html.theme--catppuccin-mocha .button.is-link.is-light:hover,html.theme--catppuccin-mocha .button.is-link.is-light.is-hovered{background-color:#dfebfe;border-color:transparent;color:#063c93}html.theme--catppuccin-mocha .button.is-link.is-light:active,html.theme--catppuccin-mocha .button.is-link.is-light.is-active{background-color:#d3e3fd;border-color:transparent;color:#063c93}html.theme--catppuccin-mocha .button.is-info{background-color:#94e2d5;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info:hover,html.theme--catppuccin-mocha .button.is-info.is-hovered{background-color:#8adfd1;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info:focus,html.theme--catppuccin-mocha .button.is-info.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info:focus:not(:active),html.theme--catppuccin-mocha .button.is-info.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(148,226,213,0.25)}html.theme--catppuccin-mocha .button.is-info:active,html.theme--catppuccin-mocha .button.is-info.is-active{background-color:#80ddcd;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-info{background-color:#94e2d5;border-color:#94e2d5;box-shadow:none}html.theme--catppuccin-mocha .button.is-info.is-inverted{background-color:rgba(0,0,0,0.7);color:#94e2d5}html.theme--catppuccin-mocha .button.is-info.is-inverted:hover,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-info.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#94e2d5}html.theme--catppuccin-mocha .button.is-info.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-info.is-outlined{background-color:transparent;border-color:#94e2d5;color:#94e2d5}html.theme--catppuccin-mocha .button.is-info.is-outlined:hover,html.theme--catppuccin-mocha .button.is-info.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-info.is-outlined:focus,html.theme--catppuccin-mocha .button.is-info.is-outlined.is-focused{background-color:#94e2d5;border-color:#94e2d5;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info.is-outlined.is-loading::after{border-color:transparent transparent #94e2d5 #94e2d5 !important}html.theme--catppuccin-mocha .button.is-info.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-info.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-info.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-info.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-info.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-info.is-outlined{background-color:transparent;border-color:#94e2d5;box-shadow:none;color:#94e2d5}html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#94e2d5}html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #94e2d5 #94e2d5 !important}html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info.is-light{background-color:#effbf9;color:#207466}html.theme--catppuccin-mocha .button.is-info.is-light:hover,html.theme--catppuccin-mocha .button.is-info.is-light.is-hovered{background-color:#e5f8f5;border-color:transparent;color:#207466}html.theme--catppuccin-mocha .button.is-info.is-light:active,html.theme--catppuccin-mocha .button.is-info.is-light.is-active{background-color:#dbf5f1;border-color:transparent;color:#207466}html.theme--catppuccin-mocha .button.is-success{background-color:#a6e3a1;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success:hover,html.theme--catppuccin-mocha .button.is-success.is-hovered{background-color:#9de097;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success:focus,html.theme--catppuccin-mocha .button.is-success.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success:focus:not(:active),html.theme--catppuccin-mocha .button.is-success.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(166,227,161,0.25)}html.theme--catppuccin-mocha .button.is-success:active,html.theme--catppuccin-mocha .button.is-success.is-active{background-color:#93dd8d;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-success{background-color:#a6e3a1;border-color:#a6e3a1;box-shadow:none}html.theme--catppuccin-mocha .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-inverted:hover,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-success.is-outlined{background-color:transparent;border-color:#a6e3a1;color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-outlined:hover,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-success.is-outlined:focus,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-focused{background-color:#a6e3a1;border-color:#a6e3a1;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading::after{border-color:transparent transparent #a6e3a1 #a6e3a1 !important}html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-success.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-success.is-outlined{background-color:transparent;border-color:#a6e3a1;box-shadow:none;color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #a6e3a1 #a6e3a1 !important}html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success.is-light{background-color:#f0faef;color:#287222}html.theme--catppuccin-mocha .button.is-success.is-light:hover,html.theme--catppuccin-mocha .button.is-success.is-light.is-hovered{background-color:#e7f7e5;border-color:transparent;color:#287222}html.theme--catppuccin-mocha .button.is-success.is-light:active,html.theme--catppuccin-mocha .button.is-success.is-light.is-active{background-color:#def4dc;border-color:transparent;color:#287222}html.theme--catppuccin-mocha .button.is-warning{background-color:#f9e2af;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning:hover,html.theme--catppuccin-mocha .button.is-warning.is-hovered{background-color:#f8dea3;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning:focus,html.theme--catppuccin-mocha .button.is-warning.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning:focus:not(:active),html.theme--catppuccin-mocha .button.is-warning.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(249,226,175,0.25)}html.theme--catppuccin-mocha .button.is-warning:active,html.theme--catppuccin-mocha .button.is-warning.is-active{background-color:#f7d997;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-warning{background-color:#f9e2af;border-color:#f9e2af;box-shadow:none}html.theme--catppuccin-mocha .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-inverted:hover,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-warning.is-outlined{background-color:transparent;border-color:#f9e2af;color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-outlined:hover,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-warning.is-outlined:focus,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-focused{background-color:#f9e2af;border-color:#f9e2af;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading::after{border-color:transparent transparent #f9e2af #f9e2af !important}html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-warning.is-outlined{background-color:transparent;border-color:#f9e2af;box-shadow:none;color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #f9e2af #f9e2af !important}html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning.is-light{background-color:#fef8ec;color:#8a620a}html.theme--catppuccin-mocha .button.is-warning.is-light:hover,html.theme--catppuccin-mocha .button.is-warning.is-light.is-hovered{background-color:#fdf4e0;border-color:transparent;color:#8a620a}html.theme--catppuccin-mocha .button.is-warning.is-light:active,html.theme--catppuccin-mocha .button.is-warning.is-light.is-active{background-color:#fcf0d4;border-color:transparent;color:#8a620a}html.theme--catppuccin-mocha .button.is-danger{background-color:#f38ba8;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-danger:hover,html.theme--catppuccin-mocha .button.is-danger.is-hovered{background-color:#f27f9f;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-danger:focus,html.theme--catppuccin-mocha .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-danger:focus:not(:active),html.theme--catppuccin-mocha .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(243,139,168,0.25)}html.theme--catppuccin-mocha .button.is-danger:active,html.theme--catppuccin-mocha .button.is-danger.is-active{background-color:#f17497;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-danger[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-danger{background-color:#f38ba8;border-color:#f38ba8;box-shadow:none}html.theme--catppuccin-mocha .button.is-danger.is-inverted{background-color:#fff;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-inverted:hover,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-mocha .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-mocha .button.is-danger.is-outlined{background-color:transparent;border-color:#f38ba8;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-outlined:hover,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-danger.is-outlined:focus,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-focused{background-color:#f38ba8;border-color:#f38ba8;color:#fff}html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #f38ba8 #f38ba8 !important}html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-mocha .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-danger.is-outlined{background-color:transparent;border-color:#f38ba8;box-shadow:none;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #f38ba8 #f38ba8 !important}html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-mocha .button.is-danger.is-light{background-color:#fdedf1;color:#991036}html.theme--catppuccin-mocha .button.is-danger.is-light:hover,html.theme--catppuccin-mocha .button.is-danger.is-light.is-hovered{background-color:#fce1e8;border-color:transparent;color:#991036}html.theme--catppuccin-mocha .button.is-danger.is-light:active,html.theme--catppuccin-mocha .button.is-danger.is-light.is-active{background-color:#fbd5e0;border-color:transparent;color:#991036}html.theme--catppuccin-mocha .button.is-small,html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--catppuccin-mocha .button.is-small:not(.is-rounded),html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--catppuccin-mocha .button.is-normal{font-size:1rem}html.theme--catppuccin-mocha .button.is-medium{font-size:1.25rem}html.theme--catppuccin-mocha .button.is-large{font-size:1.5rem}html.theme--catppuccin-mocha .button[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button{background-color:#6c7086;border-color:#585b70;box-shadow:none;opacity:.5}html.theme--catppuccin-mocha .button.is-fullwidth{display:flex;width:100%}html.theme--catppuccin-mocha .button.is-loading{color:transparent !important;pointer-events:none}html.theme--catppuccin-mocha .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--catppuccin-mocha .button.is-static{background-color:#181825;border-color:#585b70;color:#7f849c;box-shadow:none;pointer-events:none}html.theme--catppuccin-mocha .button.is-rounded,html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--catppuccin-mocha .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--catppuccin-mocha .buttons .button{margin-bottom:0.5rem}html.theme--catppuccin-mocha .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--catppuccin-mocha .buttons:last-child{margin-bottom:-0.5rem}html.theme--catppuccin-mocha .buttons:not(:last-child){margin-bottom:1rem}html.theme--catppuccin-mocha .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--catppuccin-mocha .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--catppuccin-mocha .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--catppuccin-mocha .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--catppuccin-mocha .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-mocha .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--catppuccin-mocha .buttons.has-addons .button:last-child{margin-right:0}html.theme--catppuccin-mocha .buttons.has-addons .button:hover,html.theme--catppuccin-mocha .buttons.has-addons .button.is-hovered{z-index:2}html.theme--catppuccin-mocha .buttons.has-addons .button:focus,html.theme--catppuccin-mocha .buttons.has-addons .button.is-focused,html.theme--catppuccin-mocha .buttons.has-addons .button:active,html.theme--catppuccin-mocha .buttons.has-addons .button.is-active,html.theme--catppuccin-mocha .buttons.has-addons .button.is-selected{z-index:3}html.theme--catppuccin-mocha .buttons.has-addons .button:focus:hover,html.theme--catppuccin-mocha .buttons.has-addons .button.is-focused:hover,html.theme--catppuccin-mocha .buttons.has-addons .button:active:hover,html.theme--catppuccin-mocha .buttons.has-addons .button.is-active:hover,html.theme--catppuccin-mocha .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--catppuccin-mocha .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-mocha .buttons.is-centered{justify-content:center}html.theme--catppuccin-mocha .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--catppuccin-mocha .buttons.is-right{justify-content:flex-end}html.theme--catppuccin-mocha .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--catppuccin-mocha .button.is-responsive.is-small,html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--catppuccin-mocha .button.is-responsive,html.theme--catppuccin-mocha .button.is-responsive.is-normal{font-size:.65625rem}html.theme--catppuccin-mocha .button.is-responsive.is-medium{font-size:.75rem}html.theme--catppuccin-mocha .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--catppuccin-mocha .button.is-responsive.is-small,html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--catppuccin-mocha .button.is-responsive,html.theme--catppuccin-mocha .button.is-responsive.is-normal{font-size:.75rem}html.theme--catppuccin-mocha .button.is-responsive.is-medium{font-size:1rem}html.theme--catppuccin-mocha .button.is-responsive.is-large{font-size:1.25rem}}html.theme--catppuccin-mocha .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--catppuccin-mocha .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--catppuccin-mocha .container{max-width:992px}}@media screen and (max-width: 1215px){html.theme--catppuccin-mocha .container.is-widescreen:not(.is-max-desktop){max-width:1152px}}@media screen and (max-width: 1407px){html.theme--catppuccin-mocha .container.is-fullhd:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}@media screen and (min-width: 1216px){html.theme--catppuccin-mocha .container:not(.is-max-desktop){max-width:1152px}}@media screen and (min-width: 1408px){html.theme--catppuccin-mocha .container:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}html.theme--catppuccin-mocha .content li+li{margin-top:0.25em}html.theme--catppuccin-mocha .content p:not(:last-child),html.theme--catppuccin-mocha .content dl:not(:last-child),html.theme--catppuccin-mocha .content ol:not(:last-child),html.theme--catppuccin-mocha .content ul:not(:last-child),html.theme--catppuccin-mocha .content blockquote:not(:last-child),html.theme--catppuccin-mocha .content pre:not(:last-child),html.theme--catppuccin-mocha .content table:not(:last-child){margin-bottom:1em}html.theme--catppuccin-mocha .content h1,html.theme--catppuccin-mocha .content h2,html.theme--catppuccin-mocha .content h3,html.theme--catppuccin-mocha .content h4,html.theme--catppuccin-mocha .content h5,html.theme--catppuccin-mocha .content h6{color:#cdd6f4;font-weight:600;line-height:1.125}html.theme--catppuccin-mocha .content h1{font-size:2em;margin-bottom:0.5em}html.theme--catppuccin-mocha .content h1:not(:first-child){margin-top:1em}html.theme--catppuccin-mocha .content h2{font-size:1.75em;margin-bottom:0.5714em}html.theme--catppuccin-mocha .content h2:not(:first-child){margin-top:1.1428em}html.theme--catppuccin-mocha .content h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--catppuccin-mocha .content h3:not(:first-child){margin-top:1.3333em}html.theme--catppuccin-mocha .content h4{font-size:1.25em;margin-bottom:0.8em}html.theme--catppuccin-mocha .content h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--catppuccin-mocha .content h6{font-size:1em;margin-bottom:1em}html.theme--catppuccin-mocha .content blockquote{background-color:#181825;border-left:5px solid #585b70;padding:1.25em 1.5em}html.theme--catppuccin-mocha .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-mocha .content ol:not([type]){list-style-type:decimal}html.theme--catppuccin-mocha .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--catppuccin-mocha .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--catppuccin-mocha .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--catppuccin-mocha .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--catppuccin-mocha .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-mocha .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--catppuccin-mocha .content ul ul ul{list-style-type:square}html.theme--catppuccin-mocha .content dd{margin-left:2em}html.theme--catppuccin-mocha .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--catppuccin-mocha .content figure:not(:first-child){margin-top:2em}html.theme--catppuccin-mocha .content figure:not(:last-child){margin-bottom:2em}html.theme--catppuccin-mocha .content figure img{display:inline-block}html.theme--catppuccin-mocha .content figure figcaption{font-style:italic}html.theme--catppuccin-mocha .content pre{-webkit-overflow-scrolling:touch;overflow-x:auto;padding:0;white-space:pre;word-wrap:normal}html.theme--catppuccin-mocha .content sup,html.theme--catppuccin-mocha .content sub{font-size:75%}html.theme--catppuccin-mocha .content table{width:100%}html.theme--catppuccin-mocha .content table 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.content.is-normal{font-size:1rem}html.theme--catppuccin-mocha .content.is-medium{font-size:1.25rem}html.theme--catppuccin-mocha .content.is-large{font-size:1.5rem}html.theme--catppuccin-mocha .icon{align-items:center;display:inline-flex;justify-content:center;height:1.5rem;width:1.5rem}html.theme--catppuccin-mocha .icon.is-small,html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.icon{height:1rem;width:1rem}html.theme--catppuccin-mocha .icon.is-medium{height:2rem;width:2rem}html.theme--catppuccin-mocha .icon.is-large{height:3rem;width:3rem}html.theme--catppuccin-mocha .icon-text{align-items:flex-start;color:inherit;display:inline-flex;flex-wrap:wrap;line-height:1.5rem;vertical-align:top}html.theme--catppuccin-mocha .icon-text .icon{flex-grow:0;flex-shrink:0}html.theme--catppuccin-mocha .icon-text .icon:not(:last-child){margin-right:.25em}html.theme--catppuccin-mocha .icon-text .icon:not(:first-child){margin-left:.25em}html.theme--catppuccin-mocha div.icon-text{display:flex}html.theme--catppuccin-mocha .image,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img{display:block;position:relative}html.theme--catppuccin-mocha .image img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img img{display:block;height:auto;width:100%}html.theme--catppuccin-mocha .image img.is-rounded,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img img.is-rounded{border-radius:9999px}html.theme--catppuccin-mocha .image.is-fullwidth,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-fullwidth{width:100%}html.theme--catppuccin-mocha .image.is-square img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-square img,html.theme--catppuccin-mocha .image.is-square .has-ratio,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-square .has-ratio,html.theme--catppuccin-mocha .image.is-1by1 img,html.theme--catppuccin-mocha #documenter .docs-sidebar 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.has-ratio,html.theme--catppuccin-mocha .image.is-5by3 img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-5by3 img,html.theme--catppuccin-mocha .image.is-5by3 .has-ratio,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-5by3 .has-ratio,html.theme--catppuccin-mocha .image.is-16by9 img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-16by9 img,html.theme--catppuccin-mocha .image.is-16by9 .has-ratio,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-16by9 .has-ratio,html.theme--catppuccin-mocha .image.is-2by1 img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-2by1 img,html.theme--catppuccin-mocha .image.is-2by1 .has-ratio,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-2by1 .has-ratio,html.theme--catppuccin-mocha .image.is-3by1 img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-3by1 img,html.theme--catppuccin-mocha 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img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-3by5 img,html.theme--catppuccin-mocha .image.is-3by5 .has-ratio,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-3by5 .has-ratio,html.theme--catppuccin-mocha .image.is-9by16 img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-9by16 img,html.theme--catppuccin-mocha .image.is-9by16 .has-ratio,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-9by16 .has-ratio,html.theme--catppuccin-mocha .image.is-1by2 img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-1by2 img,html.theme--catppuccin-mocha .image.is-1by2 .has-ratio,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-1by2 .has-ratio,html.theme--catppuccin-mocha .image.is-1by3 img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-1by3 img,html.theme--catppuccin-mocha .image.is-1by3 .has-ratio,html.theme--catppuccin-mocha 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html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-danger.is-light{background-color:#fbefef;color:#c03930}html.theme--documenter-dark .button.is-danger.is-light:hover,html.theme--documenter-dark .button.is-danger.is-light.is-hovered{background-color:#f8e6e5;border-color:transparent;color:#c03930}html.theme--documenter-dark .button.is-danger.is-light:active,html.theme--documenter-dark .button.is-danger.is-light.is-active{background-color:#f6dcda;border-color:transparent;color:#c03930}html.theme--documenter-dark .button.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--documenter-dark .button.is-small:not(.is-rounded),html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--documenter-dark .button.is-normal{font-size:1rem}html.theme--documenter-dark .button.is-medium{font-size:1.25rem}html.theme--documenter-dark .button.is-large{font-size:1.5rem}html.theme--documenter-dark .button[disabled],fieldset[disabled] html.theme--documenter-dark .button{background-color:#8c9b9d;border-color:#5e6d6f;box-shadow:none;opacity:.5}html.theme--documenter-dark .button.is-fullwidth{display:flex;width:100%}html.theme--documenter-dark .button.is-loading{color:transparent !important;pointer-events:none}html.theme--documenter-dark .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--documenter-dark .button.is-static{background-color:#282f2f;border-color:#5e6d6f;color:#dbdee0;box-shadow:none;pointer-events:none}html.theme--documenter-dark .button.is-rounded,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 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.button:active:hover,html.theme--documenter-dark .buttons.has-addons .button.is-active:hover,html.theme--documenter-dark .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--documenter-dark .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--documenter-dark .buttons.is-centered{justify-content:center}html.theme--documenter-dark .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--documenter-dark .buttons.is-right{justify-content:flex-end}html.theme--documenter-dark .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--documenter-dark .button.is-responsive.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--documenter-dark .button.is-responsive,html.theme--documenter-dark .button.is-responsive.is-normal{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--documenter-dark .button.is-responsive.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive,html.theme--documenter-dark .button.is-responsive.is-normal{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:1rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1.25rem}}html.theme--documenter-dark .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--documenter-dark .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--documenter-dark 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pre:not(:last-child),html.theme--documenter-dark .content table:not(:last-child){margin-bottom:1em}html.theme--documenter-dark .content h1,html.theme--documenter-dark .content h2,html.theme--documenter-dark .content h3,html.theme--documenter-dark .content h4,html.theme--documenter-dark .content h5,html.theme--documenter-dark .content h6{color:#f2f2f2;font-weight:600;line-height:1.125}html.theme--documenter-dark .content h1{font-size:2em;margin-bottom:0.5em}html.theme--documenter-dark .content h1:not(:first-child){margin-top:1em}html.theme--documenter-dark .content h2{font-size:1.75em;margin-bottom:0.5714em}html.theme--documenter-dark .content h2:not(:first-child){margin-top:1.1428em}html.theme--documenter-dark .content h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--documenter-dark .content h3:not(:first-child){margin-top:1.3333em}html.theme--documenter-dark .content h4{font-size:1.25em;margin-bottom:0.8em}html.theme--documenter-dark .content h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--documenter-dark .content h6{font-size:1em;margin-bottom:1em}html.theme--documenter-dark .content blockquote{background-color:#282f2f;border-left:5px solid #5e6d6f;padding:1.25em 1.5em}html.theme--documenter-dark .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--documenter-dark .content ol:not([type]){list-style-type:decimal}html.theme--documenter-dark .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--documenter-dark .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--documenter-dark .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--documenter-dark .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--documenter-dark .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--documenter-dark .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--documenter-dark .content ul ul ul{list-style-type:square}html.theme--documenter-dark .content dd{margin-left:2em}html.theme--documenter-dark .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--documenter-dark .content figure:not(:first-child){margin-top:2em}html.theme--documenter-dark .content figure:not(:last-child){margin-bottom:2em}html.theme--documenter-dark .content figure img{display:inline-block}html.theme--documenter-dark .content figure figcaption{font-style:italic}html.theme--documenter-dark .content pre{-webkit-overflow-scrolling:touch;overflow-x:auto;padding:0;white-space:pre;word-wrap:normal}html.theme--documenter-dark .content sup,html.theme--documenter-dark .content sub{font-size:75%}html.theme--documenter-dark .content table{width:100%}html.theme--documenter-dark .content table td,html.theme--documenter-dark .content table th{border:1px solid #5e6d6f;border-width:0 0 1px;padding:0.5em 0.75em;vertical-align:top}html.theme--documenter-dark .content table th{color:#f2f2f2}html.theme--documenter-dark .content table th:not([align]){text-align:inherit}html.theme--documenter-dark .content table thead td,html.theme--documenter-dark .content table thead th{border-width:0 0 2px;color:#f2f2f2}html.theme--documenter-dark .content table tfoot td,html.theme--documenter-dark .content table tfoot th{border-width:2px 0 0;color:#f2f2f2}html.theme--documenter-dark .content table tbody tr:last-child td,html.theme--documenter-dark .content table tbody tr:last-child th{border-bottom-width:0}html.theme--documenter-dark .content .tabs li+li{margin-top:0}html.theme--documenter-dark .content.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.content{font-size:.75rem}html.theme--documenter-dark .content.is-normal{font-size:1rem}html.theme--documenter-dark .content.is-medium{font-size:1.25rem}html.theme--documenter-dark .content.is-large{font-size:1.5rem}html.theme--documenter-dark 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img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img img{display:block;height:auto;width:100%}html.theme--documenter-dark .image img.is-rounded,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img img.is-rounded{border-radius:9999px}html.theme--documenter-dark .image.is-fullwidth,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-fullwidth{width:100%}html.theme--documenter-dark .image.is-square img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-square img,html.theme--documenter-dark .image.is-square .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-square .has-ratio,html.theme--documenter-dark .image.is-1by1 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-1by1 img,html.theme--documenter-dark .image.is-1by1 .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-1by1 .has-ratio,html.theme--documenter-dark .image.is-5by4 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GitHub

Index

diff --git a/previews/PR114/index.html b/previews/PR114/index.html new file mode 100644 index 00000000..db33dd60 --- /dev/null +++ b/previews/PR114/index.html @@ -0,0 +1,2 @@ + +Home · ModelPredictiveControl.jl
GitHub

ModelPredictiveControl.jl

An open source model predictive control package for Julia.

The package depends on ControlSystemsBase.jl for the linear systems and JuMP.jl for the solving.

The objective is to provide a simple, clear and modular framework to quickly design model predictive controllers (MPCs) in Julia, while preserving the flexibility for advanced real-time optimization. Modern MPCs based on closed-loop state estimators are the main focus of the package, but classical approaches that rely on internal models are also possible. The JuMP.jl interface allows the user to test different solvers easily if the performance of the default settings is not satisfactory.

The documentation is divided in two parts:

  • Manual — This section includes step-by-step guides to design predictive controllers on multiple case studies.
  • Functions — Documentation of methods and types exported by the package. The "Internals" section provides implementation details of functions that are not exported.

Manual

Functions: Public

Functions: Internals

diff --git a/previews/PR114/internals/predictive_control/index.html b/previews/PR114/internals/predictive_control/index.html new file mode 100644 index 00000000..fc649c33 --- /dev/null +++ b/previews/PR114/internals/predictive_control/index.html @@ -0,0 +1,112 @@ + +Predictive Controllers · ModelPredictiveControl.jl
GitHub

Functions: PredictiveController Internals

The prediction methodology of this module is mainly based on Maciejowski textbook [1].

Controller Construction

ModelPredictiveControl.init_ΔUtoU — Function
init_ΔUtoU(model, Hp, Hc) -> S, T

Init manipulated input increments to inputs conversion matrices.

The conversion from the input increments $\mathbf{ΔU}$ to manipulated inputs over $H_p$ are calculated by:

\[\mathbf{U} = \mathbf{S} \mathbf{ΔU} + \mathbf{T} \mathbf{u}(k-1) \\\]

source
ModelPredictiveControl.init_predmat — Function
init_predmat(estim, model::LinModel, Hp, Hc) -> E, G, J, K, V, ex̂, gx̂, jx̂, kx̂, vx̂

Construct the prediction matrices for LinModel model.

The model predictions are evaluated from the deviation vectors (see setop!) and:

\[\begin{aligned} + \mathbf{Ŷ_0} &= \mathbf{E ΔU} + \mathbf{G d_0}(k) + \mathbf{J D̂_0} + + \mathbf{K x̂_0}(k) + \mathbf{V u_0}(k-1) + + \mathbf{B} + \mathbf{Ŷ_s} \\ + &= \mathbf{E ΔU} + \mathbf{F} +\end{aligned}\]

in which $\mathbf{x̂_0}(k) = \mathbf{x̂}_i(k) - \mathbf{x̂_{op}}$, with $i = k$ if estim.direct==true, otherwise $i = k - 1$. The predicted outputs $\mathbf{Ŷ_0}$ and measured disturbances $\mathbf{D̂_0}$ respectively include $\mathbf{ŷ_0}(k+j)$ and $\mathbf{d̂_0}(k+j)$ values with $j=1$ to $H_p$, and input increments $\mathbf{ΔU}$, $\mathbf{Δu}(k+j)$ from $j=0$ to $H_c-1$. The vector $\mathbf{B}$ contains the contribution for non-zero state $\mathbf{x̂_{op}}$ and state update $\mathbf{f̂_{op}}$ operating points (for linearization at non-equilibrium point, see linearize). The stochastic predictions $\mathbf{Ŷ_s=0}$ if estim is not a InternalModel, see init_stochpred. The method also computes similar matrices for the predicted terminal states at $k+H_p$:

\[\begin{aligned} + \mathbf{x̂_0}(k+H_p) &= \mathbf{e_x̂ ΔU} + \mathbf{g_x̂ d_0}(k) + \mathbf{j_x̂ D̂_0} + + \mathbf{k_x̂ x̂_0}(k) + \mathbf{v_x̂ u_0}(k-1) + + \mathbf{b_x̂} \\ + &= \mathbf{e_x̂ ΔU} + \mathbf{f_x̂} +\end{aligned}\]

The matrices $\mathbf{E, G, J, K, V, B, e_x̂, g_x̂, j_x̂, k_x̂, v_x̂, b_x̂}$ are defined in the Extended Help section. The $\mathbf{F}$ and $\mathbf{f_x̂}$ vectors are recalculated at each control period $k$, see initpred! and linconstraint!.

Extended Help

Extended Help

Using the augmented matrices $\mathbf{Â, B̂_u, Ĉ, B̂_d, D̂_d}$ in estim (see augment_model), and the function $\mathbf{W}(j) = ∑_{i=0}^j \mathbf{Â}^i$, the prediction matrices are computed by :

\[\begin{aligned} +\mathbf{E} &= \begin{bmatrix} + \mathbf{Ĉ W}(0)\mathbf{B̂_u} & \mathbf{0} & \cdots & \mathbf{0} \\ + \mathbf{Ĉ W}(1)\mathbf{B̂_u} & \mathbf{Ĉ W}(0)\mathbf{B̂_u} & \cdots & \mathbf{0} \\ + \vdots & \vdots & \ddots & \vdots \\ + \mathbf{Ĉ W}(H_p-1)\mathbf{B̂_u} & \mathbf{Ĉ W}(H_p-2)\mathbf{B̂_u} & \cdots & \mathbf{Ĉ W}(H_p-H_c+1)\mathbf{B̂_u} \end{bmatrix} \\ +\mathbf{G} &= \begin{bmatrix} + \mathbf{Ĉ}\mathbf{Â}^{0} \mathbf{B̂_d} \\ + \mathbf{Ĉ}\mathbf{Â}^{1} \mathbf{B̂_d} \\ + \vdots \\ + \mathbf{Ĉ}\mathbf{Â}^{H_p-1} \mathbf{B̂_d} \end{bmatrix} \\ +\mathbf{J} &= \begin{bmatrix} + \mathbf{D̂_d} & \mathbf{0} & \cdots & \mathbf{0} \\ + \mathbf{Ĉ}\mathbf{Â}^{0} \mathbf{B̂_d} & \mathbf{D̂_d} & \cdots & \mathbf{0} \\ + \vdots & \vdots & \ddots & \vdots \\ + \mathbf{Ĉ}\mathbf{Â}^{H_p-2} \mathbf{B̂_d} & \mathbf{Ĉ}\mathbf{Â}^{H_p-3} \mathbf{B̂_d} & \cdots & \mathbf{D̂_d} \end{bmatrix} \\ +\mathbf{K} &= \begin{bmatrix} + \mathbf{Ĉ}\mathbf{Â}^{1} \\ + \mathbf{Ĉ}\mathbf{Â}^{2} \\ + \vdots \\ + \mathbf{Ĉ}\mathbf{Â}^{H_p} \end{bmatrix} \\ +\mathbf{V} &= \begin{bmatrix} + \mathbf{Ĉ W}(0)\mathbf{B̂_u} \\ + \mathbf{Ĉ W}(1)\mathbf{B̂_u} \\ + \vdots \\ + \mathbf{Ĉ W}(H_p-1)\mathbf{B̂_u} \end{bmatrix} \\ +\mathbf{B} &= \begin{bmatrix} + \mathbf{Ĉ W}(0) \\ + \mathbf{Ĉ W}(1) \\ + \vdots \\ + \mathbf{Ĉ W}(H_p-1) \end{bmatrix} \mathbf{\big(f̂_{op} - x̂_{op}\big)} +\end{aligned}\]

For the terminal constraints, the matrices are computed with:

\[\begin{aligned} +\mathbf{e_x̂} &= \begin{bmatrix} + \mathbf{W}(H_p-1)\mathbf{B̂_u} & + \mathbf{W}(H_p-2)\mathbf{B̂_u} & + \cdots & + \mathbf{W}(H_p-H_c+1)\mathbf{B̂_u} \end{bmatrix} \\ +\mathbf{g_x̂} &= \mathbf{Â}^{H_p-1} \mathbf{B̂_d} \\ +\mathbf{j_x̂} &= \begin{bmatrix} + \mathbf{Â}^{H_p-2} \mathbf{B̂_d} & + \mathbf{Â}^{H_p-3} \mathbf{B̂_d} & + \cdots & + \mathbf{0} + \end{bmatrix} \\ +\mathbf{k_x̂} &= \mathbf{Â}^{H_p} \\ +\mathbf{v_x̂} &= \mathbf{W}(H_p-1)\mathbf{B̂_u} \\ +\mathbf{b_x̂} &= \mathbf{W}(H_p-1) \mathbf{\big(f̂_{op} - x̂_{op}\big)} +\end{aligned}\]

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Return empty matrices if model is not a LinModel

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ModelPredictiveControl.relaxU — Function
relaxU(model, nϵ, C_umin, C_umax, S) -> A_Umin, A_Umax, S̃

Augment manipulated inputs constraints with slack variable ϵ for softening.

Denoting the input increments augmented with the slack variable $\mathbf{ΔŨ} = [\begin{smallmatrix} \mathbf{ΔU} \\ ϵ \end{smallmatrix}]$, it returns the augmented conversion matrix $\mathbf{S̃}$, similar to the one described at init_ΔUtoU. It also returns the $\mathbf{A}$ matrices for the inequality constraints:

\[\begin{bmatrix} + \mathbf{A_{U_{min}}} \\ + \mathbf{A_{U_{max}}} +\end{bmatrix} \mathbf{ΔŨ} ≤ +\begin{bmatrix} + - \mathbf{(U_{min} - U_{op}) + T} \mathbf{u_0}(k-1) \\ + + \mathbf{(U_{max} - U_{op}) - T} \mathbf{u_0}(k-1) +\end{bmatrix}\]

in which $\mathbf{U_{min}, U_{max}}$ and $\mathbf{U_{op}}$ vectors respectively contains $\mathbf{u_{min}, u_{max}}$ and $\mathbf{u_{op}}$ repeated $H_p$ times.

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ModelPredictiveControl.relaxΔU — Function
relaxΔU(
+    model, nϵ, C, C_Δumin, C_Δumax, ΔUmin, ΔUmax, N_Hc
+) -> A_ΔŨmin, A_ΔŨmax, ΔŨmin, ΔŨmax, Ñ_Hc

Augment input increments constraints with slack variable ϵ for softening.

Denoting the input increments augmented with the slack variable $\mathbf{ΔŨ} = [\begin{smallmatrix} \mathbf{ΔU} \\ ϵ \end{smallmatrix}]$, it returns the augmented input increment weights $\mathbf{Ñ}_{H_c}$ (that incorporate $C$). It also returns the augmented constraints $\mathbf{ΔŨ_{min}}$ and $\mathbf{ΔŨ_{max}}$ and the $\mathbf{A}$ matrices for the inequality constraints:

\[\begin{bmatrix} + \mathbf{A_{ΔŨ_{min}}} \\ + \mathbf{A_{ΔŨ_{max}}} +\end{bmatrix} \mathbf{ΔŨ} ≤ +\begin{bmatrix} + - \mathbf{ΔŨ_{min}} \\ + + \mathbf{ΔŨ_{max}} +\end{bmatrix}\]

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ModelPredictiveControl.relaxŶ — Function
relaxŶ(::LinModel, nϵ, C_ymin, C_ymax, E) -> A_Ymin, A_Ymax, Ẽ

Augment linear output prediction constraints with slack variable ϵ for softening.

Denoting the input increments augmented with the slack variable $\mathbf{ΔŨ} = [\begin{smallmatrix} \mathbf{ΔU} \\ ϵ \end{smallmatrix}]$, it returns the $\mathbf{Ẽ}$ matrix that appears in the linear model prediction equation $\mathbf{Ŷ_0 = Ẽ ΔŨ + F}$, and the $\mathbf{A}$ matrices for the inequality constraints:

\[\begin{bmatrix} + \mathbf{A_{Y_{min}}} \\ + \mathbf{A_{Y_{max}}} +\end{bmatrix} \mathbf{ΔŨ} ≤ +\begin{bmatrix} + - \mathbf{(Y_{min} - Y_{op}) + F} \\ + + \mathbf{(Y_{max} - Y_{op}) - F} +\end{bmatrix}\]

in which $\mathbf{Y_{min}, Y_{max}}$ and $\mathbf{Y_{op}}$ vectors respectively contains $\mathbf{y_{min}, y_{max}}$ and $\mathbf{y_{op}}$ repeated $H_p$ times.

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Return empty matrices if model is not a LinModel

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ModelPredictiveControl.relaxterminal — Function
relaxterminal(::LinModel, nϵ, c_x̂min, c_x̂max, ex̂) -> A_x̂min, A_x̂max, ẽx̂

Augment terminal state constraints with slack variable ϵ for softening.

Denoting the input increments augmented with the slack variable $\mathbf{ΔŨ} = [\begin{smallmatrix} \mathbf{ΔU} \\ ϵ \end{smallmatrix}]$, it returns the $\mathbf{ẽ_{x̂}}$ matrix that appears in the terminal state equation $\mathbf{x̂_0}(k + H_p) = \mathbf{ẽ_x̂ ΔŨ + f_x̂}$, and the $\mathbf{A}$ matrices for the inequality constraints:

\[\begin{bmatrix} + \mathbf{A_{x̂_{min}}} \\ + \mathbf{A_{x̂_{max}}} +\end{bmatrix} \mathbf{ΔŨ} ≤ +\begin{bmatrix} + - \mathbf{(x̂_{min} - x̂_{op}) + f_x̂} \\ + + \mathbf{(x̂_{max} - x̂_{op}) - f_x̂} +\end{bmatrix}\]

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Return empty matrices if model is not a LinModel

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ModelPredictiveControl.init_quadprog — Function
init_quadprog(model::LinModel, Ẽ, S, M_Hp, N_Hc, L_Hp) -> H̃

Init the quadratic programming Hessian H̃ for MPC.

The matrix appear in the quadratic general form:

\[ J = \min_{\mathbf{ΔŨ}} \frac{1}{2}\mathbf{(ΔŨ)'H̃(ΔŨ)} + \mathbf{q̃'(ΔŨ)} + r \]

The Hessian matrix is constant if the model and weights are linear and time invariant (LTI):

\[ \mathbf{H̃} = 2 ( \mathbf{Ẽ}'\mathbf{M}_{H_p}\mathbf{Ẽ} + \mathbf{Ñ}_{H_c} + + \mathbf{S̃}'\mathbf{L}_{H_p}\mathbf{S̃} )\]

The vector $\mathbf{q̃}$ and scalar $r$ need recalculation each control period $k$, see initpred!. $r$ does not impact the minima position. It is thus useless at optimization but required to evaluate the minimal $J$ value.

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Return empty matrices if model is not a LinModel.

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ModelPredictiveControl.init_stochpred — Function
init_stochpred(estim::InternalModel, Hp) -> Ks, Ps

Init the stochastic prediction matrices for InternalModel.

Ks and Ps matrices are defined as:

\[ \mathbf{Ŷ_s} = \mathbf{K_s x̂_s}(k) + \mathbf{P_s ŷ_s}(k)\]

Current stochastic outputs $\mathbf{ŷ_s}(k)$ comprises the measured outputs $\mathbf{ŷ_s^m}(k) = \mathbf{y^m}(k) - \mathbf{ŷ_d^m}(k)$ and unmeasured $\mathbf{ŷ_s^u}(k) = \mathbf{0}$. See [2].

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Return empty matrices if estim is not a InternalModel.

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ModelPredictiveControl.init_matconstraint_mpc — Function
init_matconstraint_mpc(model::LinModel,
+    i_Umin, i_Umax, i_ΔŨmin, i_ΔŨmax, i_Ymin, i_Ymax, i_x̂min, i_x̂max, args...
+) -> i_b, i_g, A

Init i_b, i_g and A matrices for the linear and nonlinear inequality constraints.

The linear and nonlinear inequality constraints are respectively defined as:

\[\begin{aligned} + \mathbf{A ΔŨ } &≤ \mathbf{b} \\ + \mathbf{g(ΔŨ)} &≤ \mathbf{0} +\end{aligned}\]

i_b is a BitVector including the indices of $\mathbf{b}$ that are finite numbers. i_g is a similar vector but for the indices of $\mathbf{g}$ (empty if model is a LinModel). The method also returns the $\mathbf{A}$ matrix if args is provided. In such a case, args needs to contain all the inequality constraint matrices: A_Umin, A_Umax, A_ΔŨmin, A_ΔŨmax, A_Ymin, A_Ymax, A_x̂min, A_x̂max.

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Init i_b, A without outputs and terminal constraints if model is not a LinModel.

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Update Quadratic Optimization

ModelPredictiveControl.initpred! — Method
initpred!(mpc::PredictiveController, model::LinModel, d, D̂, R̂y, R̂u) -> nothing

Init linear model prediction matrices F, q̃, r and current estimated output ŷ.

See init_predmat and init_quadprog for the definition of the matrices. They are computed with these equations using in-place operations:

\[\begin{aligned} + \mathbf{F} &= \mathbf{G d_0}(k) + \mathbf{J D̂_0} + \mathbf{K x̂_0}(k) + + \mathbf{V u_0}(k-1) + \mathbf{B} + \mathbf{Ŷ_s} \\ + \mathbf{C_y} &= \mathbf{F} - (\mathbf{R̂_y - Y_{op}}) \\ + \mathbf{C_u} &= \mathbf{T} \mathbf{u_0}(k-1) - (\mathbf{R̂_u - U_{op}}) \\ + \mathbf{q̃} &= 2[(\mathbf{M}_{H_p} \mathbf{Ẽ})' \mathbf{C_y} + + (\mathbf{L}_{H_p} \mathbf{S̃})' \mathbf{C_u}] \\ + r &= \mathbf{C_y}' \mathbf{M}_{H_p} \mathbf{C_y} + + \mathbf{C_u}' \mathbf{L}_{H_p} \mathbf{C_u} +\end{aligned}\]

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ModelPredictiveControl.linconstraint! — Method
linconstraint!(mpc::PredictiveController, model::LinModel)

Set b vector for the linear model inequality constraints ($\mathbf{A ΔŨ ≤ b}$).

Also init $\mathbf{f_x̂} = \mathbf{g_x̂ d}(k) + \mathbf{j_x̂ D̂} + \mathbf{k_x̂ x̂_0}(k) + \mathbf{v_x̂ u}(k-1) + \mathbf{b_x̂}$ vector for the terminal constraints, see init_predmat.

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Solve Optimization Problem

ModelPredictiveControl.optim_objective! — Method
optim_objective!(mpc::PredictiveController) -> ΔŨ

Optimize the objective function of mpc PredictiveController and return the solution ΔŨ.

If supported by mpc.optim, it warm-starts the solver at:

\[\mathbf{ΔŨ} = +\begin{bmatrix} + \mathbf{Δu}_{k-1}(k+0) \\ + \mathbf{Δu}_{k-1}(k+1) \\ + \vdots \\ + \mathbf{Δu}_{k-1}(k+H_c-2) \\ + \mathbf{0} \\ + ϵ_{k-1} +\end{bmatrix}\]

where $\mathbf{Δu}_{k-1}(k+j)$ is the input increment for time $k+j$ computed at the last control period $k-1$. It then calls JuMP.optimize!(mpc.optim) and extract the solution. A failed optimization prints an @error log in the REPL and returns the warm-start value.

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ModelPredictiveControl.getinput — Function
getinput(mpc::PredictiveController, ΔŨ) -> u

Get current manipulated input u from a PredictiveController solution ΔŨ.

The first manipulated input $\mathbf{u}(k)$ is extracted from the input increments vector $\mathbf{ΔŨ}$ and applied on the plant (from the receding horizon principle).

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  • 1Maciejowski, J. 2000, "Predictive control : with constraints", 1st ed., Prentice Hall, ISBN 978-0201398236.
  • 2Desbiens, A., D. Hodouin & É. Plamondon. 2000, "Global predictive control: a unified control structure for decoupling setpoint tracking, feedforward compensation and disturbance rejection dynamics", IEE Proceedings - Control Theory and Applications, vol. 147, no 4, https://doi.org/10.1049/ip-cta:20000443, p. 465–475, ISSN 1350-2379.
diff --git a/previews/PR114/internals/sim_model/index.html b/previews/PR114/internals/sim_model/index.html new file mode 100644 index 00000000..c7510e4d --- /dev/null +++ b/previews/PR114/internals/sim_model/index.html @@ -0,0 +1,2 @@ + +Plant Models · ModelPredictiveControl.jl
GitHub

Functions: SimModel Internals

State-Space Functions

Steady-State Calculation

ModelPredictiveControl.steadystate! — Function
steadystate!(model::LinModel, u0, d0)

Set model.x0 to u0 and d0 steady-state if model is a LinModel.

Following setop! notation, the method evaluates the equilibrium from:

\[ \mathbf{x_0} = \mathbf{(I - A)^{-1}(B_u u_0 + B_d d_0 + f_{op} - x_{op})}\]

with constant manipulated inputs $\mathbf{u_0 = u - u_{op}}$ and measured disturbances $\mathbf{d_0 = d - d_{op}}$. The Moore-Penrose pseudo-inverse computes $\mathbf{(I - A)^{-1}}$ to support integrating model (integrator states will be 0).

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Do nothing if model is a NonLinModel.

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diff --git a/previews/PR114/internals/state_estim/index.html b/previews/PR114/internals/state_estim/index.html new file mode 100644 index 00000000..092e97e3 --- /dev/null +++ b/previews/PR114/internals/state_estim/index.html @@ -0,0 +1,215 @@ + +State Estimators · ModelPredictiveControl.jl
GitHub

Functions: StateEstimator Internals

Augmented Model

ModelPredictiveControl.f̂! — Function
f̂!(x̂next0, û0, estim::StateEstimator, model::SimModel, x̂0, u0, d0) -> nothing

Mutating state function $\mathbf{f̂}$ of the augmented model.

By introducing an augmented state vector $\mathbf{x̂_0}$ like in augment_model, the function returns the next state of the augmented model, defined as:

\[\begin{aligned} + \mathbf{x̂_0}(k+1) &= \mathbf{f̂}\Big(\mathbf{x̂_0}(k), \mathbf{u_0}(k), \mathbf{d_0}(k)\Big) \\ + \mathbf{ŷ_0}(k) &= \mathbf{ĥ}\Big(\mathbf{x̂_0}(k), \mathbf{d_0}(k)\Big) +\end{aligned}\]

where $\mathbf{x̂_0}(k+1)$ is stored in x̂next0 argument. The method mutates x̂next0 and û0 in place, the latter stores the input vector of the augmented model $\mathbf{u_0 + ŷ_{s_u}}$. The model parameter vector model.p is not included in the function signature for conciseness.

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f̂!(x̂next0, _ , estim::StateEstimator, model::LinModel, x̂0, u0, d0) -> nothing

Use the augmented model matrices if model is a LinModel.

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f̂!(x̂next0, _ , estim::InternalModel, model::NonLinModel, x̂0, u0, d0)

State function $\mathbf{f̂}$ of InternalModel for NonLinModel.

It calls model.f!(x̂next0, x̂0, u0 ,d0, model.p) since this estimator does not augment the states.

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ModelPredictiveControl.ĥ! — Function
ĥ!(ŷ0, estim::StateEstimator, model::SimModel, x̂0, d0) -> nothing

Mutating output function $\mathbf{ĥ}$ of the augmented model, see f̂!.

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ĥ!(ŷ0, estim::StateEstimator, model::LinModel, x̂0, d0) -> nothing

Use the augmented model matrices if model is a LinModel.

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ĥ!(ŷ0, estim::InternalModel, model::NonLinModel, x̂0, d0)

Output function $\mathbf{ĥ}$ of InternalModel, it calls model.h!.

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Estimator Construction

ModelPredictiveControl.init_estimstoch — Function
init_estimstoch(model, i_ym, nint_u, nint_ym) -> As, Cs_u, Cs_y, nxs, nint_u, nint_ym

Init stochastic model matrices from integrator specifications for state estimation.

The arguments nint_u and nint_ym specify how many integrators are added to each manipulated input and measured outputs. The function returns the state-space matrices As, Cs_u and Cs_y of the stochastic model:

\[\begin{aligned} +\mathbf{x_{s}}(k+1) &= \mathbf{A_s x_s}(k) + \mathbf{B_s e}(k) \\ +\mathbf{y_{s_{u}}}(k) &= \mathbf{C_{s_{u}} x_s}(k) \\ +\mathbf{y_{s_{y}}}(k) &= \mathbf{C_{s_{y}} x_s}(k) +\end{aligned}\]

where $\mathbf{e}(k)$ is an unknown zero mean white noise and $\mathbf{A_s} = \mathrm{diag}(\mathbf{A_{s_{u}}, A_{s_{y}}})$. The estimations does not use $\mathbf{B_s}$, it is thus ignored. The function init_integrators builds the state-space matrices.

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ModelPredictiveControl.init_integrators — Function
init_integrators(nint, ny, varname::String) -> A, C, nint

Calc A, C state-space matrices from integrator specifications nint.

This function is used to initialize the stochastic part of the augmented model for the design of state estimators. The vector nint provides how many integrators (in series) should be incorporated for each output. The argument should have ny element, except for nint=0 which is an alias for no integrator at all. The specific case of one integrator per output results in A = I and C = I. The estimation does not use the B matrix, it is thus ignored. This function is called twice :

  1. for the unmeasured disturbances at manipulated inputs $\mathbf{u}$
  2. for the unmeasured disturbances at measured outputs $\mathbf{y^m}$
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ModelPredictiveControl.augment_model — Function
augment_model(
+    model::LinModel, As, Cs_u, Cs_y; verify_obsv=true
+) -> Â, B̂u, Ĉ, B̂d, D̂d, x̂op, f̂op

Augment LinModel state-space matrices with stochastic ones As, Cs_u, Cs_y.

If $\mathbf{x_0}$ are model.x0 states, and $\mathbf{x_s}$, the states defined at init_estimstoch, we define an augmented state vector $\mathbf{x̂} = [ \begin{smallmatrix} \mathbf{x_0} \\ \mathbf{x_s} \end{smallmatrix} ]$. The method returns the augmented matrices Â, B̂u, Ĉ, B̂d and D̂d:

\[\begin{aligned} + \mathbf{x̂_0}(k+1) &= \mathbf{Â x̂_0}(k) + \mathbf{B̂_u u_0}(k) + \mathbf{B̂_d d_0}(k) \\ + \mathbf{ŷ_0}(k) &= \mathbf{Ĉ x̂_0}(k) + \mathbf{D̂_d d_0}(k) +\end{aligned}\]

An error is thrown if the augmented model is not observable and verify_obsv == true. The augmented operating points x̂op and f̂op are simply $\mathbf{x_{op}}$ and $\mathbf{f_{op}}$ vectors appended with zeros (see setop!). See Extended Help for a detailed definition of the augmented matrices.

Extended Help

Extended Help

Using the As, Cs_u and Cs_y matrices of the stochastic model provided in argument and the model.A, model.Bu, model.Bd, model.C, model.Dd matrices, the state-space matrices of the augmented model are defined as follows:

\[\begin{aligned} +\mathbf{Â} &= \begin{bmatrix} + \mathbf{A} & \mathbf{B_u C_{s_u}} \\ + \mathbf{0} & \mathbf{A_s} \end{bmatrix} \\ +\mathbf{B̂_u} &= \begin{bmatrix} + \mathbf{B_u} \\ + \mathbf{0} \end{bmatrix} \\ +\mathbf{Ĉ} &= \begin{bmatrix} + \mathbf{C} & \mathbf{C_{s_y}} \end{bmatrix} \\ +\mathbf{B̂_d} &= \begin{bmatrix} + \mathbf{B_d} \\ + \mathbf{0} \end{bmatrix} \\ +\mathbf{D̂_d} &= \mathbf{D_d} +\end{aligned}\]

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augment_model(
+    model::SimModel, As, Cs_u, Cs_y; verify_obsv=false
+) -> Â, B̂u, Ĉ, B̂d, D̂d, x̂op, f̂op

Return empty matrices, and x̂op & f̂op vectors, if model is not a LinModel.

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ModelPredictiveControl.init_ukf — Function
init_ukf(model, nx̂, α, β, κ) -> nσ, γ, m̂, Ŝ

Compute the UnscentedKalmanFilter constants from $α, β$ and $κ$.

With $n_\mathbf{x̂}$ elements in the state vector $\mathbf{x̂}$ and $n_σ = 2 n_\mathbf{x̂} + 1$ sigma points, the scaling factor applied on standard deviation matrices $\sqrt{\mathbf{P̂}}$ is:

\[ γ = α \sqrt{ n_\mathbf{x̂} + κ }\]

The weight vector $(n_σ × 1)$ for the mean and the weight matrix $(n_σ × n_σ)$ for the covariance are respectively:

\[\begin{aligned} + \mathbf{m̂} &= \begin{bmatrix} 1 - \tfrac{n_\mathbf{x̂}}{γ^2} & \tfrac{1}{2γ^2} & \tfrac{1}{2γ^2} & \cdots & \tfrac{1}{2γ^2} \end{bmatrix}' \\ + \mathbf{Ŝ} &= \mathrm{diag}\big( 2 - α^2 + β - \tfrac{n_\mathbf{x̂}}{γ^2} \:,\; \tfrac{1}{2γ^2} \:,\; \tfrac{1}{2γ^2} \:,\; \cdots \:,\; \tfrac{1}{2γ^2} \big) +\end{aligned}\]

See update_estimate!(::UnscentedKalmanFilter) for other details.

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ModelPredictiveControl.init_internalmodel — Function
init_internalmodel(As, Bs, Cs, Ds) -> Âs, B̂s

Calc stochastic model update matrices Âs and B̂s for InternalModel estimator.

As, Bs, Cs and Ds are the stochastic model matrices :

\[\begin{aligned} + \mathbf{x_s}(k+1) &= \mathbf{A_s x_s}(k) + \mathbf{B_s e}(k) \\ + \mathbf{y_s}(k) &= \mathbf{C_s x_s}(k) + \mathbf{D_s e}(k) +\end{aligned}\]

where $\mathbf{e}(k)$ is conceptual and unknown zero mean white noise. Its optimal estimation is $\mathbf{ê=0}$, the expected value. Thus, the Âs and B̂s matrices that optimally update the stochastic estimate $\mathbf{x̂_s}$ are:

\[\begin{aligned} + \mathbf{x̂_s}(k+1) + &= \mathbf{(A_s - B_s D_s^{-1} C_s) x̂_s}(k) + \mathbf{(B_s D_s^{-1}) ŷ_s}(k) \\ + &= \mathbf{Â_s x̂_s}(k) + \mathbf{B̂_s ŷ_s}(k) +\end{aligned}\]

with current stochastic outputs estimation $\mathbf{ŷ_s}(k)$, composed of the measured $\mathbf{ŷ_s^m}(k) = \mathbf{y^m}(k) - \mathbf{ŷ_d^m}(k)$ and unmeasured $\mathbf{ŷ_s^u = 0}$ outputs. See [1].

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ModelPredictiveControl.init_predmat_mhe — Function
init_predmat_mhe(
+    model::LinModel, He, i_ym, Â, B̂u, Ĉm, B̂d, D̂dm, x̂op, f̂op, p
+) -> E, G, J, B, ex̄, Ex̂, Gx̂, Jx̂, Bx̂

Construct the MovingHorizonEstimator prediction matrices for LinModel model.

We first introduce the deviation vector of the estimated state at arrival $\mathbf{x̂_0}(k-N_k+p) = \mathbf{x̂}_k(k-N_k+p) - \mathbf{x̂_{op}}$ (see setop!), and the vector $\mathbf{Z} = [\begin{smallmatrix} \mathbf{x̂_0}(k-N_k+p) \\ \mathbf{Ŵ} \end{smallmatrix}]$ with the decision variables. Setting the constant $p=0$ produces an estimator in the current form, while the prediction form is obtained with $p=1$. The estimated sensor noises from time $k-N_k+1$ to $k$ are computed by:

\[\begin{aligned} + \mathbf{V̂} = \mathbf{Y_0^m - Ŷ_0^m} &= \mathbf{E Z + G U_0 + J D_0 + Y_0^m + B} \\ + &= \mathbf{E Z + F} +\end{aligned}\]

in which $\mathbf{U_0}$ and $\mathbf{Y_0^m}$ respectively include the deviation values of the manipulated inputs $\mathbf{u_0}(k-j+p)$ from $j=N_k$ to $1$ and measured outputs $\mathbf{y_0^m}(k-j+1)$ from $j=N_k$ to $1$. The vector $\mathbf{D_0}$ comprises one additional measured disturbance if $p=0$, that is, it includes the deviation vectors $\mathbf{d_0}(k-j+1)$ from $j=N_k+1-p$ to $1$. The constant $\mathbf{B}$ is the contribution for non-zero state $\mathbf{x̂_{op}}$ and state update $\mathbf{f̂_{op}}$ operating points (for linearization, see augment_model and linearize). The method also returns the matrices for the estimation error at arrival:

\[ \mathbf{x̄} = \mathbf{x̂_0^†}(k-N_k+p) - \mathbf{x̂_0}(k-N_k+p) = \mathbf{e_x̄ Z + f_x̄}\]

in which $\mathbf{e_x̄} = [\begin{smallmatrix} -\mathbf{I} & \mathbf{0} & \cdots & \mathbf{0} \end{smallmatrix}]$, and $\mathbf{f_x̄} = \mathbf{x̂_0^†}(k-N_k+p)$. The latter is the deviation vector of the state at arrival, estimated at time $k-N_k$, i.e. $\mathbf{x̂_0^†}(k-N_k+p) = \mathbf{x̂}_{k-N_k}(k-N_k+p) - \mathbf{x̂_{op}}$. Lastly, the estimates $\mathbf{x̂_0}(k-j+p)$ from $j=N_k-1$ to $0$, also in deviation form, are computed with:

\[\begin{aligned} + \mathbf{X̂_0} &= \mathbf{E_x̂ Z + G_x̂ U_0 + J_x̂ D_0 + B_x̂} \\ + &= \mathbf{E_x̂ Z + F_x̂} +\end{aligned}\]

The matrices $\mathbf{E, G, J, B, E_x̂, G_x̂, J_x̂, B_x̂}$ are defined in the Extended Help section. The vectors $\mathbf{F, F_x̂, f_x̄}$ are recalculated at each discrete time step, see initpred!(::MovingHorizonEstimator, ::LinModel) and linconstraint!(::MovingHorizonEstimator, ::LinModel).

Extended Help

Extended Help

Using the augmented process model matrices $\mathbf{Â, B̂_u, Ĉ^m, B̂_d, D̂_d^m}$, and the function $\mathbf{S}(j) = ∑_{i=0}^j \mathbf{Â}^i$, the prediction matrices for the sensor noises depend on the constant $p$. For $p=0$, the matrices are computed by:

\[\begin{aligned} +\mathbf{E} &= - \begin{bmatrix} + \mathbf{Ĉ^m}\mathbf{Â}^{1} & \mathbf{Ĉ^m}\mathbf{Â}^{0} & \cdots & \mathbf{0} \\ + \mathbf{Ĉ^m}\mathbf{Â}^{2} & \mathbf{Ĉ^m}\mathbf{Â}^{1} & \cdots & \mathbf{0} \\ + \vdots & \vdots & \ddots & \vdots \\ + \mathbf{Ĉ^m}\mathbf{Â}^{H_e} & \mathbf{Ĉ^m}\mathbf{Â}^{H_e-1} & \cdots & \mathbf{Ĉ^m}\mathbf{Â}^{0} \end{bmatrix} \\ +\mathbf{G} &= - \begin{bmatrix} + \mathbf{Ĉ^m}\mathbf{Â}^{0}\mathbf{B̂_u} & \mathbf{0} & \cdots & \mathbf{0} \\ + \mathbf{Ĉ^m}\mathbf{Â}^{1}\mathbf{B̂_u} & \mathbf{Ĉ^m}\mathbf{Â}^{0}\mathbf{B̂_u} & \cdots & \mathbf{0} \\ + \vdots & \vdots & \ddots & \vdots \\ + \mathbf{Ĉ^m}\mathbf{Â}^{H_e-1}\mathbf{B̂_u} & \mathbf{Ĉ^m}\mathbf{Â}^{H_e-2}\mathbf{B̂_u} & \cdots & \mathbf{Ĉ^m}\mathbf{Â}^{0}\mathbf{B̂_u} \end{bmatrix} \\ +\mathbf{J} &= - \begin{bmatrix} + \mathbf{Ĉ^m}\mathbf{Â}^{0}\mathbf{B̂_d} & \mathbf{D̂_d^m} & \cdots & \mathbf{0} \\ + \mathbf{Ĉ^m}\mathbf{Â}^{1}\mathbf{B̂_d} & \mathbf{Ĉ^m}\mathbf{Â}^{0}\mathbf{B̂_d} & \cdots & \mathbf{0} \\ + \vdots & \vdots & \ddots & \vdots \\ + \mathbf{Ĉ^m}\mathbf{Â}^{H_e-1}\mathbf{B̂_d} & \mathbf{Ĉ^m}\mathbf{Â}^{H_e-2}\mathbf{B̂_d} & \cdots & \mathbf{D̂_d^m} \end{bmatrix} \\ +\mathbf{B} &= - \begin{bmatrix} + \mathbf{Ĉ^m S}(0) \\ + \mathbf{Ĉ^m S}(1) \\ + \vdots \\ + \mathbf{Ĉ^m S}(H_e-1) \end{bmatrix} \mathbf{\big(f̂_{op} - x̂_{op}\big)} +\end{aligned}\]

or, for $p=1$, the matrices are given by:

\[\begin{aligned} +\mathbf{E} &= - \begin{bmatrix} + \mathbf{Ĉ^m}\mathbf{Â}^{0} & \mathbf{0} & \cdots & \mathbf{0} \\ + \mathbf{Ĉ^m}\mathbf{Â}^{1} & \mathbf{Ĉ^m}\mathbf{Â}^{0} & \cdots & \mathbf{0} \\ + \vdots & \vdots & \ddots & \vdots \\ + \mathbf{Ĉ^m}\mathbf{Â}^{H_e-1} & \mathbf{Ĉ^m}\mathbf{Â}^{H_e-2} & \cdots & \mathbf{0} \end{bmatrix} \\ +\mathbf{G} &= - \begin{bmatrix} + \mathbf{0} & \mathbf{0} & \cdots & \mathbf{0} \\ + \mathbf{Ĉ^m}\mathbf{Â}^{0}\mathbf{B̂_u} & \mathbf{0} & \cdots & \mathbf{0} \\ + \vdots & \vdots & \ddots & \vdots \\ + \mathbf{Ĉ^m}\mathbf{Â}^{H_e-2}\mathbf{B̂_u} & \mathbf{Ĉ^m}\mathbf{Â}^{H_e-3}\mathbf{B̂_u} & \cdots & \mathbf{0} \end{bmatrix} \\ +\mathbf{J} &= - \begin{bmatrix} + \mathbf{D̂_d^m} & \mathbf{0} & \cdots & \mathbf{0} \\ + \mathbf{Ĉ^m}\mathbf{Â}^{0}\mathbf{B̂_d} & \mathbf{D̂_d^m} & \cdots & \mathbf{0} \\ + \vdots & \vdots & \ddots & \vdots \\ + \mathbf{Ĉ^m}\mathbf{Â}^{H_e-2}\mathbf{B̂_d} & \mathbf{Ĉ^m}\mathbf{Â}^{H_e-3}\mathbf{B̂_d} & \cdots & \mathbf{D̂_d^m} \end{bmatrix} \\ +\mathbf{B} &= - \begin{bmatrix} + \mathbf{0} \\ + \mathbf{Ĉ^m S}(0) \\ + \vdots \\ + \mathbf{Ĉ^m S}(H_e-2) \end{bmatrix} \mathbf{\big(f̂_{op} - x̂_{op}\big)} +\end{aligned}\]

The matrices for the estimated states are computed by:

\[\begin{aligned} +\mathbf{E_x̂} &= \begin{bmatrix} + \mathbf{Â}^{1} & \mathbf{A}^{0} & \cdots & \mathbf{0} \\ + \mathbf{Â}^{2} & \mathbf{Â}^{1} & \cdots & \mathbf{0} \\ + \vdots & \vdots & \ddots & \vdots \\ + \mathbf{Â}^{H_e} & \mathbf{Â}^{H_e-1} & \cdots & \mathbf{Â}^{0} \end{bmatrix} \\ +\mathbf{G_x̂} &= \begin{bmatrix} + \mathbf{Â}^{0}\mathbf{B̂_u} & \mathbf{0} & \cdots & \mathbf{0} \\ + \mathbf{Â}^{1}\mathbf{B̂_u} & \mathbf{Â}^{0}\mathbf{B̂_u} & \cdots & \mathbf{0} \\ + \vdots & \vdots & \ddots & \vdots \\ + \mathbf{Â}^{H_e-1}\mathbf{B̂_u} & \mathbf{Â}^{H_e-2}\mathbf{B̂_u} & \cdots & \mathbf{Â}^{0}\mathbf{B̂_u} \end{bmatrix} \\ +\mathbf{J_x̂^†} &= \begin{bmatrix} + \mathbf{Â}^{0}\mathbf{B̂_d} & \mathbf{0} & \cdots & \mathbf{0} \\ + \mathbf{Â}^{1}\mathbf{B̂_d} & \mathbf{Â}^{0}\mathbf{B̂_d} & \cdots & \mathbf{0} \\ + \vdots & \vdots & \ddots & \vdots \\ + \mathbf{Â}^{H_e-1}\mathbf{B̂_d} & \mathbf{Â}^{H_e-2}\mathbf{B̂_d} & \cdots & \mathbf{Â}^{0}\mathbf{B̂_d} \end{bmatrix} \ , \quad +\mathbf{J_x̂} = \begin{cases} + [\begin{smallmatrix} \mathbf{J_x̂^†} & \mathbf{0} \end{smallmatrix}] & p=0 \\ + \mathbf{J_x̂^†} & p=1 \end{cases} \\ +\mathbf{B_x̂} &= \begin{bmatrix} + \mathbf{S}(0) \\ + \mathbf{S}(1) \\ + \vdots \\ + \mathbf{S}(H_e-1) \end{bmatrix} \mathbf{\big(f̂_{op} - x̂_{op}\big)} +\end{aligned}\]

All these matrices are truncated when $N_k < H_e$ (at the beginning).

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Return empty matrices if model is not a LinModel, except for ex̄.

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ModelPredictiveControl.relaxarrival — Function
relaxarrival(
+    model::SimModel, nϵ, c_x̂min, c_x̂max, x̂min, x̂max, ex̄
+) -> A_x̃min, A_x̃max, x̃min, x̃max, ẽx̄

Augment arrival state constraints with slack variable ϵ for softening the MHE.

Denoting the MHE decision variable augmented with the slack variable $\mathbf{Z̃} = [\begin{smallmatrix} ϵ \\ \mathbf{Z} \end{smallmatrix}]$, it returns the $\mathbf{ẽ_x̄}$ matrix that appears in the estimation error at arrival equation $\mathbf{x̄} = \mathbf{ẽ_x̄ Z̃ + f_x̄}$. It also returns the augmented constraints $\mathbf{x̃_{min}}$ and $\mathbf{x̃_{max}}$, and the $\mathbf{A}$ matrices for the inequality constraints:

\[\begin{bmatrix} + \mathbf{A_{x̃_{min}}} \\ + \mathbf{A_{x̃_{max}}} +\end{bmatrix} \mathbf{Z̃} ≤ +\begin{bmatrix} + - \mathbf{(x̃_{min} - x̃_{op})} \\ + + \mathbf{(x̃_{max} - x̃_{op})} +\end{bmatrix}\]

in which $\mathbf{x̃_{min}} = [\begin{smallmatrix} 0 \\ \mathbf{x̂_{min}} \end{smallmatrix}]$, $\mathbf{x̃_{max}} = [\begin{smallmatrix} ∞ \\ \mathbf{x̂_{max}} \end{smallmatrix}]$ and $\mathbf{x̃_{op}} = [\begin{smallmatrix} 0 \\ \mathbf{x̂_{op}} \end{smallmatrix}]$

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ModelPredictiveControl.relaxX̂ — Function
relaxX̂(model::SimModel, nϵ, C_x̂min, C_x̂max, Ex̂) -> A_X̂min, A_X̂max, Ẽx̂

Augment estimated state constraints with slack variable ϵ for softening the MHE.

Denoting the MHE decision variable augmented with the slack variable $\mathbf{Z̃} = [\begin{smallmatrix} ϵ \\ \mathbf{Z} \end{smallmatrix}]$, it returns the $\mathbf{Ẽ_x̂}$ matrix that appears in estimated states equation $\mathbf{X̂} = \mathbf{Ẽ_x̂ Z̃ + F_x̂}$. It also returns the $\mathbf{A}$ matrices for the inequality constraints:

\[\begin{bmatrix} + \mathbf{A_{X̂_{min}}} \\ + \mathbf{A_{X̂_{max}}} +\end{bmatrix} \mathbf{Z̃} ≤ +\begin{bmatrix} + - \mathbf{(X̂_{min} - X̂_{op}) + F_x̂} \\ + + \mathbf{(X̂_{max} - X̂_{op}) - F_x̂} +\end{bmatrix}\]

in which $\mathbf{X̂_{min}, X̂_{max}}$ and $\mathbf{X̂_{op}}$ vectors respectively contains $\mathbf{x̂_{min}, x̂_{max}}$ and $\mathbf{x̂_{op}}$ repeated $H_e$ times.

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Return empty matrices if model is not a LinModel

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ModelPredictiveControl.relaxŴ — Function
relaxŴ(model::SimModel, nϵ, C_ŵmin, C_ŵmax, nx̂) -> A_Ŵmin, A_Ŵmax

Augment estimated process noise constraints with slack variable ϵ for softening the MHE.

Denoting the MHE decision variable augmented with the slack variable $\mathbf{Z̃} = [\begin{smallmatrix} ϵ \\ \mathbf{Z} \end{smallmatrix}]$, it returns the $\mathbf{A}$ matrices for the inequality constraints:

\[\begin{bmatrix} + \mathbf{A_{Ŵ_{min}}} \\ + \mathbf{A_{Ŵ_{max}}} +\end{bmatrix} \mathbf{Z̃} ≤ +\begin{bmatrix} + - \mathbf{Ŵ_{min}} \\ + + \mathbf{Ŵ_{max}} +\end{bmatrix}\]

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ModelPredictiveControl.relaxV̂ — Function
relaxV̂(model::SimModel, nϵ, C_v̂min, C_v̂max, E) -> A_V̂min, A_V̂max, Ẽ

Augment estimated sensor noise constraints with slack variable ϵ for softening the MHE.

Denoting the MHE decision variable augmented with the slack variable $\mathbf{Z̃} = [\begin{smallmatrix} ϵ \\ \mathbf{Z} \end{smallmatrix}]$, it returns the $\mathbf{Ẽ}$ matrix that appears in estimated sensor noise equation $\mathbf{V̂} = \mathbf{Ẽ Z̃ + F}$. It also returns the $\mathbf{A}$ matrices for the inequality constraints:

\[\begin{bmatrix} + \mathbf{A_{V̂_{min}}} \\ + \mathbf{A_{V̂_{max}}} +\end{bmatrix} \mathbf{Z̃} ≤ +\begin{bmatrix} + - \mathbf{V̂_{min} + F} \\ + + \mathbf{V̂_{max} - F} +\end{bmatrix}\]

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Return empty matrices if model is not a LinModel

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ModelPredictiveControl.init_matconstraint_mhe — Function
init_matconstraint_mhe(model::LinModel, 
+    i_x̃min, i_x̃max, i_X̂min, i_X̂max, i_Ŵmin, i_Ŵmax, i_V̂min, i_V̂max, args...
+) -> i_b, i_g, A

Init i_b, i_g and A matrices for the MHE linear inequality constraints.

The linear and nonlinear inequality constraints are respectively defined as:

\[\begin{aligned} + \mathbf{A Z̃ } &≤ \mathbf{b} \\ + \mathbf{g(Z̃)} &≤ \mathbf{0} +\end{aligned}\]

i_b is a BitVector including the indices of $\mathbf{b}$ that are finite numbers. i_g is a similar vector but for the indices of $\mathbf{g}$ (empty if model is a LinModel). The method also returns the $\mathbf{A}$ matrix if args is provided. In such a case, args needs to contain all the inequality constraint matrices: A_x̃min, A_x̃max, A_X̂min, A_X̂max, A_Ŵmin, A_Ŵmax, A_V̂min, A_V̂max.

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Init i_b, A without state and sensor noise constraints if model is not a LinModel.

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Update Quadratic Optimization

ModelPredictiveControl.initpred! — Method
initpred!(estim::MovingHorizonEstimator, model::LinModel) -> nothing

Init quadratic optimization matrices F, fx̄, H̃, q̃, r for MovingHorizonEstimator.

See init_predmat_mhe for the definition of the vectors $\mathbf{F, f_x̄}$. It also inits estim.optim objective function, expressed as the quadratic general form:

\[ J = \min_{\mathbf{Z̃}} \frac{1}{2}\mathbf{Z̃' H̃ Z̃} + \mathbf{q̃' Z̃} + r \]

in which $\mathbf{Z̃} = [\begin{smallmatrix} ϵ \\ \mathbf{Z} \end{smallmatrix}]$. Note that $r$ is useless at optimization but required to evaluate the objective minima $J$. The Hessian $\mathbf{H̃}$ matrix of the quadratic general form is not constant here because of the time-varying $\mathbf{P̄}$ covariance . The computed variables are:

\[\begin{aligned} + \mathbf{F} &= \mathbf{G U_0} + \mathbf{J D_0} + \mathbf{Y_0^m} + \mathbf{B} \\ + \mathbf{f_x̄} &= \mathbf{x̂_0^†}(k-N_k+1) \\ + \mathbf{F_Z̃} &= [\begin{smallmatrix}\mathbf{f_x̄} \\ \mathbf{F} \end{smallmatrix}] \\ + \mathbf{Ẽ_Z̃} &= [\begin{smallmatrix}\mathbf{ẽ_x̄} \\ \mathbf{Ẽ} \end{smallmatrix}] \\ + \mathbf{M}_{N_k} &= \mathrm{diag}(\mathbf{P̄}^{-1}, \mathbf{R̂}_{N_k}^{-1}) \\ + \mathbf{Ñ}_{N_k} &= \mathrm{diag}(C, \mathbf{0}, \mathbf{Q̂}_{N_k}^{-1}) \\ + \mathbf{H̃} &= 2(\mathbf{Ẽ_Z̃}' \mathbf{M}_{N_k} \mathbf{Ẽ_Z̃} + \mathbf{Ñ}_{N_k}) \\ + \mathbf{q̃} &= 2(\mathbf{M}_{N_k} \mathbf{Ẽ_Z̃})' \mathbf{F_Z̃} \\ + r &= \mathbf{F_Z̃}' \mathbf{M}_{N_k} \mathbf{F_Z̃} +\end{aligned}\]

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ModelPredictiveControl.linconstraint! — Method
linconstraint!(estim::MovingHorizonEstimator, model::LinModel)

Set b vector for the linear model inequality constraints ($\mathbf{A Z̃ ≤ b}$) of MHE.

Also init $\mathbf{F_x̂ = G_x̂ U_0 + J_x̂ D_0 + B_x̂}$ vector for the state constraints, see init_predmat_mhe.

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Solve Optimization Problem

ModelPredictiveControl.optim_objective! — Method
optim_objective!(estim::MovingHorizonEstimator) -> Z̃

Optimize objective of estim MovingHorizonEstimator and return the solution Z̃.

If supported by estim.optim, it warm-starts the solver at:

\[\mathbf{Z̃} = +\begin{bmatrix} + ϵ_{k-1} \\ + \mathbf{x̂}_{k-1}(k-N_k+p) \\ + \mathbf{ŵ}_{k-1}(k-N_k+p+0) \\ + \mathbf{ŵ}_{k-1}(k-N_k+p+1) \\ + \vdots \\ + \mathbf{ŵ}_{k-1}(k-p-2) \\ + \mathbf{0} \\ +\end{bmatrix}\]

where $\mathbf{ŵ}_{k-1}(k-j)$ is the input increment for time $k-j$ computed at the last time step $k-1$. It then calls JuMP.optimize!(estim.optim) and extract the solution. A failed optimization prints an @error log in the REPL and returns the warm-start value.

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Remove Operating Points

ModelPredictiveControl.remove_op! — Function
remove_op!(estim::StateEstimator, ym, d, u=nothing) -> y0m, d0, u0

Remove operating pts on measured outputs ym, disturbances d and inputs u (if provided).

If u is provided, also store current inputs without operating points u0 in estim.lastu0. This field is used for PredictiveController computations.

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Init Estimate

ModelPredictiveControl.init_estimate! — Function
init_estimate!(estim::StateEstimator, model::LinModel, y0m, d0, u0)

Init estim.x̂0 estimate with the steady-state solution if model is a LinModel.

Using u0, y0m and d0 arguments (deviation values, see setop!), the steadystate problem combined to the equality constraint $\mathbf{ŷ_0^m} = \mathbf{y_0^m}$ engenders the following system to solve:

\[\begin{bmatrix} + \mathbf{I} - \mathbf{Â} \\ + \mathbf{Ĉ^m} +\end{bmatrix} \mathbf{x̂_0} = +\begin{bmatrix} + \mathbf{B̂_u u_0 + B̂_d d_0 + f̂_{op} - x̂_{op}} \\ + \mathbf{y_0^m - D̂_d^m d_0} +\end{bmatrix}\]

in which $\mathbf{Ĉ^m, D̂_d^m}$ are the rows of estim.Ĉ, estim.D̂d that correspond to measured outputs $\mathbf{y^m}$.

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init_estimate!(estim::StateEstimator, model::SimModel, _ , _ , _ )

Left estim.x̂0 estimate unchanged if model is not a LinModel.

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init_estimate!(estim::InternalModel, model::LinModel, y0m, d0, u0)

Init estim.x̂0/x̂d/x̂s estimate at steady-state for InternalModel.

The deterministic estimates estim.x̂d start at steady-state using u0 and d0 arguments:

\[ \mathbf{x̂_d} = \mathbf{(I - A)^{-1} (B_u u_0 + B_d d_0 + f_{op} - x_{op})}\]

Based on y0m argument and current stochastic outputs estimation $\mathbf{ŷ_s}$, composed of the measured $\mathbf{ŷ_s^m} = \mathbf{y_0^m} - \mathbf{ŷ_{d0}^m}$ and unmeasured $\mathbf{ŷ_s^u = 0}$ outputs, the stochastic estimates also start at steady-state:

\[ \mathbf{x̂_s} = \mathbf{(I - Â_s)^{-1} B̂_s ŷ_s}\]

This estimator does not augment the state vector, thus $\mathbf{x̂ = x̂_d}$. See init_internalmodel for details.

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Correct Estimate

ModelPredictiveControl.correct_estimate! — Function
correct_estimate!(estim::SteadyKalmanFilter, y0m, d0)

Correct estim.x̂0 with measured outputs y0m and disturbances d0 for current time step.

It computes the corrected state estimate $\mathbf{x̂}_{k}(k)$. See the docstring of update_estimate!(::SteadyKalmanFilter, ::Any, ::Any) for the equations.

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correct_estimate!(estim::KalmanFilter, y0m, d0)

Correct estim.x̂0 and estim.P̂ using the time-varying KalmanFilter.

It computes the corrected state estimate $\mathbf{x̂}_{k}(k)$ estimation covariance $\mathbf{P̂}_{k}(k)$.

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correct_estimate!(estim::UnscentedKalmanFilter, y0m, d0)

Do the same but for the UnscentedKalmanFilter.

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correct_estimate!(estim::ExtendedKalmanFilter, y0m, d0)

Do the same but for the ExtendedKalmanFilter.

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correct_estimate!(estim::Luenberger, y0m, d0, _ )

Identical to correct_estimate!(::SteadyKalmanFilter) but using Luenberger.

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correct_estimate!(estim::MovingHorizonEstimator, y0m, d0)

Do the same but for MovingHorizonEstimator objects.

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correct_estimate!(estim::InternalModel, y0m, d0)

Compute the current stochastic output estimation ŷs for InternalModel.

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Update Estimate

Info

All these methods assume that the u0, y0m and d0 arguments are deviation vectors from their respective operating points (see setop!). The associated equations in the documentation drops the $\mathbf{0}$ in subscript to simplify the notation. Strictly speaking, the manipulated inputs, measured outputs, measured disturbances and estimated states should be denoted with $\mathbf{u_0, y_0^m, d_0}$ and $\mathbf{x̂_0}$, respectively.

ModelPredictiveControl.update_estimate! — Function
update_estimate!(estim::SteadyKalmanFilter, y0m, d0, u0)

Update estim.x̂0 estimate with current inputs u0, measured outputs y0m and dist. d0.

If estim.direct == false, the SteadyKalmanFilter first corrects the state estimate with the precomputed Kalman gain $\mathbf{K̂}$. Afterward, it predicts the next state with the augmented process model. The correction step is skipped if direct == true since it is already done by the user through the preparestate! function (that calls correct_estimate!). The correction and prediction step equations are provided below.

Correction Step

\[\mathbf{x̂}_k(k) = \mathbf{x̂}_{k-1}(k) + \mathbf{K̂}[\mathbf{y^m}(k) - \mathbf{Ĉ^m x̂}_{k-1}(k) + - \mathbf{D̂_d^m d}(k) ]\]

Prediction Step

\[\mathbf{x̂}_{k}(k+1) = \mathbf{Â x̂}_{k}(k) + \mathbf{B̂_u u}(k) + \mathbf{B̂_d d}(k) \]

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update_estimate!(estim::KalmanFilter, y0m, d0, u0)

Update KalmanFilter state estim.x̂0 and estimation error covariance estim.P̂.

It implements the classical time-varying Kalman Filter based on the process model described in SteadyKalmanFilter. If estim.direct == false, it first corrects the estimate before predicting the next state. The correction step is skipped if estim.direct == true since it's already done by the user. The correction and prediction step equations are provided below, see [2] for details.

Correction Step

\[\begin{aligned} + \mathbf{M̂}(k) &= \mathbf{Ĉ^m P̂}_{k-1}(k)\mathbf{Ĉ^m}' + \mathbf{R̂} \\ + \mathbf{K̂}(k) &= \mathbf{P̂}_{k-1}(k)\mathbf{Ĉ^m}'\mathbf{M̂^{-1}}(k) \\ + \mathbf{ŷ^m}(k) &= \mathbf{Ĉ^m x̂}_{k-1}(k) + \mathbf{D̂_d^m d}(k) \\ + \mathbf{x̂}_{k}(k) &= \mathbf{x̂}_{k-1}(k) + \mathbf{K̂}(k)[\mathbf{y^m}(k) - \mathbf{ŷ^m}(k)] \\ + \mathbf{P̂}_{k}(k) &= [\mathbf{I - K̂}(k)\mathbf{Ĉ^m}]\mathbf{P̂}_{k-1}(k) +\end{aligned}\]

Prediction Step

\[\begin{aligned} + \mathbf{x̂}_{k}(k+1) &= \mathbf{Â x̂}_{k}(k) + \mathbf{B̂_u u}(k) + \mathbf{B̂_d d}(k) \\ + \mathbf{P̂}_{k}(k+1) &= \mathbf{Â P̂}_{k}(k)\mathbf{Â}' + \mathbf{Q̂} +\end{aligned}\]

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update_estimate!(estim::UnscentedKalmanFilter, y0m, d0, u0)

Update UnscentedKalmanFilter state estim.x̂0 and covariance estimate estim.P̂.

It implements the unscented Kalman Filter based on the generalized unscented transform[3]. See init_ukf for the definition of the constants $\mathbf{m̂, Ŝ}$ and $γ$. The superscript in e.g. $\mathbf{X̂}_{k-1}^j(k)$ refers the vector at the $j$th column of $\mathbf{X̂}_{k-1}(k)$. The symbol $\mathbf{0}$ is a vector with zeros. The number of sigma points is $n_σ = 2 n_\mathbf{x̂} + 1$. The matrices $\sqrt{\mathbf{P̂}_{k-1}(k)}$ and $\sqrt{\mathbf{P̂}_{k}(k)}$ are the the lower triangular factors of cholesky results. The correction and prediction step equations are provided below. The correction step is skipped if estim.direct == true since it's already done by the user.

Correction Step

\[\begin{aligned} + \mathbf{X̂}_{k-1}(k) &= \bigg[\begin{matrix} \mathbf{x̂}_{k-1}(k) & \mathbf{x̂}_{k-1}(k) & \cdots & \mathbf{x̂}_{k-1}(k) \end{matrix}\bigg] + \bigg[\begin{matrix} \mathbf{0} & γ \sqrt{\mathbf{P̂}_{k-1}(k)} & -γ \sqrt{\mathbf{P̂}_{k-1}(k)} \end{matrix}\bigg] \\ + \mathbf{Ŷ^m}(k) &= \bigg[\begin{matrix} \mathbf{ĥ^m}\Big( \mathbf{X̂}_{k-1}^{1}(k) \Big) & \mathbf{ĥ^m}\Big( \mathbf{X̂}_{k-1}^{2}(k) \Big) & \cdots & \mathbf{ĥ^m}\Big( \mathbf{X̂}_{k-1}^{n_σ}(k) \Big) \end{matrix}\bigg] \\ + \mathbf{ŷ^m}(k) &= \mathbf{Ŷ^m}(k) \mathbf{m̂} \\ + \mathbf{X̄}_{k-1}(k) &= \begin{bmatrix} \mathbf{X̂}_{k-1}^{1}(k) - \mathbf{x̂}_{k-1}(k) & \mathbf{X̂}_{k-1}^{2}(k) - \mathbf{x̂}_{k-1}(k) & \cdots & \mathbf{X̂}_{k-1}^{n_σ}(k) - \mathbf{x̂}_{k-1}(k) \end{bmatrix} \\ + \mathbf{Ȳ^m}(k) &= \begin{bmatrix} \mathbf{Ŷ^m}^{1}(k) - \mathbf{ŷ^m}(k) & \mathbf{Ŷ^m}^{2}(k) - \mathbf{ŷ^m}(k) & \cdots & \mathbf{Ŷ^m}^{n_σ}(k) - \mathbf{ŷ^m}(k) \end{bmatrix} \\ + \mathbf{M̂}(k) &= \mathbf{Ȳ^m}(k) \mathbf{Ŝ} \mathbf{Ȳ^m}'(k) + \mathbf{R̂} \\ + \mathbf{K̂}(k) &= \mathbf{X̄}_{k-1}(k) \mathbf{Ŝ} \mathbf{Ȳ^m}'(k) \mathbf{M̂^{-1}}(k) \\ + \mathbf{x̂}_k(k) &= \mathbf{x̂}_{k-1}(k) + \mathbf{K̂}(k) \big[ \mathbf{y^m}(k) - \mathbf{ŷ^m}(k) \big] \\ + \mathbf{P̂}_k(k) &= \mathbf{P̂}_{k-1}(k) - \mathbf{K̂}(k) \mathbf{M̂}(k) \mathbf{K̂}'(k) \\ +\end{aligned} \]

Prediction Step

\[\begin{aligned} + \mathbf{X̂}_k(k) &= \bigg[\begin{matrix} \mathbf{x̂}_{k}(k) & \mathbf{x̂}_{k}(k) & \cdots & \mathbf{x̂}_{k}(k) \end{matrix}\bigg] + \bigg[\begin{matrix} \mathbf{0} & \gamma \sqrt{\mathbf{P̂}_{k}(k)} & - \gamma \sqrt{\mathbf{P̂}_{k}(k)} \end{matrix}\bigg] \\ + \mathbf{X̂}_{k}(k+1) &= \bigg[\begin{matrix} \mathbf{f̂}\Big( \mathbf{X̂}_{k}^{1}(k), \mathbf{u}(k), \mathbf{d}(k) \Big) & \mathbf{f̂}\Big( \mathbf{X̂}_{k}^{2}(k), \mathbf{u}(k), \mathbf{d}(k) \Big) & \cdots & \mathbf{f̂}\Big( \mathbf{X̂}_{k}^{n_σ}(k), \mathbf{u}(k), \mathbf{d}(k) \Big) \end{matrix}\bigg] \\ + \mathbf{x̂}_{k}(k+1) &= \mathbf{X̂}_{k}(k+1)\mathbf{m̂} \\ + \mathbf{X̄}_k(k+1) &= \begin{bmatrix} \mathbf{X̂}_{k}^{1}(k+1) - \mathbf{x̂}_{k}(k+1) & \mathbf{X̂}_{k}^{2}(k+1) - \mathbf{x̂}_{k}(k+1) & \cdots &\, \mathbf{X̂}_{k}^{n_σ}(k+1) - \mathbf{x̂}_{k}(k+1) \end{bmatrix} \\ + \mathbf{P̂}_k(k+1) &= \mathbf{X̄}_k(k+1) \mathbf{Ŝ} \mathbf{X̄}_k'(k+1) + \mathbf{Q̂} +\end{aligned}\]

source
update_estimate!(estim::ExtendedKalmanFilter, y0m, d0, u0)

Update ExtendedKalmanFilter state estim.x̂0 and covariance estim.P̂.

The equations are similar to update_estimate!(::KalmanFilter) but with the substitutions $\mathbf{Ĉ^m = Ĥ^m}(k)$ and $\mathbf{Â = F̂}(k)$, the Jacobians of the augmented process model:

\[\begin{aligned} + \mathbf{Ĥ}(k) &= \left. \frac{∂\mathbf{ĥ}(\mathbf{x̂}, \mathbf{d})}{∂\mathbf{x̂}} \right|_{\mathbf{x̂ = x̂}_{k-1}(k),\, \mathbf{d = d}(k)} \\ + \mathbf{F̂}(k) &= \left. \frac{∂\mathbf{f̂}(\mathbf{x̂}, \mathbf{u}, \mathbf{d})}{∂\mathbf{x̂}} \right|_{\mathbf{x̂ = x̂}_{k}(k), \, \mathbf{u = u}(k),\, \mathbf{d = d}(k)} +\end{aligned}\]

The matrix $\mathbf{Ĥ^m}$ is the rows of $\mathbf{Ĥ}$ that are measured outputs. The function ForwardDiff.jacobian automatically computes them. The correction and prediction step equations are provided below. The correction step is skipped if estim.direct == true since it's already done by the user.

Correction Step

\[\begin{aligned} + \mathbf{Ŝ}(k) &= \mathbf{Ĥ^m}(k)\mathbf{P̂}_{k-1}(k)\mathbf{Ĥ^m}'(k) + \mathbf{R̂} \\ + \mathbf{K̂}(k) &= \mathbf{P̂}_{k-1}(k)\mathbf{Ĥ^m}'(k)\mathbf{Ŝ^{-1}}(k) \\ + \mathbf{ŷ^m}(k) &= \mathbf{ĥ^m}\Big( \mathbf{x̂}_{k-1}(k), \mathbf{d}(k) \Big) \\ + \mathbf{x̂}_{k}(k) &= \mathbf{x̂}_{k-1}(k) + \mathbf{K̂}(k)[\mathbf{y^m}(k) - \mathbf{ŷ^m}(k)] \\ + \mathbf{P̂}_{k}(k) &= [\mathbf{I - K̂}(k)\mathbf{Ĥ^m}(k)]\mathbf{P̂}_{k-1}(k) +\end{aligned}\]

Prediction Step

\[\begin{aligned} + \mathbf{x̂}_{k}(k+1) &= \mathbf{f̂}\Big( \mathbf{x̂}_{k}(k), \mathbf{u}(k), \mathbf{d}(k) \Big) \\ + \mathbf{P̂}_{k}(k+1) &= \mathbf{F̂}(k)\mathbf{P̂}_{k}(k)\mathbf{F̂}'(k) + \mathbf{Q̂} +\end{aligned}\]

source
update_estimate!(estim::Luenberger, y0m, d0, u0)

Same than update_estimate!(::SteadyKalmanFilter) but using Luenberger.

source
update_estimate!(estim::MovingHorizonEstimator, y0m, d0, u0)

Update MovingHorizonEstimator state estim.x̂0.

The optimization problem of MovingHorizonEstimator documentation is solved at each discrete time $k$. The prediction matrices are provided at init_predmat_mhe documentation. Once solved, the optimal estimate $\mathbf{x̂}_k(k+p)$ is computed by inserting the optimal values of $\mathbf{x̂}_k(k-N_k+p)$ and $\mathbf{Ŵ}$ in the augmented model from $j = N_k-1$ to $0$ inclusively. Afterward, if $N_k = H_e$, the arrival covariance for the next time step $\mathbf{P̂}_{k-N_k}(k-N_k+1)$ is estimated using estim.covestim object.

source
update_estimate!(estim::InternalModel, _ , d0, u0)

Update estim.x̂0/x̂d/x̂s with current inputs u0, measured outputs y0m and dist. d0.

The InternalModel updates the deterministic x̂d and stochastic x̂s estimates with:

\[\begin{aligned} + \mathbf{x̂_d}(k+1) &= \mathbf{f}\Big( \mathbf{x̂_d}(k), \mathbf{u}(k), \mathbf{d}(k) \Big) \\ + \mathbf{x̂_s}(k+1) &= \mathbf{Â_s x̂_s}(k) + \mathbf{B̂_s ŷ_s}(k) +\end{aligned}\]

This estimator does not augment the state vector, thus $\mathbf{x̂ = x̂_d}$. See init_internalmodel for details.

source
  • 1Desbiens, A., D. Hodouin & É. Plamondon. 2000, "Global predictive control : a unified control structure for decoupling setpoint tracking, feedforward compensation and disturbance rejection dynamics", IEE Proceedings - Control Theory and Applications, vol. 147, no 4, https://doi.org/10.1049/ip-cta:20000443, p. 465–475, ISSN 1350-2379.
  • 2"Kalman Filter", Wikipedia: The Free Encyclopedia, https://en.wikipedia.org/wiki/Kalman_filter, Accessed 2024-08-08.
  • 3Simon, D. 2006, "Chapter 14: The unscented Kalman filter" in "Optimal State Estimation: Kalman, H∞, and Nonlinear Approaches", John Wiley & Sons, p. 433–459, https://doi.org/10.1002/0470045345.ch14, ISBN9780470045343.
diff --git a/previews/PR114/manual/installation/index.html b/previews/PR114/manual/installation/index.html new file mode 100644 index 00000000..2934af4b --- /dev/null +++ b/previews/PR114/manual/installation/index.html @@ -0,0 +1,2 @@ + +Installation · ModelPredictiveControl.jl
GitHub

Manual: Installation

To install the ModelPredictiveControl package, run this command in the Julia REPL:

using Pkg; Pkg.activate(); Pkg.add("ModelPredictiveControl")

Doing so will install the package to default Julia environnement, that is, accessible anywhere. To facilitate sharing of code and reproducibility of results, it is recommended to install packages in a project environnement. To generate a new project named MPCproject with this package in the current working directory, write this in the REPL:

using Pkg; Pkg.generate("MPCproject"); Pkg.activate("."); Pkg.add("ModelPredictiveControl")

Note that that the construction of linear models typically requires ss or tf functions, it is thus advised to load the package with:

using ModelPredictiveControl, ControlSystemsBase
diff --git a/previews/PR114/manual/linmpc/index.html b/previews/PR114/manual/linmpc/index.html new file mode 100644 index 00000000..f38ec99f --- /dev/null +++ b/previews/PR114/manual/linmpc/index.html @@ -0,0 +1,134 @@ + +Linear Design · ModelPredictiveControl.jl
GitHub

Manual: Linear Design

Linear Model

The example considers a continuously stirred-tank reactor (CSTR) with a cold and hot water inlet as a plant. The water flows out of an opening at the bottom of the tank. The manipulated inputs are the cold $u_c$ and hot $u_h$ water flow rates, and the measured outputs are the liquid level $y_L$ and temperature $y_T$:

\[\begin{aligned} + \mathbf{u} &= \begin{bmatrix} u_c \\ u_h \end{bmatrix} \\ + \mathbf{y} &= \begin{bmatrix} y_L \\ y_T \end{bmatrix} +\end{aligned}\]

The following figure depicts the instrumentation installed on the CSTR:

cstr

At the steady-state operating points:

\[\begin{aligned} + \mathbf{u_{op}} &= \begin{bmatrix} 20 \\ 20 \end{bmatrix} \\ + \mathbf{y_{op}} &= \begin{bmatrix} 50 \\ 30 \end{bmatrix} +\end{aligned}\]

the following linear model accurately describes the plant dynamics:

\[\mathbf{G}(s) = \frac{\mathbf{y}(s)}{\mathbf{u}(s)} = +\begin{bmatrix} + \frac{1.90}{18s + 1} & \frac{1.90}{18s + 1} \\[3pt] + \frac{-0.74}{8s + 1} & \frac{0.74}{8s + 1} +\end{bmatrix}\]

We first need to construct a LinModel objet with setop! to handle the operating points:

using ModelPredictiveControl, ControlSystemsBase
+G = [ tf(1.90, [18, 1]) tf(1.90, [18, 1]);
+      tf(-0.74,[8, 1])  tf(0.74, [8, 1]) ]
+Ts = 2.0
+model = setop!(LinModel(G, Ts), uop=[20, 20], yop=[50, 30])
LinModel with a sample time Ts = 2.0 s and:
+ 2 manipulated inputs u
+ 2 states x
+ 2 outputs y
+ 0 measured disturbances d

The model object will be used for two purposes : to construct our controller, and as a plant simulator to test the design. Its sampling time is 2 s thus the control period will be 2 s as well.

Linear Model Predictive Controller

A linear model predictive controller (MPC) will control both the water level $y_L$ and temperature $y_T$ in the tank. The tank level should also never fall below 48:

\[y_L ≥ 48\]

We design our LinMPC controllers by including the linear level constraint with setconstraint! (±Inf values should be used when there is no bound):

mpc = LinMPC(model, Hp=10, Hc=2, Mwt=[1, 1], Nwt=[0.1, 0.1])
+mpc = setconstraint!(mpc, ymin=[48, -Inf])
LinMPC controller with a sample time Ts = 2.0 s, OSQP optimizer, SteadyKalmanFilter estimator and:
+ 10 prediction steps Hp
+  2 control steps Hc
+  1 slack variable ϵ (control constraints)
+  2 manipulated inputs u (0 integrating states)
+  4 estimated states x̂
+  2 measured outputs ym (2 integrating states)
+  0 unmeasured outputs yu
+  0 measured disturbances d

in which Hp and Hc keyword arguments are respectively the predictive and control horizons, and Mwt and Nwt, the output setpoint tracking and move suppression weights. By default, LinMPC controllers use OSQP to solve the problem, soft constraints on output predictions $\mathbf{ŷ}$ to ensure feasibility, and a SteadyKalmanFilter to estimate the plant states[1]. An attentive reader will also notice that the Kalman filter estimates two additional states compared to the plant model. These are the integrating states for the unmeasured plant disturbances, and they are automatically added to the model outputs by default if observability is preserved (see SteadyKalmanFilter for details).

Before closing the loop, the actual plant input $\mathbf{u}$ and measurement $\mathbf{y}$ should initialize the estimates $\mathbf{x̂}$ at the steady-state solution that leads to $\mathbf{ŷ}(0) = \mathbf{y}(0)$. This approach results in a bumpless transfer. The initstate! function finds this solution for LinModel. Since model simulates our plant here, its output will initialize the states. LinModel objects are callable for this purpose (an alias for evaloutput):

u, y = model.uop, model() # or equivalently : y = evaloutput(model)
+initstate!(mpc, u, y)

We can then close the loop and test mpc performance on the simulator by imposing step changes on output setpoints $\mathbf{r_y}$ and on a load disturbance $u_l$:

function test_mpc(mpc, model)
+    N = 200
+    ry, ul = [50, 30], 0
+    u_data, y_data, ry_data = zeros(model.nu, N), zeros(model.ny, N), zeros(model.ny, N)
+    for i = 1:N
+        i == 51  && (ry = [50, 35])
+        i == 101 && (ry = [54, 30])
+        i == 151 && (ul = -20)
+        y = model() # simulated measurements
+        preparestate!(mpc, y) # prepare mpc state estimate for current iteration
+        u = mpc(ry) # or equivalently : u = moveinput!(mpc, ry)
+        u_data[:,i], y_data[:,i], ry_data[:,i] = u, y, ry
+        updatestate!(mpc, u, y) # update mpc state estimate for next iteration
+        updatestate!(model, u + [0; ul]) # update simulator with load disturbance
+    end
+    return u_data, y_data, ry_data
+end
+u_data, y_data, ry_data = test_mpc(mpc, model)
+t_data = Ts*(0:(size(y_data,2)-1))

The LinMPC objects are also callable as an alternative syntax for moveinput!. It is worth mentioning that additional information like the optimal output predictions $\mathbf{Ŷ}$ can be retrieved by calling getinfo after solving the problem. Also, calling preparestate! on the mpc object prepares the estimates for the current control period, and updatestate! updates them for the next one (the same logic applies for model). This is why preparestate! is called before the controller, and updatestate!, after.

Lastly, we plot the closed-loop test with the Plots package:

using Plots
+function plot_data(t_data, u_data, y_data, ry_data)
+    p1 = plot(t_data, y_data[1,:], label="meas.", ylabel="level")
+    plot!(p1, t_data, ry_data[1,:], label="setpoint", linestyle=:dash, linetype=:steppost)
+    plot!(p1, t_data, fill(48,size(t_data)), label="min", linestyle=:dot, linewidth=1.5)
+    p2 = plot(t_data, y_data[2,:], label="meas.", legend=:topleft, ylabel="temp.")
+    plot!(p2, t_data, ry_data[2,:],label="setpoint", linestyle=:dash, linetype=:steppost)
+    p3 = plot(t_data,u_data[1,:],label="cold", linetype=:steppost, ylabel="flow rate")
+    plot!(p3, t_data,u_data[2,:],label="hot", linetype=:steppost, xlabel="time (s)")
+    return plot(p1, p2, p3, layout=(3,1))
+end
+plot_data(t_data, u_data, y_data, ry_data)

plot1_LinMPC

Compared to the default setting, adding the integrating states at the model inputs may improve the closed-loop performance. Load disturbances are indeed very common in many real-life control problems. Constructing a LinMPC with input integrators:

mpc2 = LinMPC(model, Hp=10, Hc=2, Mwt=[1, 1], Nwt=[0.1, 0.1], nint_u=[1, 1])
+mpc2 = setconstraint!(mpc2, ymin=[48, -Inf])
LinMPC controller with a sample time Ts = 2.0 s, OSQP optimizer, SteadyKalmanFilter estimator and:
+ 10 prediction steps Hp
+  2 control steps Hc
+  1 slack variable ϵ (control constraints)
+  2 manipulated inputs u (2 integrating states)
+  4 estimated states x̂
+  2 measured outputs ym (0 integrating states)
+  0 unmeasured outputs yu
+  0 measured disturbances d

does accelerate the rejection of the load disturbance and almost eliminates the level constraint violation:

setstate!(model, zeros(model.nx))
+u, y = model.uop, model()
+initstate!(mpc2, u, y)
+u_data, y_data, ry_data = test_mpc(mpc2, model)
+plot_data(t_data, u_data, y_data, ry_data)

plot2_LinMPC

Moving Horizon Estimation

The SteadyKalmanFilter is simple but it is not able to handle constraints at estimation. The MovingHorizonEstimator (MHE) can improve the accuracy of the state estimate $\mathbf{x̂}$. It solves a quadratic optimization problem under a past time window $H_e$. Bounds on the estimated plant state $\mathbf{x̂}$, estimated process noise $\mathbf{ŵ}$ and estimated sensor noise $\mathbf{v̂}$ can be included in the problem. This can be useful to add physical knowledge on the plant and its disturbances, and it does not require the installation of new physical sensors (e.g. a strictly positive concentration). The closed-loop performance of any state feedback controller, like here, depends on the accuracy of the plant state estimate.

For the CSTR, we will bound the innovation term $\mathbf{y}(k) - \mathbf{ŷ}(k) = \mathbf{v̂}(k)$, and increase the hot water unmeasured disturbance covariance in $\mathbf{Q_{int_u}}$ to accelerate the estimation of the load disturbance:

estim = MovingHorizonEstimator(model, He=10, nint_u=[1, 1], σQint_u = [1, 2])
+estim = setconstraint!(estim, v̂min=[-1, -0.5], v̂max=[+1, +0.5])
+mpc_mhe = LinMPC(estim, Hp=10, Hc=2, Mwt=[1, 1], Nwt=[0.1, 0.1])
+mpc_mhe = setconstraint!(mpc_mhe, ymin=[45, -Inf])
LinMPC controller with a sample time Ts = 2.0 s, OSQP optimizer, MovingHorizonEstimator estimator and:
+ 10 prediction steps Hp
+  2 control steps Hc
+  1 slack variable ϵ (control constraints)
+ 10 estimation steps He
+  0 slack variable ϵ (estimation constraints)
+  2 manipulated inputs u (2 integrating states)
+  4 estimated states x̂
+  2 measured outputs ym (0 integrating states)
+  0 unmeasured outputs yu
+  0 measured disturbances d

The rejection is indeed improved:

setstate!(model, zeros(model.nx))
+u, y, d = model.uop, model(), mpc_mhe.estim.model.dop
+initstate!(mpc_mhe, u, y, d)
+u_data, y_data, ry_data = test_mpc(mpc_mhe, model)
+plot_data(t_data, u_data, y_data, ry_data)

plot3_LinMPC

Adding Feedforward Compensation

Suppose that the load disturbance $u_l$ of the last section is in fact caused by a separate hot water pipe that discharges into the tank. Adding a new sensor to measure this flow rate allows us to incorporate feedforward compensation in the controller. The new plant model is:

\[\begin{bmatrix} + y_L(s) \\ y_T(s) +\end{bmatrix} = +\begin{bmatrix} + \frac{1.90}{18s + 1} & \frac{1.90}{18s + 1} & \frac{1.90}{18s + 1} \\[3pt] + \frac{-0.74}{8s + 1} & \frac{0.74}{8s + 1} & \frac{0.74}{8s + 1} +\end{bmatrix} +\begin{bmatrix} + u_c(s) \\ u_h(s) \\ u_l(s) +\end{bmatrix}\]

We need to construct a new LinModel that includes the measured disturbance $\mathbf{d} = u_l$ and the operating point $\mathbf{d_{op}} = 20$:

model_d = setop!(LinModel([G G[1:2, 2]], Ts, i_d=[3]), uop=[20, 20], yop=[50, 30], dop=[20])
LinModel with a sample time Ts = 2.0 s and:
+ 2 manipulated inputs u
+ 4 states x
+ 2 outputs y
+ 1 measured disturbances d

A LinMPC controller is constructed on this model:

mpc_d = LinMPC(model_d, Hp=10, Hc=2, Mwt=[1, 1], Nwt=[0.1, 0.1])
+mpc_d = setconstraint!(mpc_d, ymin=[48, -Inf])
LinMPC controller with a sample time Ts = 2.0 s, OSQP optimizer, SteadyKalmanFilter estimator and:
+ 10 prediction steps Hp
+  2 control steps Hc
+  1 slack variable ϵ (control constraints)
+  2 manipulated inputs u (0 integrating states)
+  6 estimated states x̂
+  2 measured outputs ym (2 integrating states)
+  0 unmeasured outputs yu
+  1 measured disturbances d

A new test function that feeds the measured disturbance $\mathbf{d}$ to the controller is also required:

function test_mpc_d(mpc_d, model)
+    N = 200
+    ry, ul = [50, 30], 0
+    dop = mpc_d.estim.model.dop
+    u_data, y_data, ry_data = zeros(model.nu, N), zeros(model.ny, N), zeros(model.ny, N)
+    for i = 1:N
+        i == 51  && (ry = [50, 35])
+        i == 101 && (ry = [54, 30])
+        i == 151 && (ul = -20)
+        d = ul .+ dop   # simulated measured disturbance
+        y = model()     # simulated measurements
+        preparestate!(mpc_d, y, d) # prepare estimate with the measured disturbance d
+        u = mpc_d(ry, d) # also feed the measured disturbance d to the controller
+        u_data[:,i], y_data[:,i], ry_data[:,i] = u, y, ry
+        updatestate!(mpc_d, u, y, d)    # update estimate with the measured disturbance d
+        updatestate!(model, u + [0; ul]) # update simulator
+    end
+    return u_data, y_data, ry_data
+end

The new feedforward compensation is able to almost perfectly reject the load disturbance:

setstate!(model, zeros(model.nx))
+u, y, d = model.uop, model(), mpc_d.estim.model.dop
+initstate!(mpc_d, u, y, d)
+u_data, y_data, ry_data = test_mpc_d(mpc_d, model)
+plot_data(t_data, u_data, y_data, ry_data)

plot4_LinMPC

Note that measured disturbances are assumed constant in the future by default but custom $\mathbf{D̂}$ predictions are possible. The same applies for the setpoint predictions $\mathbf{R̂_y}$.

  • 1As an alternative to state observer, we could have use an InternalModel structure with mpc = LinMPC(InternalModel(model), Hp=15, Hc=2, Mwt=[1, 1], Nwt=[0.1, 0.1]). It was tested on the example of this page and it gave similar results.
diff --git a/previews/PR114/manual/mtk/index.html b/previews/PR114/manual/mtk/index.html new file mode 100644 index 00000000..7c234ffa --- /dev/null +++ b/previews/PR114/manual/mtk/index.html @@ -0,0 +1,102 @@ + +ModelingToolkit · ModelPredictiveControl.jl
GitHub

Manual: ModelingToolkit Integration

Pendulum Model

This example integrates the simple pendulum model of the last section in the ModelingToolkit.jl (MTK) framework and extracts appropriate f! and h! functions to construct a NonLinModel. An NonLinMPC is designed from this model and simulated to reproduce the results of the last section.

Disclaimer

This simple example is not an official interface to ModelingToolkit.jl. It is provided as a basic starting template to combine both packages. There is no guarantee that it will work for all corner cases.

We first construct and instantiate the pendulum model:

using ModelPredictiveControl, ModelingToolkit
+using ModelingToolkit: D_nounits as D, t_nounits as t, varmap_to_vars
+@mtkmodel Pendulum begin
+    @parameters begin
+        g = 9.8
+        L = 0.4
+        K = 1.2
+        m = 0.3
+    end
+    @variables begin
+        θ(t) # state
+        ω(t) # state
+        τ(t) # input
+        y(t) # output
+    end
+    @equations begin
+        D(θ)    ~ ω
+        D(ω)    ~ -g/L*sin(θ) - K/m*ω + τ/m/L^2
+        y       ~ θ * 180 / π
+    end
+end
+@named mtk_model = Pendulum()
+mtk_model = complete(mtk_model)

\[ \begin{align} +\frac{\mathrm{d} \theta\left( t \right)}{\mathrm{d}t} &= \omega\left( t \right) \\ +\frac{\mathrm{d} \omega\left( t \right)}{\mathrm{d}t} &= \frac{ - g \sin\left( \theta\left( t \right) \right)}{L} + \frac{\tau\left( t \right)}{L^{2} m} + \frac{ - K \omega\left( t \right)}{m} \\ +y\left( t \right) &= 57.296 \theta\left( t \right) +\end{align} + \]

We than convert the MTK model to an input-output system:

function generate_f_h(model, inputs, outputs)
+    (_, f_ip), dvs, psym, io_sys = ModelingToolkit.generate_control_function(
+        model, inputs, split=false; outputs
+    )
+    if any(ModelingToolkit.is_alg_equation, equations(io_sys))
+        error("Systems with algebraic equations are not supported")
+    end
+    nu, nx, ny = length(inputs), length(dvs), length(outputs)
+    vx = string.(dvs)
+    p = varmap_to_vars(defaults(io_sys), psym)
+    function f!(ẋ, x, u, _ , p)
+        try
+            f_ip(ẋ, x, u, p, nothing)
+        catch err
+            if err isa MethodError
+                error("NonLinModel does not support a time argument t in the f function, "*
+                      "see the constructor docstring for a workaround.")
+            else
+                rethrow()
+            end
+        end
+        return nothing
+    end
+    h_ = ModelingToolkit.build_explicit_observed_function(io_sys, outputs; inputs)
+    u_nothing = fill(nothing, nu)
+    function h!(y, x, _ , p)
+        y .= try
+            # MTK.jl supports a `u` argument in `h_` function but not this package. We set
+            # `u` as a vector of nothing and `h_` function will presumably throw an
+            # MethodError it this argument is used inside the function
+            h_(x, u_nothing, p, nothing)
+        catch err
+            if err isa MethodError
+                error("NonLinModel only support strictly proper systems (no manipulated "*
+                      "input argument u in the output function h)")
+            else
+                rethrow()
+            end
+        end
+        return nothing
+    end
+    return f!, h!, p, nu, nx, ny, vx
+end
+inputs, outputs = [mtk_model.τ], [mtk_model.y]
+f!, h!, p, nu, nx, ny, vx = generate_f_h(mtk_model, inputs, outputs)
+Ts = 0.1
+vu, vy = ["\$τ\$ (Nm)"], ["\$θ\$ (°)"]

A NonLinModel can now be constructed:

model = setname!(NonLinModel(f!, h!, Ts, nu, nx, ny; p); u=vu, x=vx, y=vy)
NonLinModel with a sample time Ts = 0.1 s, RungeKutta solver and:
+ 1 manipulated inputs u
+ 2 states x
+ 1 outputs y
+ 0 measured disturbances d

We also instantiate a plant model with a 25 % larger friction coefficient $K$:

mtk_model.K = defaults(mtk_model)[mtk_model.K] * 1.25
+f_plant, h_plant, p = generate_f_h(mtk_model, inputs, outputs)
+plant = setname!(NonLinModel(f_plant, h_plant, Ts, nu, nx, ny; p); u=vu, x=vx, y=vy)
NonLinModel with a sample time Ts = 0.1 s, RungeKutta solver and:
+ 1 manipulated inputs u
+ 2 states x
+ 1 outputs y
+ 0 measured disturbances d

Controller Design

We can than reproduce the Kalman filter and the controller design of the last section:

α=0.01; σQ=[0.1, 1.0]; σR=[5.0]; nint_u=[1]; σQint_u=[0.1]
+estim = UnscentedKalmanFilter(model; α, σQ, σR, nint_u, σQint_u)
+Hp, Hc, Mwt, Nwt = 20, 2, [0.5], [2.5]
+nmpc = NonLinMPC(estim; Hp, Hc, Mwt, Nwt, Cwt=Inf)
+umin, umax = [-1.5], [+1.5]
+nmpc = setconstraint!(nmpc; umin, umax)
NonLinMPC controller with a sample time Ts = 0.1 s, Ipopt optimizer, UnscentedKalmanFilter estimator and:
+ 20 prediction steps Hp
+  2 control steps Hc
+  0 slack variable ϵ (control constraints)
+  1 manipulated inputs u (1 integrating states)
+  3 estimated states x̂
+  1 measured outputs ym (0 integrating states)
+  0 unmeasured outputs yu
+  0 measured disturbances d

The 180° setpoint response is identical:

using Plots
+N = 35
+res_ry = sim!(nmpc, N, [180.0], plant=plant, x_0=[0, 0], x̂_0=[0, 0, 0])
+plot(res_ry)

plot1_MTK

and also the output disturbance rejection:

res_yd = sim!(nmpc, N, [180.0], plant=plant, x_0=[π, 0], x̂_0=[π, 0, 0], y_step=[10])
+plot(res_yd)

plot2_MTK

Acknowledgement

Authored by 1-Bart-1 and baggepinnen, thanks for the contribution.

diff --git a/previews/PR114/manual/nonlinmpc/index.html b/previews/PR114/manual/nonlinmpc/index.html new file mode 100644 index 00000000..cb1ef21f --- /dev/null +++ b/previews/PR114/manual/nonlinmpc/index.html @@ -0,0 +1,148 @@ + +Nonlinear Design · ModelPredictiveControl.jl
GitHub

Manual: Nonlinear Design

Nonlinear Model

In this example, the goal is to control the angular position $θ$ of a pendulum attached to a motor. Knowing that the manipulated input is the motor torque $τ$ in Nm, the I/O vectors are:

\[\begin{aligned} + \mathbf{u} &= τ \\ + \mathbf{y} &= θ +\end{aligned}\]

The following figure presents the system:

pendulum

The plant model is nonlinear:

\[\begin{aligned} + \dot{θ}(t) &= ω(t) \\ + \dot{ω}(t) &= -\frac{g}{L}\sin\big( θ(t) \big) - \frac{K}{m} ω(t) + \frac{1}{m L^2} τ(t) +\end{aligned}\]

in which $g$ is the gravitational acceleration in m/s², $L$, the pendulum length in m, $K$, the friction coefficient at the pivot point in kg/s, and $m$, the mass attached at the end of the pendulum in kg, all bundled in the parameter vector $\mathbf{p} = [\begin{smallmatrix} g & L & K & m \end{smallmatrix}]'$. The NonLinModel constructor assumes by default that the state function f is continuous in time, that is, an ordinary differential equation system (like here):

using ModelPredictiveControl
+function f(x, u, _ , p)
+    g, L, K, m = p          # [m/s²], [m], [kg/s], [kg]
+    θ, ω = x[1], x[2]       # [rad], [rad/s]
+    τ  = u[1]               # [Nm]
+    dθ = ω
+    dω = -g/L*sin(θ) - K/m*ω + τ/m/L^2
+    return [dθ, dω]
+end
+h(x, _ , _ ) = [180/π*x[1]] # [°]
+p_model = [9.8, 0.4, 1.2, 0.3]
+nu, nx, ny, Ts = 1, 2, 1, 0.1
+vu, vx, vy = ["\$τ\$ (Nm)"], ["\$θ\$ (rad)", "\$ω\$ (rad/s)"], ["\$θ\$ (°)"]
+model = setname!(NonLinModel(f, h, Ts, nu, nx, ny; p=p_model); u=vu, x=vx, y=vy)
NonLinModel with a sample time Ts = 0.1 s, RungeKutta solver and:
+ 1 manipulated inputs u
+ 2 states x
+ 1 outputs y
+ 0 measured disturbances d

The output function $\mathbf{h}$ converts the $θ$ angle to degrees. Note that special characters like $θ$ can be typed in the Julia REPL or VS Code by typing \theta and pressing the <TAB> key. The parameter p can be of any type but use a mutable type like a vector of you want to modify it later. A 4th order RungeKutta method solves the differential equations by default. It is good practice to first simulate model using sim! as a quick sanity check:

using Plots
+u = [0.5]
+N = 35
+res = sim!(model, N, u)
+plot(res, plotu=false)

plot1_NonLinMPC

The setname! function allows customized Y-axis labels. The available plotting options are detailed in the documentation of the corresponding plot method.

Nonlinear Model Predictive Controller

An UnscentedKalmanFilter estimates the plant state :

α=0.01; σQ=[0.1, 1.0]; σR=[5.0]; nint_u=[1]; σQint_u=[0.1]
+estim = UnscentedKalmanFilter(model; α, σQ, σR, nint_u, σQint_u)
UnscentedKalmanFilter estimator with a sample time Ts = 0.1 s, NonLinModel and:
+ 1 manipulated inputs u (1 integrating states)
+ 3 estimated states x̂
+ 1 measured outputs ym (0 integrating states)
+ 0 unmeasured outputs yu
+ 0 measured disturbances d

The vectors σQ and σR are the standard deviations of the process and sensor noises, respectively. The value for the velocity $ω$ is higher here (σQ second value) since $\dot{ω}(t)$ equation includes an uncertain parameter: the friction coefficient $K$. Also, the argument nint_u explicitly adds one integrating state at the model input, the motor torque $τ$, with an associated standard deviation σQint_u of 0.1 N m. The estimator tuning is tested on a plant with a 25 % larger friction coefficient $K$:

p_plant = copy(p_model)
+p_plant[3] = 1.25*p_model[3]
+plant = setname!(NonLinModel(f, h, Ts, nu, nx, ny; p=p_plant); u=vu, x=vx, y=vy)
+res = sim!(estim, N, [0.5], plant=plant, y_noise=[0.5])
+plot(res, plotu=false, plotxwithx̂=true)

plot2_NonLinMPC

The estimate $x̂_3$ is the integrating state on the torque $τ$ that compensates for static errors. The Kalman filter performance seems sufficient for control.

As the motor torque is limited to -1.5 to 1.5 N m, we incorporate the input constraints in a NonLinMPC:

Hp, Hc, Mwt, Nwt = 20, 2, [0.5], [2.5]
+nmpc = NonLinMPC(estim; Hp, Hc, Mwt, Nwt, Cwt=Inf)
+umin, umax = [-1.5], [+1.5]
+nmpc = setconstraint!(nmpc; umin, umax)
NonLinMPC controller with a sample time Ts = 0.1 s, Ipopt optimizer, UnscentedKalmanFilter estimator and:
+ 20 prediction steps Hp
+  2 control steps Hc
+  0 slack variable ϵ (control constraints)
+  1 manipulated inputs u (1 integrating states)
+  3 estimated states x̂
+  1 measured outputs ym (0 integrating states)
+  0 unmeasured outputs yu
+  0 measured disturbances d

The option Cwt=Inf disables the slack variable ϵ for constraint softening. We test mpc performance on plant by imposing an angular setpoint of 180° (inverted position):

res_ry = sim!(nmpc, N, [180.0], plant=plant, x_0=[0, 0], x̂_0=[0, 0, 0])
+plot(res_ry)

plot3_NonLinMPC

The controller seems robust enough to variations on $K$ coefficient. Starting from this inverted position, the closed-loop response to a step disturbances of 10° is also satisfactory:

res_yd = sim!(nmpc, N, [180.0], plant=plant, x_0=[π, 0], x̂_0=[π, 0, 0], y_step=[10])
+plot(res_yd)

plot4_NonLinMPC

See sim! documentation for details on the possible simulation scenarios.

Economic Model Predictive Controller

Economic MPC can achieve the same objective but with lower economical costs. For this case study, the controller will aim to reduce the energy consumed by the motor. The power in watt transmitted by the motor to the pendulum is:

\[P(t) = τ(t) ω(t)\]

Thus, the work in joule done by the motor from $t = t_0$ to $t_{end}$ is:

\[W = \int_{t_0}^{t_{end}} P(t) \mathrm{d}t = \int_{t_0}^{t_{end}} τ(t) ω(t) \mathrm{d}t\]

With the sampling time $T_s$ in s, the prediction horizon $H_p$, the limits defined as $t_0 = k T_s$ and $t_{end} = (k+H_p) T_s$, and the left-endpoint rectangle method for the integral, we get:

\[W ≈ T_s \sum_{j=0}^{H_p-1} τ(k + j) ω(k + j)\]

The objective function will now include an additive term that penalizes the work done by the motor $W$ to reduce the energy consumption. Notice that $W$ is a function of the manipulated input $τ$ and the angular speed $ω$, a state that is not measured (only the angle $θ$ is measured here). As the arguments of NonLinMPC economic function JE do not include the states, the speed is now defined as an unmeasured output to design a Kalman Filter similar to the previous one ($\mathbf{y^m} = θ$ and $\mathbf{y^u} = ω$):

h2(x, _ , _ ) = [180/π*x[1], x[2]]
+nu, nx, ny = 1, 2, 2
+model2 = setname!(NonLinModel(f, h2, Ts, nu, nx, ny; p=p_model), u=vu, x=vx, y=[vy; vx[2]])
+plant2 = setname!(NonLinModel(f, h2, Ts, nu, nx, ny; p=p_plant), u=vu, x=vx, y=[vy; vx[2]])
+estim2 = UnscentedKalmanFilter(model2; σQ, σR, nint_u, σQint_u, i_ym=[1])
UnscentedKalmanFilter estimator with a sample time Ts = 0.1 s, NonLinModel and:
+ 1 manipulated inputs u (1 integrating states)
+ 3 estimated states x̂
+ 1 measured outputs ym (0 integrating states)
+ 1 unmeasured outputs yu
+ 0 measured disturbances d

The plant2 object based on h2 is also required since sim! expects that the output vector of plant argument corresponds to the model output vector in mpc argument. We can now define the $J_E$ function and the empc controller:

function JE(UE, ŶE, _ , p)
+    Ts = p
+    τ, ω = UE[1:end-1], ŶE[2:2:end-1]
+    return Ts*sum(τ.*ω)
+end
+empc = NonLinMPC(estim2; Hp, Hc, Nwt, Mwt=[0.5, 0], Cwt=Inf, Ewt=3.5e3, JE=JE, p=Ts)
+empc = setconstraint!(empc; umin, umax)
NonLinMPC controller with a sample time Ts = 0.1 s, Ipopt optimizer, UnscentedKalmanFilter estimator and:
+ 20 prediction steps Hp
+  2 control steps Hc
+  0 slack variable ϵ (control constraints)
+  1 manipulated inputs u (1 integrating states)
+  3 estimated states x̂
+  1 measured outputs ym (0 integrating states)
+  1 unmeasured outputs yu
+  0 measured disturbances d

The keyword argument Ewt weights the economic costs relative to the other terms in the objective function. The term must be large enough to be significant but a too high value can lead to a static error on the angle setpoint. The second element of Mwt is zero since the speed $ω$ is not requested to track a setpoint. The closed-loop response to a 180° setpoint is similar:

res2_ry = sim!(empc, N, [180, 0], plant=plant2, x_0=[0, 0], x̂_0=[0, 0, 0])
+plot(res2_ry, ploty=[1])

plot5_NonLinMPC

and the energy consumption is slightly lower:

function calcW(res)
+    τ, ω = res.U_data[1, 1:end-1], res.X_data[2, 1:end-1]
+    return Ts*sum(τ.*ω)
+end
+Dict(:W_nmpc => calcW(res_ry), :W_empc => calcW(res2_ry))
Dict{Symbol, Float64} with 2 entries:
+  :W_empc => 3.90053
+  :W_nmpc => 3.92078

Also, for a 10° step disturbance:

res2_yd = sim!(empc, N, [180; 0]; plant=plant2, x_0=[π, 0], x̂_0=[π, 0, 0], y_step=[10, 0])
+plot(res2_yd, ploty=[1])

plot6_NonLinMPC

the new controller is able to recuperate a little more energy from the pendulum (i.e. negative work):

Dict(:W_nmpc => calcW(res_yd), :W_empc => calcW(res2_yd))
Dict{Symbol, Float64} with 2 entries:
+  :W_empc => -0.0542454
+  :W_nmpc => -0.0440395

Of course, this gain is only exploitable if the motor electronic includes some kind of regenerative circuitry.

Model Linearization

Nonlinear MPC is more computationally expensive than LinMPC. Solving the problem should always be faster than the sampling time $T_s = 0.1$ s for real-time operation. This requirement is sometimes hard to meet on electronics or mechanical systems because of the fast dynamics. To ease the design and comparison with LinMPC, the linearize function allows automatic linearization of NonLinModel based on ForwardDiff.jl. We first linearize model at the point $θ = π$ rad and $ω = τ = 0$ (inverted position):

linmodel = linearize(model, x=[π, 0], u=[0])
LinModel with a sample time Ts = 0.1 s and:
+ 1 manipulated inputs u
+ 2 states x
+ 1 outputs y
+ 0 measured disturbances d

A SteadyKalmanFilter and a LinMPC are designed from linmodel:

skf = SteadyKalmanFilter(linmodel; σQ, σR, nint_u, σQint_u)
+mpc = LinMPC(skf; Hp, Hc, Mwt, Nwt, Cwt=Inf)
+mpc = setconstraint!(mpc, umin=[-1.5], umax=[+1.5])
LinMPC controller with a sample time Ts = 0.1 s, OSQP optimizer, SteadyKalmanFilter estimator and:
+ 20 prediction steps Hp
+  2 control steps Hc
+  0 slack variable ϵ (control constraints)
+  1 manipulated inputs u (1 integrating states)
+  3 estimated states x̂
+  1 measured outputs ym (0 integrating states)
+  0 unmeasured outputs yu
+  0 measured disturbances d

The linear controller satisfactorily rejects the 10° step disturbance:

res_lin = sim!(mpc, N, [180.0]; plant, x_0=[π, 0], y_step=[10])
+plot(res_lin)

plot7_NonLinMPC

Solving the optimization problem of mpc with DAQP optimizer instead of the default OSQP solver can improve the performance here. It is indeed documented that DAQP can perform better on small/medium dense matrices and unstable poles[1], which is obviously the case here (absolute value of unstable poles are greater than one):

using LinearAlgebra; poles = eigvals(linmodel.A)
2-element Vector{Float64}:
+ 0.48163285653799215
+ 1.3963025601286745

To install the solver, run:

using Pkg; Pkg.add("DAQP")

Constructing a LinMPC with DAQP:

using JuMP, DAQP
+daqp = Model(DAQP.Optimizer, add_bridges=false)
+mpc2 = LinMPC(skf; Hp, Hc, Mwt, Nwt, Cwt=Inf, optim=daqp)
+mpc2 = setconstraint!(mpc2; umin, umax)
LinMPC controller with a sample time Ts = 0.1 s, DAQP optimizer, SteadyKalmanFilter estimator and:
+ 20 prediction steps Hp
+  2 control steps Hc
+  0 slack variable ϵ (control constraints)
+  1 manipulated inputs u (1 integrating states)
+  3 estimated states x̂
+  1 measured outputs ym (0 integrating states)
+  0 unmeasured outputs yu
+  0 measured disturbances d

does slightly improve the rejection of the step disturbance:

res_lin2 = sim!(mpc2, N, [180.0]; plant, x_0=[π, 0], y_step=[10])
+plot(res_lin2)

plot8_NonLinMPC

The closed-loop performance is still lower than the nonlinear controller, as expected, but computations are about 210 times faster (0.000071 s versus 0.015 s per time steps, on average). However, remember that linmodel is only valid for angular positions near 180°. For example, the 180° setpoint response from 0° is unsatisfactory since the predictions are poor in the first quadrant:

res_lin3 = sim!(mpc2, N, [180.0]; plant, x_0=[0, 0])
+plot(res_lin3)

plot9_NonLinMPC

Multiple linearized model and controller objects are required for large deviations from this operating point. This is known as gain scheduling. Another approach is adapting the model of the LinMPC instance based on repeated online linearization.

Adapting the Model via Successive Linearization

The setmodel! method allows online adaptation of a linear plant model. Combined with the automatic linearization of linearize, a successive linearization MPC can be designed with minimal efforts. The SteadyKalmanFilter does not support setmodel! so we need to use the time-varying KalmanFilter, and we initialize it with a linearization at $θ = ω = τ = 0$:

linmodel = linearize(model, x=[0, 0], u=[0])
+kf = KalmanFilter(linmodel; σQ, σR, nint_u, σQint_u)
+mpc3 = LinMPC(kf; Hp, Hc, Mwt, Nwt, Cwt=Inf, optim=daqp)
+mpc3 = setconstraint!(mpc3; umin, umax)
LinMPC controller with a sample time Ts = 0.1 s, DAQP optimizer, KalmanFilter estimator and:
+ 20 prediction steps Hp
+  2 control steps Hc
+  0 slack variable ϵ (control constraints)
+  1 manipulated inputs u (1 integrating states)
+  3 estimated states x̂
+  1 measured outputs ym (0 integrating states)
+  0 unmeasured outputs yu
+  0 measured disturbances d

We create a function that simulates the plant and the adaptive controller:

function sim_adapt!(mpc, nonlinmodel, N, ry, plant, x_0, x̂_0, y_step=[0])
+    U_data, Y_data, Ry_data = zeros(plant.nu, N), zeros(plant.ny, N), zeros(plant.ny, N)
+    setstate!(plant, x_0)
+    initstate!(mpc, [0], plant())
+    setstate!(mpc, x̂_0)
+    for i = 1:N
+        y = plant() + y_step
+        x̂ = preparestate!(mpc, y)
+        u = moveinput!(mpc, ry)
+        linmodel = linearize(nonlinmodel; u, x=x̂[1:2])
+        setmodel!(mpc, linmodel)
+        U_data[:,i], Y_data[:,i], Ry_data[:,i] = u, y, ry
+        updatestate!(mpc, u, y) # update mpc state estimate
+        updatestate!(plant, u)  # update plant simulator
+    end
+    res = SimResult(mpc, U_data, Y_data; Ry_data)
+    return res
+end

The setmodel! method must be called after solving the optimization problem with moveinput! and before updating the state estimate with updatestate!, that is, when both $\mathbf{u}(k)$ and $\mathbf{x̂}_{k}(k)$ are available as the new operating point. The SimResult object is for plotting purposes only. The adaptive LinMPC performances are similar to the nonlinear MPC, both for the 180° setpoint:

x_0 = [0, 0]; x̂_0 = [0, 0, 0]; ry = [180]
+res_slin = sim_adapt!(mpc3, model, N, ry, plant, x_0, x̂_0)
+plot(res_slin)

plot10_NonLinMPC

and the 10° step disturbance:

x_0 = [π, 0]; x̂_0 = [π, 0, 0]; y_step = [10]
+res_slin = sim_adapt!(mpc3, model, N, ry, plant, x_0, x̂_0, y_step)
+plot(res_slin)

plot11_NonLinMPC

The computations of the successive linearization MPC are about 125 times faster than the nonlinear MPC (0.00012 s per time steps versus 0.015 s per time steps, on average), an impressive gain for similar closed-loop performances!

  • 1Arnström, D., Bemporad, A., and Axehill, D. (2022). A dual active-set solver for embedded quadratic programming using recursive LDLᵀ updates. IEEE Trans. Autom. Contr., 67(8). https://doi.org/doi:10.1109/TAC.2022.3176430.
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GitHub

Functions: Generic Functions

This page contains the documentation of functions that are generic to SimModel, StateEstimator and PredictiveController types.

Set Constraint

ModelPredictiveControl.setconstraint! — Function
setconstraint!(estim::MovingHorizonEstimator; <keyword arguments>) -> estim

Set the constraint parameters of the MovingHorizonEstimator estim.

It supports both soft and hard constraints on the estimated state $\mathbf{x̂}$, process noise $\mathbf{ŵ}$ and sensor noise $\mathbf{v̂}$:

\[\begin{alignat*}{3} + \mathbf{x̂_{min} - c_{x̂_{min}}} ϵ ≤&&\ \mathbf{x̂}_k(k-j+p) &≤ \mathbf{x̂_{max} + c_{x̂_{max}}} ϵ &&\qquad j = N_k, N_k - 1, ... , 0 \\ + \mathbf{ŵ_{min} - c_{ŵ_{min}}} ϵ ≤&&\ \mathbf{ŵ}(k-j+p) &≤ \mathbf{ŵ_{max} + c_{ŵ_{max}}} ϵ &&\qquad j = N_k, N_k - 1, ... , 1 \\ + \mathbf{v̂_{min} - c_{v̂_{min}}} ϵ ≤&&\ \mathbf{v̂}(k-j+1) &≤ \mathbf{v̂_{max} + c_{v̂_{max}}} ϵ &&\qquad j = N_k, N_k - 1, ... , 1 +\end{alignat*}\]

and also $ϵ ≥ 0$. All the constraint parameters are vector. Use ±Inf values when there is no bound. The constraint softness parameters $\mathbf{c}$, also called equal concern for relaxation, are non-negative values that specify the softness of the associated bound. Use 0.0 values for hard constraints (default for all of them). Notice that constraining the estimated sensor noises is equivalent to bounding the innovation term, since $\mathbf{v̂}(k) = \mathbf{y^m}(k) - \mathbf{ŷ^m}(k)$. See Extended Help for details on the constant $p$, on model augmentation and on time-varying constraints.

Arguments

Info

All the keyword arguments have non-Unicode alternatives e.g. xhatmin or Vhatmax.

The default constraints are mentioned here for clarity but omitting a keyword argument will not re-assign to its default value (defaults are set at construction only).

  • estim::MovingHorizonEstimator : moving horizon estimator to set constraints
  • x̂min=fill(-Inf,nx̂) / x̂max=fill(+Inf,nx̂) : estimated state bound $\mathbf{x̂_{min/max}}$
  • ŵmin=fill(-Inf,nx̂) / ŵmax=fill(+Inf,nx̂) : estimated process noise bound $\mathbf{ŵ_{min/max}}$
  • v̂min=fill(-Inf,nym) / v̂max=fill(+Inf,nym) : estimated sensor noise bound $\mathbf{v̂_{min/max}}$
  • c_x̂min=fill(0.0,nx̂) / c_x̂max=fill(0.0,nx̂) : x̂min / x̂max softness weight $\mathbf{c_{x̂_{min/max}}}$
  • c_ŵmin=fill(0.0,nx̂) / c_ŵmax=fill(0.0,nx̂) : ŵmin / ŵmax softness weight $\mathbf{c_{ŵ_{min/max}}}$
  • c_v̂min=fill(0.0,nym) / c_v̂max=fill(0.0,nym) : v̂min / v̂max softness weight $\mathbf{c_{v̂_{min/max}}}$
  • all the keyword arguments above but with a first capital letter, e.g. X̂max or C_ŵmax: for time-varying constraints (see Extended Help)

Examples

julia> estim = MovingHorizonEstimator(LinModel(ss(0.5,1,1,0,1)), He=3);
+
+julia> estim = setconstraint!(estim, x̂min=[-50, -50], x̂max=[50, 50])
+MovingHorizonEstimator estimator with a sample time Ts = 1.0 s, OSQP optimizer, LinModel and:
+ 3 estimation steps He
+ 0 slack variable ϵ (estimation constraints)
+ 1 manipulated inputs u (0 integrating states)
+ 2 estimated states x̂
+ 1 measured outputs ym (1 integrating states)
+ 0 unmeasured outputs yu
+ 0 measured disturbances d

Extended Help

Extended Help

The constant $p=0$ if estim.direct==true (current form), else $p=1$ (prediction form). Note that the state $\mathbf{x̂}$ and process noise $\mathbf{ŵ}$ constraints are applied on the augmented model, detailed in SteadyKalmanFilter Extended Help. For variable constraints, the bounds can be modified after calling updatestate!, that is, at runtime, except for ±Inf bounds. Time-varying constraints over the estimation horizon $H_e$ are also possible, mathematically defined as:

\[\begin{alignat*}{3} + \mathbf{X̂_{min} - C_{x̂_{min}}} ϵ ≤&&\ \mathbf{X̂} &≤ \mathbf{X̂_{max} + C_{x̂_{max}}} ϵ \\ + \mathbf{Ŵ_{min} - C_{ŵ_{min}}} ϵ ≤&&\ \mathbf{Ŵ} &≤ \mathbf{Ŵ_{max} + C_{ŵ_{max}}} ϵ \\ + \mathbf{V̂_{min} - C_{v̂_{min}}} ϵ ≤&&\ \mathbf{V̂} &≤ \mathbf{V̂_{max} + C_{v̂_{max}}} ϵ +\end{alignat*}\]

For this, use the same keyword arguments as above but with a first capital letter:

  • X̂min / X̂max / C_x̂min / C_x̂max : $\mathbf{X̂}$ constraints (nx̂*(He+1),).
  • Ŵmin / Ŵmax / C_ŵmin / C_ŵmax : $\mathbf{Ŵ}$ constraints (nx̂*He,).
  • V̂min / V̂max / C_v̂min / C_v̂max : $\mathbf{V̂}$ constraints (nym*He,).
source
setconstraint!(mpc::PredictiveController; <keyword arguments>) -> mpc

Set the constraint parameters of the PredictiveController mpc.

The predictive controllers support both soft and hard constraints, defined by:

\[\begin{alignat*}{3} + \mathbf{u_{min} - c_{u_{min}}} ϵ ≤&&\ \mathbf{u}(k+j) &≤ \mathbf{u_{max} + c_{u_{max}}} ϵ &&\qquad j = 0, 1 ,..., H_p - 1 \\ + \mathbf{Δu_{min} - c_{Δu_{min}}} ϵ ≤&&\ \mathbf{Δu}(k+j) &≤ \mathbf{Δu_{max} + c_{Δu_{max}}} ϵ &&\qquad j = 0, 1 ,..., H_c - 1 \\ + \mathbf{y_{min} - c_{y_{min}}} ϵ ≤&&\ \mathbf{ŷ}(k+j) &≤ \mathbf{y_{max} + c_{y_{max}}} ϵ &&\qquad j = 1, 2 ,..., H_p \\ + \mathbf{x̂_{min} - c_{x̂_{min}}} ϵ ≤&&\ \mathbf{x̂}_i(k+j) &≤ \mathbf{x̂_{max} + c_{x̂_{max}}} ϵ &&\qquad j = H_p +\end{alignat*}\]

and also $ϵ ≥ 0$. The last line is the terminal constraints applied on the states at the end of the horizon (see Extended Help). See MovingHorizonEstimator constraints for details on bounds and softness parameters $\mathbf{c}$. The output and terminal constraints are all soft by default. See Extended Help for time-varying constraints.

Arguments

Info

The keyword arguments Δumin, Δumax, c_Δumin, c_Δumax, x̂min, x̂max, c_x̂min, c_x̂max and their capital letter versions have non-Unicode alternatives e.g. Deltaumin, xhatmax and C_Deltaumin

The default constraints are mentioned here for clarity but omitting a keyword argument will not re-assign to its default value (defaults are set at construction only).

  • mpc::PredictiveController : predictive controller to set constraints
  • umin=fill(-Inf,nu) / umax=fill(+Inf,nu) : manipulated input bound $\mathbf{u_{min/max}}$
  • Δumin=fill(-Inf,nu) / Δumax=fill(+Inf,nu) : manipulated input increment bound $\mathbf{Δu_{min/max}}$
  • ymin=fill(-Inf,ny) / ymax=fill(+Inf,ny) : predicted output bound $\mathbf{y_{min/max}}$
  • x̂min=fill(-Inf,nx̂) / x̂max=fill(+Inf,nx̂) : terminal constraint bound $\mathbf{x̂_{min/max}}$
  • c_umin=fill(0.0,nu) / c_umax=fill(0.0,nu) : umin / umax softness weight $\mathbf{c_{u_{min/max}}}$
  • c_Δumin=fill(0.0,nu) / c_Δumax=fill(0.0,nu) : Δumin / Δumax softness weight $\mathbf{c_{Δu_{min/max}}}$
  • c_ymin=fill(1.0,ny) / c_ymax=fill(1.0,ny) : ymin / ymax softness weight $\mathbf{c_{y_{min/max}}}$
  • c_x̂min=fill(1.0,nx̂) / c_x̂max=fill(1.0,nx̂) : x̂min / x̂max softness weight $\mathbf{c_{x̂_{min/max}}}$
  • all the keyword arguments above but with a first capital letter, except for the terminal constraints, e.g. Ymax or C_Δumin: for time-varying constraints (see Extended Help)

Examples

julia> mpc = LinMPC(setop!(LinModel(tf(3, [30, 1]), 4), uop=[50], yop=[25]));
+
+julia> mpc = setconstraint!(mpc, umin=[0], umax=[100], Δumin=[-10], Δumax=[+10])
+LinMPC controller with a sample time Ts = 4.0 s, OSQP optimizer, SteadyKalmanFilter estimator and:
+ 10 prediction steps Hp
+  2 control steps Hc
+  1 slack variable ϵ (control constraints)
+  1 manipulated inputs u (0 integrating states)
+  2 estimated states x̂
+  1 measured outputs ym (1 integrating states)
+  0 unmeasured outputs yu
+  0 measured disturbances d

Extended Help

Extended Help

Terminal constraints provide closed-loop stability guarantees on the nominal plant model. They can render an unfeasible problem however. In practice, a sufficiently large prediction horizon $H_p$ without terminal constraints is typically enough for stability. If mpc.estim.direct==true, the estimator computes the states at $i = k$ (the current time step), otherwise at $i = k - 1$. Note that terminal constraints are applied on the augmented state vector $\mathbf{x̂}$ (see SteadyKalmanFilter for details on augmentation).

For variable constraints, the bounds can be modified after calling moveinput!, that is, at runtime, but not the softness parameters $\mathbf{c}$. It is not possible to modify ±Inf bounds at runtime.

Tip

To keep a variable unconstrained while maintaining the ability to add a constraint later at runtime, set the bound to an absolute value sufficiently large when you create the controller (but different than ±Inf).

It is also possible to specify time-varying constraints over $H_p$ and $H_c$ horizons. In such a case, they are defined by:

\[\begin{alignat*}{3} + \mathbf{U_{min} - C_{u_{min}}} ϵ ≤&&\ \mathbf{U} &≤ \mathbf{U_{max} + C_{u_{max}}} ϵ \\ + \mathbf{ΔU_{min} - C_{Δu_{min}}} ϵ ≤&&\ \mathbf{ΔU} &≤ \mathbf{ΔU_{max} + C_{Δu_{max}}} ϵ \\ + \mathbf{Y_{min} - C_{y_{min}}} ϵ ≤&&\ \mathbf{Ŷ} &≤ \mathbf{Y_{max} + C_{y_{max}}} ϵ +\end{alignat*}\]

For this, use the same keyword arguments as above but with a first capital letter:

  • Umin / Umax / C_umin / C_umax : $\mathbf{U}$ constraints (nu*Hp,).
  • ΔUmin / ΔUmax / C_Δumin / C_Δumax : $\mathbf{ΔU}$ constraints (nu*Hc,).
  • Ymin / Ymax / C_ymin / C_ymax : $\mathbf{Ŷ}$ constraints (ny*Hp,).
source

Evaluate Output y

ModelPredictiveControl.evaloutput — Function
evaloutput(model::SimModel, d=[]) -> y

Evaluate SimModel outputs y from model.x0 states and measured disturbances d.

It returns model output at the current time step $\mathbf{y}(k)$. Calling a SimModel object calls this evaloutput method.

Examples

julia> model = setop!(LinModel(tf(2, [10, 1]), 5.0), yop=[20]);
+
+julia> y = evaloutput(model)
+1-element Vector{Float64}:
+ 20.0
source
evaloutput(estim::StateEstimator, d=[]) -> ŷ

Evaluate StateEstimator outputs ŷ from estim.x̂0 states and disturbances d.

It returns estim output at the current time step $\mathbf{ŷ}(k)$. If estim.direct is true, the method preparestate! should be called beforehand to correct the state estimate.

Calling a StateEstimator object calls this evaloutput method.

Examples

julia> kf = SteadyKalmanFilter(setop!(LinModel(tf(2, [10, 1]), 5), yop=[20]), direct=false);
+
+julia> ŷ = evaloutput(kf)
+1-element Vector{Float64}:
+ 20.0
source

Change State x

Prepare State x

ModelPredictiveControl.preparestate! — Function
preparestate!(model::SimModel, _ , _ ) -> x

Do nothing for SimModel and return the current model state $\mathbf{x}(k)$.

source
preparestate!(estim::StateEstimator, ym, d=estim.model.dop) -> x̂

Prepare estim.x̂0 estimate with meas. outputs ym and dist. d for the current time step.

This function should be called at the beginning of each discrete time step. Its behavior depends if estim is a StateEstimator in the current/filter (1.) or delayed/predictor (2.) formulation:

  1. If estim.direct is true, it removes the operating points with remove_op!, calls correct_estimate!, and returns the corrected state estimate $\mathbf{x̂}_k(k)$.
  2. Else, it does nothing and returns the current best estimate $\mathbf{x̂}_{k-1}(k)$.

Examples

julia> estim2 = SteadyKalmanFilter(LinModel(ss(0.1, 0.5, 1, 0, 4)), nint_ym=0, direct=true);
+
+julia> x̂ = round.(preparestate!(estim2, [1]), digits=3)
+1-element Vector{Float64}:
+ 0.01
+
+julia> estim1 = SteadyKalmanFilter(LinModel(ss(0.1, 0.5, 1, 0, 4)), nint_ym=0, direct=false);
+
+julia> x̂ = preparestate!(estim1, [1])
+1-element Vector{Float64}:
+ 0.0
source
preparestate!(mpc::PredictiveController, ym, d=[]) -> x̂

Call preparestate! on mpc.estim StateEstimator.

source

Update State x

ModelPredictiveControl.updatestate! — Function
updatestate!(model::SimModel, u, d=[]) -> xnext

Update model.x0 states with current inputs u and meas. dist. d for the next time step.

The method computes and returns the model state for the next time step $\mathbf{x}(k+1)$.

Examples

julia> model = LinModel(ss(1.0, 1.0, 1.0, 0, 1.0));
+
+julia> x = updatestate!(model, [1])
+1-element Vector{Float64}:
+ 1.0
source
updatestate!(estim::StateEstimator, u, ym, d=[]) -> x̂next

Update estim.x̂0 estimate with current inputs u, measured outputs ym and dist. d.

This function should be called at the end of each discrete time step. It removes the operating points with remove_op!, calls update_estimate! and returns the state estimate for the next time step $\mathbf{x̂}_k(k+1)$. The method preparestate! should be called prior to this one to correct the estimate when applicable (if estim.direct == true). Note that the MovingHorizonEstimator with the default direct=true option is not able to estimate $\mathbf{x̂}_k(k+1)$, the returned value is therefore the current corrected state $\mathbf{x̂}_k(k)$.

Examples

julia> kf = SteadyKalmanFilter(LinModel(ss(0.1, 0.5, 1, 0, 4.0))); u = [1]; ym = [0];
+
+julia> preparestate!(kf, ym);
+
+julia> x̂ = updatestate!(kf, u, ym) # x̂[2] is the integrator state (nint_ym argument)
+2-element Vector{Float64}:
+ 0.5
+ 0.0
source
updatestate!(mpc::PredictiveController, u, ym, d=[]) -> x̂next

Call updatestate! on mpc.estim StateEstimator.

source

Init State x

ModelPredictiveControl.initstate! — Function
initstate!(model::SimModel, u, d=[]) -> x

Init model.x0 with manipulated inputs u and meas. dist. d steady-state and reset time.

The method tries to initialize the model state $\mathbf{x}$ at steady-state. It removes the operating points on u and d and calls steadystate!:

  • If model is a LinModel, the method computes the steady-state of current inputs u and measured disturbances d.
  • Else, model.x0 is left unchanged. Use setstate! to manually modify it.

Examples

julia> model = LinModel(tf(6, [10, 1]), 2.0);
+
+julia> u = [1]; x = initstate!(model, u); y = round.(evaloutput(model), digits=3)
+1-element Vector{Float64}:
+ 6.0
+ 
+julia> x ≈ updatestate!(model, u)
+true
source
initstate!(estim::StateEstimator, u, ym, d=[]) -> x̂

Init estim.x̂0 states from current inputs u, measured outputs ym and disturbances d.

The method tries to find a good steady-state for the initial estimate $\mathbf{x̂}$. It stores u - estim.model.uop at estim.lastu0 and removes the operating points with remove_op!, and call init_estimate!:

  • If estim.model is a LinModel, it finds the steady-state of the augmented model using u and d arguments, and uses the ym argument to enforce that $\mathbf{ŷ^m}(0) = \mathbf{y^m}(0)$. For control applications, this solution produces a bumpless manual to automatic transfer. See init_estimate! for details.
  • Else, estim.x̂0 is left unchanged. Use setstate! to manually modify it.

If applicable, it also sets the error covariance estim.P̂ to estim.P̂_0.

Examples

julia> estim = SteadyKalmanFilter(LinModel(tf(3, [10, 1]), 0.5), nint_ym=[2], direct=false);
+
+julia> u = [1]; y = [3 - 0.1]; x̂ = round.(initstate!(estim, u, y), digits=3)
+3-element Vector{Float64}:
+  5.0
+  0.0
+ -0.1
+
+julia> x̂ ≈ updatestate!(estim, u, y)
+true
+
+julia> evaloutput(estim) ≈ y
+true
source
initstate!(mpc::PredictiveController, u, ym, d=[]) -> x̂

Init the states of mpc.estim StateEstimator and warm start mpc.ΔŨ at zero.

source

Set State x

ModelPredictiveControl.setstate! — Function
setstate!(model::SimModel, x) -> model

Set model.x0 to x - model.xop from the argument x.

source
setstate!(estim::StateEstimator, x̂) -> estim

Set estim.x̂0 to x̂ - estim.x̂op from the argument x̂.

source
setstate!(mpc::PredictiveController, x̂) -> mpc

Set mpc.estim.x̂0 to x̂ - estim.x̂op from the argument x̂.

source

Set Model and Weights

ModelPredictiveControl.setmodel! — Function
setmodel!(estim::StateEstimator, model=estim.model; <keyword arguments>) -> estim

Set model and covariance matrices of estim StateEstimator.

Allows model adaptation of estimators based on LinModel at runtime. Modification of NonLinModel state-space functions is not supported. New covariance matrices can be specified with the keyword arguments (see SteadyKalmanFilter documentation for the nomenclature). Not supported by Luenberger and SteadyKalmanFilter, use the time-varying KalmanFilter instead. The MovingHorizonEstimator model is kept constant over the estimation horizon $H_e$. The matrix dimensions and sample time must stay the same. Note that the observability and controllability of the new augmented model is not verified (see Extended Help for more info).

Arguments

Info

Keyword arguments with emphasis are non-Unicode alternatives.

  • estim::StateEstimator : estimator to set model and covariances.
  • model=estim.model : new plant model (not supported by NonLinModel).
  • Q̂=nothing or Qhat : new augmented model $\mathbf{Q̂}$ covariance matrix.
  • R̂=nothing or Rhat : new augmented model $\mathbf{R̂}$ covariance matrix.

Examples

julia> kf = KalmanFilter(LinModel(ss(0.1, 0.5, 1, 0, 4.0)), σQ=[√4.0], σQint_ym=[√0.25]);
+
+julia> kf.model.A[], kf.Q̂[1, 1], kf.Q̂[2, 2] 
+(0.1, 4.0, 0.25)
+
+julia> setmodel!(kf, LinModel(ss(0.42, 0.5, 1, 0, 4.0)), Q̂=[1 0;0 0.5]);
+
+julia> kf.model.A[], kf.Q̂[1, 1], kf.Q̂[2, 2] 
+(0.42, 1.0, 0.5)

Extended Help

Extended Help

Using the default model augmentation computed by the default_nint method, switching from a non-integrating plant model to an integrating one will produce an augmented model that is not observable. Moving the unmeasured disturbances at the model input (nint_u parameter) can fix this issue.

source
setmodel!(mpc::PredictiveController, model=mpc.estim.model; <keyword arguments>) -> mpc

Set model and objective function weights of mpc PredictiveController.

Allows model adaptation of controllers based on LinModel at runtime. Modification of NonLinModel state-space functions is not supported. New weight matrices in the objective function can be specified with the keyword arguments (see LinMPC for the nomenclature). If Cwt ≠ Inf, the augmented move suppression weight is $\mathbf{Ñ}_{H_c} = \mathrm{diag}(\mathbf{N}_{H_c}, C)$, else $\mathbf{Ñ}_{H_c} = \mathbf{N}_{H_c}$. The StateEstimator mpc.estim cannot be a Luenberger observer or a SteadyKalmanFilter (the default estimator). Construct the mpc object with a time-varying KalmanFilter instead. Note that the model is constant over the prediction horizon $H_p$.

Arguments

Info

Keyword arguments with emphasis are non-Unicode alternatives.

  • mpc::PredictiveController : controller to set model and weights.
  • model=mpc.estim.model : new plant model (not supported by NonLinModel).
  • Mwt=nothing : new main diagonal in $\mathbf{M}$ weight matrix (vector).
  • Nwt=nothing : new main diagonal in $\mathbf{N}$ weight matrix (vector).
  • Lwt=nothing : new main diagonal in $\mathbf{L}$ weight matrix (vector).
  • M_Hp=nothing : new $\mathbf{M}_{H_p}$ weight matrix.
  • Ñ_Hc=nothing or Ntilde_Hc : new $\mathbf{Ñ}_{H_c}$ weight matrix (see def. above).
  • L_Hp=nothing : new $\mathbf{L}_{H_p}$ weight matrix.
  • additional keyword arguments are passed to setmodel!(mpc.estim).

Examples

julia> mpc = LinMPC(KalmanFilter(LinModel(ss(0.1, 0.5, 1, 0, 4.0)), σR=[√25]), Hp=1, Hc=1);
+
+julia> mpc.estim.model.A[1], mpc.estim.R̂[1], mpc.M_Hp[1], mpc.Ñ_Hc[1]
+(0.1, 25.0, 1.0, 0.1)
+
+julia> setmodel!(mpc, LinModel(ss(0.42, 0.5, 1, 0, 4.0)); R̂=[9], M_Hp=[10], Nwt=[0.666]);
+
+julia> mpc.estim.model.A[1], mpc.estim.R̂[1], mpc.M_Hp[1], mpc.Ñ_Hc[1]
+(0.42, 9.0, 10.0, 0.666)
source

Get Additional Information

ModelPredictiveControl.getinfo — Function
getinfo(estim::MovingHorizonEstimator) -> info

Get additional info on estim MovingHorizonEstimator optimum for troubleshooting.

If estim.direct==true, the function should be called after calling preparestate!. Otherwise, call it after updatestate!. It returns the dictionary info with the following fields:

Info

Fields with emphasis are non-Unicode alternatives.

  • :Ŵ or :What : optimal estimated process noise over $N_k$, $\mathbf{Ŵ}$
  • :ϵ or :epsilon : optimal slack variable, $ϵ$
  • :X̂ or :Xhat : optimal estimated states over $N_k+1$, $\mathbf{X̂}$
  • :x̂ or :xhat : optimal estimated state for the next time step, $\mathbf{x̂}_k(k+1)$
  • :V̂ or :Vhat : optimal estimated sensor noise over $N_k$, $\mathbf{V̂}$
  • :P̄ or :Pbar : estimation error covariance at arrival, $\mathbf{P̄}$
  • :x̄ or :xbar : optimal estimation error at arrival, $\mathbf{x̄}$
  • :Ŷ or :Yhat : optimal estimated outputs over $N_k$, $\mathbf{Ŷ}$
  • :Ŷm or :Yhatm : optimal estimated measured outputs over $N_k$, $\mathbf{Ŷ^m}$
  • :x̂arr or :xhatarr : optimal estimated state at arrival, $\mathbf{x̂}_k(k-N_k+p)$
  • :J : objective value optimum, $J$
  • :Ym : measured outputs over $N_k$, $\mathbf{Y^m}$
  • :U : manipulated inputs over $N_k$, $\mathbf{U}$
  • :D : measured disturbances over $N_k$, $\mathbf{D}$
  • :sol : solution summary of the optimizer for printing

Examples

julia> model = LinModel(ss(1.0, 1.0, 1.0, 0, 5.0));
+
+julia> estim = MovingHorizonEstimator(model, He=1, nint_ym=0, direct=false);
+
+julia> updatestate!(estim, [0], [1]);
+
+julia> round.(getinfo(estim)[:Ŷ], digits=3)
+1-element Vector{Float64}:
+ 0.5
source
getinfo(mpc::PredictiveController) -> info

Get additional info about mpc PredictiveController optimum for troubleshooting.

The function should be called after calling moveinput!. It returns the dictionary info with the following fields:

Info

Fields with emphasis are non-Unicode alternatives.

  • :ΔU or :DeltaU : optimal manipulated input increments over $H_c$, $\mathbf{ΔU}$
  • :ϵ or :epsilon : optimal slack variable, $ϵ$
  • :D̂ or :Dhat : predicted measured disturbances over $H_p$, $\mathbf{D̂}$
  • :ŷ or :yhat : current estimated output, $\mathbf{ŷ}(k)$
  • :Ŷ or :Yhat : optimal predicted outputs over $H_p$, $\mathbf{Ŷ}$
  • :Ŷs or :Yhats : predicted stochastic output over $H_p$ of InternalModel, $\mathbf{Ŷ_s}$
  • :R̂y or :Rhaty : predicted output setpoint over $H_p$, $\mathbf{R̂_y}$
  • :R̂u or :Rhatu : predicted manipulated input setpoint over $H_p$, $\mathbf{R̂_u}$
  • :x̂end or :xhatend : optimal terminal states, $\mathbf{x̂}_i(k+H_p)$
  • :J : objective value optimum, $J$
  • :U : optimal manipulated inputs over $H_p$, $\mathbf{U}$
  • :u : current optimal manipulated input, $\mathbf{u}(k)$
  • :d : current measured disturbance, $\mathbf{d}(k)$

For LinMPC and NonLinMPC, the field :sol also contains the optimizer solution summary that can be printed. Lastly, the optimal economic cost :JE is also available for NonLinMPC.

Examples

julia> mpc = LinMPC(LinModel(tf(5, [2, 1]), 3), Nwt=[0], Hp=1, Hc=1);
+
+julia> preparestate!(mpc, [0]); u = moveinput!(mpc, [10]);
+
+julia> round.(getinfo(mpc)[:Ŷ], digits=3)
+1-element Vector{Float64}:
+ 10.0
source

Real-Time Simulate and Control

Disclaimer

These utilities are for soft real-time applications. They are not suitable for hard real-time environnement like safety-critical processes.

Save current time t

ModelPredictiveControl.savetime! — Function
savetime!(model::SimModel) -> t

Set model.t to time() and return the value.

Used in conjunction with periodsleep for simple soft real-time simulations. Call this function before any other in the simulation loop.

source
savetime!(estim::StateEstimator) -> t

Call savetime!(estim.model) and return the time t.

source
savetime!(mpc::PredictiveController) -> t

Call savetime!(mpc.estim.model) and return the time t.

source

Period Sleep

ModelPredictiveControl.periodsleep — Function
periodsleep(model::SimModel, busywait=false) -> nothing

Sleep for model.Ts s minus the time elapsed since the last call to savetime!.

It calls sleep if busywait is false. Else, a simple while loop implements busy-waiting. As a rule-of-thumb, busy-waiting should be used if model.Ts < 0.1 s, since the accuracy of sleep is around 1 ms. Can be used to implement simple soft real-time simulations, see the example below.

Warning

The allocations in Julia are garbage-collected (GC) automatically. This can affect the timing. In such cases, you can temporarily stop the GC with GC.enable(false), and restart it at a convenient time e.g.: just before calling periodsleep.

Examples

julia> model = LinModel(tf(2, [0.3, 1]), 0.1);
+
+julia> function sim_realtime!(model)
+           t_0 = time()
+           for i=1:3
+               t = savetime!(model)      # first function called
+               println(round(t - t_0, digits=3))
+               updatestate!(model, [1])
+               periodsleep(model, true)  # last function called
+           end
+       end;
+
+julia> sim_realtime!(model)
+0.0
+0.1
+0.2
source
periodsleep(estim::StateEstimator) -> nothing

Call periodsleep(estim.model).

source
periodsleep(mpc::PredictiveController) -> nothing

Call periodsleep(mpc.estim.model).

source
diff --git a/previews/PR114/public/plot_sim/index.html b/previews/PR114/public/plot_sim/index.html new file mode 100644 index 00000000..09e84e9c --- /dev/null +++ b/previews/PR114/public/plot_sim/index.html @@ -0,0 +1,47 @@ + +Simulations and Plots · ModelPredictiveControl.jl
GitHub

Functions: Simulations and Plots

This page documents the functions for quick plotting of open- and closed-loop simulations. They are generic to SimModel, StateEstimator and PredictiveController types. A SimResult instance must be created first with its constructor or by calling sim!. The results are then visualized with plot function from Plots.jl.

Quick Simulations

ModelPredictiveControl.sim! — Function
sim!(plant::SimModel, N::Int, u=plant.uop.+1, d=plant.dop; x_0=plant.xop) -> res

Open-loop simulation of plant for N time steps, default to unit bump test on all inputs.

The manipulated inputs $\mathbf{u}$ and measured disturbances $\mathbf{d}$ are held constant at u and d values, respectively. The plant initial state $\mathbf{x}(0)$ is specified by x_0 keyword arguments. The function returns SimResult instances that can be visualized by calling plot on them. Note that the method mutates plant internal states.

Examples

julia> plant = NonLinModel((x,u,d,_)->0.1x+u+d, (x,_,_)->2x, 5, 1, 1, 1, 1, solver=nothing);
+
+julia> res = sim!(plant, 15, [0], [0], x_0=[1])
+Simulation results of NonLinModel with 15 time steps.
source
sim!(
+    estim::StateEstimator,
+    N::Int,
+    u = estim.model.uop .+ 1,
+    d = estim.model.dop;
+    <keyword arguments>
+) -> res

Closed-loop simulation of estim estimator for N steps, default to input bumps.

See Arguments for the available options. The noises are provided as standard deviations σ vectors. The simulated sensor and process noises of plant are specified by y_noise and x_noise arguments, respectively.

Arguments

Info

Keyword arguments with emphasis are non-Unicode alternatives.

  • estim::StateEstimator : state estimator to simulate
  • N::Int : simulation length in time steps
  • u = estim.model.uop .+ 1 : manipulated input $\mathbf{u}$ value
  • d = estim.model.dop : plant measured disturbance $\mathbf{d}$ value
  • plant::SimModel = estim.model : simulated plant model
  • u_step = zeros(plant.nu) : step load disturbance on plant inputs $\mathbf{u}$
  • u_noise = zeros(plant.nu) : gaussian load disturbance on plant inputs $\mathbf{u}$
  • y_step = zeros(plant.ny) : step disturbance on plant outputs $\mathbf{y}$
  • y_noise = zeros(plant.ny) : additive gaussian noise on plant outputs $\mathbf{y}$
  • d_step = zeros(plant.nd) : step on measured disturbances $\mathbf{d}$
  • d_noise = zeros(plant.nd) : additive gaussian noise on measured dist. $\mathbf{d}$
  • x_noise = zeros(plant.nx) : additive gaussian noise on plant states $\mathbf{x}$
  • x_0 = plant.xop : plant initial state $\mathbf{x}(0)$
  • x̂_0 = nothing or xhat_0 : initial estimate $\mathbf{x̂}(0)$, initstate! is used if nothing
  • lastu = plant.uop : last plant input $\mathbf{u}$ for $\mathbf{x̂}$ initialization

Examples

julia> model = LinModel(tf(3, [30, 1]), 0.5);
+
+julia> estim = KalmanFilter(model, σR=[0.5], σQ=[0.25], σQint_ym=[0.01], σPint_ym_0=[0.1]);
+
+julia> res = sim!(estim, 50, [0], y_noise=[0.5], x_noise=[0.25], x_0=[-10], x̂_0=[0, 0])
+Simulation results of KalmanFilter with 50 time steps.
source
sim!(
+    mpc::PredictiveController, 
+    N::Int,
+    ry = mpc.estim.model.yop .+ 1, 
+    d  = mpc.estim.model.dop,
+    ru = mpc.estim.model.uop;
+    <keyword arguments>
+) -> res

Closed-loop simulation of mpc controller for N steps, default to output setpoint bumps.

The output and manipulated input setpoints are held constant at ry and ru, respectively. The keyword arguments are identical to sim!(::StateEstimator, ::Int).

Examples

julia> model = LinModel([tf(3, [30, 1]); tf(2, [5, 1])], 4);
+
+julia> mpc = setconstraint!(LinMPC(model, Mwt=[0, 1], Nwt=[0.01], Hp=30), ymin=[0, -Inf]);
+
+julia> res = sim!(mpc, 25, [0, 0], y_noise=[0.1], y_step=[-10, 0])
+Simulation results of LinMPC with 25 time steps.
source

Simulation Results

ModelPredictiveControl.SimResult — Type
SimResult(obj::SimModel,             U_data, Y_data, D_data=[]; <keyword arguments>)
+SimResult(obj::StateEstimator,       U_data, Y_data, D_data=[]; <keyword arguments>)
+SimResult(obj::PredictiveController, U_data, Y_data, D_data=[]; <keyword arguments>)

Manually construct a SimResult to quickly plot obj simulations.

Except for obj, all the arguments should be matrices of N columns, where N is the number of time steps. SimResult objects allow to quickly plot simulation results. Simply call plot on them.

Arguments

Info

Keyword arguments with emphasis are non-Unicode alternatives.

  • obj : simulated SimModel/StateEstimator/PredictiveController
  • U_data : manipulated inputs
  • Y_data : plant outputs
  • D_data=[] : measured disturbances
  • X_data=nothing : plant states
  • X̂_data=nothing or Xhat_data : estimated states
  • Ŷ_data=nothing or Yhat_data : estimated outputs
  • Ry_data=nothing : plant output setpoints
  • Ru_data=nothing : manipulated input setpoints
  • plant=get_model(obj) : simulated plant model, default to obj internal plant model

Examples

julia> model = LinModel(tf(1, [1, 1]), 1.0);
+
+julia> N = 5; U_data = fill(1.0, 1, N); Y_data = zeros(1, N);
+
+julia> for i=1:N; updatestate!(model, U_data[:, i]); Y_data[:, i] = model(); end; Y_data
+1×5 Matrix{Float64}:
+ 0.632121  0.864665  0.950213  0.981684  0.993262
+
+julia> res = SimResult(model, U_data, Y_data)
+Simulation results of LinModel with 5 time steps.
source

Plotting Results

The plot methods are based on Plots.jl package. To install it run using Pkg; Pkg.add("Plots") in the Julia REPL.

ModelPredictiveControl.plot — Function
plot(res::SimResult{<:Real, <:SimModel}; <keyword arguments>)

Plot the simulation results of a SimModel.

Arguments

Info

The keyword arguments can be Bools, index ranges (2:4) or vectors ([1, 3]), to select the variables to plot.

  • res::SimResult{<:Real, <:SimModel} : simulation results to plot
  • ploty=true : plot plant outputs $\mathbf{y}$
  • plotu=true : plot manipulated inputs $\mathbf{u}$
  • plotd=true : plot measured disturbances $\mathbf{d}$ if applicable
  • plotx=false : plot plant states $\mathbf{x}$

Examples

julia> res = sim!(LinModel(tf(2, [10, 1]), 2.0), 25);
+
+julia> using Plots; plot(res, plotu=false)

plot_model

source
plot(res::SimResult{<:Real, <:StateEstimator}; <keyword arguments>)

Plot the simulation results of a StateEstimator.

Arguments

Info

The keyword arguments can be Bools, index ranges (2:4) or vectors ([1, 3]), to select the variables to plot. Keywords in emphasis are non-Unicode alternatives.

  • res::SimResult{<:Real, <:StateEstimator} : simulation results to plot
  • plotŷ=true or plotyhat : plot estimated outputs $\mathbf{ŷ}$
  • plotx̂=false or plotxhat : plot estimated states $\mathbf{x̂}$
  • plotxwithx̂=false or plotxwithxhat : plot plant states $\mathbf{x}$ and estimated states $\mathbf{x̂}$ together
  • plotx̂min=true or plotxhatmin : plot estimated state lower bounds $\mathbf{x̂_{min}}$ if applicable
  • plotx̂max=true or plotxhatmax : plot estimated state upper bounds $\mathbf{x̂_{max}}$ if applicable
  • <keyword arguments> of plot(::SimResult{<:Real, <:SimModel})

Examples

julia> res = sim!(KalmanFilter(LinModel(tf(3, [2.0, 1]), 1.0)), 25, [0], y_step=[1]);
+
+julia> using Plots; plot(res, plotu=false, plotŷ=true, plotxwithx̂=true)

plot_estimator

source
plot(res::SimResult{<:Real, <:PredictiveController}; <keyword arguments>)

Plot the simulation results of a PredictiveController.

Arguments

Info

The keyword arguments can be Bools, index ranges (2:4) or vectors ([1, 3]), to select the variables to plot.

  • res::SimResult{<:Real, <:PredictiveController} : simulation results to plot
  • plotry=true : plot plant output setpoints $\mathbf{r_y}$ if applicable
  • plotymin=true : plot predicted output lower bounds $\mathbf{y_{min}}$ if applicable
  • plotymax=true : plot predicted output upper bounds $\mathbf{y_{max}}$ if applicable
  • plotru=true : plot manipulated input setpoints $\mathbf{r_u}$ if applicable
  • plotumin=true : plot manipulated input lower bounds $\mathbf{u_{min}}$ if applicable
  • plotumax=true : plot manipulated input upper bounds $\mathbf{u_{max}}$ if applicable
  • <keyword arguments> of plot(::SimResult{<:Real, <:SimModel})
  • <keyword arguments> of plot(::SimResult{<:Real, <:StateEstimator})

Examples

julia> model = LinModel(tf(2, [5.0, 1]), 1.0);
+
+julia> res = sim!(setconstraint!(LinMPC(model), umax=[1.0]), 25, [0], u_step=[-1]);
+
+julia> using Plots; plot(res, plotŷ=true, plotry=true, plotumax=true, plotx̂=[2])

plot_controller

source
diff --git a/previews/PR114/public/predictive_control/index.html b/previews/PR114/public/predictive_control/index.html new file mode 100644 index 00000000..4f3d3d17 --- /dev/null +++ b/previews/PR114/public/predictive_control/index.html @@ -0,0 +1,105 @@ + +Predictive Controllers · ModelPredictiveControl.jl
GitHub

Functions: Predictive Controllers

All the predictive controllers in this module rely on a state estimator to compute the predictions. The default LinMPC estimator is a SteadyKalmanFilter, and NonLinMPC with nonlinear models, an UnscentedKalmanFilter. For simpler and more classical designs, an InternalModel structure is also available, that assumes by default that the current model mismatch estimation is constant in the future (same approach as dynamic matrix control, DMC).

Info

The nomenclature uses capital letters for time series (and matrices) and hats for the predictions (and estimations, for state estimators).

To be precise, at the $k$th control period, the vectors that encompass the future measured disturbances $\mathbf{d̂}$, model outputs $\mathbf{ŷ}$ and setpoints $\mathbf{r̂_y}$ over the prediction horizon $H_p$ are defined as:

\[ \mathbf{D̂} = \begin{bmatrix} + \mathbf{d̂}(k+1) \\ \mathbf{d̂}(k+2) \\ \vdots \\ \mathbf{d̂}(k+H_p) + \end{bmatrix} \: , \quad + \mathbf{Ŷ} = \begin{bmatrix} + \mathbf{ŷ}(k+1) \\ \mathbf{ŷ}(k+2) \\ \vdots \\ \mathbf{ŷ}(k+H_p) + \end{bmatrix} \quad \text{and} \quad + \mathbf{R̂_y} = \begin{bmatrix} + \mathbf{r̂_y}(k+1) \\ \mathbf{r̂_y}(k+2) \\ \vdots \\ \mathbf{r̂_y}(k+H_p) + \end{bmatrix}\]

The vectors for the manipulated input $\mathbf{u}$ are shifted by one time step:

\[ \mathbf{U} = \begin{bmatrix} + \mathbf{u}(k+0) \\ \mathbf{u}(k+1) \\ \vdots \\ \mathbf{u}(k+H_p-1) + \end{bmatrix} \quad \text{and} \quad + \mathbf{R̂_u} = \begin{bmatrix} + \mathbf{r̂_u}(k+0) \\ \mathbf{r̂_u}(k+1) \\ \vdots \\ \mathbf{r̂_u}(k+H_p-1) + \end{bmatrix}\]

Defining the manipulated input increment as $\mathbf{Δu}(k+j) = \mathbf{u}(k+j) - \mathbf{u}(k+j-1)$, the control horizon $H_c$ enforces that $\mathbf{Δu}(k+j) = \mathbf{0}$ for $j ≥ H_c$. For this reason, the vector that collects them is truncated up to $k+H_c-1$:

\[ \mathbf{ΔU} = + \begin{bmatrix} + \mathbf{Δu}(k+0) \\ \mathbf{Δu}(k+1) \\ \vdots \\ \mathbf{Δu}(k+H_c-1) + \end{bmatrix}\]

PredictiveController

ModelPredictiveControl.PredictiveController — Type

Abstract supertype of all predictive controllers.

(mpc::PredictiveController)(ry, d=[]; kwargs...) -> u

Functor allowing callable PredictiveController object as an alias for moveinput!.

Examples

julia> mpc = LinMPC(LinModel(tf(5, [2, 1]), 3), Nwt=[0], Hp=1000, Hc=1, direct=false);
+
+julia> u = mpc([5]); round.(u, digits=3)
+1-element Vector{Float64}:
+ 1.0
source

LinMPC

ModelPredictiveControl.LinMPC — Type
LinMPC(model::LinModel; <keyword arguments>)

Construct a linear predictive controller based on LinModel model.

The controller minimizes the following objective function at each discrete time $k$:

\[\begin{aligned} +\min_{\mathbf{ΔU}, ϵ} \mathbf{(R̂_y - Ŷ)}' \mathbf{M}_{H_p} \mathbf{(R̂_y - Ŷ)} + + \mathbf{(ΔU)}' \mathbf{N}_{H_c} \mathbf{(ΔU)} \\ + + \mathbf{(R̂_u - U)}' \mathbf{L}_{H_p} \mathbf{(R̂_u - U)} + + C ϵ^2 +\end{aligned}\]

in which the weight matrices are repeated $H_p$ or $H_c$ times by default:

\[\begin{aligned} + \mathbf{M}_{H_p} &= \text{diag}\mathbf{(M,M,...,M)} \\ + \mathbf{N}_{H_c} &= \text{diag}\mathbf{(N,N,...,N)} \\ + \mathbf{L}_{H_p} &= \text{diag}\mathbf{(L,L,...,L)} +\end{aligned}\]

Time-varying and non-diagonal weights are also supported. Modify the last block in $\mathbf{M}_{H_p}$ to specify a terminal weight. The $\mathbf{ΔU}$ includes the input increments $\mathbf{Δu}(k+j) = \mathbf{u}(k+j) - \mathbf{u}(k+j-1)$ from $j=0$ to $H_c-1$, the $\mathbf{Ŷ}$ vector, the output predictions $\mathbf{ŷ}(k+j)$ from $j=1$ to $H_p$, and the $\mathbf{U}$ vector, the manipulated inputs $\mathbf{u}(k+j)$ from $j=0$ to $H_p-1$. The slack variable $ϵ$ relaxes the constraints, as described in setconstraint! documentation. See Extended Help for a detailed nomenclature.

This method uses the default state estimator, a SteadyKalmanFilter with default arguments.

Arguments

  • model::LinModel : model used for controller predictions and state estimations.

  • Hp=10+nk: prediction horizon $H_p$, nk is the number of delays in model.

  • Hc=2 : control horizon $H_c$.

  • Mwt=fill(1.0,model.ny) : main diagonal of $\mathbf{M}$ weight matrix (vector).

  • Nwt=fill(0.1,model.nu) : main diagonal of $\mathbf{N}$ weight matrix (vector).

  • Lwt=fill(0.0,model.nu) : main diagonal of $\mathbf{L}$ weight matrix (vector).

  • M_Hp=diagm(repeat(Mwt,Hp)) : positive semidefinite symmetric matrix $\mathbf{M}_{H_p}$.

  • N_Hc=diagm(repeat(Nwt,Hc)) : positive semidefinite symmetric matrix $\mathbf{N}_{H_c}$.

  • L_Hp=diagm(repeat(Lwt,Hp)) : positive semidefinite symmetric matrix $\mathbf{L}_{H_p}$.

  • Cwt=1e5 : slack variable weight $C$ (scalar), use Cwt=Inf for hard constraints only.

  • optim=JuMP.Model(OSQP.MathOptInterfaceOSQP.Optimizer) : quadratic optimizer used in the predictive controller, provided as a JuMP.Model (default to OSQP optimizer).

  • additional keyword arguments are passed to SteadyKalmanFilter constructor.

Examples

julia> model = LinModel([tf(3, [30, 1]); tf(-2, [5, 1])], 4);
+
+julia> mpc = LinMPC(model, Mwt=[0, 1], Nwt=[0.5], Hp=30, Hc=1)
+LinMPC controller with a sample time Ts = 4.0 s, OSQP optimizer, SteadyKalmanFilter estimator and:
+ 30 prediction steps Hp
+  1 control steps Hc
+  1 slack variable ϵ (control constraints)
+  1 manipulated inputs u (0 integrating states)
+  4 estimated states x̂
+  2 measured outputs ym (2 integrating states)
+  0 unmeasured outputs yu
+  0 measured disturbances d

Extended Help

Extended Help

Manipulated inputs setpoints $\mathbf{r_u}$ are not common but they can be interesting for over-actuated systems, when nu > ny (e.g. prioritize solutions with lower economical costs). The default Lwt value implies that this feature is disabled by default.

The objective function follows this nomenclature:

VARIABLEDESCRIPTIONSIZE
$H_p$prediction horizon (integer)()
$H_c$control horizon (integer)()
$\mathbf{ΔU}$manipulated input increments over $H_c$(nu*Hc,)
$\mathbf{Ŷ}$predicted outputs over $H_p$(ny*Hp,)
$\mathbf{U}$manipulated inputs over $H_p$(nu*Hp,)
$\mathbf{R̂_y}$predicted output setpoints over $H_p$(ny*Hp,)
$\mathbf{R̂_u}$predicted manipulated input setpoints over $H_p$(nu*Hp,)
$\mathbf{M}_{H_p}$output setpoint tracking weights over $H_p$(ny*Hp, ny*Hp)
$\mathbf{N}_{H_c}$manipulated input increment weights over $H_c$(nu*Hc, nu*Hc)
$\mathbf{L}_{H_p}$manipulated input setpoint tracking weights over $H_p$(nu*Hp, nu*Hp)
$C$slack variable weight()
$ϵ$slack variable for constraint softening()
source
LinMPC(estim::StateEstimator; <keyword arguments>)

Use custom state estimator estim to construct LinMPC.

estim.model must be a LinModel. Else, a NonLinMPC is required.

Examples

julia> estim = KalmanFilter(LinModel([tf(3, [30, 1]); tf(-2, [5, 1])], 4), i_ym=[2]);
+
+julia> mpc = LinMPC(estim, Mwt=[0, 1], Nwt=[0.5], Hp=30, Hc=1)
+LinMPC controller with a sample time Ts = 4.0 s, OSQP optimizer, KalmanFilter estimator and:
+ 30 prediction steps Hp
+  1 control steps Hc
+  1 slack variable ϵ (control constraints)
+  1 manipulated inputs u (0 integrating states)
+  3 estimated states x̂
+  1 measured outputs ym (1 integrating states)
+  1 unmeasured outputs yu
+  0 measured disturbances d
source

ExplicitMPC

ModelPredictiveControl.ExplicitMPC — Type
ExplicitMPC(model::LinModel; <keyword arguments>)

Construct an explicit linear predictive controller based on LinModel model.

The controller minimizes the following objective function at each discrete time $k$:

\[\begin{aligned} +\min_{\mathbf{ΔU}} \mathbf{(R̂_y - Ŷ)}' \mathbf{M}_{H_p} \mathbf{(R̂_y - Ŷ)} + + \mathbf{(ΔU)}' \mathbf{N}_{H_c} \mathbf{(ΔU)} \\ + + \mathbf{(R̂_u - U)}' \mathbf{L}_{H_p} \mathbf{(R̂_u - U)} +\end{aligned}\]

See LinMPC for the variable definitions. This controller does not support constraints but the computational costs are extremely low (array division), therefore suitable for applications that require small sample times. The keyword arguments are identical to LinMPC, except for Cwt and optim which are not supported.

This method uses the default state estimator, a SteadyKalmanFilter with default arguments.

Examples

julia> model = LinModel([tf(3, [30, 1]); tf(-2, [5, 1])], 4);
+
+julia> mpc = ExplicitMPC(model, Mwt=[0, 1], Nwt=[0.5], Hp=30, Hc=1)
+ExplicitMPC controller with a sample time Ts = 4.0 s, SteadyKalmanFilter estimator and:
+ 30 prediction steps Hp
+  1 control steps Hc
+  1 manipulated inputs u (0 integrating states)
+  4 estimated states x̂
+  2 measured outputs ym (2 integrating states)
+  0 unmeasured outputs yu
+  0 measured disturbances d
source
ExplicitMPC(estim::StateEstimator; <keyword arguments>)

Use custom state estimator estim to construct ExplicitMPC.

estim.model must be a LinModel. Else, a NonLinMPC is required.

source

NonLinMPC

ModelPredictiveControl.NonLinMPC — Type
NonLinMPC(model::SimModel; <keyword arguments>)

Construct a nonlinear predictive controller based on SimModel model.

Both NonLinModel and LinModel are supported (see Extended Help). The controller minimizes the following objective function at each discrete time $k$:

\[\begin{aligned} +\min_{\mathbf{ΔU}, ϵ}\ & \mathbf{(R̂_y - Ŷ)}' \mathbf{M}_{H_p} \mathbf{(R̂_y - Ŷ)} + + \mathbf{(ΔU)}' \mathbf{N}_{H_c} \mathbf{(ΔU)} \\& + + \mathbf{(R̂_u - U)}' \mathbf{L}_{H_p} \mathbf{(R̂_u - U)} + + C ϵ^2 + + E J_E(\mathbf{U}_E, \mathbf{Ŷ}_E, \mathbf{D̂}_E, \mathbf{p}) +\end{aligned}\]

See LinMPC for the variable definitions. The custom economic function $J_E$ can penalizes solutions with high economic costs. Setting all the weights to 0 except $E$ creates a pure economic model predictive controller (EMPC). The arguments of $J_E$ include the manipulated inputs, the predicted outputs and measured disturbances from $k$ to $k+H_p$ inclusively:

\[ \mathbf{U}_E = \begin{bmatrix} \mathbf{U} \\ \mathbf{u}(k+H_p-1) \end{bmatrix} , \quad + \mathbf{Ŷ}_E = \begin{bmatrix} \mathbf{ŷ}(k) \\ \mathbf{Ŷ} \end{bmatrix} , \quad + \mathbf{D̂}_E = \begin{bmatrix} \mathbf{d}(k) \\ \mathbf{D̂} \end{bmatrix}\]

since $H_c ≤ H_p$ implies that $\mathbf{Δu}(k+H_p) = \mathbf{0}$ or $\mathbf{u}(k+H_p)= \mathbf{u}(k+H_p-1)$. The vector $\mathbf{D̂}$ includes the predicted measured disturbance over $H_p$. The argument $\mathbf{p}$ is a custom parameter object of any type but use a mutable one if you want to modify it later e.g.: a vector.

Tip

Replace any of the 4 arguments with _ if not needed (see JE default value below).

This method uses the default state estimator :

Warning

See Extended Help if you get an error like: MethodError: no method matching Float64(::ForwardDiff.Dual).

Arguments

  • model::SimModel : model used for controller predictions and state estimations.
  • Hp=nothing: prediction horizon $H_p$, must be specified for NonLinModel.
  • Hc=2 : control horizon $H_c$.
  • Mwt=fill(1.0,model.ny) : main diagonal of $\mathbf{M}$ weight matrix (vector).
  • Nwt=fill(0.1,model.nu) : main diagonal of $\mathbf{N}$ weight matrix (vector).
  • Lwt=fill(0.0,model.nu) : main diagonal of $\mathbf{L}$ weight matrix (vector).
  • M_Hp=diagm(repeat(Mwt,Hp)) : positive semidefinite symmetric matrix $\mathbf{M}_{H_p}$.
  • N_Hc=diagm(repeat(Nwt,Hc)) : positive semidefinite symmetric matrix $\mathbf{N}_{H_c}$.
  • L_Hp=diagm(repeat(Lwt,Hp)) : positive semidefinite symmetric matrix $\mathbf{L}_{H_p}$.
  • Cwt=1e5 : slack variable weight $C$ (scalar), use Cwt=Inf for hard constraints only.
  • Ewt=0.0 : economic costs weight $E$ (scalar).
  • JE=(_,_,_,_)->0.0 : economic function $J_E(\mathbf{U}_E, \mathbf{Ŷ}_E, \mathbf{D̂}_E, \mathbf{p})$.
  • p=model.p : $J_E$ function parameter $\mathbf{p}$ (any type).
  • optim=JuMP.Model(Ipopt.Optimizer) : nonlinear optimizer used in the predictive controller, provided as a JuMP.Model (default to Ipopt optimizer).
  • additional keyword arguments are passed to UnscentedKalmanFilter constructor (or SteadyKalmanFilter, for LinModel).

Examples

julia> model = NonLinModel((x,u,_,_)->0.5x+u, (x,_,_)->2x, 10.0, 1, 1, 1, solver=nothing);
+
+julia> mpc = NonLinMPC(model, Hp=20, Hc=1, Cwt=1e6)
+NonLinMPC controller with a sample time Ts = 10.0 s, Ipopt optimizer, UnscentedKalmanFilter estimator and:
+ 20 prediction steps Hp
+  1 control steps Hc
+  1 slack variable ϵ (control constraints)
+  1 manipulated inputs u (0 integrating states)
+  2 estimated states x̂
+  1 measured outputs ym (1 integrating states)
+  0 unmeasured outputs yu
+  0 measured disturbances d

Extended Help

Extended Help

NonLinMPC controllers based on LinModel compute the predictions with matrix algebra instead of a for loop. This feature can accelerate the optimization, especially for the constraint handling, and is not available in any other package, to my knowledge.

The optimization relies on JuMP automatic differentiation (AD) to compute the objective and constraint derivatives. Optimizers generally benefit from exact derivatives like AD. However, the NonLinModel state-space functions must be compatible with this feature. See Automatic differentiation for common mistakes when writing these functions.

Note that if Cwt≠Inf, the attribute nlp_scaling_max_gradient of Ipopt is set to 10/Cwt (if not already set), to scale the small values of $ϵ$.

source
NonLinMPC(estim::StateEstimator; <keyword arguments>)

Use custom state estimator estim to construct NonLinMPC.

Examples

julia> model = NonLinModel((x,u,_,_)->0.5x+u, (x,_,_)->2x, 10.0, 1, 1, 1, solver=nothing);
+
+julia> estim = UnscentedKalmanFilter(model, σQint_ym=[0.05]);
+
+julia> mpc = NonLinMPC(estim, Hp=20, Hc=1, Cwt=1e6)
+NonLinMPC controller with a sample time Ts = 10.0 s, Ipopt optimizer, UnscentedKalmanFilter estimator and:
+ 20 prediction steps Hp
+  1 control steps Hc
+  1 slack variable ϵ (control constraints)
+  1 manipulated inputs u (0 integrating states)
+  2 estimated states x̂
+  1 measured outputs ym (1 integrating states)
+  0 unmeasured outputs yu
+  0 measured disturbances d
source

Move Manipulated Input u

ModelPredictiveControl.moveinput! — Function
moveinput!(mpc::PredictiveController, ry=mpc.estim.model.yop, d=[]; <keyword args>) -> u

Compute the optimal manipulated input value u for the current control period.

Solve the optimization problem of mpc PredictiveController and return the results $\mathbf{u}(k)$. Following the receding horizon principle, the algorithm discards the optimal future manipulated inputs $\mathbf{u}(k+1), \mathbf{u}(k+2), ...$ Note that the method mutates mpc internal data but it does not modifies mpc.estim states. Call preparestate!(mpc, ym, d) before moveinput!, and updatestate!(mpc, u, ym, d) after, to update mpc state estimates.

Calling a PredictiveController object calls this method.

See also LinMPC, ExplicitMPC, NonLinMPC.

Arguments

Info

Keyword arguments with emphasis are non-Unicode alternatives.

  • mpc::PredictiveController : solve optimization problem of mpc.
  • ry=mpc.estim.model.yop : current output setpoints $\mathbf{r_y}(k)$.
  • d=[] : current measured disturbances $\mathbf{d}(k)$.
  • D̂=repeat(d, mpc.Hp) or Dhat : predicted measured disturbances $\mathbf{D̂}$, constant in the future by default or $\mathbf{d̂}(k+j)=\mathbf{d}(k)$ for $j=1$ to $H_p$.
  • R̂y=repeat(ry, mpc.Hp) or Rhaty : predicted output setpoints $\mathbf{R̂_y}$, constant in the future by default or $\mathbf{r̂_y}(k+j)=\mathbf{r_y}(k)$ for $j=1$ to $H_p$.
  • R̂u=mpc.Uop or Rhatu : predicted manipulated input setpoints, constant in the future by default or $\mathbf{r̂_u}(k+j)=\mathbf{u_{op}}$ for $j=0$ to $H_p-1$.

Examples

julia> mpc = LinMPC(LinModel(tf(5, [2, 1]), 3), Nwt=[0], Hp=1000, Hc=1);
+
+julia> preparestate!(mpc, [0]); ry = [5];
+
+julia> u = moveinput!(mpc, ry); round.(u, digits=3)
+1-element Vector{Float64}:
+ 1.0
source
diff --git a/previews/PR114/public/sim_model/index.html b/previews/PR114/public/sim_model/index.html new file mode 100644 index 00000000..033f3c93 --- /dev/null +++ b/previews/PR114/public/sim_model/index.html @@ -0,0 +1,116 @@ + +Plant Models · ModelPredictiveControl.jl
GitHub

Functions: Plant Models

The SimModel types represents discrete state-space models that can be used to construct StateEstimators and PredictiveControllers, or as plant simulators by calling evaloutput and updatestate! methods on SimModel instances (to test estimator/controller designs). For time simulations, the states $\mathbf{x}$ are stored inside SimModel instances. Use setstate! method to manually modify them.

SimModel

ModelPredictiveControl.SimModel — Type

Abstract supertype of LinModel and NonLinModel types.

(model::SimModel)(d=[]) -> y

Functor allowing callable SimModel object as an alias for evaloutput.

Examples

julia> model = NonLinModel((x,u,_,_)->-x + u, (x,_,_)->x .+ 20, 4, 1, 1, 1, solver=nothing);
+
+julia> y = model()
+1-element Vector{Float64}:
+ 20.0
source

LinModel

ModelPredictiveControl.LinModel — Type
LinModel(sys::StateSpace[, Ts]; i_u=1:size(sys,2), i_d=Int[])

Construct a linear model from state-space model sys with sampling time Ts in second.

The system sys can be continuous or discrete-time (Ts can be omitted for the latter). For continuous dynamics, its state-space equations are (discrete case in Extended Help):

\[\begin{aligned} + \mathbf{ẋ}(t) &= \mathbf{A x}(t) + \mathbf{B z}(t) \\ + \mathbf{y}(t) &= \mathbf{C x}(t) + \mathbf{D z}(t) +\end{aligned}\]

with the state $\mathbf{x}$ and output $\mathbf{y}$ vectors. The $\mathbf{z}$ vector comprises the manipulated inputs $\mathbf{u}$ and measured disturbances $\mathbf{d}$, in any order. i_u provides the indices of $\mathbf{z}$ that are manipulated, and i_d, the measured disturbances. The constructor automatically discretizes continuous systems, resamples discrete ones if Ts ≠ sys.Ts, computes a new realization if not minimal, and separates the $\mathbf{z}$ terms in two parts (details in Extended Help). The rest of the documentation assumes discrete dynamics since all systems end up in this form.

See also ss

Examples

julia> model1 = LinModel(ss(-0.1, 1.0, 1.0, 0), 2.0) # continuous-time StateSpace
+LinModel with a sample time Ts = 2.0 s and:
+ 1 manipulated inputs u
+ 1 states x
+ 1 outputs y
+ 0 measured disturbances d
+
+julia> model2 = LinModel(ss(0.4, 0.2, 0.3, 0, 0.1)) # discrete-time StateSpace
+LinModel with a sample time Ts = 0.1 s and:
+ 1 manipulated inputs u
+ 1 states x
+ 1 outputs y
+ 0 measured disturbances d

Extended Help

Extended Help

The state-space equations are similar if sys is discrete-time:

\[\begin{aligned} + \mathbf{x}(k+1) &= \mathbf{A x}(k) + \mathbf{B z}(k) \\ + \mathbf{y}(k) &= \mathbf{C x}(k) + \mathbf{D z}(k) +\end{aligned}\]

Continuous dynamics are internally discretized using c2d and :zoh for manipulated inputs, and :tustin, for measured disturbances. Lastly, if sys is discrete and the provided argument Ts ≠ sys.Ts, the system is resampled by using the aforementioned discretization methods.

Note that the constructor transforms the system to its minimal realization using minreal for controllability/observability. As a consequence, the final state-space representation may be different from the one provided in sys. It is also converted into a more practical form ($\mathbf{D_u=0}$ because of the zero-order hold):

\[\begin{aligned} + \mathbf{x}(k+1) &= \mathbf{A x}(k) + \mathbf{B_u u}(k) + \mathbf{B_d d}(k) \\ + \mathbf{y}(k) &= \mathbf{C x}(k) + \mathbf{D_d d}(k) +\end{aligned}\]

Use the syntax LinModel{NT}(A, Bu, C, Bd, Dd, Ts) to force a specific state-space representation.

source
LinModel(sys::TransferFunction[, Ts]; i_u=1:size(sys,2), i_d=Int[])

Convert to minimal realization state-space when sys is a transfer function.

sys is equal to $\frac{\mathbf{y}(s)}{\mathbf{z}(s)}$ for continuous-time, and $\frac{\mathbf{y}(z)}{\mathbf{z}(z)}$, for discrete-time.

See also tf

Examples

julia> model = LinModel([tf(3, [30, 1]) tf(-2, [5, 1])], 0.5, i_d=[2])
+LinModel with a sample time Ts = 0.5 s and:
+ 1 manipulated inputs u
+ 2 states x
+ 1 outputs y
+ 1 measured disturbances d
source
LinModel(sys::DelayLtiSystem, Ts; i_u=1:size(sys,2), i_d=Int[])

Discretize with zero-order hold when sys is a continuous system with delays.

The delays must be multiples of the sample time Ts.

source
LinModel{NT}(A, Bu, C, Bd, Dd, Ts)

Construct the model from the discrete state-space matrices A, Bu, C, Bd, Dd directly.

See LinModel(::StateSpace) Extended Help for the meaning of the matrices. The arguments Bd and Dd can be the scalar 0 if there is no measured disturbance. This syntax do not modify the state-space representation provided in argument (minreal is not called). Care must be taken to ensure that the model is controllable and observable. The optional parameter NT explicitly set the number type of vectors (default to Float64).

source
LinModel(model::NonLinModel; x=model.x0+model.xop, u=model.uop, d=model.dop)

Call linearize(model; x, u, d) and return the resulting linear model.

source

NonLinModel

ModelPredictiveControl.NonLinModel — Type
NonLinModel{NT}(f::Function,  h::Function,  Ts, nu, nx, ny, nd=0; p=[], solver=RungeKutta(4))
+NonLinModel{NT}(f!::Function, h!::Function, Ts, nu, nx, ny, nd=0; p=[], solver=RungeKutta(4))

Construct a nonlinear model from state-space functions f/f! and h/h!.

Both continuous and discrete-time models are supported. The default arguments assume continuous dynamics. Use solver=nothing for the discrete case (see Extended Help). The functions are defined as:

\[\begin{aligned} + \mathbf{ẋ}(t) &= \mathbf{f}\Big( \mathbf{x}(t), \mathbf{u}(t), \mathbf{d}(t), \mathbf{p} \Big) \\ + \mathbf{y}(t) &= \mathbf{h}\Big( \mathbf{x}(t), \mathbf{d}(t), \mathbf{p} \Big) +\end{aligned}\]

where $\mathbf{x}$, $\mathbf{y}$, $\mathbf{u}$, $\mathbf{d}$ and $\mathbf{p}$ are respectively the state, output, manipulated input, measured disturbance and parameter vectors. If the dynamics is a function of the time, simply add a measured disturbance defined as $d(t) = t$. The functions can be implemented in two possible ways:

  1. Non-mutating functions (out-of-place): define them as f(x, u, d, p) -> ẋ and h(x, d, p) -> y. This syntax is simple and intuitive but it allocates more memory.
  2. Mutating functions (in-place): define them as f!(ẋ, x, u, d, p) -> nothing and h!(y, x, d, p) -> nothing. This syntax reduces the allocations and potentially the computational burden as well.

Ts is the sampling time in second. nu, nx, ny and nd are the respective number of manipulated inputs, states, outputs and measured disturbances. The keyword argument p is the parameters of the model passed to the two functions. It can be of any Julia object but use a mutable type if you want to change them later e.g.: a vector.

Tip

Replace the d or p argument with _ in your functions if not needed (see Examples below).

A 4th order RungeKutta solver discretizes the differential equations by default. The rest of the documentation assumes discrete dynamics since all models end up in this form. The optional parameter NT explicitly set the number type of vectors (default to Float64).

Warning

The two functions must be in pure Julia to use the model in NonLinMPC, ExtendedKalmanFilter, MovingHorizonEstimator and linearize.

See also LinModel.

Examples

julia> f!(ẋ, x, u, _ , p) = (ẋ .= p*x .+ u; nothing);
+
+julia> h!(y, x, _ , _ ) = (y .= 0.1x; nothing);
+
+julia> model1 = NonLinModel(f!, h!, 5.0, 1, 1, 1, p=-0.2)       # continuous dynamics
+NonLinModel with a sample time Ts = 5.0 s, RungeKutta solver and:
+ 1 manipulated inputs u
+ 1 states x
+ 1 outputs y
+ 0 measured disturbances d
+
+julia> f(x, u, _ , _ ) = 0.1x + u;
+
+julia> h(x, _ , _ ) = 2x;
+
+julia> model2 = NonLinModel(f, h, 2.0, 1, 1, 1, solver=nothing) # discrete dynamics
+NonLinModel with a sample time Ts = 2.0 s, empty solver and:
+ 1 manipulated inputs u
+ 1 states x
+ 1 outputs y
+ 0 measured disturbances d

Extended Help

Extended Help

State-space functions are similar for discrete dynamics:

\[\begin{aligned} + \mathbf{x}(k+1) &= \mathbf{f}\Big( \mathbf{x}(k), \mathbf{u}(k), \mathbf{d}(k), \mathbf{p} \Big) \\ + \mathbf{y}(k) &= \mathbf{h}\Big( \mathbf{x}(k), \mathbf{d}(k), \mathbf{p} \Big) +\end{aligned}\]

with two possible implementations as well:

  1. Non-mutating functions: define them as f(x, u, d, p) -> xnext and h(x, d, p) -> y.
  2. Mutating functions: define them as f!(xnext, x, u, d, p) -> nothing and h!(y, x, d, p) -> nothing.
source

Set Variable Names

ModelPredictiveControl.setname! — Function
setname!(model::SimModel; u=nothing, y=nothing, d=nothing, x=nothing) -> model

Set the names of model inputs u, outputs y, disturbances d, and states x.

The keyword arguments u, y, d, and x must be vectors of strings. The strings are used in the plotting functions.

Examples

julia> model = setname!(LinModel(tf(3, [10, 1]), 2.0), u=["\$A\$ (%)"], y=["\$T\$ (∘C)"])
+LinModel with a sample time Ts = 2.0 s and:
+ 1 manipulated inputs u
+ 1 states x
+ 1 outputs y
+ 0 measured disturbances d
source

Set Operating Points

ModelPredictiveControl.setop! — Function
setop!(model; uop=nothing, yop=nothing, dop=nothing, xop=nothing, fop=nothing) -> model

Set the operating points of model (both LinModel and NonLinModel).

Introducing deviations vectors around manipulated input uop, model output yop, measured disturbance dop, and model state xop operating points (a.k.a. nominal values):

\[\begin{align*} + \mathbf{u_0}(k) &= \mathbf{u}(k) - \mathbf{u_{op}} \\ + \mathbf{d_0}(k) &= \mathbf{d}(k) - \mathbf{d_{op}} \\ + \mathbf{y_0}(k) &= \mathbf{y}(k) - \mathbf{y_{op}} \\ + \mathbf{x_0}(k) &= \mathbf{x}(k) - \mathbf{x_{op}} \\ +\end{align*}\]

The state-space description of LinModel around the operating points is:

\[\begin{aligned} + \mathbf{x_0}(k+1) &= \mathbf{A x_0}(k) + \mathbf{B_u u_0}(k) + \mathbf{B_d d_0}(k) + + \mathbf{f_{op}} - \mathbf{x_{op}} \\ + \mathbf{y_0}(k) &= \mathbf{C x_0}(k) + \mathbf{D_d d_0}(k) +\end{aligned}\]

and, for NonLinModel:

\[\begin{aligned} + \mathbf{x_0}(k+1) &= \mathbf{f}\Big(\mathbf{x_0}(k), \mathbf{u_0}(k), \mathbf{d_0}(k), \mathbf{p}\Big) + + \mathbf{f_{op}} - \mathbf{x_{op}} \\ + \mathbf{y_0}(k) &= \mathbf{h}\Big(\mathbf{x_0}(k), \mathbf{d_0}(k), \mathbf{p}\Big) +\end{aligned}\]

The state xop and the additional fop operating points are frequently zero e.g.: when model comes from system identification. The two vectors are internally used by linearize for non-equilibrium points.

Examples

julia> model = setop!(LinModel(tf(3, [10, 1]), 2.0), uop=[50], yop=[20])
+LinModel with a sample time Ts = 2.0 s and:
+ 1 manipulated inputs u
+ 1 states x
+ 1 outputs y
+ 0 measured disturbances d
+
+julia> y = model()
+1-element Vector{Float64}:
+ 20.0
source

Linearize

ModelPredictiveControl.linearize — Function
linearize(model::SimModel; x=model.x0+model.xop, u=model.uop, d=model.dop) -> linmodel

Linearize model at the operating points x, u, d and return the LinModel.

The arguments x, u and d are the linearization points for the state $\mathbf{x}$, manipulated input $\mathbf{u}$ and measured disturbance $\mathbf{d}$, respectively (not necessarily an equilibrium, details in Extended Help). The Jacobians of $\mathbf{f}$ and $\mathbf{h}$ functions are automatically computed with ForwardDiff.jl.

Warning

See Extended Help if you get an error like: MethodError: no method matching (::var"##")(::Vector{ForwardDiff.Dual}).

Examples

julia> model = NonLinModel((x,u,_,_)->x.^3 + u, (x,_,_)->x, 0.1, 1, 1, 1, solver=nothing);
+
+julia> linmodel = linearize(model, x=[10.0], u=[0.0]); 
+
+julia> linmodel.A
+1×1 Matrix{Float64}:
+ 300.0

Extended Help

Extended Help

With the nonlinear state-space model:

\[\begin{aligned} + \mathbf{x}(k+1) &= \mathbf{f}\Big(\mathbf{x}(k), \mathbf{u}(k), \mathbf{d}(k), \mathbf{p}\Big) \\ + \mathbf{y}(k) &= \mathbf{h}\Big(\mathbf{x}(k), \mathbf{d}(k), \mathbf{p}\Big) +\end{aligned}\]

its linearization at the operating point $\mathbf{x_{op}, u_{op}, d_{op}}$ is:

\[\begin{aligned} + \mathbf{x_0}(k+1) &≈ \mathbf{A x_0}(k) + \mathbf{B_u u_0}(k) + \mathbf{B_d d_0}(k) + + \mathbf{f(x_{op}, u_{op}, d_{op}, p)} - \mathbf{x_{op}} \\ + \mathbf{y_0}(k) &≈ \mathbf{C x_0}(k) + \mathbf{D_d d_0}(k) +\end{aligned}\]

based on the deviation vectors $\mathbf{x_0, u_0, d_0, y_0}$ introduced in setop! documentation, and the Jacobian matrices:

\[\begin{aligned} + \mathbf{A} &= \left. \frac{∂\mathbf{f(x, u, d, p)}}{∂\mathbf{x}} \right|_{\mathbf{x=x_{op},\, u=u_{op},\, d=d_{op}}} \\ + \mathbf{B_u} &= \left. \frac{∂\mathbf{f(x, u, d, p)}}{∂\mathbf{u}} \right|_{\mathbf{x=x_{op},\, u=u_{op},\, d=d_{op}}} \\ + \mathbf{B_d} &= \left. \frac{∂\mathbf{f(x, u, d, p)}}{∂\mathbf{d}} \right|_{\mathbf{x=x_{op},\, u=u_{op},\, d=d_{op}}} \\ + \mathbf{C} &= \left. \frac{∂\mathbf{h(x, d, p)}}{∂\mathbf{x}} \right|_{\mathbf{x=x_{op},\, d=d_{op}}} \\ + \mathbf{D_d} &= \left. \frac{∂\mathbf{h(x, d, p)}}{∂\mathbf{d}} \right|_{\mathbf{x=x_{op},\, d=d_{op}}} +\end{aligned}\]

Following setop! notation, we find:

\[\begin{aligned} + \mathbf{f_{op}} &= \mathbf{f(x_{op}, u_{op}, d_{op}, p)} \\ + \mathbf{y_{op}} &= \mathbf{h(x_{op}, d_{op}, p)} +\end{aligned}\]

Notice that $\mathbf{f_{op} - x_{op} = 0}$ if the point is an equilibrium. The equations are similar if the nonlinear model has nonzero operating points.

Automatic differentiation (AD) allows exact Jacobians. The NonLinModel f and h functions must be compatible with this feature though. See Automatic differentiation for common mistakes when writing these functions.

source
ModelPredictiveControl.linearize! — Function
linearize!(linmodel::LinModel, model::SimModel; <keyword arguments>) -> linmodel

Linearize model and store the result in linmodel (in-place).

The keyword arguments are identical to linearize.

Examples

julia> model = NonLinModel((x,u,_,_)->x.^3 + u, (x,_,_)->x, 0.1, 1, 1, 1, solver=nothing);
+
+julia> linmodel = linearize(model, x=[10.0], u=[0.0]); linmodel.A
+1×1 Matrix{Float64}:
+ 300.0
+
+julia> linearize!(linmodel, model, x=[20.0], u=[0.0]); linmodel.A
+1×1 Matrix{Float64}:
+ 1200.0
source

Differential Equation Solvers

DiffSolver

RungeKutta

ModelPredictiveControl.RungeKutta — Type
RungeKutta(order=4; supersample=1)

Create a Runge-Kutta solver with optional super-sampling.

Only the 4th order Runge-Kutta is supported for now. The keyword argument supersample provides the number of internal steps (default to 1 step).

source
diff --git a/previews/PR114/public/state_estim/index.html b/previews/PR114/public/state_estim/index.html new file mode 100644 index 00000000..38b852bb --- /dev/null +++ b/previews/PR114/public/state_estim/index.html @@ -0,0 +1,119 @@ + +State Estimators · ModelPredictiveControl.jl
GitHub

Functions: State Estimators

This module includes many state estimators (or state observer), both for deterministic and stochastic systems. The implementations focus on control applications, that is, relying on the estimates to compute a full state feedback (predictive controllers, in this package). They all incorporates some kind of integral action by default, since it is generally desired to eliminate the steady-state error with closed-loop control (offset-free tracking).

Warning

If you plan to use the estimators for other contexts than this specific package (e.g. : filter, parameter estimation, etc.), careful must be taken at construction since the integral action is not necessarily desired. The options nint_u=0 and nint_ym=0 disable it.

The estimators are all implemented in the current form (a.k.a. as filter form) by default to improve accuracy and robustness, that is, they all estimates at each discrete time $k$ the states of the current period $\mathbf{x̂}_k(k)$[1] (using the newest measurements, see Manual for examples). The predictor form (a.k.a. delayed form) is also available with the option direct=false. This allow moving the estimator computations after solving the MPC problem with moveinput!, for when the estimations are expensive (for instance, with the MovingHorizonEstimator).

Info

All the estimators support measured $\mathbf{y^m}$ and unmeasured $\mathbf{y^u}$ model outputs, where $\mathbf{y}$ refers to all of them.

StateEstimator

ModelPredictiveControl.StateEstimator — Type

Abstract supertype of all state estimators.

(estim::StateEstimator)(d=[]) -> ŷ

Functor allowing callable StateEstimator object as an alias for evaloutput.

Examples

julia> kf = KalmanFilter(setop!(LinModel(tf(3, [10, 1]), 2), yop=[20]), direct=false);
+
+julia> ŷ = kf() 
+1-element Vector{Float64}:
+ 20.0
source

SteadyKalmanFilter

ModelPredictiveControl.SteadyKalmanFilter — Type
SteadyKalmanFilter(model::LinModel; <keyword arguments>)

Construct a steady-state Kalman Filter with the LinModel model.

The steady-state (or asymptotic) Kalman filter is based on the process model :

\[\begin{aligned} + \mathbf{x}(k+1) &= + \mathbf{Â x}(k) + \mathbf{B̂_u u}(k) + \mathbf{B̂_d d}(k) + \mathbf{w}(k) \\ + \mathbf{y^m}(k) &= \mathbf{Ĉ^m x}(k) + \mathbf{D̂_d^m d}(k) + \mathbf{v}(k) \\ + \mathbf{y^u}(k) &= \mathbf{Ĉ^u x}(k) + \mathbf{D̂_d^u d}(k) +\end{aligned}\]

with sensor $\mathbf{v}(k)$ and process $\mathbf{w}(k)$ noises as uncorrelated zero mean white noise vectors, with a respective covariance of $\mathbf{R̂}$ and $\mathbf{Q̂}$. The arguments are in standard deviations σ, i.e. same units than outputs and states. The matrices $\mathbf{Â, B̂_u, B̂_d, Ĉ, D̂_d}$ are model matrices augmented with the stochastic model, which is specified by the numbers of integrator nint_u and nint_ym (see Extended Help). Likewise, the covariance matrices are augmented with $\mathbf{Q̂ = \text{diag}(Q, Q_{int_u}, Q_{int_{ym}})}$ and $\mathbf{R̂ = R}$. The Extended Help provide some guidelines on the covariance tuning. The matrices $\mathbf{Ĉ^m, D̂_d^m}$ are the rows of $\mathbf{Ĉ, D̂_d}$ that correspond to measured outputs $\mathbf{y^m}$ (and unmeasured ones, for $\mathbf{Ĉ^u, D̂_d^u}$). The Kalman filter will estimate the current state with the newest measurements $\mathbf{x̂}_k(k)$ if direct is true, else it will predict the state of the next time step $\mathbf{x̂}_k(k+1)$.

Arguments

Info

Keyword arguments with emphasis are non-Unicode alternatives.

  • model::LinModel : (deterministic) model for the estimations.
  • i_ym=1:model.ny : model output indices that are measured $\mathbf{y^m}$, the rest are unmeasured $\mathbf{y^u}$.
  • σQ=fill(1/model.nx,model.nx) or sigmaQ : main diagonal of the process noise covariance $\mathbf{Q}$ of model, specified as a standard deviation vector.
  • σR=fill(1,length(i_ym)) or sigmaR : main diagonal of the sensor noise covariance $\mathbf{R}$ of model measured outputs, specified as a standard deviation vector.
  • nint_u=0: integrator quantity for the stochastic model of the unmeasured disturbances at the manipulated inputs (vector), use nint_u=0 for no integrator (see Extended Help).
  • nint_ym=default_nint(model,i_ym,nint_u) : same than nint_u but for the unmeasured disturbances at the measured outputs, use nint_ym=0 for no integrator (see Extended Help).
  • σQint_u=fill(1,sum(nint_u)) or sigmaQint_u : same than σQ but for the unmeasured disturbances at manipulated inputs $\mathbf{Q_{int_u}}$ (composed of integrators).
  • σQint_ym=fill(1,sum(nint_ym)) or sigmaQint_u : same than σQ for the unmeasured disturbances at measured outputs $\mathbf{Q_{int_{ym}}}$ (composed of integrators).
  • direct=true: construct with a direct transmission from $\mathbf{y^m}$ (a.k.a. current estimator, in opposition to the delayed/predictor form).

Examples

julia> model = LinModel([tf(3, [30, 1]); tf(-2, [5, 1])], 0.5);
+
+julia> estim = SteadyKalmanFilter(model, i_ym=[2], σR=[1], σQint_ym=[0.01])
+SteadyKalmanFilter estimator with a sample time Ts = 0.5 s, LinModel and:
+ 1 manipulated inputs u (0 integrating states)
+ 3 estimated states x̂
+ 1 measured outputs ym (1 integrating states)
+ 1 unmeasured outputs yu
+ 0 measured disturbances d

Extended Help

Extended Help

The σR argument is generally fixed at the estimated standard deviations of the sensor noises. The σQ, σQint_u and σQint_ym arguments can be used to tune the filter response. Increasing them make the filter more responsive to disturbances but more sensitive to measurement noise.

The model augmentation with nint_u vector adds integrators at model manipulated inputs, and nint_ym, at measured outputs. They create the integral action when the estimator is used in a controller as state feedback. By default, the method default_nint adds one integrator per measured output if feasible. The argument nint_ym can also be tweaked by following these rules on each measured output:

  • Use 0 integrator if the model output is already integrating (else it will be unobservable)
  • Use 1 integrator if the disturbances on the output are typically "step-like"
  • Use 2 integrators if the disturbances on the output are typically "ramp-like"

The function init_estimstoch builds the stochastic model for estimation.

Tip

Increasing σQint_u and σQint_ym values increases the integral action "gain".

The constructor pre-compute the steady-state Kalman gain K̂ with the kalman function. It can sometimes fail, for example when model matrices are ill-conditioned. In such a case, you can try the alternative time-varying KalmanFilter.

source
SteadyKalmanFilter(model, i_ym, nint_u, nint_ym, Q̂, R̂; direct=true)

Construct the estimator from the augmented covariance matrices Q̂ and R̂.

This syntax allows nonzero off-diagonal elements in $\mathbf{Q̂, R̂}$.

source

KalmanFilter

ModelPredictiveControl.KalmanFilter — Type
KalmanFilter(model::LinModel; <keyword arguments>)

Construct a time-varying Kalman Filter with the LinModel model.

The process model is identical to SteadyKalmanFilter. The matrix $\mathbf{P̂}$ is the estimation error covariance of model states augmented with the stochastic ones (specified by nint_u and nint_ym). Three keyword arguments specify its initial value with $\mathbf{P̂}_{-1}(0) = \mathrm{diag}\{ \mathbf{P}(0), \mathbf{P_{int_{u}}}(0), \mathbf{P_{int_{ym}}}(0) \}$. The initial state estimate $\mathbf{x̂}_{-1}(0)$ can be manually specified with setstate!, or automatically with initstate!.

Arguments

Info

Keyword arguments with emphasis are non-Unicode alternatives.

  • model::LinModel : (deterministic) model for the estimations.
  • i_ym=1:model.ny : model output indices that are measured $\mathbf{y^m}$, the rest are unmeasured $\mathbf{y^u}$.
  • σP_0=fill(1/model.nx,model.nx) or sigmaP_0 : main diagonal of the initial estimate covariance $\mathbf{P}(0)$, specified as a standard deviation vector.
  • σQ=fill(1/model.nx,model.nx) or sigmaQ : main diagonal of the process noise covariance $\mathbf{Q}$ of model, specified as a standard deviation vector.
  • σR=fill(1,length(i_ym)) or sigmaR : main diagonal of the sensor noise covariance $\mathbf{R}$ of model measured outputs, specified as a standard deviation vector.
  • nint_u=0: integrator quantity for the stochastic model of the unmeasured disturbances at the manipulated inputs (vector), use nint_u=0 for no integrator.
  • nint_ym=default_nint(model,i_ym,nint_u) : same than nint_u but for the unmeasured disturbances at the measured outputs, use nint_ym=0 for no integrator.
  • σQint_u=fill(1,sum(nint_u)) or sigmaQint_u : same than σQ but for the unmeasured disturbances at manipulated inputs $\mathbf{Q_{int_u}}$ (composed of integrators).
  • σPint_u_0=fill(1,sum(nint_u)) or sigmaPint_u_0 : same than σP_0 but for the unmeasured disturbances at manipulated inputs $\mathbf{P_{int_u}}(0)$ (composed of integrators).
  • σQint_ym=fill(1,sum(nint_ym)) or sigmaQint_u : same than σQ for the unmeasured disturbances at measured outputs $\mathbf{Q_{int_{ym}}}$ (composed of integrators).
  • σPint_ym_0=fill(1,sum(nint_ym)) or sigmaPint_ym_0 : same than σP_0 but for the unmeasured disturbances at measured outputs $\mathbf{P_{int_{ym}}}(0)$ (composed of integrators).
  • direct=true: construct with a direct transmission from $\mathbf{y^m}$ (a.k.a. current estimator, in opposition to the delayed/predictor form).

Examples

julia> model = LinModel([tf(3, [30, 1]); tf(-2, [5, 1])], 0.5);
+
+julia> estim = KalmanFilter(model, i_ym=[2], σR=[1], σP_0=[100, 100], σQint_ym=[0.01])
+KalmanFilter estimator with a sample time Ts = 0.5 s, LinModel and:
+ 1 manipulated inputs u (0 integrating states)
+ 3 estimated states x̂
+ 1 measured outputs ym (1 integrating states)
+ 1 unmeasured outputs yu
+ 0 measured disturbances d
source
KalmanFilter(model, i_ym, nint_u, nint_ym, P̂_0, Q̂, R̂; direct=true)

Construct the estimator from the augmented covariance matrices P̂_0, Q̂ and R̂.

This syntax allows nonzero off-diagonal elements in $\mathbf{P̂}_{-1}(0), \mathbf{Q̂, R̂}$.

source

Luenberger

ModelPredictiveControl.Luenberger — Type
Luenberger(
+    model::LinModel; 
+    i_ym = 1:model.ny, 
+    nint_u  = 0,
+    nint_ym = default_nint(model, i_ym),
+    poles = 1e-3*(1:(model.nx + sum(nint_u) + sum(nint_ym))) .+ 0.5,
+    direct = true
+)

Construct a Luenberger observer with the LinModel model.

i_ym provides the model output indices that are measured $\mathbf{y^m}$, the rest are unmeasured $\mathbf{y^u}$. model matrices are augmented with the stochastic model, which is specified by the numbers of integrator nint_u and nint_ym (see SteadyKalmanFilter Extended Help). The argument poles is a vector of model.nx + sum(nint_u) + sum(nint_ym) elements specifying the observer poles/eigenvalues (near $z=0.5$ by default). The observer is constructed with a direct transmission from $\mathbf{y^m}$ if direct=true (a.k.a. current observers, in opposition to the delayed/prediction form). The method computes the observer gain K̂ with place.

Examples

julia> model = LinModel([tf(3, [30, 1]); tf(-2, [5, 1])], 0.5);
+
+julia> estim = Luenberger(model, nint_ym=[1, 1], poles=[0.61, 0.62, 0.63, 0.64])
+Luenberger estimator with a sample time Ts = 0.5 s, LinModel and:
+ 1 manipulated inputs u (0 integrating states)
+ 4 estimated states x̂
+ 2 measured outputs ym (2 integrating states)
+ 0 unmeasured outputs yu
+ 0 measured disturbances d
source

UnscentedKalmanFilter

ModelPredictiveControl.UnscentedKalmanFilter — Type
UnscentedKalmanFilter(model::SimModel; <keyword arguments>)

Construct an unscented Kalman Filter with the SimModel model.

Both LinModel and NonLinModel are supported. The unscented Kalman filter is based on the process model :

\[\begin{aligned} + \mathbf{x}(k+1) &= \mathbf{f̂}\Big(\mathbf{x}(k), \mathbf{u}(k), \mathbf{d}(k)\Big) + + \mathbf{w}(k) \\ + \mathbf{y^m}(k) &= \mathbf{ĥ^m}\Big(\mathbf{x}(k), \mathbf{d}(k)\Big) + \mathbf{v}(k) \\ + \mathbf{y^u}(k) &= \mathbf{ĥ^u}\Big(\mathbf{x}(k), \mathbf{d}(k)\Big) \\ +\end{aligned}\]

See SteadyKalmanFilter for details on $\mathbf{v}(k), \mathbf{w}(k)$ noises and $\mathbf{R̂}, \mathbf{Q̂}$ covariances. The two matrices are constructed from $\mathbf{Q̂ = \text{diag}(Q, Q_{int_u}, Q_{int_{ym}})}$ and $\mathbf{R̂ = R}$. The functions $\mathbf{f̂, ĥ}$ are model state-space functions augmented with the stochastic model of the unmeasured disturbances, which is specified by the numbers of integrator nint_u and nint_ym (see Extended Help). Model parameters $\mathbf{p}$ are not argument of $\mathbf{f̂, ĥ}$ functions for conciseness. The $\mathbf{ĥ^m}$ function represents the measured outputs of $\mathbf{ĥ}$ function (and unmeasured ones, for $\mathbf{ĥ^u}$). The matrix $\mathbf{P̂}$ is the estimation error covariance of model state augmented with the stochastic ones. Three keyword arguments specify its initial value with $\mathbf{P̂}_{-1}(0) = \mathrm{diag}\{ \mathbf{P}(0), \mathbf{P_{int_{u}}}(0), \mathbf{P_{int_{ym}}}(0) \}$. The initial state estimate $\mathbf{x̂}_{-1}(0)$ can be manually specified with setstate!.

Arguments

Info

Keyword arguments with emphasis are non-Unicode alternatives.

  • model::SimModel : (deterministic) model for the estimations.
  • i_ym=1:model.ny : model output indices that are measured $\mathbf{y^m}$, the rest are unmeasured $\mathbf{y^u}$.
  • σP_0=fill(1/model.nx,model.nx) or sigmaP_0 : main diagonal of the initial estimate covariance $\mathbf{P}(0)$, specified as a standard deviation vector.
  • σQ=fill(1/model.nx,model.nx) or sigmaQ : main diagonal of the process noise covariance $\mathbf{Q}$ of model, specified as a standard deviation vector.
  • σR=fill(1,length(i_ym)) or sigmaR : main diagonal of the sensor noise covariance $\mathbf{R}$ of model measured outputs, specified as a standard deviation vector.
  • nint_u=0: integrator quantity for the stochastic model of the unmeasured disturbances at the manipulated inputs (vector), use nint_u=0 for no integrator (see Extended Help).
  • nint_ym=default_nint(model,i_ym,nint_u) : same than nint_u but for the unmeasured disturbances at the measured outputs, use nint_ym=0 for no integrator (see Extended Help).
  • σQint_u=fill(1,sum(nint_u)) or sigmaQint_u : same than σQ but for the unmeasured disturbances at manipulated inputs $\mathbf{Q_{int_u}}$ (composed of integrators).
  • σPint_u_0=fill(1,sum(nint_u)) or sigmaPint_u_0 : same than σP_0 but for the unmeasured disturbances at manipulated inputs $\mathbf{P_{int_u}}(0)$ (composed of integrators).
  • σQint_ym=fill(1,sum(nint_ym)) or sigmaQint_u : same than σQ for the unmeasured disturbances at measured outputs $\mathbf{Q_{int_{ym}}}$ (composed of integrators).
  • σPint_ym_0=fill(1,sum(nint_ym)) or sigmaPint_ym_0 : same than σP_0 but for the unmeasured disturbances at measured outputs $\mathbf{P_{int_{ym}}}(0)$ (composed of integrators).
  • direct=true: construct with a direct transmission from $\mathbf{y^m}$ (a.k.a. current estimator, in opposition to the delayed/predictor form).
  • α=1e-3 or alpha : alpha parameter, spread of the state distribution $(0 < α ≤ 1)$.
  • β=2 or beta : beta parameter, skewness and kurtosis of the states distribution $(β ≥ 0)$.
  • κ=0 or kappa : kappa parameter, another spread parameter $(0 ≤ κ ≤ 3)$.

Examples

julia> model = NonLinModel((x,u,_,_)->0.1x+u, (x,_,_)->2x, 10.0, 1, 1, 1, solver=nothing);
+
+julia> estim = UnscentedKalmanFilter(model, σR=[1], nint_ym=[2], σPint_ym_0=[1, 1])
+UnscentedKalmanFilter estimator with a sample time Ts = 10.0 s, NonLinModel and:
+ 1 manipulated inputs u (0 integrating states)
+ 3 estimated states x̂
+ 1 measured outputs ym (2 integrating states)
+ 0 unmeasured outputs yu
+ 0 measured disturbances d

Extended Help

Extended Help

The Extended Help of SteadyKalmanFilter details the tuning of the covariances and the augmentation with nint_ym and nint_u arguments. The default augmentation scheme is identical, that is nint_u=0 and nint_ym computed by default_nint. Note that the constructor does not validate the observability of the resulting augmented NonLinModel. In such cases, it is the user's responsibility to ensure that it is still observable.

source
UnscentedKalmanFilter(
+    model, i_ym, nint_u, nint_ym, P̂_0, Q̂, R̂, α=1e-3, β=2, κ=0; direct=true
+)

Construct the estimator from the augmented covariance matrices P̂_0, Q̂ and R̂.

This syntax allows nonzero off-diagonal elements in $\mathbf{P̂}_{-1}(0), \mathbf{Q̂, R̂}$.

source

ExtendedKalmanFilter

ModelPredictiveControl.ExtendedKalmanFilter — Type
ExtendedKalmanFilter(model::SimModel; <keyword arguments>)

Construct an extended Kalman Filter with the SimModel model.

Both LinModel and NonLinModel are supported. The process model and the keyword arguments are identical to UnscentedKalmanFilter, except for α, β and κ which do not apply to the extended Kalman Filter. The Jacobians of the augmented model $\mathbf{f̂, ĥ}$ are computed with ForwardDiff.jl automatic differentiation.

Warning

See the Extended Help of linearize function if you get an error like: MethodError: no method matching (::var"##")(::Vector{ForwardDiff.Dual}).

Examples

julia> model = NonLinModel((x,u,_,_)->0.2x+u, (x,_,_)->-3x, 5.0, 1, 1, 1, solver=nothing);
+
+julia> estim = ExtendedKalmanFilter(model, σQ=[2], σQint_ym=[2], σP_0=[0.1], σPint_ym_0=[0.1])
+ExtendedKalmanFilter estimator with a sample time Ts = 5.0 s, NonLinModel and:
+ 1 manipulated inputs u (0 integrating states)
+ 2 estimated states x̂
+ 1 measured outputs ym (1 integrating states)
+ 0 unmeasured outputs yu
+ 0 measured disturbances d
source
ExtendedKalmanFilter(model, i_ym, nint_u, nint_ym, P̂_0, Q̂, R̂; direct=true)

Construct the estimator from the augmented covariance matrices P̂_0, Q̂ and R̂.

This syntax allows nonzero off-diagonal elements in $\mathbf{P̂}_{-1}(0), \mathbf{Q̂, R̂}$.

source

MovingHorizonEstimator

ModelPredictiveControl.MovingHorizonEstimator — Type
MovingHorizonEstimator(model::SimModel; <keyword arguments>)

Construct a moving horizon estimator (MHE) based on model (LinModel or NonLinModel).

It can handle constraints on the estimates, see setconstraint!. Additionally, model is not linearized like the ExtendedKalmanFilter, and the probability distribution is not approximated like the UnscentedKalmanFilter. The computational costs are drastically higher, however, since it minimizes the following objective function at each discrete time $k$:

\[\min_{\mathbf{x̂}_k(k-N_k+p), \mathbf{Ŵ}, ϵ} \mathbf{x̄}' \mathbf{P̄}^{-1} \mathbf{x̄} + + \mathbf{Ŵ}' \mathbf{Q̂}_{N_k}^{-1} \mathbf{Ŵ} + + \mathbf{V̂}' \mathbf{R̂}_{N_k}^{-1} \mathbf{V̂} + + C ϵ^2\]

in which the arrival costs are evaluated from the states estimated at time $k-N_k$:

\[\begin{aligned} + \mathbf{x̄} &= \mathbf{x̂}_{k-N_k}(k-N_k+p) - \mathbf{x̂}_k(k-N_k+p) \\ + \mathbf{P̄} &= \mathbf{P̂}_{k-N_k}(k-N_k+p) +\end{aligned}\]

and the covariances are repeated $N_k$ times:

\[\begin{aligned} + \mathbf{Q̂}_{N_k} &= \text{diag}\mathbf{(Q̂,Q̂,...,Q̂)} \\ + \mathbf{R̂}_{N_k} &= \text{diag}\mathbf{(R̂,R̂,...,R̂)} +\end{aligned}\]

The estimation horizon $H_e$ limits the window length:

\[N_k = \begin{cases} + k + 1 & k < H_e \\ + H_e & k ≥ H_e \end{cases}\]

The vectors $\mathbf{Ŵ}$ and $\mathbf{V̂}$ respectively encompass the estimated process noises $\mathbf{ŵ}(k-j+p)$ from $j=N_k$ to $1$ and sensor noises $\mathbf{v̂}(k-j+1)$ from $j=N_k$ to $1$. The Extended Help defines the two vectors, the slack variable $ϵ$, and the estimation of the covariance at arrival $\mathbf{P̂}_{k-N_k}(k-N_k+p)$. If the keyword argument direct=true (default value), the constant $p=0$ in the equations above, and the MHE is in the current form. Else $p=1$, leading to the prediction form.

See UnscentedKalmanFilter for details on the augmented process model and $\mathbf{R̂}, \mathbf{Q̂}$ covariances.

Warning

See the Extended Help if you get an error like: MethodError: no method matching (::var"##")(::Vector{ForwardDiff.Dual}).

Arguments

Info

Keyword arguments with emphasis are non-Unicode alternatives.

  • model::SimModel : (deterministic) model for the estimations.
  • He=nothing : estimation horizon $H_e$, must be specified.
  • i_ym=1:model.ny : model output indices that are measured $\mathbf{y^m}$, the rest are unmeasured $\mathbf{y^u}$.
  • σP_0=fill(1/model.nx,model.nx) or sigmaP_0 : main diagonal of the initial estimate covariance $\mathbf{P}(0)$, specified as a standard deviation vector.
  • σQ=fill(1/model.nx,model.nx) or sigmaQ : main diagonal of the process noise covariance $\mathbf{Q}$ of model, specified as a standard deviation vector.
  • σR=fill(1,length(i_ym)) or sigmaR : main diagonal of the sensor noise covariance $\mathbf{R}$ of model measured outputs, specified as a standard deviation vector.
  • nint_u=0: integrator quantity for the stochastic model of the unmeasured disturbances at the manipulated inputs (vector), use nint_u=0 for no integrator (see Extended Help).
  • nint_ym=default_nint(model,i_ym,nint_u) : same than nint_u but for the unmeasured disturbances at the measured outputs, use nint_ym=0 for no integrator (see Extended Help).
  • σQint_u=fill(1,sum(nint_u)) or sigmaQint_u : same than σQ but for the unmeasured disturbances at manipulated inputs $\mathbf{Q_{int_u}}$ (composed of integrators).
  • σPint_u_0=fill(1,sum(nint_u)) or sigmaPint_u_0 : same than σP_0 but for the unmeasured disturbances at manipulated inputs $\mathbf{P_{int_u}}(0)$ (composed of integrators).
  • σQint_ym=fill(1,sum(nint_ym)) or sigmaQint_u : same than σQ for the unmeasured disturbances at measured outputs $\mathbf{Q_{int_{ym}}}$ (composed of integrators).
  • σPint_ym_0=fill(1,sum(nint_ym)) or sigmaPint_ym_0 : same than σP_0 but for the unmeasured disturbances at measured outputs $\mathbf{P_{int_{ym}}}(0)$ (composed of integrators).
  • Cwt=Inf : slack variable weight $C$, default to Inf meaning hard constraints only.
  • optim=default_optim_mhe(model) : a JuMP.Model with a quadratic/nonlinear optimizer for solving (default to Ipopt, or OSQP if model is a LinModel).
  • direct=true: construct with a direct transmission from $\mathbf{y^m}$ (a.k.a. current estimator, in opposition to the delayed/predictor form).

Examples

julia> model = NonLinModel((x,u,_,_)->0.1x+u, (x,_,_)->2x, 10.0, 1, 1, 1, solver=nothing);
+
+julia> estim = MovingHorizonEstimator(model, He=5, σR=[1], σP_0=[0.01])
+MovingHorizonEstimator estimator with a sample time Ts = 10.0 s, Ipopt optimizer, NonLinModel and:
+ 5 estimation steps He
+ 0 slack variable ϵ (estimation constraints)
+ 1 manipulated inputs u (0 integrating states)
+ 2 estimated states x̂
+ 1 measured outputs ym (1 integrating states)
+ 0 unmeasured outputs yu
+ 0 measured disturbances d

Extended Help

Extended Help

The estimated process and sensor noises are defined as:

\[\mathbf{Ŵ} = +\begin{bmatrix} + \mathbf{ŵ}(k-N_k+p+0) \\ + \mathbf{ŵ}(k-N_k+p+1) \\ + \vdots \\ + \mathbf{ŵ}(k+p-1) +\end{bmatrix} , \quad +\mathbf{V̂} = +\begin{bmatrix} + \mathbf{v̂}(k-N_k+1) \\ + \mathbf{v̂}(k-N_k+2) \\ + \vdots \\ + \mathbf{v̂}(k) +\end{bmatrix}\]

based on the augmented model functions $\mathbf{f̂, ĥ^m}$:

\[\begin{aligned} + \mathbf{v̂}(k-j) &= \mathbf{y^m}(k-j) - \mathbf{ĥ^m}\Big(\mathbf{x̂}_k(k-j), \mathbf{d}(k-j)\Big) \\ + \mathbf{x̂}_k(k-j+1) &= \mathbf{f̂}\Big(\mathbf{x̂}_k(k-j), \mathbf{u}(k-j), \mathbf{d}(k-j)\Big) + \mathbf{ŵ}(k-j) +\end{aligned}\]

The constant $p$ equals to !direct. In other words, $\mathbf{Ŵ}$ and $\mathbf{V̂}$ are shifted by one time step if direct==true. The non-default prediction form with $p=1$ is particularly useful for the MHE since it moves its expensive computations after the MPC optimization. That is, preparestate! will solve the optimization by default, but it can be postponed to updatestate! with direct=false.

The Extended Help of SteadyKalmanFilter details the tuning of the covariances and the augmentation with nint_ym and nint_u arguments. The default augmentation scheme is identical, that is nint_u=0 and nint_ym computed by default_nint. Note that the constructor does not validate the observability of the resulting augmented NonLinModel. In such cases, it is the user's responsibility to ensure that it is still observable.

The estimation covariance at arrival $\mathbf{P̂}_{k-N_k}(k-N_k+p)$ gives an uncertainty on the state estimate at the beginning of the window $k-N_k+p$, that is, in the past. It is not the same as the current estimate covariance $\mathbf{P̂}_k(k)$, a value not computed by the MHE (contrarily to e.g. the KalmanFilter). Three keyword arguments specify its initial value with $\mathbf{P̂_i} = \mathrm{diag}\{ \mathbf{P}(0), \mathbf{P_{int_{u}}}(0), \mathbf{P_{int_{ym}}}(0) \}$. The initial state estimate $\mathbf{x̂_i}$ can be manually specified with setstate!, or automatically with initstate! for LinModel. Note the MHE with $p=0$ is slightly inconsistent with all the other estimators here. It interprets the initial values as $\mathbf{x̂_i} = \mathbf{x̂}_{-1}(-1)$ and $\mathbf{P̂_i} = \mathbf{P̂}_{-1}(-1)$, an a posteriori estimate[2] from the last time step. The MHE with $p=1$ is consistent, interpreting them as $\mathbf{x̂_i} = \mathbf{x̂}_{-1}(0)$ and $\mathbf{P̂_i} = \mathbf{P̂}_{-1}(0)$.

The optimization and the update of the arrival covariance depend on model:

  • If model is a LinModel, the optimization is treated as a quadratic program with a time-varying Hessian, which is generally cheaper than nonlinear programming. By default, a KalmanFilter estimates the arrival covariance (customizable).
  • Else, a nonlinear program with automatic differentiation (AD) solves the optimization. Optimizers generally benefit from exact derivatives like AD. However, the f and h functions must be compatible with this feature. See Automatic differentiation for common mistakes when writing these functions. An UnscentedKalmanFilter estimates the arrival covariance by default.

The slack variable $ϵ$ relaxes the constraints if enabled, see setconstraint!. It is disabled by default for the MHE (from Cwt=Inf) but it should be activated for problems with two or more types of bounds, to ensure feasibility (e.g. on the estimated state $\mathbf{x̂}$ and sensor noise $\mathbf{v̂}$). Note that if Cwt≠Inf, the attribute nlp_scaling_max_gradient of Ipopt is set to 10/Cwt (if not already set), to scale the small values of $ϵ$. Use the second constructor to specify the arrival covariance estimation method.

source
MovingHorizonEstimator(
+    model, He, i_ym, nint_u, nint_ym, P̂_0, Q̂, R̂, Cwt=Inf;
+    optim=default_optim_mhe(model), 
+    direct=true,
+    covestim=default_covestim_mhe(model, i_ym, nint_u, nint_ym, P̂_0, Q̂, R̂; direct)
+)

Construct the estimator from the augmented covariance matrices P̂_0, Q̂ and R̂.

This syntax allows nonzero off-diagonal elements in $\mathbf{P̂_i}, \mathbf{Q̂, R̂}$, where $\mathbf{P̂_i}$ is the initial estimation covariance, provided by P̂_0 argument. The keyword argument covestim also allows specifying a custom StateEstimator object for the estimation of covariance at the arrival $\mathbf{P̂}_{k-N_k}(k-N_k+p)$. The supported types are KalmanFilter, UnscentedKalmanFilter and ExtendedKalmanFilter.

source

InternalModel

ModelPredictiveControl.InternalModel — Type
InternalModel(model::SimModel; i_ym=1:model.ny, stoch_ym=ss(I,I,I,I,model.Ts))

Construct an internal model estimator based on model (LinModel or NonLinModel).

i_ym provides the model output indices that are measured $\mathbf{y^m}$, the rest are unmeasured $\mathbf{y^u}$. model evaluates the deterministic predictions $\mathbf{ŷ_d}$, and stoch_ym, the stochastic predictions of the measured outputs $\mathbf{ŷ_s^m}$ (the unmeasured ones being $\mathbf{ŷ_s^u=0}$). The predicted outputs sum both values : $\mathbf{ŷ = ŷ_d + ŷ_s}$. See the Extended Help for more details.

Warning

InternalModel estimator does not work if model is integrating or unstable. The constructor verifies these aspects for LinModel but not for NonLinModel. Uses any other state estimator in such cases.

Examples

julia> estim = InternalModel(LinModel([tf(3, [30, 1]); tf(-2, [5, 1])], 0.5), i_ym=[2])
+InternalModel estimator with a sample time Ts = 0.5 s, LinModel and:
+ 1 manipulated inputs u
+ 2 estimated states x̂
+ 1 measured outputs ym
+ 1 unmeasured outputs yu
+ 0 measured disturbances d

Extended Help

Extended Help

stoch_ym is a TransferFunction or StateSpace object that models disturbances on $\mathbf{y^m}$. Its input is a hypothetical zero mean white noise vector. stoch_ym supposes 1 integrator per measured outputs by default, assuming that the current stochastic estimate $\mathbf{ŷ_s^m}(k) = \mathbf{y^m}(k) - \mathbf{ŷ_d^m}(k)$ is constant in the future. This is the dynamic matrix control (DMC) strategy, which is simple but sometimes too aggressive. Additional poles and zeros in stoch_ym can mitigate this. The following block diagram summarizes the internal model structure.

block diagram of the internal model structure

source

Default Model Augmentation

ModelPredictiveControl.default_nint — Function
default_nint(model::LinModel, i_ym=1:model.ny, nint_u=0) -> nint_ym

Get default integrator quantity per measured outputs nint_ym for LinModel.

The arguments i_ym and nint_u are the measured output indices and the integrator quantity on each manipulated input, respectively. By default, one integrator is added on each measured outputs. If $\mathbf{Â, Ĉ}$ matrices of the augmented model become unobservable, the integrator is removed. This approach works well for stable, integrating and unstable model (see Examples).

Examples

julia> model = LinModel(append(tf(3, [10, 1]), tf(2, [1, 0]), tf(4,[-5, 1])), 1.0);
+
+julia> nint_ym = default_nint(model)
+3-element Vector{Int64}:
+ 1
+ 0
+ 1
source
default_nint(model::SimModel, i_ym=1:model.ny, nint_u=0)

One integrator on each measured output by default if model is not a LinModel.

Theres is no verification the augmented model remains observable. If the integrator quantity per manipulated input nint_u ≠ 0, the method returns zero integrator on each measured output.

source
  • 1also denoted $\mathbf{x̂}_{k|k}$ elsewhere.
  • 2M. Hovd (2012), "A Note On The Smoothing Formulation Of Moving Horizon Estimation", Facta Universitatis, Vol. 11 №2.
diff --git a/previews/PR114/search_index.js b/previews/PR114/search_index.js new file mode 100644 index 00000000..9edff9f4 --- /dev/null +++ b/previews/PR114/search_index.js @@ -0,0 +1,3 @@ +var documenterSearchIndex = {"docs": +[{"location":"manual/installation/#Manual:-Installation","page":"Installation","title":"Manual: Installation","text":"","category":"section"},{"location":"manual/installation/","page":"Installation","title":"Installation","text":"To install the ModelPredictiveControl package, run this command in the Julia REPL:","category":"page"},{"location":"manual/installation/","page":"Installation","title":"Installation","text":"using Pkg; Pkg.activate(); Pkg.add(\"ModelPredictiveControl\")","category":"page"},{"location":"manual/installation/","page":"Installation","title":"Installation","text":"Doing so will install the package to default Julia environnement, that is, accessible anywhere. To facilitate sharing of code and reproducibility of results, it is recommended to install packages in a project environnement. To generate a new project named MPCproject with this package in the current working directory, write this in the REPL:","category":"page"},{"location":"manual/installation/","page":"Installation","title":"Installation","text":"using Pkg; Pkg.generate(\"MPCproject\"); Pkg.activate(\".\"); Pkg.add(\"ModelPredictiveControl\")","category":"page"},{"location":"manual/installation/","page":"Installation","title":"Installation","text":"Note that that the construction of linear models typically requires ss or tf functions, it is thus advised to load the package with:","category":"page"},{"location":"manual/installation/","page":"Installation","title":"Installation","text":"using ModelPredictiveControl, ControlSystemsBase","category":"page"},{"location":"internals/predictive_control/#Functions:-PredictiveController-Internals","page":"Predictive Controllers","title":"Functions: PredictiveController Internals","text":"","category":"section"},{"location":"internals/predictive_control/","page":"Predictive Controllers","title":"Predictive Controllers","text":"Pages = [\"predictive_control.md\"]","category":"page"},{"location":"internals/predictive_control/","page":"Predictive Controllers","title":"Predictive Controllers","text":"The prediction methodology of this module is mainly based on Maciejowski textbook [1].","category":"page"},{"location":"internals/predictive_control/","page":"Predictive Controllers","title":"Predictive Controllers","text":"[1]: Maciejowski, J. 2000, \"Predictive control : with constraints\", 1st ed., Prentice Hall, ISBN 978-0201398236.","category":"page"},{"location":"internals/predictive_control/#Controller-Construction","page":"Predictive Controllers","title":"Controller Construction","text":"","category":"section"},{"location":"internals/predictive_control/","page":"Predictive Controllers","title":"Predictive Controllers","text":"ModelPredictiveControl.init_ΔUtoU\nModelPredictiveControl.init_predmat\nModelPredictiveControl.relaxU\nModelPredictiveControl.relaxΔU\nModelPredictiveControl.relaxŶ\nModelPredictiveControl.relaxterminal\nModelPredictiveControl.init_quadprog\nModelPredictiveControl.init_stochpred\nModelPredictiveControl.init_matconstraint_mpc","category":"page"},{"location":"internals/predictive_control/#ModelPredictiveControl.init_ΔUtoU","page":"Predictive Controllers","title":"ModelPredictiveControl.init_ΔUtoU","text":"init_ΔUtoU(model, Hp, Hc) -> S, T\n\nInit manipulated input increments to inputs conversion matrices.\n\nThe conversion from the input increments mathbfΔU to manipulated inputs over H_p are calculated by:\n\nmathbfU = mathbfS mathbfΔU + mathbfT mathbfu(k-1) \n\n\n\n\n\n","category":"function"},{"location":"internals/predictive_control/#ModelPredictiveControl.init_predmat","page":"Predictive Controllers","title":"ModelPredictiveControl.init_predmat","text":"init_predmat(estim, model::LinModel, Hp, Hc) -> E, G, J, K, V, ex̂, gx̂, jx̂, kx̂, vx̂\n\nConstruct the prediction matrices for LinModel model.\n\nThe model predictions are evaluated from the deviation vectors (see setop!) and:\n\nbeginaligned\n mathbfY_0 = mathbfE ΔU + mathbfG d_0(k) + mathbfJ D_0 \n + mathbfK x_0(k) + mathbfV u_0(k-1) \n + mathbfB + mathbfY_s \n = mathbfE ΔU + mathbfF\nendaligned\n\nin which mathbfx_0(k) = mathbfx_i(k) - mathbfx_op, with i = k if estim.direct==true, otherwise i = k - 1. The predicted outputs mathbfY_0 and measured disturbances mathbfD_0 respectively include mathbfy_0(k+j) and mathbfd_0(k+j) values with j=1 to H_p, and input increments mathbfΔU, mathbfΔu(k+j) from j=0 to H_c-1. The vector mathbfB contains the contribution for non-zero state mathbfx_op and state update mathbff_op operating points (for linearization at non-equilibrium point, see linearize). The stochastic predictions mathbfY_s=0 if estim is not a InternalModel, see init_stochpred. The method also computes similar matrices for the predicted terminal states at k+H_p:\n\nbeginaligned\n mathbfx_0(k+H_p) = mathbfe_x ΔU + mathbfg_x d_0(k) + mathbfj_x D_0 \n + mathbfk_x x_0(k) + mathbfv_x u_0(k-1)\n + mathbfb_x \n = mathbfe_x ΔU + mathbff_x\nendaligned\n\nThe matrices mathbfE G J K V B e_x g_x j_x k_x v_x b_x are defined in the Extended Help section. The mathbfF and mathbff_x vectors are recalculated at each control period k, see initpred! and linconstraint!.\n\nExtended Help\n\ndetails: Extended Help\nUsing the augmented matrices mathbfA B_u C B_d D_d in estim (see augment_model), and the function mathbfW(j) = _i=0^j mathbfA^i, the prediction matrices are computed by :beginaligned\nmathbfE = beginbmatrix\n mathbfC W(0)mathbfB_u mathbf0 cdots mathbf0 \n mathbfC W(1)mathbfB_u mathbfC W(0)mathbfB_u cdots mathbf0 \n vdots vdots ddots vdots \n mathbfC W(H_p-1)mathbfB_u mathbfC W(H_p-2)mathbfB_u cdots mathbfC W(H_p-H_c+1)mathbfB_u endbmatrix \nmathbfG = beginbmatrix\n mathbfCmathbfA^0 mathbfB_d \n mathbfCmathbfA^1 mathbfB_d \n vdots \n mathbfCmathbfA^H_p-1 mathbfB_d endbmatrix \nmathbfJ = beginbmatrix\n mathbfD_d mathbf0 cdots mathbf0 \n mathbfCmathbfA^0 mathbfB_d mathbfD_d cdots mathbf0 \n vdots vdots ddots vdots \n mathbfCmathbfA^H_p-2 mathbfB_d mathbfCmathbfA^H_p-3 mathbfB_d cdots mathbfD_d endbmatrix \nmathbfK = beginbmatrix\n mathbfCmathbfA^1 \n mathbfCmathbfA^2 \n vdots \n mathbfCmathbfA^H_p endbmatrix \nmathbfV = beginbmatrix\n mathbfC W(0)mathbfB_u \n mathbfC W(1)mathbfB_u \n vdots \n mathbfC W(H_p-1)mathbfB_u endbmatrix \nmathbfB = beginbmatrix\n mathbfC W(0) \n mathbfC W(1) \n vdots \n mathbfC W(H_p-1) endbmatrix mathbfbig(f_op - x_opbig) \nendalignedFor the terminal constraints, the matrices are computed with:beginaligned\nmathbfe_x = beginbmatrix \n mathbfW(H_p-1)mathbfB_u \n mathbfW(H_p-2)mathbfB_u \n cdots \n mathbfW(H_p-H_c+1)mathbfB_u endbmatrix \nmathbfg_x = mathbfA^H_p-1 mathbfB_d \nmathbfj_x = beginbmatrix \n mathbfA^H_p-2 mathbfB_d \n mathbfA^H_p-3 mathbfB_d \n cdots \n mathbf0 \n endbmatrix \nmathbfk_x = mathbfA^H_p \nmathbfv_x = mathbfW(H_p-1)mathbfB_u \nmathbfb_x = mathbfW(H_p-1) mathbfbig(f_op - x_opbig)\nendaligned\n\n\n\n\n\nReturn empty matrices if model is not a LinModel\n\n\n\n\n\n","category":"function"},{"location":"internals/predictive_control/#ModelPredictiveControl.relaxU","page":"Predictive Controllers","title":"ModelPredictiveControl.relaxU","text":"relaxU(model, nϵ, C_umin, C_umax, S) -> A_Umin, A_Umax, S̃\n\nAugment manipulated inputs constraints with slack variable ϵ for softening.\n\nDenoting the input increments augmented with the slack variable mathbfΔU = beginsmallmatrix mathbfΔU ϵ endsmallmatrix, it returns the augmented conversion matrix mathbfS, similar to the one described at init_ΔUtoU. It also returns the mathbfA matrices for the inequality constraints:\n\nbeginbmatrix \n mathbfA_U_min \n mathbfA_U_max \nendbmatrix mathbfΔU \nbeginbmatrix\n - mathbf(U_min - U_op) + T mathbfu_0(k-1) \n + mathbf(U_max - U_op) - T mathbfu_0(k-1)\nendbmatrix\n\nin which mathbfU_min U_max and mathbfU_op vectors respectively contains mathbfu_min u_max and mathbfu_op repeated H_p times.\n\n\n\n\n\n","category":"function"},{"location":"internals/predictive_control/#ModelPredictiveControl.relaxΔU","page":"Predictive Controllers","title":"ModelPredictiveControl.relaxΔU","text":"relaxΔU(\n model, nϵ, C, C_Δumin, C_Δumax, ΔUmin, ΔUmax, N_Hc\n) -> A_ΔŨmin, A_ΔŨmax, ΔŨmin, ΔŨmax, Ñ_Hc\n\nAugment input increments constraints with slack variable ϵ for softening.\n\nDenoting the input increments augmented with the slack variable mathbfΔU = beginsmallmatrix mathbfΔU ϵ endsmallmatrix, it returns the augmented input increment weights mathbfN_H_c (that incorporate C). It also returns the augmented constraints mathbfΔU_min and mathbfΔU_max and the mathbfA matrices for the inequality constraints:\n\nbeginbmatrix \n mathbfA_ΔU_min \n mathbfA_ΔU_max\nendbmatrix mathbfΔU \nbeginbmatrix\n - mathbfΔU_min \n + mathbfΔU_max\nendbmatrix\n\n\n\n\n\n","category":"function"},{"location":"internals/predictive_control/#ModelPredictiveControl.relaxŶ","page":"Predictive Controllers","title":"ModelPredictiveControl.relaxŶ","text":"relaxŶ(::LinModel, nϵ, C_ymin, C_ymax, E) -> A_Ymin, A_Ymax, Ẽ\n\nAugment linear output prediction constraints with slack variable ϵ for softening.\n\nDenoting the input increments augmented with the slack variable mathbfΔU = beginsmallmatrix mathbfΔU ϵ endsmallmatrix, it returns the mathbfE matrix that appears in the linear model prediction equation mathbfY_0 = E ΔU + F, and the mathbfA matrices for the inequality constraints:\n\nbeginbmatrix \n mathbfA_Y_min \n mathbfA_Y_max\nendbmatrix mathbfΔU \nbeginbmatrix\n - mathbf(Y_min - Y_op) + F \n + mathbf(Y_max - Y_op) - F \nendbmatrix\n\nin which mathbfY_min Y_max and mathbfY_op vectors respectively contains mathbfy_min y_max and mathbfy_op repeated H_p times.\n\n\n\n\n\nReturn empty matrices if model is not a LinModel\n\n\n\n\n\n","category":"function"},{"location":"internals/predictive_control/#ModelPredictiveControl.relaxterminal","page":"Predictive Controllers","title":"ModelPredictiveControl.relaxterminal","text":"relaxterminal(::LinModel, nϵ, c_x̂min, c_x̂max, ex̂) -> A_x̂min, A_x̂max, ẽx̂\n\nAugment terminal state constraints with slack variable ϵ for softening.\n\nDenoting the input increments augmented with the slack variable mathbfΔU = beginsmallmatrix mathbfΔU ϵ endsmallmatrix, it returns the mathbfe_x matrix that appears in the terminal state equation mathbfx_0(k + H_p) = mathbfe_x ΔU + f_x, and the mathbfA matrices for the inequality constraints:\n\nbeginbmatrix \n mathbfA_x_min \n mathbfA_x_max\nendbmatrix mathbfΔU \nbeginbmatrix\n - mathbf(x_min - x_op) + f_x \n + mathbf(x_max - x_op) - f_x\nendbmatrix\n\n\n\n\n\nReturn empty matrices if model is not a LinModel\n\n\n\n\n\n","category":"function"},{"location":"internals/predictive_control/#ModelPredictiveControl.init_quadprog","page":"Predictive Controllers","title":"ModelPredictiveControl.init_quadprog","text":"init_quadprog(model::LinModel, Ẽ, S, M_Hp, N_Hc, L_Hp) -> H̃\n\nInit the quadratic programming Hessian H̃ for MPC.\n\nThe matrix appear in the quadratic general form:\n\n J = min_mathbfΔU frac12mathbf(ΔU)H(ΔU) + mathbfq(ΔU) + r \n\nThe Hessian matrix is constant if the model and weights are linear and time invariant (LTI): \n\n mathbfH = 2 ( mathbfEmathbfM_H_pmathbfE + mathbfN_H_c \n + mathbfSmathbfL_H_pmathbfS )\n\nThe vector mathbfq and scalar r need recalculation each control period k, see initpred!. r does not impact the minima position. It is thus useless at optimization but required to evaluate the minimal J value.\n\n\n\n\n\nReturn empty matrices if model is not a LinModel.\n\n\n\n\n\n","category":"function"},{"location":"internals/predictive_control/#ModelPredictiveControl.init_stochpred","page":"Predictive Controllers","title":"ModelPredictiveControl.init_stochpred","text":"init_stochpred(estim::InternalModel, Hp) -> Ks, Ps\n\nInit the stochastic prediction matrices for InternalModel.\n\nKs and Ps matrices are defined as:\n\n mathbfY_s = mathbfK_s x_s(k) + mathbfP_s y_s(k)\n\nCurrent stochastic outputs mathbfy_s(k) comprises the measured outputs mathbfy_s^m(k) = mathbfy^m(k) - mathbfy_d^m(k) and unmeasured mathbfy_s^u(k) = mathbf0. See [2].\n\n[2]: Desbiens, A., D. Hodouin & É. Plamondon. 2000, \"Global predictive control: a unified control structure for decoupling setpoint tracking, feedforward compensation and disturbance rejection dynamics\", IEE Proceedings - Control Theory and Applications, vol. 147, no 4, https://doi.org/10.1049/ip-cta:20000443, p. 465–475, ISSN 1350-2379.\n\n\n\n\n\nReturn empty matrices if estim is not a InternalModel.\n\n\n\n\n\n","category":"function"},{"location":"internals/predictive_control/#ModelPredictiveControl.init_matconstraint_mpc","page":"Predictive Controllers","title":"ModelPredictiveControl.init_matconstraint_mpc","text":"init_matconstraint_mpc(model::LinModel,\n i_Umin, i_Umax, i_ΔŨmin, i_ΔŨmax, i_Ymin, i_Ymax, i_x̂min, i_x̂max, args...\n) -> i_b, i_g, A\n\nInit i_b, i_g and A matrices for the linear and nonlinear inequality constraints.\n\nThe linear and nonlinear inequality constraints are respectively defined as:\n\nbeginaligned \n mathbfA ΔU mathbfb \n mathbfg(ΔU) mathbf0\nendaligned\n\ni_b is a BitVector including the indices of mathbfb that are finite numbers. i_g is a similar vector but for the indices of mathbfg (empty if model is a LinModel). The method also returns the mathbfA matrix if args is provided. In such a case, args needs to contain all the inequality constraint matrices: A_Umin, A_Umax, A_ΔŨmin, A_ΔŨmax, A_Ymin, A_Ymax, A_x̂min, A_x̂max.\n\n\n\n\n\nInit i_b, A without outputs and terminal constraints if model is not a LinModel.\n\n\n\n\n\n","category":"function"},{"location":"internals/predictive_control/#Update-Quadratic-Optimization","page":"Predictive Controllers","title":"Update Quadratic Optimization","text":"","category":"section"},{"location":"internals/predictive_control/","page":"Predictive Controllers","title":"Predictive Controllers","text":"ModelPredictiveControl.initpred!(::PredictiveController, ::LinModel, ::Any, ::Any, ::Any, ::Any)\nModelPredictiveControl.linconstraint!(::PredictiveController, ::LinModel)","category":"page"},{"location":"internals/predictive_control/#ModelPredictiveControl.initpred!-Tuple{PredictiveController, LinModel, Vararg{Any, 4}}","page":"Predictive Controllers","title":"ModelPredictiveControl.initpred!","text":"initpred!(mpc::PredictiveController, model::LinModel, d, D̂, R̂y, R̂u) -> nothing\n\nInit linear model prediction matrices F, q̃, r and current estimated output ŷ.\n\nSee init_predmat and init_quadprog for the definition of the matrices. They are computed with these equations using in-place operations:\n\nbeginaligned\n mathbfF = mathbfG d_0(k) + mathbfJ D_0 + mathbfK x_0(k) \n + mathbfV u_0(k-1) + mathbfB + mathbfY_s \n mathbfC_y = mathbfF - (mathbfR_y - Y_op) \n mathbfC_u = mathbfT mathbfu_0(k-1) - (mathbfR_u - U_op) \n mathbfq = 2(mathbfM_H_p mathbfE) mathbfC_y \n + (mathbfL_H_p mathbfS) mathbfC_u \n r = mathbfC_y mathbfM_H_p mathbfC_y \n + mathbfC_u mathbfL_H_p mathbfC_u\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"internals/predictive_control/#ModelPredictiveControl.linconstraint!-Tuple{PredictiveController, LinModel}","page":"Predictive Controllers","title":"ModelPredictiveControl.linconstraint!","text":"linconstraint!(mpc::PredictiveController, model::LinModel)\n\nSet b vector for the linear model inequality constraints (mathbfA ΔU b).\n\nAlso init mathbff_x = mathbfg_x d(k) + mathbfj_x D + mathbfk_x x_0(k) + mathbfv_x u(k-1) + mathbfb_x vector for the terminal constraints, see init_predmat.\n\n\n\n\n\n","category":"method"},{"location":"internals/predictive_control/#Solve-Optimization-Problem","page":"Predictive Controllers","title":"Solve Optimization Problem","text":"","category":"section"},{"location":"internals/predictive_control/","page":"Predictive Controllers","title":"Predictive Controllers","text":"ModelPredictiveControl.optim_objective!(::PredictiveController)\nModelPredictiveControl.getinput","category":"page"},{"location":"internals/predictive_control/#ModelPredictiveControl.optim_objective!-Tuple{PredictiveController}","page":"Predictive Controllers","title":"ModelPredictiveControl.optim_objective!","text":"optim_objective!(mpc::PredictiveController) -> ΔŨ\n\nOptimize the objective function of mpc PredictiveController and return the solution ΔŨ.\n\nIf supported by mpc.optim, it warm-starts the solver at:\n\nmathbfΔU = \nbeginbmatrix\n mathbfΔu_k-1(k+0) \n mathbfΔu_k-1(k+1) \n vdots \n mathbfΔu_k-1(k+H_c-2) \n mathbf0 \n ϵ_k-1\nendbmatrix\n\nwhere mathbfΔu_k-1(k+j) is the input increment for time k+j computed at the last control period k-1. It then calls JuMP.optimize!(mpc.optim) and extract the solution. A failed optimization prints an @error log in the REPL and returns the warm-start value.\n\n\n\n\n\n","category":"method"},{"location":"internals/predictive_control/#ModelPredictiveControl.getinput","page":"Predictive Controllers","title":"ModelPredictiveControl.getinput","text":"getinput(mpc::PredictiveController, ΔŨ) -> u\n\nGet current manipulated input u from a PredictiveController solution ΔŨ.\n\nThe first manipulated input mathbfu(k) is extracted from the input increments vector mathbfΔU and applied on the plant (from the receding horizon principle).\n\n\n\n\n\n","category":"function"},{"location":"manual/mtk/#man_mtk","page":"ModelingToolkit","title":"Manual: ModelingToolkit Integration","text":"","category":"section"},{"location":"manual/mtk/","page":"ModelingToolkit","title":"ModelingToolkit","text":"Pages = [\"mtk.md\"]","category":"page"},{"location":"manual/mtk/","page":"ModelingToolkit","title":"ModelingToolkit","text":"using Logging; errlogger = ConsoleLogger(stderr, Error);\nold_logger = global_logger(); global_logger(errlogger);","category":"page"},{"location":"manual/mtk/#Pendulum-Model","page":"ModelingToolkit","title":"Pendulum Model","text":"","category":"section"},{"location":"manual/mtk/","page":"ModelingToolkit","title":"ModelingToolkit","text":"This example integrates the simple pendulum model of the last section in the ModelingToolkit.jl (MTK) framework and extracts appropriate f! and h! functions to construct a NonLinModel. An NonLinMPC is designed from this model and simulated to reproduce the results of the last section.","category":"page"},{"location":"manual/mtk/","page":"ModelingToolkit","title":"ModelingToolkit","text":"danger: Disclaimer\nThis simple example is not an official interface to ModelingToolkit.jl. It is provided as a basic starting template to combine both packages. There is no guarantee that it will work for all corner cases.","category":"page"},{"location":"manual/mtk/","page":"ModelingToolkit","title":"ModelingToolkit","text":"We first construct and instantiate the pendulum model:","category":"page"},{"location":"manual/mtk/","page":"ModelingToolkit","title":"ModelingToolkit","text":"using ModelPredictiveControl, ModelingToolkit\nusing ModelingToolkit: D_nounits as D, t_nounits as t, varmap_to_vars\n@mtkmodel Pendulum begin\n @parameters begin\n g = 9.8\n L = 0.4\n K = 1.2\n m = 0.3\n end\n @variables begin\n θ(t) # state\n ω(t) # state\n τ(t) # input\n y(t) # output\n end\n @equations begin\n D(θ) ~ ω\n D(ω) ~ -g/L*sin(θ) - K/m*ω + τ/m/L^2\n y ~ θ * 180 / π\n end\nend\n@named mtk_model = Pendulum()\nmtk_model = complete(mtk_model)","category":"page"},{"location":"manual/mtk/","page":"ModelingToolkit","title":"ModelingToolkit","text":"We than convert the MTK model to an input-output system:","category":"page"},{"location":"manual/mtk/","page":"ModelingToolkit","title":"ModelingToolkit","text":"function generate_f_h(model, inputs, outputs)\n (_, f_ip), dvs, psym, io_sys = ModelingToolkit.generate_control_function(\n model, inputs, split=false; outputs\n )\n if any(ModelingToolkit.is_alg_equation, equations(io_sys)) \n error(\"Systems with algebraic equations are not supported\")\n end\n nu, nx, ny = length(inputs), length(dvs), length(outputs)\n vx = string.(dvs)\n p = varmap_to_vars(defaults(io_sys), psym)\n function f!(ẋ, x, u, _ , p)\n try\n f_ip(ẋ, x, u, p, nothing)\n catch err\n if err isa MethodError\n error(\"NonLinModel does not support a time argument t in the f function, \"*\n \"see the constructor docstring for a workaround.\")\n else\n rethrow()\n end\n end\n return nothing\n end\n h_ = ModelingToolkit.build_explicit_observed_function(io_sys, outputs; inputs)\n u_nothing = fill(nothing, nu)\n function h!(y, x, _ , p)\n y .= try\n # MTK.jl supports a `u` argument in `h_` function but not this package. We set\n # `u` as a vector of nothing and `h_` function will presumably throw an\n # MethodError it this argument is used inside the function\n h_(x, u_nothing, p, nothing)\n catch err\n if err isa MethodError\n error(\"NonLinModel only support strictly proper systems (no manipulated \"*\n \"input argument u in the output function h)\")\n else\n rethrow()\n end\n end\n return nothing\n end\n return f!, h!, p, nu, nx, ny, vx\nend\ninputs, outputs = [mtk_model.τ], [mtk_model.y]\nf!, h!, p, nu, nx, ny, vx = generate_f_h(mtk_model, inputs, outputs)\nTs = 0.1\nvu, vy = [\"\\$τ\\$ (Nm)\"], [\"\\$θ\\$ (°)\"]\nnothing # hide","category":"page"},{"location":"manual/mtk/","page":"ModelingToolkit","title":"ModelingToolkit","text":"A NonLinModel can now be constructed:","category":"page"},{"location":"manual/mtk/","page":"ModelingToolkit","title":"ModelingToolkit","text":"model = setname!(NonLinModel(f!, h!, Ts, nu, nx, ny; p); u=vu, x=vx, y=vy)","category":"page"},{"location":"manual/mtk/","page":"ModelingToolkit","title":"ModelingToolkit","text":"We also instantiate a plant model with a 25 % larger friction coefficient K:","category":"page"},{"location":"manual/mtk/","page":"ModelingToolkit","title":"ModelingToolkit","text":"mtk_model.K = defaults(mtk_model)[mtk_model.K] * 1.25\nf_plant, h_plant, p = generate_f_h(mtk_model, inputs, outputs)\nplant = setname!(NonLinModel(f_plant, h_plant, Ts, nu, nx, ny; p); u=vu, x=vx, y=vy)","category":"page"},{"location":"manual/mtk/#Controller-Design","page":"ModelingToolkit","title":"Controller Design","text":"","category":"section"},{"location":"manual/mtk/","page":"ModelingToolkit","title":"ModelingToolkit","text":"We can than reproduce the Kalman filter and the controller design of the last section:","category":"page"},{"location":"manual/mtk/","page":"ModelingToolkit","title":"ModelingToolkit","text":"α=0.01; σQ=[0.1, 1.0]; σR=[5.0]; nint_u=[1]; σQint_u=[0.1]\nestim = UnscentedKalmanFilter(model; α, σQ, σR, nint_u, σQint_u)\nHp, Hc, Mwt, Nwt = 20, 2, [0.5], [2.5]\nnmpc = NonLinMPC(estim; Hp, Hc, Mwt, Nwt, Cwt=Inf)\numin, umax = [-1.5], [+1.5]\nnmpc = setconstraint!(nmpc; umin, umax)","category":"page"},{"location":"manual/mtk/","page":"ModelingToolkit","title":"ModelingToolkit","text":"The 180° setpoint response is identical:","category":"page"},{"location":"manual/mtk/","page":"ModelingToolkit","title":"ModelingToolkit","text":"using Plots\nN = 35\nusing JuMP; unset_time_limit_sec(nmpc.optim) # hide\nres_ry = sim!(nmpc, N, [180.0], plant=plant, x_0=[0, 0], x̂_0=[0, 0, 0])\nplot(res_ry)\nsavefig(\"plot1_MTK.svg\"); nothing # hide","category":"page"},{"location":"manual/mtk/","page":"ModelingToolkit","title":"ModelingToolkit","text":"(Image: plot1_MTK)","category":"page"},{"location":"manual/mtk/","page":"ModelingToolkit","title":"ModelingToolkit","text":"and also the output disturbance rejection:","category":"page"},{"location":"manual/mtk/","page":"ModelingToolkit","title":"ModelingToolkit","text":"res_yd = sim!(nmpc, N, [180.0], plant=plant, x_0=[π, 0], x̂_0=[π, 0, 0], y_step=[10])\nplot(res_yd)\nsavefig(\"plot2_MTK.svg\"); nothing # hide","category":"page"},{"location":"manual/mtk/","page":"ModelingToolkit","title":"ModelingToolkit","text":"(Image: plot2_MTK)","category":"page"},{"location":"manual/mtk/#Acknowledgement","page":"ModelingToolkit","title":"Acknowledgement","text":"","category":"section"},{"location":"manual/mtk/","page":"ModelingToolkit","title":"ModelingToolkit","text":"Authored by 1-Bart-1 and baggepinnen, thanks for the contribution.","category":"page"},{"location":"manual/mtk/","page":"ModelingToolkit","title":"ModelingToolkit","text":"global_logger(old_logger);","category":"page"},{"location":"public/plot_sim/#Functions:-Simulations-and-Plots","page":"Simulations and Plots","title":"Functions: Simulations and Plots","text":"","category":"section"},{"location":"public/plot_sim/","page":"Simulations and Plots","title":"Simulations and Plots","text":"Pages = [\"plot_sim.md\"]","category":"page"},{"location":"public/plot_sim/","page":"Simulations and Plots","title":"Simulations and Plots","text":"This page documents the functions for quick plotting of open- and closed-loop simulations. They are generic to SimModel, StateEstimator and PredictiveController types. A SimResult instance must be created first with its constructor or by calling sim!. The results are then visualized with plot function from Plots.jl.","category":"page"},{"location":"public/plot_sim/#Quick-Simulations","page":"Simulations and Plots","title":"Quick Simulations","text":"","category":"section"},{"location":"public/plot_sim/","page":"Simulations and Plots","title":"Simulations and Plots","text":"sim!","category":"page"},{"location":"public/plot_sim/#ModelPredictiveControl.sim!","page":"Simulations and Plots","title":"ModelPredictiveControl.sim!","text":"sim!(plant::SimModel, N::Int, u=plant.uop.+1, d=plant.dop; x_0=plant.xop) -> res\n\nOpen-loop simulation of plant for N time steps, default to unit bump test on all inputs.\n\nThe manipulated inputs mathbfu and measured disturbances mathbfd are held constant at u and d values, respectively. The plant initial state mathbfx(0) is specified by x_0 keyword arguments. The function returns SimResult instances that can be visualized by calling plot on them. Note that the method mutates plant internal states.\n\nExamples\n\njulia> plant = NonLinModel((x,u,d,_)->0.1x+u+d, (x,_,_)->2x, 5, 1, 1, 1, 1, solver=nothing);\n\njulia> res = sim!(plant, 15, [0], [0], x_0=[1])\nSimulation results of NonLinModel with 15 time steps.\n\n\n\n\n\nsim!(\n estim::StateEstimator,\n N::Int,\n u = estim.model.uop .+ 1,\n d = estim.model.dop;\n \n) -> res\n\nClosed-loop simulation of estim estimator for N steps, default to input bumps.\n\nSee Arguments for the available options. The noises are provided as standard deviations σ vectors. The simulated sensor and process noises of plant are specified by y_noise and x_noise arguments, respectively.\n\nArguments\n\ninfo: Info\nKeyword arguments with emphasis are non-Unicode alternatives.\n\nestim::StateEstimator : state estimator to simulate\nN::Int : simulation length in time steps\nu = estim.model.uop .+ 1 : manipulated input mathbfu value\nd = estim.model.dop : plant measured disturbance mathbfd value\nplant::SimModel = estim.model : simulated plant model\nu_step = zeros(plant.nu) : step load disturbance on plant inputs mathbfu\nu_noise = zeros(plant.nu) : gaussian load disturbance on plant inputs mathbfu\ny_step = zeros(plant.ny) : step disturbance on plant outputs mathbfy\ny_noise = zeros(plant.ny) : additive gaussian noise on plant outputs mathbfy\nd_step = zeros(plant.nd) : step on measured disturbances mathbfd\nd_noise = zeros(plant.nd) : additive gaussian noise on measured dist. mathbfd\nx_noise = zeros(plant.nx) : additive gaussian noise on plant states mathbfx\nx_0 = plant.xop : plant initial state mathbfx(0)\nx̂_0 = nothing or xhat_0 : initial estimate mathbfx(0), initstate! is used if nothing\nlastu = plant.uop : last plant input mathbfu for mathbfx initialization\n\nExamples\n\njulia> model = LinModel(tf(3, [30, 1]), 0.5);\n\njulia> estim = KalmanFilter(model, σR=[0.5], σQ=[0.25], σQint_ym=[0.01], σPint_ym_0=[0.1]);\n\njulia> res = sim!(estim, 50, [0], y_noise=[0.5], x_noise=[0.25], x_0=[-10], x̂_0=[0, 0])\nSimulation results of KalmanFilter with 50 time steps.\n\n\n\n\n\nsim!(\n mpc::PredictiveController, \n N::Int,\n ry = mpc.estim.model.yop .+ 1, \n d = mpc.estim.model.dop,\n ru = mpc.estim.model.uop;\n \n) -> res\n\nClosed-loop simulation of mpc controller for N steps, default to output setpoint bumps.\n\nThe output and manipulated input setpoints are held constant at ry and ru, respectively. The keyword arguments are identical to sim!(::StateEstimator, ::Int).\n\nExamples\n\njulia> model = LinModel([tf(3, [30, 1]); tf(2, [5, 1])], 4);\n\njulia> mpc = setconstraint!(LinMPC(model, Mwt=[0, 1], Nwt=[0.01], Hp=30), ymin=[0, -Inf]);\n\njulia> res = sim!(mpc, 25, [0, 0], y_noise=[0.1], y_step=[-10, 0])\nSimulation results of LinMPC with 25 time steps.\n\n\n\n\n\n","category":"function"},{"location":"public/plot_sim/#Simulation-Results","page":"Simulations and Plots","title":"Simulation Results","text":"","category":"section"},{"location":"public/plot_sim/","page":"Simulations and Plots","title":"Simulations and Plots","text":"SimResult","category":"page"},{"location":"public/plot_sim/#ModelPredictiveControl.SimResult","page":"Simulations and Plots","title":"ModelPredictiveControl.SimResult","text":"SimResult(obj::SimModel, U_data, Y_data, D_data=[]; )\nSimResult(obj::StateEstimator, U_data, Y_data, D_data=[]; )\nSimResult(obj::PredictiveController, U_data, Y_data, D_data=[]; )\n\nManually construct a SimResult to quickly plot obj simulations.\n\nExcept for obj, all the arguments should be matrices of N columns, where N is the number of time steps. SimResult objects allow to quickly plot simulation results. Simply call plot on them.\n\nArguments\n\ninfo: Info\nKeyword arguments with emphasis are non-Unicode alternatives.\n\nobj : simulated SimModel/StateEstimator/PredictiveController\nU_data : manipulated inputs\nY_data : plant outputs\nD_data=[] : measured disturbances\nX_data=nothing : plant states\nX̂_data=nothing or Xhat_data : estimated states\nŶ_data=nothing or Yhat_data : estimated outputs\nRy_data=nothing : plant output setpoints\nRu_data=nothing : manipulated input setpoints\nplant=get_model(obj) : simulated plant model, default to obj internal plant model\n\nExamples\n\njulia> model = LinModel(tf(1, [1, 1]), 1.0);\n\njulia> N = 5; U_data = fill(1.0, 1, N); Y_data = zeros(1, N);\n\njulia> for i=1:N; updatestate!(model, U_data[:, i]); Y_data[:, i] = model(); end; Y_data\n1×5 Matrix{Float64}:\n 0.632121 0.864665 0.950213 0.981684 0.993262\n\njulia> res = SimResult(model, U_data, Y_data)\nSimulation results of LinModel with 5 time steps.\n\n\n\n\n\n","category":"type"},{"location":"public/plot_sim/#Plotting-Results","page":"Simulations and Plots","title":"Plotting Results","text":"","category":"section"},{"location":"public/plot_sim/","page":"Simulations and Plots","title":"Simulations and Plots","text":"The plot methods are based on Plots.jl package. To install it run using Pkg; Pkg.add(\"Plots\") in the Julia REPL.","category":"page"},{"location":"public/plot_sim/","page":"Simulations and Plots","title":"Simulations and Plots","text":"ModelPredictiveControl.plot","category":"page"},{"location":"public/plot_sim/#ModelPredictiveControl.plot","page":"Simulations and Plots","title":"ModelPredictiveControl.plot","text":"plot(res::SimResult{<:Real, <:SimModel}; )\n\nPlot the simulation results of a SimModel.\n\nArguments\n\ninfo: Info\nThe keyword arguments can be Bools, index ranges (2:4) or vectors ([1, 3]), to select the variables to plot.\n\nres::SimResult{<:Real, <:SimModel} : simulation results to plot\nploty=true : plot plant outputs mathbfy\nplotu=true : plot manipulated inputs mathbfu\nplotd=true : plot measured disturbances mathbfd if applicable\nplotx=false : plot plant states mathbfx\n\nExamples\n\njulia> res = sim!(LinModel(tf(2, [10, 1]), 2.0), 25);\n\njulia> using Plots; plot(res, plotu=false)\n\n(Image: plot_model)\n\n\n\n\n\nplot(res::SimResult{<:Real, <:StateEstimator}; )\n\nPlot the simulation results of a StateEstimator.\n\nArguments\n\ninfo: Info\nThe keyword arguments can be Bools, index ranges (2:4) or vectors ([1, 3]), to select the variables to plot. Keywords in emphasis are non-Unicode alternatives.\n\nres::SimResult{<:Real, <:StateEstimator} : simulation results to plot\nplotŷ=true or plotyhat : plot estimated outputs mathbfy\nplotx̂=false or plotxhat : plot estimated states mathbfx\nplotxwithx̂=false or plotxwithxhat : plot plant states mathbfx and estimated states mathbfx together\nplotx̂min=true or plotxhatmin : plot estimated state lower bounds mathbfx_min if applicable\nplotx̂max=true or plotxhatmax : plot estimated state upper bounds mathbfx_max if applicable\n of plot(::SimResult{<:Real, <:SimModel})\n\nExamples\n\njulia> res = sim!(KalmanFilter(LinModel(tf(3, [2.0, 1]), 1.0)), 25, [0], y_step=[1]);\n\njulia> using Plots; plot(res, plotu=false, plotŷ=true, plotxwithx̂=true)\n\n(Image: plot_estimator)\n\n\n\n\n\nplot(res::SimResult{<:Real, <:PredictiveController}; )\n\nPlot the simulation results of a PredictiveController.\n\nArguments\n\ninfo: Info\nThe keyword arguments can be Bools, index ranges (2:4) or vectors ([1, 3]), to select the variables to plot.\n\nres::SimResult{<:Real, <:PredictiveController} : simulation results to plot\nplotry=true : plot plant output setpoints mathbfr_y if applicable\nplotymin=true : plot predicted output lower bounds mathbfy_min if applicable\nplotymax=true : plot predicted output upper bounds mathbfy_max if applicable\nplotru=true : plot manipulated input setpoints mathbfr_u if applicable\nplotumin=true : plot manipulated input lower bounds mathbfu_min if applicable\nplotumax=true : plot manipulated input upper bounds mathbfu_max if applicable\n of plot(::SimResult{<:Real, <:SimModel})\n of plot(::SimResult{<:Real, <:StateEstimator})\n\nExamples\n\njulia> model = LinModel(tf(2, [5.0, 1]), 1.0);\n\njulia> res = sim!(setconstraint!(LinMPC(model), umax=[1.0]), 25, [0], u_step=[-1]);\n\njulia> using Plots; plot(res, plotŷ=true, plotry=true, plotumax=true, plotx̂=[2])\n\n(Image: plot_controller)\n\n\n\n\n\n","category":"function"},{"location":"internals/state_estim/#Functions:-StateEstimator-Internals","page":"State Estimators","title":"Functions: StateEstimator Internals","text":"","category":"section"},{"location":"internals/state_estim/","page":"State Estimators","title":"State Estimators","text":"Pages = [\"state_estim.md\"]","category":"page"},{"location":"internals/state_estim/#Augmented-Model","page":"State Estimators","title":"Augmented Model","text":"","category":"section"},{"location":"internals/state_estim/","page":"State Estimators","title":"State Estimators","text":"ModelPredictiveControl.f̂!\nModelPredictiveControl.ĥ!","category":"page"},{"location":"internals/state_estim/#ModelPredictiveControl.f̂!","page":"State Estimators","title":"ModelPredictiveControl.f̂!","text":"f̂!(x̂next0, û0, estim::StateEstimator, model::SimModel, x̂0, u0, d0) -> nothing\n\nMutating state function mathbff of the augmented model.\n\nBy introducing an augmented state vector mathbfx_0 like in augment_model, the function returns the next state of the augmented model, defined as:\n\nbeginaligned\n mathbfx_0(k+1) = mathbffBig(mathbfx_0(k) mathbfu_0(k) mathbfd_0(k)Big) \n mathbfy_0(k) = mathbfhBig(mathbfx_0(k) mathbfd_0(k)Big) \nendaligned\n\nwhere mathbfx_0(k+1) is stored in x̂next0 argument. The method mutates x̂next0 and û0 in place, the latter stores the input vector of the augmented model mathbfu_0 + y_s_u. The model parameter vector model.p is not included in the function signature for conciseness.\n\n\n\n\n\nf̂!(x̂next0, _ , estim::StateEstimator, model::LinModel, x̂0, u0, d0) -> nothing\n\nUse the augmented model matrices if model is a LinModel.\n\n\n\n\n\nf̂!(x̂next0, _ , estim::InternalModel, model::NonLinModel, x̂0, u0, d0)\n\nState function mathbff of InternalModel for NonLinModel.\n\nIt calls model.f!(x̂next0, x̂0, u0 ,d0, model.p) since this estimator does not augment the states.\n\n\n\n\n\n","category":"function"},{"location":"internals/state_estim/#ModelPredictiveControl.ĥ!","page":"State Estimators","title":"ModelPredictiveControl.ĥ!","text":"ĥ!(ŷ0, estim::StateEstimator, model::SimModel, x̂0, d0) -> nothing\n\nMutating output function mathbfh of the augmented model, see f̂!.\n\n\n\n\n\nĥ!(ŷ0, estim::StateEstimator, model::LinModel, x̂0, d0) -> nothing\n\nUse the augmented model matrices if model is a LinModel.\n\n\n\n\n\nĥ!(ŷ0, estim::InternalModel, model::NonLinModel, x̂0, d0)\n\nOutput function mathbfh of InternalModel, it calls model.h!.\n\n\n\n\n\n","category":"function"},{"location":"internals/state_estim/#Estimator-Construction","page":"State Estimators","title":"Estimator Construction","text":"","category":"section"},{"location":"internals/state_estim/","page":"State Estimators","title":"State Estimators","text":"ModelPredictiveControl.init_estimstoch\nModelPredictiveControl.init_integrators\nModelPredictiveControl.augment_model\nModelPredictiveControl.init_ukf\nModelPredictiveControl.init_internalmodel\nModelPredictiveControl.init_predmat_mhe\nModelPredictiveControl.relaxarrival\nModelPredictiveControl.relaxX̂\nModelPredictiveControl.relaxŴ\nModelPredictiveControl.relaxV̂\nModelPredictiveControl.init_matconstraint_mhe","category":"page"},{"location":"internals/state_estim/#ModelPredictiveControl.init_estimstoch","page":"State Estimators","title":"ModelPredictiveControl.init_estimstoch","text":"init_estimstoch(model, i_ym, nint_u, nint_ym) -> As, Cs_u, Cs_y, nxs, nint_u, nint_ym\n\nInit stochastic model matrices from integrator specifications for state estimation.\n\nThe arguments nint_u and nint_ym specify how many integrators are added to each manipulated input and measured outputs. The function returns the state-space matrices As, Cs_u and Cs_y of the stochastic model:\n\nbeginaligned\nmathbfx_s(k+1) = mathbfA_s x_s(k) + mathbfB_s e(k) \nmathbfy_s_u(k) = mathbfC_s_u x_s(k) \nmathbfy_s_y(k) = mathbfC_s_y x_s(k) \nendaligned\n\nwhere mathbfe(k) is an unknown zero mean white noise and mathbfA_s = mathrmdiag(mathbfA_s_u A_s_y). The estimations does not use mathbfB_s, it is thus ignored. The function init_integrators builds the state-space matrices.\n\n\n\n\n\n","category":"function"},{"location":"internals/state_estim/#ModelPredictiveControl.init_integrators","page":"State Estimators","title":"ModelPredictiveControl.init_integrators","text":"init_integrators(nint, ny, varname::String) -> A, C, nint\n\nCalc A, C state-space matrices from integrator specifications nint.\n\nThis function is used to initialize the stochastic part of the augmented model for the design of state estimators. The vector nint provides how many integrators (in series) should be incorporated for each output. The argument should have ny element, except for nint=0 which is an alias for no integrator at all. The specific case of one integrator per output results in A = I and C = I. The estimation does not use the B matrix, it is thus ignored. This function is called twice :\n\nfor the unmeasured disturbances at manipulated inputs mathbfu\nfor the unmeasured disturbances at measured outputs mathbfy^m\n\n\n\n\n\n","category":"function"},{"location":"internals/state_estim/#ModelPredictiveControl.augment_model","page":"State Estimators","title":"ModelPredictiveControl.augment_model","text":"augment_model(\n model::LinModel, As, Cs_u, Cs_y; verify_obsv=true\n) -> Â, B̂u, Ĉ, B̂d, D̂d, x̂op, f̂op\n\nAugment LinModel state-space matrices with stochastic ones As, Cs_u, Cs_y.\n\nIf mathbfx_0 are model.x0 states, and mathbfx_s, the states defined at init_estimstoch, we define an augmented state vector mathbfx = beginsmallmatrix mathbfx_0 mathbfx_s endsmallmatrix . The method returns the augmented matrices Â, B̂u, Ĉ, B̂d and D̂d:\n\nbeginaligned\n mathbfx_0(k+1) = mathbfA x_0(k) + mathbfB_u u_0(k) + mathbfB_d d_0(k) \n mathbfy_0(k) = mathbfC x_0(k) + mathbfD_d d_0(k)\nendaligned\n\nAn error is thrown if the augmented model is not observable and verify_obsv == true. The augmented operating points x̂op and f̂op are simply mathbfx_op and mathbff_op vectors appended with zeros (see setop!). See Extended Help for a detailed definition of the augmented matrices.\n\nExtended Help\n\ndetails: Extended Help\nUsing the As, Cs_u and Cs_y matrices of the stochastic model provided in argument and the model.A, model.Bu, model.Bd, model.C, model.Dd matrices, the state-space matrices of the augmented model are defined as follows:beginaligned\nmathbfA = beginbmatrix \n mathbfA mathbfB_u C_s_u \n mathbf0 mathbfA_s endbmatrix \nmathbfB_u = beginbmatrix\n mathbfB_u \n mathbf0 endbmatrix \nmathbfC = beginbmatrix\n mathbfC mathbfC_s_y endbmatrix \nmathbfB_d = beginbmatrix \n mathbfB_d \n mathbf0 endbmatrix \nmathbfD_d = mathbfD_d\nendaligned\n\n\n\n\n\naugment_model(\n model::SimModel, As, Cs_u, Cs_y; verify_obsv=false\n) -> Â, B̂u, Ĉ, B̂d, D̂d, x̂op, f̂op\n\nReturn empty matrices, and x̂op & f̂op vectors, if model is not a LinModel.\n\n\n\n\n\n","category":"function"},{"location":"internals/state_estim/#ModelPredictiveControl.init_ukf","page":"State Estimators","title":"ModelPredictiveControl.init_ukf","text":"init_ukf(model, nx̂, α, β, κ) -> nσ, γ, m̂, Ŝ\n\nCompute the UnscentedKalmanFilter constants from α β and κ.\n\nWith n_mathbfx elements in the state vector mathbfx and n_σ = 2 n_mathbfx + 1 sigma points, the scaling factor applied on standard deviation matrices sqrtmathbfP is:\n\n γ = α sqrt n_mathbfx + κ \n\nThe weight vector (n_σ 1) for the mean and the weight matrix (n_σ n_σ) for the covariance are respectively:\n\nbeginaligned\n mathbfm = beginbmatrix 1 - tfracn_mathbfxγ^2 tfrac12γ^2 tfrac12γ^2 cdots tfrac12γ^2 endbmatrix \n mathbfS = mathrmdiagbig( 2 - α^2 + β - tfracn_mathbfxγ^2 tfrac12γ^2 tfrac12γ^2 cdots tfrac12γ^2 big)\nendaligned\n\nSee update_estimate!(::UnscentedKalmanFilter) for other details.\n\n\n\n\n\n","category":"function"},{"location":"internals/state_estim/#ModelPredictiveControl.init_internalmodel","page":"State Estimators","title":"ModelPredictiveControl.init_internalmodel","text":"init_internalmodel(As, Bs, Cs, Ds) -> Âs, B̂s\n\nCalc stochastic model update matrices Âs and B̂s for InternalModel estimator.\n\nAs, Bs, Cs and Ds are the stochastic model matrices :\n\nbeginaligned\n mathbfx_s(k+1) = mathbfA_s x_s(k) + mathbfB_s e(k) \n mathbfy_s(k) = mathbfC_s x_s(k) + mathbfD_s e(k)\nendaligned\n\nwhere mathbfe(k) is conceptual and unknown zero mean white noise. Its optimal estimation is mathbfe=0, the expected value. Thus, the Âs and B̂s matrices that optimally update the stochastic estimate mathbfx_s are:\n\nbeginaligned\n mathbfx_s(k+1) \n = mathbf(A_s - B_s D_s^-1 C_s) x_s(k) + mathbf(B_s D_s^-1) y_s(k) \n = mathbfA_s x_s(k) + mathbfB_s y_s(k)\nendaligned\n\nwith current stochastic outputs estimation mathbfy_s(k), composed of the measured mathbfy_s^m(k) = mathbfy^m(k) - mathbfy_d^m(k) and unmeasured mathbfy_s^u = 0 outputs. See [1].\n\n[1]: Desbiens, A., D. Hodouin & É. Plamondon. 2000, \"Global predictive control : a unified control structure for decoupling setpoint tracking, feedforward compensation and disturbance rejection dynamics\", IEE Proceedings - Control Theory and Applications, vol. 147, no 4, https://doi.org/10.1049/ip-cta:20000443, p. 465–475, ISSN 1350-2379.\n\n\n\n\n\n","category":"function"},{"location":"internals/state_estim/#ModelPredictiveControl.init_predmat_mhe","page":"State Estimators","title":"ModelPredictiveControl.init_predmat_mhe","text":"init_predmat_mhe(\n model::LinModel, He, i_ym, Â, B̂u, Ĉm, B̂d, D̂dm, x̂op, f̂op, p\n) -> E, G, J, B, ex̄, Ex̂, Gx̂, Jx̂, Bx̂\n\nConstruct the MovingHorizonEstimator prediction matrices for LinModel model.\n\nWe first introduce the deviation vector of the estimated state at arrival mathbfx_0(k-N_k+p) = mathbfx_k(k-N_k+p) - mathbfx_op (see setop!), and the vector mathbfZ = beginsmallmatrix mathbfx_0(k-N_k+p) mathbfW endsmallmatrix with the decision variables. Setting the constant p=0 produces an estimator in the current form, while the prediction form is obtained with p=1. The estimated sensor noises from time k-N_k+1 to k are computed by:\n\nbeginaligned\n mathbfV = mathbfY_0^m - Y_0^m = mathbfE Z + G U_0 + J D_0 + Y_0^m + B \n = mathbfE Z + F\nendaligned\n\nin which mathbfU_0 and mathbfY_0^m respectively include the deviation values of the manipulated inputs mathbfu_0(k-j+p) from j=N_k to 1 and measured outputs mathbfy_0^m(k-j+1) from j=N_k to 1. The vector mathbfD_0 comprises one additional measured disturbance if p=0, that is, it includes the deviation vectors mathbfd_0(k-j+1) from j=N_k+1-p to 1. The constant mathbfB is the contribution for non-zero state mathbfx_op and state update mathbff_op operating points (for linearization, see augment_model and linearize). The method also returns the matrices for the estimation error at arrival:\n\n mathbfx = mathbfx_0^(k-N_k+p) - mathbfx_0(k-N_k+p) = mathbfe_x Z + f_x\n\nin which mathbfe_x = beginsmallmatrix -mathbfI mathbf0 cdots mathbf0 endsmallmatrix, and mathbff_x = mathbfx_0^(k-N_k+p). The latter is the deviation vector of the state at arrival, estimated at time k-N_k, i.e. mathbfx_0^(k-N_k+p) = mathbfx_k-N_k(k-N_k+p) - mathbfx_op. Lastly, the estimates mathbfx_0(k-j+p) from j=N_k-1 to 0, also in deviation form, are computed with:\n\nbeginaligned\n mathbfX_0 = mathbfE_x Z + G_x U_0 + J_x D_0 + B_x \n = mathbfE_x Z + F_x\nendaligned\n\nThe matrices mathbfE G J B E_x G_x J_x B_x are defined in the Extended Help section. The vectors mathbfF F_x f_x are recalculated at each discrete time step, see initpred!(::MovingHorizonEstimator, ::LinModel) and linconstraint!(::MovingHorizonEstimator, ::LinModel).\n\nExtended Help\n\ndetails: Extended Help\nUsing the augmented process model matrices mathbfA B_u C^m B_d D_d^m, and the function mathbfS(j) = _i=0^j mathbfA^i, the prediction matrices for the sensor noises depend on the constant p. For p=0, the matrices are computed by:beginaligned\nmathbfE = - beginbmatrix\n mathbfC^mmathbfA^1 mathbfC^mmathbfA^0 cdots mathbf0 \n mathbfC^mmathbfA^2 mathbfC^mmathbfA^1 cdots mathbf0 \n vdots vdots ddots vdots \n mathbfC^mmathbfA^H_e mathbfC^mmathbfA^H_e-1 cdots mathbfC^mmathbfA^0 endbmatrix \nmathbfG = - beginbmatrix\n mathbfC^mmathbfA^0mathbfB_u mathbf0 cdots mathbf0 \n mathbfC^mmathbfA^1mathbfB_u mathbfC^mmathbfA^0mathbfB_u cdots mathbf0 \n vdots vdots ddots vdots \n mathbfC^mmathbfA^H_e-1mathbfB_u mathbfC^mmathbfA^H_e-2mathbfB_u cdots mathbfC^mmathbfA^0mathbfB_u endbmatrix \nmathbfJ = - beginbmatrix\n mathbfC^mmathbfA^0mathbfB_d mathbfD_d^m cdots mathbf0 \n mathbfC^mmathbfA^1mathbfB_d mathbfC^mmathbfA^0mathbfB_d cdots mathbf0 \n vdots vdots ddots vdots \n mathbfC^mmathbfA^H_e-1mathbfB_d mathbfC^mmathbfA^H_e-2mathbfB_d cdots mathbfD_d^m endbmatrix \nmathbfB = - beginbmatrix\n mathbfC^m S(0) \n mathbfC^m S(1) \n vdots \n mathbfC^m S(H_e-1) endbmatrix mathbfbig(f_op - x_opbig)\nendalignedor, for p=1, the matrices are given by:beginaligned\nmathbfE = - beginbmatrix\n mathbfC^mmathbfA^0 mathbf0 cdots mathbf0 \n mathbfC^mmathbfA^1 mathbfC^mmathbfA^0 cdots mathbf0 \n vdots vdots ddots vdots \n mathbfC^mmathbfA^H_e-1 mathbfC^mmathbfA^H_e-2 cdots mathbf0 endbmatrix \nmathbfG = - beginbmatrix\n mathbf0 mathbf0 cdots mathbf0 \n mathbfC^mmathbfA^0mathbfB_u mathbf0 cdots mathbf0 \n vdots vdots ddots vdots \n mathbfC^mmathbfA^H_e-2mathbfB_u mathbfC^mmathbfA^H_e-3mathbfB_u cdots mathbf0 endbmatrix \nmathbfJ = - beginbmatrix\n mathbfD_d^m mathbf0 cdots mathbf0 \n mathbfC^mmathbfA^0mathbfB_d mathbfD_d^m cdots mathbf0 \n vdots vdots ddots vdots \n mathbfC^mmathbfA^H_e-2mathbfB_d mathbfC^mmathbfA^H_e-3mathbfB_d cdots mathbfD_d^m endbmatrix \nmathbfB = - beginbmatrix\n mathbf0 \n mathbfC^m S(0) \n vdots \n mathbfC^m S(H_e-2) endbmatrix mathbfbig(f_op - x_opbig)\nendalignedThe matrices for the estimated states are computed by:beginaligned\nmathbfE_x = beginbmatrix\n mathbfA^1 mathbfA^0 cdots mathbf0 \n mathbfA^2 mathbfA^1 cdots mathbf0 \n vdots vdots ddots vdots \n mathbfA^H_e mathbfA^H_e-1 cdots mathbfA^0 endbmatrix \nmathbfG_x = beginbmatrix\n mathbfA^0mathbfB_u mathbf0 cdots mathbf0 \n mathbfA^1mathbfB_u mathbfA^0mathbfB_u cdots mathbf0 \n vdots vdots ddots vdots \n mathbfA^H_e-1mathbfB_u mathbfA^H_e-2mathbfB_u cdots mathbfA^0mathbfB_u endbmatrix \nmathbfJ_x^ = beginbmatrix\n mathbfA^0mathbfB_d mathbf0 cdots mathbf0 \n mathbfA^1mathbfB_d mathbfA^0mathbfB_d cdots mathbf0 \n vdots vdots ddots vdots \n mathbfA^H_e-1mathbfB_d mathbfA^H_e-2mathbfB_d cdots mathbfA^0mathbfB_d endbmatrix quad\nmathbfJ_x = begincases\n beginsmallmatrix mathbfJ_x^ mathbf0 endsmallmatrix p=0 \n mathbfJ_x^ p=1 endcases \nmathbfB_x = beginbmatrix\n mathbfS(0) \n mathbfS(1) \n vdots \n mathbfS(H_e-1) endbmatrix mathbfbig(f_op - x_opbig)\nendalignedAll these matrices are truncated when N_k H_e (at the beginning).\n\n\n\n\n\nReturn empty matrices if model is not a LinModel, except for ex̄.\n\n\n\n\n\n","category":"function"},{"location":"internals/state_estim/#ModelPredictiveControl.relaxarrival","page":"State Estimators","title":"ModelPredictiveControl.relaxarrival","text":"relaxarrival(\n model::SimModel, nϵ, c_x̂min, c_x̂max, x̂min, x̂max, ex̄\n) -> A_x̃min, A_x̃max, x̃min, x̃max, ẽx̄\n\nAugment arrival state constraints with slack variable ϵ for softening the MHE.\n\nDenoting the MHE decision variable augmented with the slack variable mathbfZ = beginsmallmatrix ϵ mathbfZ endsmallmatrix, it returns the mathbfe_x matrix that appears in the estimation error at arrival equation mathbfx = mathbfe_x Z + f_x. It also returns the augmented constraints mathbfx_min and mathbfx_max, and the mathbfA matrices for the inequality constraints:\n\nbeginbmatrix \n mathbfA_x_min \n mathbfA_x_max\nendbmatrix mathbfZ \nbeginbmatrix\n - mathbf(x_min - x_op) \n + mathbf(x_max - x_op)\nendbmatrix\n\nin which mathbfx_min = beginsmallmatrix 0 mathbfx_min endsmallmatrix, mathbfx_max = beginsmallmatrix mathbfx_max endsmallmatrix and mathbfx_op = beginsmallmatrix 0 mathbfx_op endsmallmatrix\n\n\n\n\n\n","category":"function"},{"location":"internals/state_estim/#ModelPredictiveControl.relaxX̂","page":"State Estimators","title":"ModelPredictiveControl.relaxX̂","text":"relaxX̂(model::SimModel, nϵ, C_x̂min, C_x̂max, Ex̂) -> A_X̂min, A_X̂max, Ẽx̂\n\nAugment estimated state constraints with slack variable ϵ for softening the MHE.\n\nDenoting the MHE decision variable augmented with the slack variable mathbfZ = beginsmallmatrix ϵ mathbfZ endsmallmatrix, it returns the mathbfE_x matrix that appears in estimated states equation mathbfX = mathbfE_x Z + F_x. It also returns the mathbfA matrices for the inequality constraints:\n\nbeginbmatrix \n mathbfA_X_min \n mathbfA_X_max\nendbmatrix mathbfZ \nbeginbmatrix\n - mathbf(X_min - X_op) + F_x \n + mathbf(X_max - X_op) - F_x\nendbmatrix\n\nin which mathbfX_min X_max and mathbfX_op vectors respectively contains mathbfx_min x_max and mathbfx_op repeated H_e times.\n\n\n\n\n\nReturn empty matrices if model is not a LinModel\n\n\n\n\n\n","category":"function"},{"location":"internals/state_estim/#ModelPredictiveControl.relaxŴ","page":"State Estimators","title":"ModelPredictiveControl.relaxŴ","text":"relaxŴ(model::SimModel, nϵ, C_ŵmin, C_ŵmax, nx̂) -> A_Ŵmin, A_Ŵmax\n\nAugment estimated process noise constraints with slack variable ϵ for softening the MHE.\n\nDenoting the MHE decision variable augmented with the slack variable mathbfZ = beginsmallmatrix ϵ mathbfZ endsmallmatrix, it returns the mathbfA matrices for the inequality constraints:\n\nbeginbmatrix \n mathbfA_W_min \n mathbfA_W_max\nendbmatrix mathbfZ \nbeginbmatrix\n - mathbfW_min \n + mathbfW_max\nendbmatrix\n\n\n\n\n\n","category":"function"},{"location":"internals/state_estim/#ModelPredictiveControl.relaxV̂","page":"State Estimators","title":"ModelPredictiveControl.relaxV̂","text":"relaxV̂(model::SimModel, nϵ, C_v̂min, C_v̂max, E) -> A_V̂min, A_V̂max, Ẽ\n\nAugment estimated sensor noise constraints with slack variable ϵ for softening the MHE.\n\nDenoting the MHE decision variable augmented with the slack variable mathbfZ = beginsmallmatrix ϵ mathbfZ endsmallmatrix, it returns the mathbfE matrix that appears in estimated sensor noise equation mathbfV = mathbfE Z + F. It also returns the mathbfA matrices for the inequality constraints:\n\nbeginbmatrix \n mathbfA_V_min \n mathbfA_V_max\nendbmatrix mathbfZ \nbeginbmatrix\n - mathbfV_min + F \n + mathbfV_max - F\nendbmatrix\n\n\n\n\n\nReturn empty matrices if model is not a LinModel\n\n\n\n\n\n","category":"function"},{"location":"internals/state_estim/#ModelPredictiveControl.init_matconstraint_mhe","page":"State Estimators","title":"ModelPredictiveControl.init_matconstraint_mhe","text":"init_matconstraint_mhe(model::LinModel, \n i_x̃min, i_x̃max, i_X̂min, i_X̂max, i_Ŵmin, i_Ŵmax, i_V̂min, i_V̂max, args...\n) -> i_b, i_g, A\n\nInit i_b, i_g and A matrices for the MHE linear inequality constraints.\n\nThe linear and nonlinear inequality constraints are respectively defined as:\n\nbeginaligned \n mathbfA Z mathbfb \n mathbfg(Z) mathbf0\nendaligned\n\ni_b is a BitVector including the indices of mathbfb that are finite numbers. i_g is a similar vector but for the indices of mathbfg (empty if model is a LinModel). The method also returns the mathbfA matrix if args is provided. In such a case, args needs to contain all the inequality constraint matrices: A_x̃min, A_x̃max, A_X̂min, A_X̂max, A_Ŵmin, A_Ŵmax, A_V̂min, A_V̂max.\n\n\n\n\n\nInit i_b, A without state and sensor noise constraints if model is not a LinModel.\n\n\n\n\n\n","category":"function"},{"location":"internals/state_estim/#Update-Quadratic-Optimization","page":"State Estimators","title":"Update Quadratic Optimization","text":"","category":"section"},{"location":"internals/state_estim/","page":"State Estimators","title":"State Estimators","text":"ModelPredictiveControl.initpred!(::MovingHorizonEstimator, ::LinModel)\nModelPredictiveControl.linconstraint!(::MovingHorizonEstimator, ::LinModel)","category":"page"},{"location":"internals/state_estim/#ModelPredictiveControl.initpred!-Tuple{MovingHorizonEstimator, LinModel}","page":"State Estimators","title":"ModelPredictiveControl.initpred!","text":"initpred!(estim::MovingHorizonEstimator, model::LinModel) -> nothing\n\nInit quadratic optimization matrices F, fx̄, H̃, q̃, r for MovingHorizonEstimator.\n\nSee init_predmat_mhe for the definition of the vectors mathbfF f_x. It also inits estim.optim objective function, expressed as the quadratic general form:\n\n J = min_mathbfZ frac12mathbfZ H Z + mathbfq Z + r \n\nin which mathbfZ = beginsmallmatrix ϵ mathbfZ endsmallmatrix. Note that r is useless at optimization but required to evaluate the objective minima J. The Hessian mathbfH matrix of the quadratic general form is not constant here because of the time-varying mathbfP covariance . The computed variables are:\n\nbeginaligned\n mathbfF = mathbfG U_0 + mathbfJ D_0 + mathbfY_0^m + mathbfB \n mathbff_x = mathbfx_0^(k-N_k+1) \n mathbfF_Z = beginsmallmatrixmathbff_x mathbfF endsmallmatrix \n mathbfE_Z = beginsmallmatrixmathbfe_x mathbfE endsmallmatrix \n mathbfM_N_k = mathrmdiag(mathbfP^-1 mathbfR_N_k^-1) \n mathbfN_N_k = mathrmdiag(C mathbf0 mathbfQ_N_k^-1) \n mathbfH = 2(mathbfE_Z mathbfM_N_k mathbfE_Z + mathbfN_N_k) \n mathbfq = 2(mathbfM_N_k mathbfE_Z) mathbfF_Z \n r = mathbfF_Z mathbfM_N_k mathbfF_Z\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"internals/state_estim/#ModelPredictiveControl.linconstraint!-Tuple{MovingHorizonEstimator, LinModel}","page":"State Estimators","title":"ModelPredictiveControl.linconstraint!","text":"linconstraint!(estim::MovingHorizonEstimator, model::LinModel)\n\nSet b vector for the linear model inequality constraints (mathbfA Z b) of MHE.\n\nAlso init mathbfF_x = G_x U_0 + J_x D_0 + B_x vector for the state constraints, see init_predmat_mhe.\n\n\n\n\n\n","category":"method"},{"location":"internals/state_estim/#Solve-Optimization-Problem","page":"State Estimators","title":"Solve Optimization Problem","text":"","category":"section"},{"location":"internals/state_estim/","page":"State Estimators","title":"State Estimators","text":"ModelPredictiveControl.optim_objective!(::MovingHorizonEstimator)","category":"page"},{"location":"internals/state_estim/#ModelPredictiveControl.optim_objective!-Tuple{MovingHorizonEstimator}","page":"State Estimators","title":"ModelPredictiveControl.optim_objective!","text":"optim_objective!(estim::MovingHorizonEstimator) -> Z̃\n\nOptimize objective of estim MovingHorizonEstimator and return the solution Z̃.\n\nIf supported by estim.optim, it warm-starts the solver at:\n\nmathbfZ = \nbeginbmatrix\n ϵ_k-1 \n mathbfx_k-1(k-N_k+p) \n mathbfw_k-1(k-N_k+p+0) \n mathbfw_k-1(k-N_k+p+1) \n vdots \n mathbfw_k-1(k-p-2) \n mathbf0 \nendbmatrix\n\nwhere mathbfw_k-1(k-j) is the input increment for time k-j computed at the last time step k-1. It then calls JuMP.optimize!(estim.optim) and extract the solution. A failed optimization prints an @error log in the REPL and returns the warm-start value.\n\n\n\n\n\n","category":"method"},{"location":"internals/state_estim/#Remove-Operating-Points","page":"State Estimators","title":"Remove Operating Points","text":"","category":"section"},{"location":"internals/state_estim/","page":"State Estimators","title":"State Estimators","text":"ModelPredictiveControl.remove_op!","category":"page"},{"location":"internals/state_estim/#ModelPredictiveControl.remove_op!","page":"State Estimators","title":"ModelPredictiveControl.remove_op!","text":"remove_op!(estim::StateEstimator, ym, d, u=nothing) -> y0m, d0, u0\n\nRemove operating pts on measured outputs ym, disturbances d and inputs u (if provided).\n\nIf u is provided, also store current inputs without operating points u0 in estim.lastu0. This field is used for PredictiveController computations.\n\n\n\n\n\n","category":"function"},{"location":"internals/state_estim/#Init-Estimate","page":"State Estimators","title":"Init Estimate","text":"","category":"section"},{"location":"internals/state_estim/","page":"State Estimators","title":"State Estimators","text":"ModelPredictiveControl.init_estimate!","category":"page"},{"location":"internals/state_estim/#ModelPredictiveControl.init_estimate!","page":"State Estimators","title":"ModelPredictiveControl.init_estimate!","text":"init_estimate!(estim::StateEstimator, model::LinModel, y0m, d0, u0)\n\nInit estim.x̂0 estimate with the steady-state solution if model is a LinModel.\n\nUsing u0, y0m and d0 arguments (deviation values, see setop!), the steadystate problem combined to the equality constraint mathbfy_0^m = mathbfy_0^m engenders the following system to solve:\n\nbeginbmatrix\n mathbfI - mathbfA \n mathbfC^m\nendbmatrix mathbfx_0 =\nbeginbmatrix\n mathbfB_u u_0 + B_d d_0 + f_op - x_op \n mathbfy_0^m - D_d^m d_0\nendbmatrix\n\nin which mathbfC^m D_d^m are the rows of estim.Ĉ, estim.D̂d that correspond to measured outputs mathbfy^m.\n\n\n\n\n\ninit_estimate!(estim::StateEstimator, model::SimModel, _ , _ , _ )\n\nLeft estim.x̂0 estimate unchanged if model is not a LinModel.\n\n\n\n\n\ninit_estimate!(estim::InternalModel, model::LinModel, y0m, d0, u0)\n\nInit estim.x̂0/x̂d/x̂s estimate at steady-state for InternalModel.\n\nThe deterministic estimates estim.x̂d start at steady-state using u0 and d0 arguments:\n\n mathbfx_d = mathbf(I - A)^-1 (B_u u_0 + B_d d_0 + f_op - x_op)\n\nBased on y0m argument and current stochastic outputs estimation mathbfy_s, composed of the measured mathbfy_s^m = mathbfy_0^m - mathbfy_d0^m and unmeasured mathbfy_s^u = 0 outputs, the stochastic estimates also start at steady-state:\n\n mathbfx_s = mathbf(I - A_s)^-1 B_s y_s\n\nThis estimator does not augment the state vector, thus mathbfx = x_d. See init_internalmodel for details.\n\n\n\n\n\n","category":"function"},{"location":"internals/state_estim/#Correct-Estimate","page":"State Estimators","title":"Correct Estimate","text":"","category":"section"},{"location":"internals/state_estim/","page":"State Estimators","title":"State Estimators","text":"ModelPredictiveControl.correct_estimate!","category":"page"},{"location":"internals/state_estim/#ModelPredictiveControl.correct_estimate!","page":"State Estimators","title":"ModelPredictiveControl.correct_estimate!","text":"correct_estimate!(estim::SteadyKalmanFilter, y0m, d0)\n\nCorrect estim.x̂0 with measured outputs y0m and disturbances d0 for current time step.\n\nIt computes the corrected state estimate mathbfx_k(k). See the docstring of update_estimate!(::SteadyKalmanFilter, ::Any, ::Any) for the equations.\n\n\n\n\n\ncorrect_estimate!(estim::KalmanFilter, y0m, d0)\n\nCorrect estim.x̂0 and estim.P̂ using the time-varying KalmanFilter.\n\nIt computes the corrected state estimate mathbfx_k(k) estimation covariance mathbfP_k(k).\n\n\n\n\n\ncorrect_estimate!(estim::UnscentedKalmanFilter, y0m, d0)\n\nDo the same but for the UnscentedKalmanFilter.\n\n\n\n\n\ncorrect_estimate!(estim::ExtendedKalmanFilter, y0m, d0)\n\nDo the same but for the ExtendedKalmanFilter.\n\n\n\n\n\ncorrect_estimate!(estim::Luenberger, y0m, d0, _ )\n\nIdentical to correct_estimate!(::SteadyKalmanFilter) but using Luenberger.\n\n\n\n\n\ncorrect_estimate!(estim::MovingHorizonEstimator, y0m, d0)\n\nDo the same but for MovingHorizonEstimator objects.\n\n\n\n\n\ncorrect_estimate!(estim::InternalModel, y0m, d0)\n\nCompute the current stochastic output estimation ŷs for InternalModel.\n\n\n\n\n\n","category":"function"},{"location":"internals/state_estim/#Update-Estimate","page":"State Estimators","title":"Update Estimate","text":"","category":"section"},{"location":"internals/state_estim/","page":"State Estimators","title":"State Estimators","text":"info: Info\nAll these methods assume that the u0, y0m and d0 arguments are deviation vectors from their respective operating points (see setop!). The associated equations in the documentation drops the mathbf0 in subscript to simplify the notation. Strictly speaking, the manipulated inputs, measured outputs, measured disturbances and estimated states should be denoted with mathbfu_0 y_0^m d_0 and mathbfx_0, respectively.","category":"page"},{"location":"internals/state_estim/","page":"State Estimators","title":"State Estimators","text":"ModelPredictiveControl.update_estimate!","category":"page"},{"location":"internals/state_estim/#ModelPredictiveControl.update_estimate!","page":"State Estimators","title":"ModelPredictiveControl.update_estimate!","text":"update_estimate!(estim::SteadyKalmanFilter, y0m, d0, u0)\n\nUpdate estim.x̂0 estimate with current inputs u0, measured outputs y0m and dist. d0.\n\nIf estim.direct == false, the SteadyKalmanFilter first corrects the state estimate with the precomputed Kalman gain mathbfK. Afterward, it predicts the next state with the augmented process model. The correction step is skipped if direct == true since it is already done by the user through the preparestate! function (that calls correct_estimate!). The correction and prediction step equations are provided below.\n\nCorrection Step\n\nmathbfx_k(k) = mathbfx_k-1(k) + mathbfKmathbfy^m(k) - mathbfC^m x_k-1(k)\n - mathbfD_d^m d(k) \n\nPrediction Step\n\nmathbfx_k(k+1) = mathbfA x_k(k) + mathbfB_u u(k) + mathbfB_d d(k) \n\n\n\n\n\nupdate_estimate!(estim::KalmanFilter, y0m, d0, u0)\n\nUpdate KalmanFilter state estim.x̂0 and estimation error covariance estim.P̂.\n\nIt implements the classical time-varying Kalman Filter based on the process model described in SteadyKalmanFilter. If estim.direct == false, it first corrects the estimate before predicting the next state. The correction step is skipped if estim.direct == true since it's already done by the user. The correction and prediction step equations are provided below, see [2] for details.\n\nCorrection Step\n\nbeginaligned\n mathbfM(k) = mathbfC^m P_k-1(k)mathbfC^m + mathbfR \n mathbfK(k) = mathbfP_k-1(k)mathbfC^mmathbfM^-1(k) \n mathbfy^m(k) = mathbfC^m x_k-1(k) + mathbfD_d^m d(k) \n mathbfx_k(k) = mathbfx_k-1(k) + mathbfK(k)mathbfy^m(k) - mathbfy^m(k) \n mathbfP_k(k) = mathbfI - K(k)mathbfC^mmathbfP_k-1(k)\nendaligned\n\nPrediction Step\n\nbeginaligned\n mathbfx_k(k+1) = mathbfA x_k(k) + mathbfB_u u(k) + mathbfB_d d(k) \n mathbfP_k(k+1) = mathbfA P_k(k)mathbfA + mathbfQ\nendaligned\n\n[2]: \"Kalman Filter\", Wikipedia: The Free Encyclopedia, https://en.wikipedia.org/wiki/Kalman_filter, Accessed 2024-08-08.\n\n\n\n\n\nupdate_estimate!(estim::UnscentedKalmanFilter, y0m, d0, u0)\n\nUpdate UnscentedKalmanFilter state estim.x̂0 and covariance estimate estim.P̂.\n\nIt implements the unscented Kalman Filter based on the generalized unscented transform[3]. See init_ukf for the definition of the constants mathbfm S and γ. The superscript in e.g. mathbfX_k-1^j(k) refers the vector at the jth column of mathbfX_k-1(k). The symbol mathbf0 is a vector with zeros. The number of sigma points is n_σ = 2 n_mathbfx + 1. The matrices sqrtmathbfP_k-1(k) and sqrtmathbfP_k(k) are the the lower triangular factors of cholesky results. The correction and prediction step equations are provided below. The correction step is skipped if estim.direct == true since it's already done by the user.\n\nCorrection Step\n\nbeginaligned\n mathbfX_k-1(k) = biggbeginmatrix mathbfx_k-1(k) mathbfx_k-1(k) cdots mathbfx_k-1(k) endmatrixbigg + biggbeginmatrix mathbf0 γ sqrtmathbfP_k-1(k) -γ sqrtmathbfP_k-1(k) endmatrixbigg \n mathbfY^m(k) = biggbeginmatrix mathbfh^mBig( mathbfX_k-1^1(k) Big) mathbfh^mBig( mathbfX_k-1^2(k) Big) cdots mathbfh^mBig( mathbfX_k-1^n_σ(k) Big) endmatrixbigg \n mathbfy^m(k) = mathbfY^m(k) mathbfm \n mathbfX_k-1(k) = beginbmatrix mathbfX_k-1^1(k) - mathbfx_k-1(k) mathbfX_k-1^2(k) - mathbfx_k-1(k) cdots mathbfX_k-1^n_σ(k) - mathbfx_k-1(k) endbmatrix \n mathbfY^m(k) = beginbmatrix mathbfY^m^1(k) - mathbfy^m(k) mathbfY^m^2(k) - mathbfy^m(k) cdots mathbfY^m^n_σ(k) - mathbfy^m(k) endbmatrix \n mathbfM(k) = mathbfY^m(k) mathbfS mathbfY^m(k) + mathbfR \n mathbfK(k) = mathbfX_k-1(k) mathbfS mathbfY^m(k) mathbfM^-1(k) \n mathbfx_k(k) = mathbfx_k-1(k) + mathbfK(k) big mathbfy^m(k) - mathbfy^m(k) big \n mathbfP_k(k) = mathbfP_k-1(k) - mathbfK(k) mathbfM(k) mathbfK(k) \nendaligned \n\nPrediction Step\n\nbeginaligned\n mathbfX_k(k) = biggbeginmatrix mathbfx_k(k) mathbfx_k(k) cdots mathbfx_k(k) endmatrixbigg + biggbeginmatrix mathbf0 gamma sqrtmathbfP_k(k) - gamma sqrtmathbfP_k(k) endmatrixbigg \n mathbfX_k(k+1) = biggbeginmatrix mathbffBig( mathbfX_k^1(k) mathbfu(k) mathbfd(k) Big) mathbffBig( mathbfX_k^2(k) mathbfu(k) mathbfd(k) Big) cdots mathbffBig( mathbfX_k^n_σ(k) mathbfu(k) mathbfd(k) Big) endmatrixbigg \n mathbfx_k(k+1) = mathbfX_k(k+1)mathbfm \n mathbfX_k(k+1) = beginbmatrix mathbfX_k^1(k+1) - mathbfx_k(k+1) mathbfX_k^2(k+1) - mathbfx_k(k+1) cdots mathbfX_k^n_σ(k+1) - mathbfx_k(k+1) endbmatrix \n mathbfP_k(k+1) = mathbfX_k(k+1) mathbfS mathbfX_k(k+1) + mathbfQ\nendaligned\n\n[3]: Simon, D. 2006, \"Chapter 14: The unscented Kalman filter\" in \"Optimal State Estimation: Kalman, H∞, and Nonlinear Approaches\", John Wiley & Sons, p. 433–459, https://doi.org/10.1002/0470045345.ch14, ISBN9780470045343.\n\n\n\n\n\nupdate_estimate!(estim::ExtendedKalmanFilter, y0m, d0, u0)\n\nUpdate ExtendedKalmanFilter state estim.x̂0 and covariance estim.P̂.\n\nThe equations are similar to update_estimate!(::KalmanFilter) but with the substitutions mathbfC^m = H^m(k) and mathbfA = F(k), the Jacobians of the augmented process model:\n\nbeginaligned\n mathbfH(k) = left fracmathbfh(mathbfx mathbfd)mathbfx right_mathbfx = x_k-1(k) mathbfd = d(k) \n mathbfF(k) = left fracmathbff(mathbfx mathbfu mathbfd)mathbfx right_mathbfx = x_k(k) mathbfu = u(k) mathbfd = d(k)\nendaligned\n\nThe matrix mathbfH^m is the rows of mathbfH that are measured outputs. The function ForwardDiff.jacobian automatically computes them. The correction and prediction step equations are provided below. The correction step is skipped if estim.direct == true since it's already done by the user.\n\nCorrection Step\n\nbeginaligned\n mathbfS(k) = mathbfH^m(k)mathbfP_k-1(k)mathbfH^m(k) + mathbfR \n mathbfK(k) = mathbfP_k-1(k)mathbfH^m(k)mathbfS^-1(k) \n mathbfy^m(k) = mathbfh^mBig( mathbfx_k-1(k) mathbfd(k) Big) \n mathbfx_k(k) = mathbfx_k-1(k) + mathbfK(k)mathbfy^m(k) - mathbfy^m(k) \n mathbfP_k(k) = mathbfI - K(k)mathbfH^m(k)mathbfP_k-1(k)\nendaligned\n\nPrediction Step\n\nbeginaligned\n mathbfx_k(k+1) = mathbffBig( mathbfx_k(k) mathbfu(k) mathbfd(k) Big) \n mathbfP_k(k+1) = mathbfF(k)mathbfP_k(k)mathbfF(k) + mathbfQ\nendaligned\n\n\n\n\n\nupdate_estimate!(estim::Luenberger, y0m, d0, u0)\n\nSame than update_estimate!(::SteadyKalmanFilter) but using Luenberger.\n\n\n\n\n\nupdate_estimate!(estim::MovingHorizonEstimator, y0m, d0, u0)\n\nUpdate MovingHorizonEstimator state estim.x̂0.\n\nThe optimization problem of MovingHorizonEstimator documentation is solved at each discrete time k. The prediction matrices are provided at init_predmat_mhe documentation. Once solved, the optimal estimate mathbfx_k(k+p) is computed by inserting the optimal values of mathbfx_k(k-N_k+p) and mathbfW in the augmented model from j = N_k-1 to 0 inclusively. Afterward, if N_k = H_e, the arrival covariance for the next time step mathbfP_k-N_k(k-N_k+1) is estimated using estim.covestim object.\n\n\n\n\n\nupdate_estimate!(estim::InternalModel, _ , d0, u0)\n\nUpdate estim.x̂0/x̂d/x̂s with current inputs u0, measured outputs y0m and dist. d0.\n\nThe InternalModel updates the deterministic x̂d and stochastic x̂s estimates with:\n\nbeginaligned\n mathbfx_d(k+1) = mathbffBig( mathbfx_d(k) mathbfu(k) mathbfd(k) Big) \n mathbfx_s(k+1) = mathbfA_s x_s(k) + mathbfB_s y_s(k)\nendaligned\n\nThis estimator does not augment the state vector, thus mathbfx = x_d. See init_internalmodel for details. \n\n\n\n\n\n","category":"function"},{"location":"manual/nonlinmpc/#man_nonlin","page":"Nonlinear Design","title":"Manual: Nonlinear Design","text":"","category":"section"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"Pages = [\"nonlinmpc.md\"]","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"using Logging; errlogger = ConsoleLogger(stderr, Error);\nold_logger = global_logger(); global_logger(errlogger);","category":"page"},{"location":"manual/nonlinmpc/#Nonlinear-Model","page":"Nonlinear Design","title":"Nonlinear Model","text":"","category":"section"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"In this example, the goal is to control the angular position θ of a pendulum attached to a motor. Knowing that the manipulated input is the motor torque τ in Nm, the I/O vectors are:","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"beginaligned\n mathbfu = τ \n mathbfy = θ\nendaligned","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"The following figure presents the system:","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"

\"pendulum\"

","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"The plant model is nonlinear:","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"beginaligned\n dotθ(t) = ω(t) \n dotω(t) = -fracgLsinbig( θ(t) big) - fracKm ω(t) + frac1m L^2 τ(t)\nendaligned","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"in which g is the gravitational acceleration in m/s², L, the pendulum length in m, K, the friction coefficient at the pivot point in kg/s, and m, the mass attached at the end of the pendulum in kg, all bundled in the parameter vector mathbfp = beginsmallmatrix g L K m endsmallmatrix. The NonLinModel constructor assumes by default that the state function f is continuous in time, that is, an ordinary differential equation system (like here):","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"using ModelPredictiveControl\nfunction f(x, u, _ , p)\n g, L, K, m = p # [m/s²], [m], [kg/s], [kg]\n θ, ω = x[1], x[2] # [rad], [rad/s]\n τ = u[1] # [Nm]\n dθ = ω\n dω = -g/L*sin(θ) - K/m*ω + τ/m/L^2\n return [dθ, dω]\nend\nh(x, _ , _ ) = [180/π*x[1]] # [°]\np_model = [9.8, 0.4, 1.2, 0.3]\nnu, nx, ny, Ts = 1, 2, 1, 0.1\nvu, vx, vy = [\"\\$τ\\$ (Nm)\"], [\"\\$θ\\$ (rad)\", \"\\$ω\\$ (rad/s)\"], [\"\\$θ\\$ (°)\"]\nmodel = setname!(NonLinModel(f, h, Ts, nu, nx, ny; p=p_model); u=vu, x=vx, y=vy)","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"The output function mathbfh converts the θ angle to degrees. Note that special characters like θ can be typed in the Julia REPL or VS Code by typing \\theta and pressing the key. The parameter p can be of any type but use a mutable type like a vector of you want to modify it later. A 4th order RungeKutta method solves the differential equations by default. It is good practice to first simulate model using sim! as a quick sanity check:","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"using Plots\nu = [0.5]\nN = 35\nres = sim!(model, N, u)\nplot(res, plotu=false)\nsavefig(\"plot1_NonLinMPC.svg\"); nothing # hide","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"(Image: plot1_NonLinMPC)","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"The setname! function allows customized Y-axis labels. The available plotting options are detailed in the documentation of the corresponding plot method.","category":"page"},{"location":"manual/nonlinmpc/#Nonlinear-Model-Predictive-Controller","page":"Nonlinear Design","title":"Nonlinear Model Predictive Controller","text":"","category":"section"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"An UnscentedKalmanFilter estimates the plant state :","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"α=0.01; σQ=[0.1, 1.0]; σR=[5.0]; nint_u=[1]; σQint_u=[0.1]\nestim = UnscentedKalmanFilter(model; α, σQ, σR, nint_u, σQint_u)","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"The vectors σQ and σR are the standard deviations of the process and sensor noises, respectively. The value for the velocity ω is higher here (σQ second value) since dotω(t) equation includes an uncertain parameter: the friction coefficient K. Also, the argument nint_u explicitly adds one integrating state at the model input, the motor torque τ, with an associated standard deviation σQint_u of 0.1 N m. The estimator tuning is tested on a plant with a 25 % larger friction coefficient K:","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"p_plant = copy(p_model)\np_plant[3] = 1.25*p_model[3]\nplant = setname!(NonLinModel(f, h, Ts, nu, nx, ny; p=p_plant); u=vu, x=vx, y=vy)\nres = sim!(estim, N, [0.5], plant=plant, y_noise=[0.5])\nplot(res, plotu=false, plotxwithx̂=true)\nsavefig(\"plot2_NonLinMPC.svg\"); nothing # hide","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"(Image: plot2_NonLinMPC)","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"The estimate x_3 is the integrating state on the torque τ that compensates for static errors. The Kalman filter performance seems sufficient for control.","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"As the motor torque is limited to -1.5 to 1.5 N m, we incorporate the input constraints in a NonLinMPC:","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"Hp, Hc, Mwt, Nwt = 20, 2, [0.5], [2.5]\nnmpc = NonLinMPC(estim; Hp, Hc, Mwt, Nwt, Cwt=Inf)\numin, umax = [-1.5], [+1.5]\nnmpc = setconstraint!(nmpc; umin, umax)","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"The option Cwt=Inf disables the slack variable ϵ for constraint softening. We test mpc performance on plant by imposing an angular setpoint of 180° (inverted position):","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"using JuMP; unset_time_limit_sec(nmpc.optim) # hide\nres_ry = sim!(nmpc, N, [180.0], plant=plant, x_0=[0, 0], x̂_0=[0, 0, 0])\nplot(res_ry)\nsavefig(\"plot3_NonLinMPC.svg\"); nothing # hide","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"(Image: plot3_NonLinMPC)","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"The controller seems robust enough to variations on K coefficient. Starting from this inverted position, the closed-loop response to a step disturbances of 10° is also satisfactory:","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"res_yd = sim!(nmpc, N, [180.0], plant=plant, x_0=[π, 0], x̂_0=[π, 0, 0], y_step=[10])\nplot(res_yd)\nsavefig(\"plot4_NonLinMPC.svg\"); nothing # hide","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"(Image: plot4_NonLinMPC)","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"See sim! documentation for details on the possible simulation scenarios.","category":"page"},{"location":"manual/nonlinmpc/#Economic-Model-Predictive-Controller","page":"Nonlinear Design","title":"Economic Model Predictive Controller","text":"","category":"section"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"Economic MPC can achieve the same objective but with lower economical costs. For this case study, the controller will aim to reduce the energy consumed by the motor. The power in watt transmitted by the motor to the pendulum is:","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"P(t) = τ(t) ω(t)","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"Thus, the work in joule done by the motor from t = t_0 to t_end is:","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"W = int_t_0^t_end P(t) mathrmdt = int_t_0^t_end τ(t) ω(t) mathrmdt","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"With the sampling time T_s in s, the prediction horizon H_p, the limits defined as t_0 = k T_s and t_end = (k+H_p) T_s, and the left-endpoint rectangle method for the integral, we get:","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"W T_s sum_j=0^H_p-1 τ(k + j) ω(k + j)","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"The objective function will now include an additive term that penalizes the work done by the motor W to reduce the energy consumption. Notice that W is a function of the manipulated input τ and the angular speed ω, a state that is not measured (only the angle θ is measured here). As the arguments of NonLinMPC economic function JE do not include the states, the speed is now defined as an unmeasured output to design a Kalman Filter similar to the previous one (mathbfy^m = θ and mathbfy^u = ω):","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"h2(x, _ , _ ) = [180/π*x[1], x[2]]\nnu, nx, ny = 1, 2, 2\nmodel2 = setname!(NonLinModel(f, h2, Ts, nu, nx, ny; p=p_model), u=vu, x=vx, y=[vy; vx[2]])\nplant2 = setname!(NonLinModel(f, h2, Ts, nu, nx, ny; p=p_plant), u=vu, x=vx, y=[vy; vx[2]])\nestim2 = UnscentedKalmanFilter(model2; σQ, σR, nint_u, σQint_u, i_ym=[1])","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"The plant2 object based on h2 is also required since sim! expects that the output vector of plant argument corresponds to the model output vector in mpc argument. We can now define the J_E function and the empc controller:","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"function JE(UE, ŶE, _ , p)\n Ts = p\n τ, ω = UE[1:end-1], ŶE[2:2:end-1]\n return Ts*sum(τ.*ω)\nend\nempc = NonLinMPC(estim2; Hp, Hc, Nwt, Mwt=[0.5, 0], Cwt=Inf, Ewt=3.5e3, JE=JE, p=Ts)\nempc = setconstraint!(empc; umin, umax)","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"The keyword argument Ewt weights the economic costs relative to the other terms in the objective function. The term must be large enough to be significant but a too high value can lead to a static error on the angle setpoint. The second element of Mwt is zero since the speed ω is not requested to track a setpoint. The closed-loop response to a 180° setpoint is similar:","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"unset_time_limit_sec(empc.optim) # hide\nres2_ry = sim!(empc, N, [180, 0], plant=plant2, x_0=[0, 0], x̂_0=[0, 0, 0])\nplot(res2_ry, ploty=[1])\nsavefig(\"plot5_NonLinMPC.svg\"); nothing # hide","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"(Image: plot5_NonLinMPC)","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"and the energy consumption is slightly lower:","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"function calcW(res)\n τ, ω = res.U_data[1, 1:end-1], res.X_data[2, 1:end-1]\n return Ts*sum(τ.*ω)\nend\nDict(:W_nmpc => calcW(res_ry), :W_empc => calcW(res2_ry))","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"Also, for a 10° step disturbance:","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"res2_yd = sim!(empc, N, [180; 0]; plant=plant2, x_0=[π, 0], x̂_0=[π, 0, 0], y_step=[10, 0])\nplot(res2_yd, ploty=[1])\nsavefig(\"plot6_NonLinMPC.svg\"); nothing # hide","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"(Image: plot6_NonLinMPC)","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"the new controller is able to recuperate a little more energy from the pendulum (i.e. negative work):","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"Dict(:W_nmpc => calcW(res_yd), :W_empc => calcW(res2_yd))","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"Of course, this gain is only exploitable if the motor electronic includes some kind of regenerative circuitry.","category":"page"},{"location":"manual/nonlinmpc/#Model-Linearization","page":"Nonlinear Design","title":"Model Linearization","text":"","category":"section"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"Nonlinear MPC is more computationally expensive than LinMPC. Solving the problem should always be faster than the sampling time T_s = 01 s for real-time operation. This requirement is sometimes hard to meet on electronics or mechanical systems because of the fast dynamics. To ease the design and comparison with LinMPC, the linearize function allows automatic linearization of NonLinModel based on ForwardDiff.jl. We first linearize model at the point θ = π rad and ω = τ = 0 (inverted position):","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"linmodel = linearize(model, x=[π, 0], u=[0])","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"A SteadyKalmanFilter and a LinMPC are designed from linmodel:","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"\nskf = SteadyKalmanFilter(linmodel; σQ, σR, nint_u, σQint_u)\nmpc = LinMPC(skf; Hp, Hc, Mwt, Nwt, Cwt=Inf)\nmpc = setconstraint!(mpc, umin=[-1.5], umax=[+1.5])","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"The linear controller satisfactorily rejects the 10° step disturbance:","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"res_lin = sim!(mpc, N, [180.0]; plant, x_0=[π, 0], y_step=[10])\nplot(res_lin)\nsavefig(\"plot7_NonLinMPC.svg\"); nothing # hide","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"(Image: plot7_NonLinMPC)","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"Solving the optimization problem of mpc with DAQP optimizer instead of the default OSQP solver can improve the performance here. It is indeed documented that DAQP can perform better on small/medium dense matrices and unstable poles[1], which is obviously the case here (absolute value of unstable poles are greater than one):","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"[1]: Arnström, D., Bemporad, A., and Axehill, D. (2022). A dual active-set solver for embedded quadratic programming using recursive LDLᵀ updates. IEEE Trans. Autom. Contr., 67(8). https://doi.org/doi:10.1109/TAC.2022.3176430.","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"using LinearAlgebra; poles = eigvals(linmodel.A)","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"To install the solver, run:","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"using Pkg; Pkg.add(\"DAQP\")","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"Constructing a LinMPC with DAQP:","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"using JuMP, DAQP\ndaqp = Model(DAQP.Optimizer, add_bridges=false)\nmpc2 = LinMPC(skf; Hp, Hc, Mwt, Nwt, Cwt=Inf, optim=daqp)\nmpc2 = setconstraint!(mpc2; umin, umax)","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"does slightly improve the rejection of the step disturbance:","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"res_lin2 = sim!(mpc2, N, [180.0]; plant, x_0=[π, 0], y_step=[10])\nplot(res_lin2)\nsavefig(\"plot8_NonLinMPC.svg\"); nothing # hide","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"(Image: plot8_NonLinMPC)","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"The closed-loop performance is still lower than the nonlinear controller, as expected, but computations are about 210 times faster (0.000071 s versus 0.015 s per time steps, on average). However, remember that linmodel is only valid for angular positions near 180°. For example, the 180° setpoint response from 0° is unsatisfactory since the predictions are poor in the first quadrant:","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"res_lin3 = sim!(mpc2, N, [180.0]; plant, x_0=[0, 0])\nplot(res_lin3)\nsavefig(\"plot9_NonLinMPC.svg\"); nothing # hide","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"(Image: plot9_NonLinMPC)","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"Multiple linearized model and controller objects are required for large deviations from this operating point. This is known as gain scheduling. Another approach is adapting the model of the LinMPC instance based on repeated online linearization.","category":"page"},{"location":"manual/nonlinmpc/#Adapting-the-Model-via-Successive-Linearization","page":"Nonlinear Design","title":"Adapting the Model via Successive Linearization","text":"","category":"section"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"The setmodel! method allows online adaptation of a linear plant model. Combined with the automatic linearization of linearize, a successive linearization MPC can be designed with minimal efforts. The SteadyKalmanFilter does not support setmodel! so we need to use the time-varying KalmanFilter, and we initialize it with a linearization at θ = ω = τ = 0:","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"linmodel = linearize(model, x=[0, 0], u=[0])\nkf = KalmanFilter(linmodel; σQ, σR, nint_u, σQint_u)\nmpc3 = LinMPC(kf; Hp, Hc, Mwt, Nwt, Cwt=Inf, optim=daqp)\nmpc3 = setconstraint!(mpc3; umin, umax)","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"We create a function that simulates the plant and the adaptive controller:","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"function sim_adapt!(mpc, nonlinmodel, N, ry, plant, x_0, x̂_0, y_step=[0])\n U_data, Y_data, Ry_data = zeros(plant.nu, N), zeros(plant.ny, N), zeros(plant.ny, N)\n setstate!(plant, x_0)\n initstate!(mpc, [0], plant())\n setstate!(mpc, x̂_0)\n for i = 1:N\n y = plant() + y_step\n x̂ = preparestate!(mpc, y)\n u = moveinput!(mpc, ry)\n linmodel = linearize(nonlinmodel; u, x=x̂[1:2])\n setmodel!(mpc, linmodel)\n U_data[:,i], Y_data[:,i], Ry_data[:,i] = u, y, ry\n updatestate!(mpc, u, y) # update mpc state estimate\n updatestate!(plant, u) # update plant simulator\n end\n res = SimResult(mpc, U_data, Y_data; Ry_data)\n return res\nend\nnothing # hide","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"The setmodel! method must be called after solving the optimization problem with moveinput! and before updating the state estimate with updatestate!, that is, when both mathbfu(k) and mathbfx_k(k) are available as the new operating point. The SimResult object is for plotting purposes only. The adaptive LinMPC performances are similar to the nonlinear MPC, both for the 180° setpoint:","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"x_0 = [0, 0]; x̂_0 = [0, 0, 0]; ry = [180]\nres_slin = sim_adapt!(mpc3, model, N, ry, plant, x_0, x̂_0)\nplot(res_slin)\nsavefig(\"plot10_NonLinMPC.svg\"); nothing # hide","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"(Image: plot10_NonLinMPC)","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"and the 10° step disturbance:","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"x_0 = [π, 0]; x̂_0 = [π, 0, 0]; y_step = [10]\nres_slin = sim_adapt!(mpc3, model, N, ry, plant, x_0, x̂_0, y_step)\nplot(res_slin)\nsavefig(\"plot11_NonLinMPC.svg\"); nothing # hide","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"(Image: plot11_NonLinMPC)","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"The computations of the successive linearization MPC are about 125 times faster than the nonlinear MPC (0.00012 s per time steps versus 0.015 s per time steps, on average), an impressive gain for similar closed-loop performances!","category":"page"},{"location":"manual/nonlinmpc/","page":"Nonlinear Design","title":"Nonlinear Design","text":"global_logger(old_logger);","category":"page"},{"location":"public/state_estim/#Functions:-State-Estimators","page":"State Estimators","title":"Functions: State Estimators","text":"","category":"section"},{"location":"public/state_estim/","page":"State Estimators","title":"State Estimators","text":"Pages = [\"state_estim.md\"]","category":"page"},{"location":"public/state_estim/","page":"State Estimators","title":"State Estimators","text":"This module includes many state estimators (or state observer), both for deterministic and stochastic systems. The implementations focus on control applications, that is, relying on the estimates to compute a full state feedback (predictive controllers, in this package). They all incorporates some kind of integral action by default, since it is generally desired to eliminate the steady-state error with closed-loop control (offset-free tracking).","category":"page"},{"location":"public/state_estim/","page":"State Estimators","title":"State Estimators","text":"warning: Warning\nIf you plan to use the estimators for other contexts than this specific package (e.g. : filter, parameter estimation, etc.), careful must be taken at construction since the integral action is not necessarily desired. The options nint_u=0 and nint_ym=0 disable it.","category":"page"},{"location":"public/state_estim/","page":"State Estimators","title":"State Estimators","text":"The estimators are all implemented in the current form (a.k.a. as filter form) by default to improve accuracy and robustness, that is, they all estimates at each discrete time k the states of the current period mathbfx_k(k)[1] (using the newest measurements, see Manual for examples). The predictor form (a.k.a. delayed form) is also available with the option direct=false. This allow moving the estimator computations after solving the MPC problem with moveinput!, for when the estimations are expensive (for instance, with the MovingHorizonEstimator).","category":"page"},{"location":"public/state_estim/","page":"State Estimators","title":"State Estimators","text":"[1]: also denoted mathbfx_kk elsewhere.","category":"page"},{"location":"public/state_estim/","page":"State Estimators","title":"State Estimators","text":"info: Info\nAll the estimators support measured mathbfy^m and unmeasured mathbfy^u model outputs, where mathbfy refers to all of them.","category":"page"},{"location":"public/state_estim/#StateEstimator","page":"State Estimators","title":"StateEstimator","text":"","category":"section"},{"location":"public/state_estim/","page":"State Estimators","title":"State Estimators","text":"StateEstimator","category":"page"},{"location":"public/state_estim/#ModelPredictiveControl.StateEstimator","page":"State Estimators","title":"ModelPredictiveControl.StateEstimator","text":"Abstract supertype of all state estimators.\n\n\n\n(estim::StateEstimator)(d=[]) -> ŷ\n\nFunctor allowing callable StateEstimator object as an alias for evaloutput.\n\nExamples\n\njulia> kf = KalmanFilter(setop!(LinModel(tf(3, [10, 1]), 2), yop=[20]), direct=false);\n\njulia> ŷ = kf() \n1-element Vector{Float64}:\n 20.0\n\n\n\n\n\n","category":"type"},{"location":"public/state_estim/#SteadyKalmanFilter","page":"State Estimators","title":"SteadyKalmanFilter","text":"","category":"section"},{"location":"public/state_estim/","page":"State Estimators","title":"State Estimators","text":"SteadyKalmanFilter","category":"page"},{"location":"public/state_estim/#ModelPredictiveControl.SteadyKalmanFilter","page":"State Estimators","title":"ModelPredictiveControl.SteadyKalmanFilter","text":"SteadyKalmanFilter(model::LinModel; )\n\nConstruct a steady-state Kalman Filter with the LinModel model.\n\nThe steady-state (or asymptotic) Kalman filter is based on the process model :\n\nbeginaligned\n mathbfx(k+1) = \n mathbfA x(k) + mathbfB_u u(k) + mathbfB_d d(k) + mathbfw(k) \n mathbfy^m(k) = mathbfC^m x(k) + mathbfD_d^m d(k) + mathbfv(k) \n mathbfy^u(k) = mathbfC^u x(k) + mathbfD_d^u d(k)\nendaligned\n\nwith sensor mathbfv(k) and process mathbfw(k) noises as uncorrelated zero mean white noise vectors, with a respective covariance of mathbfR and mathbfQ. The arguments are in standard deviations σ, i.e. same units than outputs and states. The matrices mathbfA B_u B_d C D_d are model matrices augmented with the stochastic model, which is specified by the numbers of integrator nint_u and nint_ym (see Extended Help). Likewise, the covariance matrices are augmented with mathbfQ = textdiag(Q Q_int_u Q_int_ym) and mathbfR = R. The Extended Help provide some guidelines on the covariance tuning. The matrices mathbfC^m D_d^m are the rows of mathbfC D_d that correspond to measured outputs mathbfy^m (and unmeasured ones, for mathbfC^u D_d^u). The Kalman filter will estimate the current state with the newest measurements mathbfx_k(k) if direct is true, else it will predict the state of the next time step mathbfx_k(k+1).\n\nArguments\n\ninfo: Info\nKeyword arguments with emphasis are non-Unicode alternatives.\n\nmodel::LinModel : (deterministic) model for the estimations.\ni_ym=1:model.ny : model output indices that are measured mathbfy^m, the rest are unmeasured mathbfy^u.\nσQ=fill(1/model.nx,model.nx) or sigmaQ : main diagonal of the process noise covariance mathbfQ of model, specified as a standard deviation vector.\nσR=fill(1,length(i_ym)) or sigmaR : main diagonal of the sensor noise covariance mathbfR of model measured outputs, specified as a standard deviation vector.\nnint_u=0: integrator quantity for the stochastic model of the unmeasured disturbances at the manipulated inputs (vector), use nint_u=0 for no integrator (see Extended Help).\nnint_ym=default_nint(model,i_ym,nint_u) : same than nint_u but for the unmeasured disturbances at the measured outputs, use nint_ym=0 for no integrator (see Extended Help).\nσQint_u=fill(1,sum(nint_u)) or sigmaQint_u : same than σQ but for the unmeasured disturbances at manipulated inputs mathbfQ_int_u (composed of integrators).\nσQint_ym=fill(1,sum(nint_ym)) or sigmaQint_u : same than σQ for the unmeasured disturbances at measured outputs mathbfQ_int_ym (composed of integrators).\ndirect=true: construct with a direct transmission from mathbfy^m (a.k.a. current estimator, in opposition to the delayed/predictor form).\n\nExamples\n\njulia> model = LinModel([tf(3, [30, 1]); tf(-2, [5, 1])], 0.5);\n\njulia> estim = SteadyKalmanFilter(model, i_ym=[2], σR=[1], σQint_ym=[0.01])\nSteadyKalmanFilter estimator with a sample time Ts = 0.5 s, LinModel and:\n 1 manipulated inputs u (0 integrating states)\n 3 estimated states x̂\n 1 measured outputs ym (1 integrating states)\n 1 unmeasured outputs yu\n 0 measured disturbances d\n\nExtended Help\n\ndetails: Extended Help\nThe σR argument is generally fixed at the estimated standard deviations of the sensor noises. The σQ, σQint_u and σQint_ym arguments can be used to tune the filter response. Increasing them make the filter more responsive to disturbances but more sensitive to measurement noise.The model augmentation with nint_u vector adds integrators at model manipulated inputs, and nint_ym, at measured outputs. They create the integral action when the estimator is used in a controller as state feedback. By default, the method default_nint adds one integrator per measured output if feasible. The argument nint_ym can also be tweaked by following these rules on each measured output:Use 0 integrator if the model output is already integrating (else it will be unobservable)\nUse 1 integrator if the disturbances on the output are typically \"step-like\"\nUse 2 integrators if the disturbances on the output are typically \"ramp-like\" The function init_estimstoch builds the stochastic model for estimation.tip: Tip\nIncreasing σQint_u and σQint_ym values increases the integral action \"gain\".The constructor pre-compute the steady-state Kalman gain K̂ with the kalman function. It can sometimes fail, for example when model matrices are ill-conditioned. In such a case, you can try the alternative time-varying KalmanFilter.\n\n\n\n\n\nSteadyKalmanFilter(model, i_ym, nint_u, nint_ym, Q̂, R̂; direct=true)\n\nConstruct the estimator from the augmented covariance matrices Q̂ and R̂.\n\nThis syntax allows nonzero off-diagonal elements in mathbfQ R.\n\n\n\n\n\n","category":"type"},{"location":"public/state_estim/#KalmanFilter","page":"State Estimators","title":"KalmanFilter","text":"","category":"section"},{"location":"public/state_estim/","page":"State Estimators","title":"State Estimators","text":"KalmanFilter","category":"page"},{"location":"public/state_estim/#ModelPredictiveControl.KalmanFilter","page":"State Estimators","title":"ModelPredictiveControl.KalmanFilter","text":"KalmanFilter(model::LinModel; )\n\nConstruct a time-varying Kalman Filter with the LinModel model.\n\nThe process model is identical to SteadyKalmanFilter. The matrix mathbfP is the estimation error covariance of model states augmented with the stochastic ones (specified by nint_u and nint_ym). Three keyword arguments specify its initial value with mathbfP_-1(0) = mathrmdiag mathbfP(0) mathbfP_int_u(0) mathbfP_int_ym(0) . The initial state estimate mathbfx_-1(0) can be manually specified with setstate!, or automatically with initstate!.\n\nArguments\n\ninfo: Info\nKeyword arguments with emphasis are non-Unicode alternatives.\n\nmodel::LinModel : (deterministic) model for the estimations.\ni_ym=1:model.ny : model output indices that are measured mathbfy^m, the rest are unmeasured mathbfy^u.\nσP_0=fill(1/model.nx,model.nx) or sigmaP_0 : main diagonal of the initial estimate covariance mathbfP(0), specified as a standard deviation vector.\nσQ=fill(1/model.nx,model.nx) or sigmaQ : main diagonal of the process noise covariance mathbfQ of model, specified as a standard deviation vector.\nσR=fill(1,length(i_ym)) or sigmaR : main diagonal of the sensor noise covariance mathbfR of model measured outputs, specified as a standard deviation vector.\nnint_u=0: integrator quantity for the stochastic model of the unmeasured disturbances at the manipulated inputs (vector), use nint_u=0 for no integrator.\nnint_ym=default_nint(model,i_ym,nint_u) : same than nint_u but for the unmeasured disturbances at the measured outputs, use nint_ym=0 for no integrator.\nσQint_u=fill(1,sum(nint_u)) or sigmaQint_u : same than σQ but for the unmeasured disturbances at manipulated inputs mathbfQ_int_u (composed of integrators).\nσPint_u_0=fill(1,sum(nint_u)) or sigmaPint_u_0 : same than σP_0 but for the unmeasured disturbances at manipulated inputs mathbfP_int_u(0) (composed of integrators).\nσQint_ym=fill(1,sum(nint_ym)) or sigmaQint_u : same than σQ for the unmeasured disturbances at measured outputs mathbfQ_int_ym (composed of integrators).\nσPint_ym_0=fill(1,sum(nint_ym)) or sigmaPint_ym_0 : same than σP_0 but for the unmeasured disturbances at measured outputs mathbfP_int_ym(0) (composed of integrators).\ndirect=true: construct with a direct transmission from mathbfy^m (a.k.a. current estimator, in opposition to the delayed/predictor form).\n\nExamples\n\njulia> model = LinModel([tf(3, [30, 1]); tf(-2, [5, 1])], 0.5);\n\njulia> estim = KalmanFilter(model, i_ym=[2], σR=[1], σP_0=[100, 100], σQint_ym=[0.01])\nKalmanFilter estimator with a sample time Ts = 0.5 s, LinModel and:\n 1 manipulated inputs u (0 integrating states)\n 3 estimated states x̂\n 1 measured outputs ym (1 integrating states)\n 1 unmeasured outputs yu\n 0 measured disturbances d\n\n\n\n\n\nKalmanFilter(model, i_ym, nint_u, nint_ym, P̂_0, Q̂, R̂; direct=true)\n\nConstruct the estimator from the augmented covariance matrices P̂_0, Q̂ and R̂.\n\nThis syntax allows nonzero off-diagonal elements in mathbfP_-1(0) mathbfQ R.\n\n\n\n\n\n","category":"type"},{"location":"public/state_estim/#Luenberger","page":"State Estimators","title":"Luenberger","text":"","category":"section"},{"location":"public/state_estim/","page":"State Estimators","title":"State Estimators","text":"Luenberger","category":"page"},{"location":"public/state_estim/#ModelPredictiveControl.Luenberger","page":"State Estimators","title":"ModelPredictiveControl.Luenberger","text":"Luenberger(\n model::LinModel; \n i_ym = 1:model.ny, \n nint_u = 0,\n nint_ym = default_nint(model, i_ym),\n poles = 1e-3*(1:(model.nx + sum(nint_u) + sum(nint_ym))) .+ 0.5,\n direct = true\n)\n\nConstruct a Luenberger observer with the LinModel model.\n\ni_ym provides the model output indices that are measured mathbfy^m, the rest are unmeasured mathbfy^u. model matrices are augmented with the stochastic model, which is specified by the numbers of integrator nint_u and nint_ym (see SteadyKalmanFilter Extended Help). The argument poles is a vector of model.nx + sum(nint_u) + sum(nint_ym) elements specifying the observer poles/eigenvalues (near z=05 by default). The observer is constructed with a direct transmission from mathbfy^m if direct=true (a.k.a. current observers, in opposition to the delayed/prediction form). The method computes the observer gain K̂ with place.\n\nExamples\n\njulia> model = LinModel([tf(3, [30, 1]); tf(-2, [5, 1])], 0.5);\n\njulia> estim = Luenberger(model, nint_ym=[1, 1], poles=[0.61, 0.62, 0.63, 0.64])\nLuenberger estimator with a sample time Ts = 0.5 s, LinModel and:\n 1 manipulated inputs u (0 integrating states)\n 4 estimated states x̂\n 2 measured outputs ym (2 integrating states)\n 0 unmeasured outputs yu\n 0 measured disturbances d\n\n\n\n\n\n","category":"type"},{"location":"public/state_estim/#UnscentedKalmanFilter","page":"State Estimators","title":"UnscentedKalmanFilter","text":"","category":"section"},{"location":"public/state_estim/","page":"State Estimators","title":"State Estimators","text":"UnscentedKalmanFilter","category":"page"},{"location":"public/state_estim/#ModelPredictiveControl.UnscentedKalmanFilter","page":"State Estimators","title":"ModelPredictiveControl.UnscentedKalmanFilter","text":"UnscentedKalmanFilter(model::SimModel; )\n\nConstruct an unscented Kalman Filter with the SimModel model.\n\nBoth LinModel and NonLinModel are supported. The unscented Kalman filter is based on the process model :\n\nbeginaligned\n mathbfx(k+1) = mathbffBig(mathbfx(k) mathbfu(k) mathbfd(k)Big) \n + mathbfw(k) \n mathbfy^m(k) = mathbfh^mBig(mathbfx(k) mathbfd(k)Big) + mathbfv(k) \n mathbfy^u(k) = mathbfh^uBig(mathbfx(k) mathbfd(k)Big) \nendaligned\n\nSee SteadyKalmanFilter for details on mathbfv(k) mathbfw(k) noises and mathbfR mathbfQ covariances. The two matrices are constructed from mathbfQ = textdiag(Q Q_int_u Q_int_ym) and mathbfR = R. The functions mathbff h are model state-space functions augmented with the stochastic model of the unmeasured disturbances, which is specified by the numbers of integrator nint_u and nint_ym (see Extended Help). Model parameters mathbfp are not argument of mathbff h functions for conciseness. The mathbfh^m function represents the measured outputs of mathbfh function (and unmeasured ones, for mathbfh^u). The matrix mathbfP is the estimation error covariance of model state augmented with the stochastic ones. Three keyword arguments specify its initial value with mathbfP_-1(0) = mathrmdiag mathbfP(0) mathbfP_int_u(0) mathbfP_int_ym(0) . The initial state estimate mathbfx_-1(0) can be manually specified with setstate!.\n\nArguments\n\ninfo: Info\nKeyword arguments with emphasis are non-Unicode alternatives.\n\nmodel::SimModel : (deterministic) model for the estimations.\ni_ym=1:model.ny : model output indices that are measured mathbfy^m, the rest are unmeasured mathbfy^u.\nσP_0=fill(1/model.nx,model.nx) or sigmaP_0 : main diagonal of the initial estimate covariance mathbfP(0), specified as a standard deviation vector.\nσQ=fill(1/model.nx,model.nx) or sigmaQ : main diagonal of the process noise covariance mathbfQ of model, specified as a standard deviation vector.\nσR=fill(1,length(i_ym)) or sigmaR : main diagonal of the sensor noise covariance mathbfR of model measured outputs, specified as a standard deviation vector.\nnint_u=0: integrator quantity for the stochastic model of the unmeasured disturbances at the manipulated inputs (vector), use nint_u=0 for no integrator (see Extended Help).\nnint_ym=default_nint(model,i_ym,nint_u) : same than nint_u but for the unmeasured disturbances at the measured outputs, use nint_ym=0 for no integrator (see Extended Help).\nσQint_u=fill(1,sum(nint_u)) or sigmaQint_u : same than σQ but for the unmeasured disturbances at manipulated inputs mathbfQ_int_u (composed of integrators).\nσPint_u_0=fill(1,sum(nint_u)) or sigmaPint_u_0 : same than σP_0 but for the unmeasured disturbances at manipulated inputs mathbfP_int_u(0) (composed of integrators).\nσQint_ym=fill(1,sum(nint_ym)) or sigmaQint_u : same than σQ for the unmeasured disturbances at measured outputs mathbfQ_int_ym (composed of integrators).\nσPint_ym_0=fill(1,sum(nint_ym)) or sigmaPint_ym_0 : same than σP_0 but for the unmeasured disturbances at measured outputs mathbfP_int_ym(0) (composed of integrators).\ndirect=true: construct with a direct transmission from mathbfy^m (a.k.a. current estimator, in opposition to the delayed/predictor form).\nα=1e-3 or alpha : alpha parameter, spread of the state distribution (0 α 1).\nβ=2 or beta : beta parameter, skewness and kurtosis of the states distribution (β 0).\nκ=0 or kappa : kappa parameter, another spread parameter (0 κ 3).\n\nExamples\n\njulia> model = NonLinModel((x,u,_,_)->0.1x+u, (x,_,_)->2x, 10.0, 1, 1, 1, solver=nothing);\n\njulia> estim = UnscentedKalmanFilter(model, σR=[1], nint_ym=[2], σPint_ym_0=[1, 1])\nUnscentedKalmanFilter estimator with a sample time Ts = 10.0 s, NonLinModel and:\n 1 manipulated inputs u (0 integrating states)\n 3 estimated states x̂\n 1 measured outputs ym (2 integrating states)\n 0 unmeasured outputs yu\n 0 measured disturbances d\n\nExtended Help\n\ndetails: Extended Help\nThe Extended Help of SteadyKalmanFilter details the tuning of the covariances and the augmentation with nint_ym and nint_u arguments. The default augmentation scheme is identical, that is nint_u=0 and nint_ym computed by default_nint. Note that the constructor does not validate the observability of the resulting augmented NonLinModel. In such cases, it is the user's responsibility to ensure that it is still observable.\n\n\n\n\n\nUnscentedKalmanFilter(\n model, i_ym, nint_u, nint_ym, P̂_0, Q̂, R̂, α=1e-3, β=2, κ=0; direct=true\n)\n\nConstruct the estimator from the augmented covariance matrices P̂_0, Q̂ and R̂.\n\nThis syntax allows nonzero off-diagonal elements in mathbfP_-1(0) mathbfQ R.\n\n\n\n\n\n","category":"type"},{"location":"public/state_estim/#ExtendedKalmanFilter","page":"State Estimators","title":"ExtendedKalmanFilter","text":"","category":"section"},{"location":"public/state_estim/","page":"State Estimators","title":"State Estimators","text":"ExtendedKalmanFilter","category":"page"},{"location":"public/state_estim/#ModelPredictiveControl.ExtendedKalmanFilter","page":"State Estimators","title":"ModelPredictiveControl.ExtendedKalmanFilter","text":"ExtendedKalmanFilter(model::SimModel; )\n\nConstruct an extended Kalman Filter with the SimModel model.\n\nBoth LinModel and NonLinModel are supported. The process model and the keyword arguments are identical to UnscentedKalmanFilter, except for α, β and κ which do not apply to the extended Kalman Filter. The Jacobians of the augmented model mathbff h are computed with ForwardDiff.jl automatic differentiation.\n\nwarning: Warning\nSee the Extended Help of linearize function if you get an error like: MethodError: no method matching (::var\"##\")(::Vector{ForwardDiff.Dual}).\n\nExamples\n\njulia> model = NonLinModel((x,u,_,_)->0.2x+u, (x,_,_)->-3x, 5.0, 1, 1, 1, solver=nothing);\n\njulia> estim = ExtendedKalmanFilter(model, σQ=[2], σQint_ym=[2], σP_0=[0.1], σPint_ym_0=[0.1])\nExtendedKalmanFilter estimator with a sample time Ts = 5.0 s, NonLinModel and:\n 1 manipulated inputs u (0 integrating states)\n 2 estimated states x̂\n 1 measured outputs ym (1 integrating states)\n 0 unmeasured outputs yu\n 0 measured disturbances d\n\n\n\n\n\nExtendedKalmanFilter(model, i_ym, nint_u, nint_ym, P̂_0, Q̂, R̂; direct=true)\n\nConstruct the estimator from the augmented covariance matrices P̂_0, Q̂ and R̂.\n\nThis syntax allows nonzero off-diagonal elements in mathbfP_-1(0) mathbfQ R.\n\n\n\n\n\n","category":"type"},{"location":"public/state_estim/#MovingHorizonEstimator","page":"State Estimators","title":"MovingHorizonEstimator","text":"","category":"section"},{"location":"public/state_estim/","page":"State Estimators","title":"State Estimators","text":"MovingHorizonEstimator","category":"page"},{"location":"public/state_estim/#ModelPredictiveControl.MovingHorizonEstimator","page":"State Estimators","title":"ModelPredictiveControl.MovingHorizonEstimator","text":"MovingHorizonEstimator(model::SimModel; )\n\nConstruct a moving horizon estimator (MHE) based on model (LinModel or NonLinModel).\n\nIt can handle constraints on the estimates, see setconstraint!. Additionally, model is not linearized like the ExtendedKalmanFilter, and the probability distribution is not approximated like the UnscentedKalmanFilter. The computational costs are drastically higher, however, since it minimizes the following objective function at each discrete time k:\n\nmin_mathbfx_k(k-N_k+p) mathbfW ϵ mathbfx mathbfP^-1 mathbfx \n + mathbfW mathbfQ_N_k^-1 mathbfW \n + mathbfV mathbfR_N_k^-1 mathbfV\n + C ϵ^2\n\nin which the arrival costs are evaluated from the states estimated at time k-N_k:\n\nbeginaligned\n mathbfx = mathbfx_k-N_k(k-N_k+p) - mathbfx_k(k-N_k+p) \n mathbfP = mathbfP_k-N_k(k-N_k+p)\nendaligned\n\nand the covariances are repeated N_k times:\n\nbeginaligned\n mathbfQ_N_k = textdiagmathbf(QQQ) \n mathbfR_N_k = textdiagmathbf(RRR) \nendaligned\n\nThe estimation horizon H_e limits the window length:\n\nN_k = begincases\n k + 1 k H_e \n H_e k H_e endcases\n\nThe vectors mathbfW and mathbfV respectively encompass the estimated process noises mathbfw(k-j+p) from j=N_k to 1 and sensor noises mathbfv(k-j+1) from j=N_k to 1. The Extended Help defines the two vectors, the slack variable ϵ, and the estimation of the covariance at arrival mathbfP_k-N_k(k-N_k+p). If the keyword argument direct=true (default value), the constant p=0 in the equations above, and the MHE is in the current form. Else p=1, leading to the prediction form.\n\nSee UnscentedKalmanFilter for details on the augmented process model and mathbfR mathbfQ covariances.\n\nwarning: Warning\nSee the Extended Help if you get an error like: MethodError: no method matching (::var\"##\")(::Vector{ForwardDiff.Dual}).\n\nArguments\n\ninfo: Info\nKeyword arguments with emphasis are non-Unicode alternatives.\n\nmodel::SimModel : (deterministic) model for the estimations.\nHe=nothing : estimation horizon H_e, must be specified.\ni_ym=1:model.ny : model output indices that are measured mathbfy^m, the rest are unmeasured mathbfy^u.\nσP_0=fill(1/model.nx,model.nx) or sigmaP_0 : main diagonal of the initial estimate covariance mathbfP(0), specified as a standard deviation vector.\nσQ=fill(1/model.nx,model.nx) or sigmaQ : main diagonal of the process noise covariance mathbfQ of model, specified as a standard deviation vector.\nσR=fill(1,length(i_ym)) or sigmaR : main diagonal of the sensor noise covariance mathbfR of model measured outputs, specified as a standard deviation vector.\nnint_u=0: integrator quantity for the stochastic model of the unmeasured disturbances at the manipulated inputs (vector), use nint_u=0 for no integrator (see Extended Help).\nnint_ym=default_nint(model,i_ym,nint_u) : same than nint_u but for the unmeasured disturbances at the measured outputs, use nint_ym=0 for no integrator (see Extended Help).\nσQint_u=fill(1,sum(nint_u)) or sigmaQint_u : same than σQ but for the unmeasured disturbances at manipulated inputs mathbfQ_int_u (composed of integrators).\nσPint_u_0=fill(1,sum(nint_u)) or sigmaPint_u_0 : same than σP_0 but for the unmeasured disturbances at manipulated inputs mathbfP_int_u(0) (composed of integrators).\nσQint_ym=fill(1,sum(nint_ym)) or sigmaQint_u : same than σQ for the unmeasured disturbances at measured outputs mathbfQ_int_ym (composed of integrators).\nσPint_ym_0=fill(1,sum(nint_ym)) or sigmaPint_ym_0 : same than σP_0 but for the unmeasured disturbances at measured outputs mathbfP_int_ym(0) (composed of integrators).\nCwt=Inf : slack variable weight C, default to Inf meaning hard constraints only.\noptim=default_optim_mhe(model) : a JuMP.Model with a quadratic/nonlinear optimizer for solving (default to Ipopt, or OSQP if model is a LinModel).\ndirect=true: construct with a direct transmission from mathbfy^m (a.k.a. current estimator, in opposition to the delayed/predictor form).\n\nExamples\n\njulia> model = NonLinModel((x,u,_,_)->0.1x+u, (x,_,_)->2x, 10.0, 1, 1, 1, solver=nothing);\n\njulia> estim = MovingHorizonEstimator(model, He=5, σR=[1], σP_0=[0.01])\nMovingHorizonEstimator estimator with a sample time Ts = 10.0 s, Ipopt optimizer, NonLinModel and:\n 5 estimation steps He\n 0 slack variable ϵ (estimation constraints)\n 1 manipulated inputs u (0 integrating states)\n 2 estimated states x̂\n 1 measured outputs ym (1 integrating states)\n 0 unmeasured outputs yu\n 0 measured disturbances d\n\nExtended Help\n\ndetails: Extended Help\nThe estimated process and sensor noises are defined as:mathbfW = \nbeginbmatrix\n mathbfw(k-N_k+p+0) \n mathbfw(k-N_k+p+1) \n vdots \n mathbfw(k+p-1)\nendbmatrix quad\nmathbfV =\nbeginbmatrix\n mathbfv(k-N_k+1) \n mathbfv(k-N_k+2) \n vdots \n mathbfv(k)\nendbmatrixbased on the augmented model functions mathbff h^m:beginaligned\n mathbfv(k-j) = mathbfy^m(k-j) - mathbfh^mBig(mathbfx_k(k-j) mathbfd(k-j)Big) \n mathbfx_k(k-j+1) = mathbffBig(mathbfx_k(k-j) mathbfu(k-j) mathbfd(k-j)Big) + mathbfw(k-j)\nendalignedThe constant p equals to !direct. In other words, mathbfW and mathbfV are shifted by one time step if direct==true. The non-default prediction form with p=1 is particularly useful for the MHE since it moves its expensive computations after the MPC optimization. That is, preparestate! will solve the optimization by default, but it can be postponed to updatestate! with direct=false.The Extended Help of SteadyKalmanFilter details the tuning of the covariances and the augmentation with nint_ym and nint_u arguments. The default augmentation scheme is identical, that is nint_u=0 and nint_ym computed by default_nint. Note that the constructor does not validate the observability of the resulting augmented NonLinModel. In such cases, it is the user's responsibility to ensure that it is still observable.The estimation covariance at arrival mathbfP_k-N_k(k-N_k+p) gives an uncertainty on the state estimate at the beginning of the window k-N_k+p, that is, in the past. It is not the same as the current estimate covariance mathbfP_k(k), a value not computed by the MHE (contrarily to e.g. the KalmanFilter). Three keyword arguments specify its initial value with mathbfP_i = mathrmdiag mathbfP(0) mathbfP_int_u(0) mathbfP_int_ym(0) . The initial state estimate mathbfx_i can be manually specified with setstate!, or automatically with initstate! for LinModel. Note the MHE with p=0 is slightly inconsistent with all the other estimators here. It interprets the initial values as mathbfx_i = mathbfx_-1(-1) and mathbfP_i = mathbfP_-1(-1), an a posteriori estimate[2] from the last time step. The MHE with p=1 is consistent, interpreting them as mathbfx_i = mathbfx_-1(0) and mathbfP_i = mathbfP_-1(0).[2]: M. Hovd (2012), \"A Note On The Smoothing Formulation Of Moving Horizon Estimation\", Facta Universitatis, Vol. 11 №2.The optimization and the update of the arrival covariance depend on model:If model is a LinModel, the optimization is treated as a quadratic program with a time-varying Hessian, which is generally cheaper than nonlinear programming. By default, a KalmanFilter estimates the arrival covariance (customizable).\nElse, a nonlinear program with automatic differentiation (AD) solves the optimization. Optimizers generally benefit from exact derivatives like AD. However, the f and h functions must be compatible with this feature. See Automatic differentiation for common mistakes when writing these functions. An UnscentedKalmanFilter estimates the arrival covariance by default.The slack variable ϵ relaxes the constraints if enabled, see setconstraint!. It is disabled by default for the MHE (from Cwt=Inf) but it should be activated for problems with two or more types of bounds, to ensure feasibility (e.g. on the estimated state mathbfx and sensor noise mathbfv). Note that if Cwt≠Inf, the attribute nlp_scaling_max_gradient of Ipopt is set to 10/Cwt (if not already set), to scale the small values of ϵ. Use the second constructor to specify the arrival covariance estimation method.\n\n\n\n\n\nMovingHorizonEstimator(\n model, He, i_ym, nint_u, nint_ym, P̂_0, Q̂, R̂, Cwt=Inf;\n optim=default_optim_mhe(model), \n direct=true,\n covestim=default_covestim_mhe(model, i_ym, nint_u, nint_ym, P̂_0, Q̂, R̂; direct)\n)\n\nConstruct the estimator from the augmented covariance matrices P̂_0, Q̂ and R̂.\n\nThis syntax allows nonzero off-diagonal elements in mathbfP_i mathbfQ R, where mathbfP_i is the initial estimation covariance, provided by P̂_0 argument. The keyword argument covestim also allows specifying a custom StateEstimator object for the estimation of covariance at the arrival mathbfP_k-N_k(k-N_k+p). The supported types are KalmanFilter, UnscentedKalmanFilter and ExtendedKalmanFilter.\n\n\n\n\n\n","category":"type"},{"location":"public/state_estim/#InternalModel","page":"State Estimators","title":"InternalModel","text":"","category":"section"},{"location":"public/state_estim/","page":"State Estimators","title":"State Estimators","text":"InternalModel","category":"page"},{"location":"public/state_estim/#ModelPredictiveControl.InternalModel","page":"State Estimators","title":"ModelPredictiveControl.InternalModel","text":"InternalModel(model::SimModel; i_ym=1:model.ny, stoch_ym=ss(I,I,I,I,model.Ts))\n\nConstruct an internal model estimator based on model (LinModel or NonLinModel).\n\ni_ym provides the model output indices that are measured mathbfy^m, the rest are unmeasured mathbfy^u. model evaluates the deterministic predictions mathbfy_d, and stoch_ym, the stochastic predictions of the measured outputs mathbfy_s^m (the unmeasured ones being mathbfy_s^u=0). The predicted outputs sum both values : mathbfy = y_d + y_s. See the Extended Help for more details.\n\nwarning: Warning\nInternalModel estimator does not work if model is integrating or unstable. The constructor verifies these aspects for LinModel but not for NonLinModel. Uses any other state estimator in such cases.\n\nExamples\n\njulia> estim = InternalModel(LinModel([tf(3, [30, 1]); tf(-2, [5, 1])], 0.5), i_ym=[2])\nInternalModel estimator with a sample time Ts = 0.5 s, LinModel and:\n 1 manipulated inputs u\n 2 estimated states x̂\n 1 measured outputs ym\n 1 unmeasured outputs yu\n 0 measured disturbances d\n\nExtended Help\n\ndetails: Extended Help\nstoch_ym is a TransferFunction or StateSpace object that models disturbances on mathbfy^m. Its input is a hypothetical zero mean white noise vector. stoch_ym supposes 1 integrator per measured outputs by default, assuming that the current stochastic estimate mathbfy_s^m(k) = mathbfy^m(k) - mathbfy_d^m(k) is constant in the future. This is the dynamic matrix control (DMC) strategy, which is simple but sometimes too aggressive. Additional poles and zeros in stoch_ym can mitigate this. The following block diagram summarizes the internal model structure.(Image: block diagram of the internal model structure)\n\n\n\n\n\n","category":"type"},{"location":"public/state_estim/#Default-Model-Augmentation","page":"State Estimators","title":"Default Model Augmentation","text":"","category":"section"},{"location":"public/state_estim/","page":"State Estimators","title":"State Estimators","text":"default_nint","category":"page"},{"location":"public/state_estim/#ModelPredictiveControl.default_nint","page":"State Estimators","title":"ModelPredictiveControl.default_nint","text":"default_nint(model::LinModel, i_ym=1:model.ny, nint_u=0) -> nint_ym\n\nGet default integrator quantity per measured outputs nint_ym for LinModel.\n\nThe arguments i_ym and nint_u are the measured output indices and the integrator quantity on each manipulated input, respectively. By default, one integrator is added on each measured outputs. If mathbfA C matrices of the augmented model become unobservable, the integrator is removed. This approach works well for stable, integrating and unstable model (see Examples).\n\nExamples\n\njulia> model = LinModel(append(tf(3, [10, 1]), tf(2, [1, 0]), tf(4,[-5, 1])), 1.0);\n\njulia> nint_ym = default_nint(model)\n3-element Vector{Int64}:\n 1\n 0\n 1\n\n\n\n\n\ndefault_nint(model::SimModel, i_ym=1:model.ny, nint_u=0)\n\nOne integrator on each measured output by default if model is not a LinModel.\n\nTheres is no verification the augmented model remains observable. If the integrator quantity per manipulated input nint_u ≠ 0, the method returns zero integrator on each measured output.\n\n\n\n\n\n","category":"function"},{"location":"func_index/#Index","page":"Index","title":"Index","text":"","category":"section"},{"location":"func_index/","page":"Index","title":"Index","text":"","category":"page"},{"location":"manual/linmpc/#man_lin","page":"Linear Design","title":"Manual: Linear Design","text":"","category":"section"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"Pages = [\"linmpc.md\"]","category":"page"},{"location":"manual/linmpc/#Linear-Model","page":"Linear Design","title":"Linear Model","text":"","category":"section"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"The example considers a continuously stirred-tank reactor (CSTR) with a cold and hot water inlet as a plant. The water flows out of an opening at the bottom of the tank. The manipulated inputs are the cold u_c and hot u_h water flow rates, and the measured outputs are the liquid level y_L and temperature y_T:","category":"page"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"beginaligned\n mathbfu = beginbmatrix u_c u_h endbmatrix \n mathbfy = beginbmatrix y_L y_T endbmatrix\nendaligned","category":"page"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"The following figure depicts the instrumentation installed on the CSTR:","category":"page"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"

\"cstr\"

","category":"page"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"At the steady-state operating points:","category":"page"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"beginaligned\n mathbfu_op = beginbmatrix 20 20 endbmatrix \n mathbfy_op = beginbmatrix 50 30 endbmatrix \nendaligned","category":"page"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"the following linear model accurately describes the plant dynamics:","category":"page"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"mathbfG(s) = fracmathbfy(s)mathbfu(s) =\nbeginbmatrix\n frac19018s + 1 frac19018s + 1 3pt\n frac-0748s + 1 frac0748s + 1\nendbmatrix","category":"page"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"We first need to construct a LinModel objet with setop! to handle the operating points:","category":"page"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"using ModelPredictiveControl, ControlSystemsBase\nG = [ tf(1.90, [18, 1]) tf(1.90, [18, 1]);\n tf(-0.74,[8, 1]) tf(0.74, [8, 1]) ]\nTs = 2.0\nmodel = setop!(LinModel(G, Ts), uop=[20, 20], yop=[50, 30])","category":"page"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"The model object will be used for two purposes : to construct our controller, and as a plant simulator to test the design. Its sampling time is 2 s thus the control period will be 2 s as well.","category":"page"},{"location":"manual/linmpc/#Linear-Model-Predictive-Controller","page":"Linear Design","title":"Linear Model Predictive Controller","text":"","category":"section"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"A linear model predictive controller (MPC) will control both the water level y_L and temperature y_T in the tank. The tank level should also never fall below 48:","category":"page"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"y_L 48","category":"page"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"We design our LinMPC controllers by including the linear level constraint with setconstraint! (±Inf values should be used when there is no bound):","category":"page"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"mpc = LinMPC(model, Hp=10, Hc=2, Mwt=[1, 1], Nwt=[0.1, 0.1])\nmpc = setconstraint!(mpc, ymin=[48, -Inf])","category":"page"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"in which Hp and Hc keyword arguments are respectively the predictive and control horizons, and Mwt and Nwt, the output setpoint tracking and move suppression weights. By default, LinMPC controllers use OSQP to solve the problem, soft constraints on output predictions mathbfy to ensure feasibility, and a SteadyKalmanFilter to estimate the plant states[1]. An attentive reader will also notice that the Kalman filter estimates two additional states compared to the plant model. These are the integrating states for the unmeasured plant disturbances, and they are automatically added to the model outputs by default if observability is preserved (see SteadyKalmanFilter for details).","category":"page"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"[1]: As an alternative to state observer, we could have use an InternalModel structure with mpc = LinMPC(InternalModel(model), Hp=15, Hc=2, Mwt=[1, 1], Nwt=[0.1, 0.1]). It was tested on the example of this page and it gave similar results.","category":"page"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"Before closing the loop, the actual plant input mathbfu and measurement mathbfy should initialize the estimates mathbfx at the steady-state solution that leads to mathbfy(0) = mathbfy(0). This approach results in a bumpless transfer. The initstate! function finds this solution for LinModel. Since model simulates our plant here, its output will initialize the states. LinModel objects are callable for this purpose (an alias for evaloutput):","category":"page"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"u, y = model.uop, model() # or equivalently : y = evaloutput(model)\ninitstate!(mpc, u, y)\nnothing # hide","category":"page"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"We can then close the loop and test mpc performance on the simulator by imposing step changes on output setpoints mathbfr_y and on a load disturbance u_l:","category":"page"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"function test_mpc(mpc, model)\n N = 200\n ry, ul = [50, 30], 0\n u_data, y_data, ry_data = zeros(model.nu, N), zeros(model.ny, N), zeros(model.ny, N)\n for i = 1:N\n i == 51 && (ry = [50, 35])\n i == 101 && (ry = [54, 30])\n i == 151 && (ul = -20)\n y = model() # simulated measurements\n preparestate!(mpc, y) # prepare mpc state estimate for current iteration\n u = mpc(ry) # or equivalently : u = moveinput!(mpc, ry)\n u_data[:,i], y_data[:,i], ry_data[:,i] = u, y, ry\n updatestate!(mpc, u, y) # update mpc state estimate for next iteration\n updatestate!(model, u + [0; ul]) # update simulator with load disturbance\n end\n return u_data, y_data, ry_data\nend\nu_data, y_data, ry_data = test_mpc(mpc, model)\nt_data = Ts*(0:(size(y_data,2)-1))\nnothing # hide","category":"page"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"The LinMPC objects are also callable as an alternative syntax for moveinput!. It is worth mentioning that additional information like the optimal output predictions mathbfY can be retrieved by calling getinfo after solving the problem. Also, calling preparestate! on the mpc object prepares the estimates for the current control period, and updatestate! updates them for the next one (the same logic applies for model). This is why preparestate! is called before the controller, and updatestate!, after.","category":"page"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"Lastly, we plot the closed-loop test with the Plots package:","category":"page"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"using Plots\nfunction plot_data(t_data, u_data, y_data, ry_data)\n p1 = plot(t_data, y_data[1,:], label=\"meas.\", ylabel=\"level\")\n plot!(p1, t_data, ry_data[1,:], label=\"setpoint\", linestyle=:dash, linetype=:steppost)\n plot!(p1, t_data, fill(48,size(t_data)), label=\"min\", linestyle=:dot, linewidth=1.5)\n p2 = plot(t_data, y_data[2,:], label=\"meas.\", legend=:topleft, ylabel=\"temp.\")\n plot!(p2, t_data, ry_data[2,:],label=\"setpoint\", linestyle=:dash, linetype=:steppost)\n p3 = plot(t_data,u_data[1,:],label=\"cold\", linetype=:steppost, ylabel=\"flow rate\")\n plot!(p3, t_data,u_data[2,:],label=\"hot\", linetype=:steppost, xlabel=\"time (s)\")\n return plot(p1, p2, p3, layout=(3,1))\nend\nplot_data(t_data, u_data, y_data, ry_data)\nsavefig(\"plot1_LinMPC.svg\"); nothing # hide","category":"page"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"(Image: plot1_LinMPC)","category":"page"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"Compared to the default setting, adding the integrating states at the model inputs may improve the closed-loop performance. Load disturbances are indeed very common in many real-life control problems. Constructing a LinMPC with input integrators:","category":"page"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"mpc2 = LinMPC(model, Hp=10, Hc=2, Mwt=[1, 1], Nwt=[0.1, 0.1], nint_u=[1, 1])\nmpc2 = setconstraint!(mpc2, ymin=[48, -Inf])","category":"page"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"does accelerate the rejection of the load disturbance and almost eliminates the level constraint violation:","category":"page"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"setstate!(model, zeros(model.nx))\nu, y = model.uop, model()\ninitstate!(mpc2, u, y)\nu_data, y_data, ry_data = test_mpc(mpc2, model)\nplot_data(t_data, u_data, y_data, ry_data)\nsavefig(\"plot2_LinMPC.svg\"); nothing # hide","category":"page"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"(Image: plot2_LinMPC)","category":"page"},{"location":"manual/linmpc/#Moving-Horizon-Estimation","page":"Linear Design","title":"Moving Horizon Estimation","text":"","category":"section"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"The SteadyKalmanFilter is simple but it is not able to handle constraints at estimation. The MovingHorizonEstimator (MHE) can improve the accuracy of the state estimate mathbfx. It solves a quadratic optimization problem under a past time window H_e. Bounds on the estimated plant state mathbfx, estimated process noise mathbfw and estimated sensor noise mathbfv can be included in the problem. This can be useful to add physical knowledge on the plant and its disturbances, and it does not require the installation of new physical sensors (e.g. a strictly positive concentration). The closed-loop performance of any state feedback controller, like here, depends on the accuracy of the plant state estimate.","category":"page"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"For the CSTR, we will bound the innovation term mathbfy(k) - mathbfy(k) = mathbfv(k), and increase the hot water unmeasured disturbance covariance in mathbfQ_int_u to accelerate the estimation of the load disturbance:","category":"page"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"estim = MovingHorizonEstimator(model, He=10, nint_u=[1, 1], σQint_u = [1, 2])\nestim = setconstraint!(estim, v̂min=[-1, -0.5], v̂max=[+1, +0.5])\nmpc_mhe = LinMPC(estim, Hp=10, Hc=2, Mwt=[1, 1], Nwt=[0.1, 0.1])\nmpc_mhe = setconstraint!(mpc_mhe, ymin=[45, -Inf])","category":"page"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"The rejection is indeed improved:","category":"page"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"setstate!(model, zeros(model.nx))\nu, y, d = model.uop, model(), mpc_mhe.estim.model.dop\ninitstate!(mpc_mhe, u, y, d)\nu_data, y_data, ry_data = test_mpc(mpc_mhe, model)\nplot_data(t_data, u_data, y_data, ry_data)\nsavefig(\"plot3_LinMPC.svg\"); nothing # hide","category":"page"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"(Image: plot3_LinMPC)","category":"page"},{"location":"manual/linmpc/#Adding-Feedforward-Compensation","page":"Linear Design","title":"Adding Feedforward Compensation","text":"","category":"section"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"Suppose that the load disturbance u_l of the last section is in fact caused by a separate hot water pipe that discharges into the tank. Adding a new sensor to measure this flow rate allows us to incorporate feedforward compensation in the controller. The new plant model is:","category":"page"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"beginbmatrix\n y_L(s) y_T(s)\nendbmatrix = \nbeginbmatrix\n frac19018s + 1 frac19018s + 1 frac19018s + 1 3pt\n frac-0748s + 1 frac0748s + 1 frac0748s + 1\nendbmatrix\nbeginbmatrix\n u_c(s) u_h(s) u_l(s)\nendbmatrix","category":"page"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"We need to construct a new LinModel that includes the measured disturbance mathbfd = u_l and the operating point mathbfd_op = 20:","category":"page"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"model_d = setop!(LinModel([G G[1:2, 2]], Ts, i_d=[3]), uop=[20, 20], yop=[50, 30], dop=[20])","category":"page"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"A LinMPC controller is constructed on this model:","category":"page"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"mpc_d = LinMPC(model_d, Hp=10, Hc=2, Mwt=[1, 1], Nwt=[0.1, 0.1])\nmpc_d = setconstraint!(mpc_d, ymin=[48, -Inf])","category":"page"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"A new test function that feeds the measured disturbance mathbfd to the controller is also required:","category":"page"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"function test_mpc_d(mpc_d, model)\n N = 200\n ry, ul = [50, 30], 0\n dop = mpc_d.estim.model.dop\n u_data, y_data, ry_data = zeros(model.nu, N), zeros(model.ny, N), zeros(model.ny, N)\n for i = 1:N\n i == 51 && (ry = [50, 35])\n i == 101 && (ry = [54, 30])\n i == 151 && (ul = -20)\n d = ul .+ dop # simulated measured disturbance\n y = model() # simulated measurements\n preparestate!(mpc_d, y, d) # prepare estimate with the measured disturbance d\n u = mpc_d(ry, d) # also feed the measured disturbance d to the controller\n u_data[:,i], y_data[:,i], ry_data[:,i] = u, y, ry\n updatestate!(mpc_d, u, y, d) # update estimate with the measured disturbance d\n updatestate!(model, u + [0; ul]) # update simulator\n end\n return u_data, y_data, ry_data\nend\nnothing # hide","category":"page"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"The new feedforward compensation is able to almost perfectly reject the load disturbance:","category":"page"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"setstate!(model, zeros(model.nx))\nu, y, d = model.uop, model(), mpc_d.estim.model.dop\ninitstate!(mpc_d, u, y, d)\nu_data, y_data, ry_data = test_mpc_d(mpc_d, model)\nplot_data(t_data, u_data, y_data, ry_data)\nsavefig(\"plot4_LinMPC.svg\"); nothing # hide","category":"page"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"(Image: plot4_LinMPC)","category":"page"},{"location":"manual/linmpc/","page":"Linear Design","title":"Linear Design","text":"Note that measured disturbances are assumed constant in the future by default but custom mathbfD predictions are possible. The same applies for the setpoint predictions mathbfR_y.","category":"page"},{"location":"public/sim_model/#func_sim_model","page":"Plant Models","title":"Functions: Plant Models","text":"","category":"section"},{"location":"public/sim_model/","page":"Plant Models","title":"Plant Models","text":"Pages = [\"sim_model.md\"]","category":"page"},{"location":"public/sim_model/","page":"Plant Models","title":"Plant Models","text":"The SimModel types represents discrete state-space models that can be used to construct StateEstimators and PredictiveControllers, or as plant simulators by calling evaloutput and updatestate! methods on SimModel instances (to test estimator/controller designs). For time simulations, the states mathbfx are stored inside SimModel instances. Use setstate! method to manually modify them.","category":"page"},{"location":"public/sim_model/#SimModel","page":"Plant Models","title":"SimModel","text":"","category":"section"},{"location":"public/sim_model/","page":"Plant Models","title":"Plant Models","text":"SimModel","category":"page"},{"location":"public/sim_model/#ModelPredictiveControl.SimModel","page":"Plant Models","title":"ModelPredictiveControl.SimModel","text":"Abstract supertype of LinModel and NonLinModel types.\n\n\n\n(model::SimModel)(d=[]) -> y\n\nFunctor allowing callable SimModel object as an alias for evaloutput.\n\nExamples\n\njulia> model = NonLinModel((x,u,_,_)->-x + u, (x,_,_)->x .+ 20, 4, 1, 1, 1, solver=nothing);\n\njulia> y = model()\n1-element Vector{Float64}:\n 20.0\n\n\n\n\n\n","category":"type"},{"location":"public/sim_model/#LinModel","page":"Plant Models","title":"LinModel","text":"","category":"section"},{"location":"public/sim_model/","page":"Plant Models","title":"Plant Models","text":"LinModel","category":"page"},{"location":"public/sim_model/#ModelPredictiveControl.LinModel","page":"Plant Models","title":"ModelPredictiveControl.LinModel","text":"LinModel(sys::StateSpace[, Ts]; i_u=1:size(sys,2), i_d=Int[])\n\nConstruct a linear model from state-space model sys with sampling time Ts in second.\n\nThe system sys can be continuous or discrete-time (Ts can be omitted for the latter). For continuous dynamics, its state-space equations are (discrete case in Extended Help):\n\nbeginaligned\n mathbfx(t) = mathbfA x(t) + mathbfB z(t) \n mathbfy(t) = mathbfC x(t) + mathbfD z(t)\nendaligned\n\nwith the state mathbfx and output mathbfy vectors. The mathbfz vector comprises the manipulated inputs mathbfu and measured disturbances mathbfd, in any order. i_u provides the indices of mathbfz that are manipulated, and i_d, the measured disturbances. The constructor automatically discretizes continuous systems, resamples discrete ones if Ts ≠ sys.Ts, computes a new realization if not minimal, and separates the mathbfz terms in two parts (details in Extended Help). The rest of the documentation assumes discrete dynamics since all systems end up in this form.\n\nSee also ss\n\nExamples\n\njulia> model1 = LinModel(ss(-0.1, 1.0, 1.0, 0), 2.0) # continuous-time StateSpace\nLinModel with a sample time Ts = 2.0 s and:\n 1 manipulated inputs u\n 1 states x\n 1 outputs y\n 0 measured disturbances d\n\njulia> model2 = LinModel(ss(0.4, 0.2, 0.3, 0, 0.1)) # discrete-time StateSpace\nLinModel with a sample time Ts = 0.1 s and:\n 1 manipulated inputs u\n 1 states x\n 1 outputs y\n 0 measured disturbances d\n\nExtended Help\n\ndetails: Extended Help\nThe state-space equations are similar if sys is discrete-time:beginaligned\n mathbfx(k+1) = mathbfA x(k) + mathbfB z(k) \n mathbfy(k) = mathbfC x(k) + mathbfD z(k)\nendalignedContinuous dynamics are internally discretized using c2d and :zoh for manipulated inputs, and :tustin, for measured disturbances. Lastly, if sys is discrete and the provided argument Ts ≠ sys.Ts, the system is resampled by using the aforementioned discretization methods.Note that the constructor transforms the system to its minimal realization using minreal for controllability/observability. As a consequence, the final state-space representation may be different from the one provided in sys. It is also converted into a more practical form (mathbfD_u=0 because of the zero-order hold):beginaligned\n mathbfx(k+1) = mathbfA x(k) + mathbfB_u u(k) + mathbfB_d d(k) \n mathbfy(k) = mathbfC x(k) + mathbfD_d d(k)\nendalignedUse the syntax LinModel{NT}(A, Bu, C, Bd, Dd, Ts) to force a specific state-space representation.\n\n\n\n\n\nLinModel(sys::TransferFunction[, Ts]; i_u=1:size(sys,2), i_d=Int[])\n\nConvert to minimal realization state-space when sys is a transfer function.\n\nsys is equal to fracmathbfy(s)mathbfz(s) for continuous-time, and fracmathbfy(z)mathbfz(z), for discrete-time.\n\nSee also tf\n\nExamples\n\njulia> model = LinModel([tf(3, [30, 1]) tf(-2, [5, 1])], 0.5, i_d=[2])\nLinModel with a sample time Ts = 0.5 s and:\n 1 manipulated inputs u\n 2 states x\n 1 outputs y\n 1 measured disturbances d\n\n\n\n\n\nLinModel(sys::DelayLtiSystem, Ts; i_u=1:size(sys,2), i_d=Int[])\n\nDiscretize with zero-order hold when sys is a continuous system with delays.\n\nThe delays must be multiples of the sample time Ts.\n\n\n\n\n\nLinModel{NT}(A, Bu, C, Bd, Dd, Ts)\n\nConstruct the model from the discrete state-space matrices A, Bu, C, Bd, Dd directly.\n\nSee LinModel(::StateSpace) Extended Help for the meaning of the matrices. The arguments Bd and Dd can be the scalar 0 if there is no measured disturbance. This syntax do not modify the state-space representation provided in argument (minreal is not called). Care must be taken to ensure that the model is controllable and observable. The optional parameter NT explicitly set the number type of vectors (default to Float64).\n\n\n\n\n\nLinModel(model::NonLinModel; x=model.x0+model.xop, u=model.uop, d=model.dop)\n\nCall linearize(model; x, u, d) and return the resulting linear model.\n\n\n\n\n\n","category":"type"},{"location":"public/sim_model/#NonLinModel","page":"Plant Models","title":"NonLinModel","text":"","category":"section"},{"location":"public/sim_model/","page":"Plant Models","title":"Plant Models","text":"NonLinModel","category":"page"},{"location":"public/sim_model/#ModelPredictiveControl.NonLinModel","page":"Plant Models","title":"ModelPredictiveControl.NonLinModel","text":"NonLinModel{NT}(f::Function, h::Function, Ts, nu, nx, ny, nd=0; p=[], solver=RungeKutta(4))\nNonLinModel{NT}(f!::Function, h!::Function, Ts, nu, nx, ny, nd=0; p=[], solver=RungeKutta(4))\n\nConstruct a nonlinear model from state-space functions f/f! and h/h!.\n\nBoth continuous and discrete-time models are supported. The default arguments assume continuous dynamics. Use solver=nothing for the discrete case (see Extended Help). The functions are defined as:\n\nbeginaligned\n mathbfx(t) = mathbffBig( mathbfx(t) mathbfu(t) mathbfd(t) mathbfp Big) \n mathbfy(t) = mathbfhBig( mathbfx(t) mathbfd(t) mathbfp Big)\nendaligned\n\nwhere mathbfx, mathbfy, mathbfu, mathbfd and mathbfp are respectively the state, output, manipulated input, measured disturbance and parameter vectors. If the dynamics is a function of the time, simply add a measured disturbance defined as d(t) = t. The functions can be implemented in two possible ways:\n\nNon-mutating functions (out-of-place): define them as f(x, u, d, p) -> ẋ and h(x, d, p) -> y. This syntax is simple and intuitive but it allocates more memory.\nMutating functions (in-place): define them as f!(ẋ, x, u, d, p) -> nothing and h!(y, x, d, p) -> nothing. This syntax reduces the allocations and potentially the computational burden as well.\n\nTs is the sampling time in second. nu, nx, ny and nd are the respective number of manipulated inputs, states, outputs and measured disturbances. The keyword argument p is the parameters of the model passed to the two functions. It can be of any Julia object but use a mutable type if you want to change them later e.g.: a vector.\n\ntip: Tip\nReplace the d or p argument with _ in your functions if not needed (see Examples below).\n\nA 4th order RungeKutta solver discretizes the differential equations by default. The rest of the documentation assumes discrete dynamics since all models end up in this form. The optional parameter NT explicitly set the number type of vectors (default to Float64).\n\nwarning: Warning\nThe two functions must be in pure Julia to use the model in NonLinMPC, ExtendedKalmanFilter, MovingHorizonEstimator and linearize.\n\nSee also LinModel.\n\nExamples\n\njulia> f!(ẋ, x, u, _ , p) = (ẋ .= p*x .+ u; nothing);\n\njulia> h!(y, x, _ , _ ) = (y .= 0.1x; nothing);\n\njulia> model1 = NonLinModel(f!, h!, 5.0, 1, 1, 1, p=-0.2) # continuous dynamics\nNonLinModel with a sample time Ts = 5.0 s, RungeKutta solver and:\n 1 manipulated inputs u\n 1 states x\n 1 outputs y\n 0 measured disturbances d\n\njulia> f(x, u, _ , _ ) = 0.1x + u;\n\njulia> h(x, _ , _ ) = 2x;\n\njulia> model2 = NonLinModel(f, h, 2.0, 1, 1, 1, solver=nothing) # discrete dynamics\nNonLinModel with a sample time Ts = 2.0 s, empty solver and:\n 1 manipulated inputs u\n 1 states x\n 1 outputs y\n 0 measured disturbances d\n\nExtended Help\n\ndetails: Extended Help\nState-space functions are similar for discrete dynamics:beginaligned\n mathbfx(k+1) = mathbffBig( mathbfx(k) mathbfu(k) mathbfd(k) mathbfp Big) \n mathbfy(k) = mathbfhBig( mathbfx(k) mathbfd(k) mathbfp Big)\nendalignedwith two possible implementations as well:Non-mutating functions: define them as f(x, u, d, p) -> xnext and h(x, d, p) -> y.\nMutating functions: define them as f!(xnext, x, u, d, p) -> nothing and h!(y, x, d, p) -> nothing.\n\n\n\n\n\n","category":"type"},{"location":"public/sim_model/#Set-Variable-Names","page":"Plant Models","title":"Set Variable Names","text":"","category":"section"},{"location":"public/sim_model/","page":"Plant Models","title":"Plant Models","text":"setname!","category":"page"},{"location":"public/sim_model/#ModelPredictiveControl.setname!","page":"Plant Models","title":"ModelPredictiveControl.setname!","text":"setname!(model::SimModel; u=nothing, y=nothing, d=nothing, x=nothing) -> model\n\nSet the names of model inputs u, outputs y, disturbances d, and states x.\n\nThe keyword arguments u, y, d, and x must be vectors of strings. The strings are used in the plotting functions.\n\nExamples\n\njulia> model = setname!(LinModel(tf(3, [10, 1]), 2.0), u=[\"\\$A\\$ (%)\"], y=[\"\\$T\\$ (∘C)\"])\nLinModel with a sample time Ts = 2.0 s and:\n 1 manipulated inputs u\n 1 states x\n 1 outputs y\n 0 measured disturbances d\n\n\n\n\n\n","category":"function"},{"location":"public/sim_model/#Set-Operating-Points","page":"Plant Models","title":"Set Operating Points","text":"","category":"section"},{"location":"public/sim_model/","page":"Plant Models","title":"Plant Models","text":"setop!","category":"page"},{"location":"public/sim_model/#ModelPredictiveControl.setop!","page":"Plant Models","title":"ModelPredictiveControl.setop!","text":"setop!(model; uop=nothing, yop=nothing, dop=nothing, xop=nothing, fop=nothing) -> model\n\nSet the operating points of model (both LinModel and NonLinModel).\n\nIntroducing deviations vectors around manipulated input uop, model output yop, measured disturbance dop, and model state xop operating points (a.k.a. nominal values):\n\nbeginalign*\n mathbfu_0(k) = mathbfu(k) - mathbfu_op \n mathbfd_0(k) = mathbfd(k) - mathbfd_op \n mathbfy_0(k) = mathbfy(k) - mathbfy_op \n mathbfx_0(k) = mathbfx(k) - mathbfx_op \nendalign*\n\nThe state-space description of LinModel around the operating points is:\n\nbeginaligned\n mathbfx_0(k+1) = mathbfA x_0(k) + mathbfB_u u_0(k) + mathbfB_d d_0(k) \n + mathbff_op - mathbfx_op \n mathbfy_0(k) = mathbfC x_0(k) + mathbfD_d d_0(k) \nendaligned\n\nand, for NonLinModel:\n\nbeginaligned\n mathbfx_0(k+1) = mathbffBig(mathbfx_0(k) mathbfu_0(k) mathbfd_0(k) mathbfpBig) \n + mathbff_op - mathbfx_op \n mathbfy_0(k) = mathbfhBig(mathbfx_0(k) mathbfd_0(k) mathbfpBig)\nendaligned\n\nThe state xop and the additional fop operating points are frequently zero e.g.: when model comes from system identification. The two vectors are internally used by linearize for non-equilibrium points.\n\nExamples\n\njulia> model = setop!(LinModel(tf(3, [10, 1]), 2.0), uop=[50], yop=[20])\nLinModel with a sample time Ts = 2.0 s and:\n 1 manipulated inputs u\n 1 states x\n 1 outputs y\n 0 measured disturbances d\n\njulia> y = model()\n1-element Vector{Float64}:\n 20.0\n\n\n\n\n\n","category":"function"},{"location":"public/sim_model/#Linearize","page":"Plant Models","title":"Linearize","text":"","category":"section"},{"location":"public/sim_model/","page":"Plant Models","title":"Plant Models","text":"linearize\nlinearize!","category":"page"},{"location":"public/sim_model/#ModelPredictiveControl.linearize","page":"Plant Models","title":"ModelPredictiveControl.linearize","text":"linearize(model::SimModel; x=model.x0+model.xop, u=model.uop, d=model.dop) -> linmodel\n\nLinearize model at the operating points x, u, d and return the LinModel.\n\nThe arguments x, u and d are the linearization points for the state mathbfx, manipulated input mathbfu and measured disturbance mathbfd, respectively (not necessarily an equilibrium, details in Extended Help). The Jacobians of mathbff and mathbfh functions are automatically computed with ForwardDiff.jl.\n\nwarning: Warning\nSee Extended Help if you get an error like: MethodError: no method matching (::var\"##\")(::Vector{ForwardDiff.Dual}).\n\nExamples\n\njulia> model = NonLinModel((x,u,_,_)->x.^3 + u, (x,_,_)->x, 0.1, 1, 1, 1, solver=nothing);\n\njulia> linmodel = linearize(model, x=[10.0], u=[0.0]); \n\njulia> linmodel.A\n1×1 Matrix{Float64}:\n 300.0\n\nExtended Help\n\ndetails: Extended Help\nWith the nonlinear state-space model:beginaligned\n mathbfx(k+1) = mathbffBig(mathbfx(k) mathbfu(k) mathbfd(k) mathbfpBig) \n mathbfy(k) = mathbfhBig(mathbfx(k) mathbfd(k) mathbfpBig)\nendalignedits linearization at the operating point mathbfx_op u_op d_op is:beginaligned\n mathbfx_0(k+1) mathbfA x_0(k) + mathbfB_u u_0(k) + mathbfB_d d_0(k) \n + mathbff(x_op u_op d_op p) - mathbfx_op \n mathbfy_0(k) mathbfC x_0(k) + mathbfD_d d_0(k) \nendalignedbased on the deviation vectors mathbfx_0 u_0 d_0 y_0 introduced in setop! documentation, and the Jacobian matrices:beginaligned\n mathbfA = left fracmathbff(x u d p)mathbfx right_mathbfx=x_op u=u_op d=d_op \n mathbfB_u = left fracmathbff(x u d p)mathbfu right_mathbfx=x_op u=u_op d=d_op \n mathbfB_d = left fracmathbff(x u d p)mathbfd right_mathbfx=x_op u=u_op d=d_op \n mathbfC = left fracmathbfh(x d p)mathbfx right_mathbfx=x_op d=d_op \n mathbfD_d = left fracmathbfh(x d p)mathbfd right_mathbfx=x_op d=d_op\nendalignedFollowing setop! notation, we find:beginaligned\n mathbff_op = mathbff(x_op u_op d_op p) \n mathbfy_op = mathbfh(x_op d_op p)\nendalignedNotice that mathbff_op - x_op = 0 if the point is an equilibrium. The equations are similar if the nonlinear model has nonzero operating points.Automatic differentiation (AD) allows exact Jacobians. The NonLinModel f and h functions must be compatible with this feature though. See Automatic differentiation for common mistakes when writing these functions.\n\n\n\n\n\n","category":"function"},{"location":"public/sim_model/#ModelPredictiveControl.linearize!","page":"Plant Models","title":"ModelPredictiveControl.linearize!","text":"linearize!(linmodel::LinModel, model::SimModel; ) -> linmodel\n\nLinearize model and store the result in linmodel (in-place).\n\nThe keyword arguments are identical to linearize.\n\nExamples\n\njulia> model = NonLinModel((x,u,_,_)->x.^3 + u, (x,_,_)->x, 0.1, 1, 1, 1, solver=nothing);\n\njulia> linmodel = linearize(model, x=[10.0], u=[0.0]); linmodel.A\n1×1 Matrix{Float64}:\n 300.0\n\njulia> linearize!(linmodel, model, x=[20.0], u=[0.0]); linmodel.A\n1×1 Matrix{Float64}:\n 1200.0\n\n\n\n\n\n","category":"function"},{"location":"public/sim_model/#Differential-Equation-Solvers","page":"Plant Models","title":"Differential Equation Solvers","text":"","category":"section"},{"location":"public/sim_model/#DiffSolver","page":"Plant Models","title":"DiffSolver","text":"","category":"section"},{"location":"public/sim_model/","page":"Plant Models","title":"Plant Models","text":"ModelPredictiveControl.DiffSolver","category":"page"},{"location":"public/sim_model/#ModelPredictiveControl.DiffSolver","page":"Plant Models","title":"ModelPredictiveControl.DiffSolver","text":"Abstract supertype of all differential equation solvers.\n\n\n\n\n\n","category":"type"},{"location":"public/sim_model/#RungeKutta","page":"Plant Models","title":"RungeKutta","text":"","category":"section"},{"location":"public/sim_model/","page":"Plant Models","title":"Plant Models","text":"RungeKutta","category":"page"},{"location":"public/sim_model/#ModelPredictiveControl.RungeKutta","page":"Plant Models","title":"ModelPredictiveControl.RungeKutta","text":"RungeKutta(order=4; supersample=1)\n\nCreate a Runge-Kutta solver with optional super-sampling.\n\nOnly the 4th order Runge-Kutta is supported for now. The keyword argument supersample provides the number of internal steps (default to 1 step).\n\n\n\n\n\n","category":"type"},{"location":"public/predictive_control/#Functions:-Predictive-Controllers","page":"Predictive Controllers","title":"Functions: Predictive Controllers","text":"","category":"section"},{"location":"public/predictive_control/","page":"Predictive Controllers","title":"Predictive Controllers","text":"Pages = [\"predictive_control.md\"]","category":"page"},{"location":"public/predictive_control/","page":"Predictive Controllers","title":"Predictive Controllers","text":"All the predictive controllers in this module rely on a state estimator to compute the predictions. The default LinMPC estimator is a SteadyKalmanFilter, and NonLinMPC with nonlinear models, an UnscentedKalmanFilter. For simpler and more classical designs, an InternalModel structure is also available, that assumes by default that the current model mismatch estimation is constant in the future (same approach as dynamic matrix control, DMC).","category":"page"},{"location":"public/predictive_control/","page":"Predictive Controllers","title":"Predictive Controllers","text":"info: Info\nThe nomenclature uses capital letters for time series (and matrices) and hats for the predictions (and estimations, for state estimators).","category":"page"},{"location":"public/predictive_control/","page":"Predictive Controllers","title":"Predictive Controllers","text":"To be precise, at the kth control period, the vectors that encompass the future measured disturbances mathbfd, model outputs mathbfy and setpoints mathbfr_y over the prediction horizon H_p are defined as:","category":"page"},{"location":"public/predictive_control/","page":"Predictive Controllers","title":"Predictive Controllers","text":" mathbfD = beginbmatrix\n mathbfd(k+1) mathbfd(k+2) vdots mathbfd(k+H_p)\n endbmatrix quad\n mathbfY = beginbmatrix\n mathbfy(k+1) mathbfy(k+2) vdots mathbfy(k+H_p)\n endbmatrix quad textand quad\n mathbfR_y = beginbmatrix\n mathbfr_y(k+1) mathbfr_y(k+2) vdots mathbfr_y(k+H_p)\n endbmatrix","category":"page"},{"location":"public/predictive_control/","page":"Predictive Controllers","title":"Predictive Controllers","text":"The vectors for the manipulated input mathbfu are shifted by one time step:","category":"page"},{"location":"public/predictive_control/","page":"Predictive Controllers","title":"Predictive Controllers","text":" mathbfU = beginbmatrix\n mathbfu(k+0) mathbfu(k+1) vdots mathbfu(k+H_p-1)\n endbmatrix quad textand quad\n mathbfR_u = beginbmatrix\n mathbfr_u(k+0) mathbfr_u(k+1) vdots mathbfr_u(k+H_p-1)\n endbmatrix","category":"page"},{"location":"public/predictive_control/","page":"Predictive Controllers","title":"Predictive Controllers","text":"Defining the manipulated input increment as mathbfΔu(k+j) = mathbfu(k+j) - mathbfu(k+j-1), the control horizon H_c enforces that mathbfΔu(k+j) = mathbf0 for j H_c. For this reason, the vector that collects them is truncated up to k+H_c-1:","category":"page"},{"location":"public/predictive_control/","page":"Predictive Controllers","title":"Predictive Controllers","text":" mathbfΔU =\n beginbmatrix\n mathbfΔu(k+0) mathbfΔu(k+1) vdots mathbfΔu(k+H_c-1)\n endbmatrix","category":"page"},{"location":"public/predictive_control/#PredictiveController","page":"Predictive Controllers","title":"PredictiveController","text":"","category":"section"},{"location":"public/predictive_control/","page":"Predictive Controllers","title":"Predictive Controllers","text":"PredictiveController","category":"page"},{"location":"public/predictive_control/#ModelPredictiveControl.PredictiveController","page":"Predictive Controllers","title":"ModelPredictiveControl.PredictiveController","text":"Abstract supertype of all predictive controllers.\n\n\n\n(mpc::PredictiveController)(ry, d=[]; kwargs...) -> u\n\nFunctor allowing callable PredictiveController object as an alias for moveinput!.\n\nExamples\n\njulia> mpc = LinMPC(LinModel(tf(5, [2, 1]), 3), Nwt=[0], Hp=1000, Hc=1, direct=false);\n\njulia> u = mpc([5]); round.(u, digits=3)\n1-element Vector{Float64}:\n 1.0\n\n\n\n\n\n","category":"type"},{"location":"public/predictive_control/#LinMPC","page":"Predictive Controllers","title":"LinMPC","text":"","category":"section"},{"location":"public/predictive_control/","page":"Predictive Controllers","title":"Predictive Controllers","text":"LinMPC","category":"page"},{"location":"public/predictive_control/#ModelPredictiveControl.LinMPC","page":"Predictive Controllers","title":"ModelPredictiveControl.LinMPC","text":"LinMPC(model::LinModel; )\n\nConstruct a linear predictive controller based on LinModel model.\n\nThe controller minimizes the following objective function at each discrete time k:\n\nbeginaligned\nmin_mathbfΔU ϵ mathbf(R_y - Y) mathbfM_H_p mathbf(R_y - Y)\n + mathbf(ΔU) mathbfN_H_c mathbf(ΔU) \n + mathbf(R_u - U) mathbfL_H_p mathbf(R_u - U) \n + C ϵ^2\nendaligned\n\nin which the weight matrices are repeated H_p or H_c times by default:\n\nbeginaligned\n mathbfM_H_p = textdiagmathbf(MMM) \n mathbfN_H_c = textdiagmathbf(NNN) \n mathbfL_H_p = textdiagmathbf(LLL) \nendaligned\n\nTime-varying and non-diagonal weights are also supported. Modify the last block in mathbfM_H_p to specify a terminal weight. The mathbfΔU includes the input increments mathbfΔu(k+j) = mathbfu(k+j) - mathbfu(k+j-1) from j=0 to H_c-1, the mathbfY vector, the output predictions mathbfy(k+j) from j=1 to H_p, and the mathbfU vector, the manipulated inputs mathbfu(k+j) from j=0 to H_p-1. The slack variable ϵ relaxes the constraints, as described in setconstraint! documentation. See Extended Help for a detailed nomenclature. \n\nThis method uses the default state estimator, a SteadyKalmanFilter with default arguments.\n\nArguments\n\nmodel::LinModel : model used for controller predictions and state estimations.\nHp=10+nk: prediction horizon H_p, nk is the number of delays in model.\nHc=2 : control horizon H_c.\nMwt=fill(1.0,model.ny) : main diagonal of mathbfM weight matrix (vector).\nNwt=fill(0.1,model.nu) : main diagonal of mathbfN weight matrix (vector).\nLwt=fill(0.0,model.nu) : main diagonal of mathbfL weight matrix (vector).\nM_Hp=diagm(repeat(Mwt,Hp)) : positive semidefinite symmetric matrix mathbfM_H_p.\nN_Hc=diagm(repeat(Nwt,Hc)) : positive semidefinite symmetric matrix mathbfN_H_c.\nL_Hp=diagm(repeat(Lwt,Hp)) : positive semidefinite symmetric matrix mathbfL_H_p.\nCwt=1e5 : slack variable weight C (scalar), use Cwt=Inf for hard constraints only.\noptim=JuMP.Model(OSQP.MathOptInterfaceOSQP.Optimizer) : quadratic optimizer used in the predictive controller, provided as a JuMP.Model (default to OSQP optimizer).\nadditional keyword arguments are passed to SteadyKalmanFilter constructor.\n\nExamples\n\njulia> model = LinModel([tf(3, [30, 1]); tf(-2, [5, 1])], 4);\n\njulia> mpc = LinMPC(model, Mwt=[0, 1], Nwt=[0.5], Hp=30, Hc=1)\nLinMPC controller with a sample time Ts = 4.0 s, OSQP optimizer, SteadyKalmanFilter estimator and:\n 30 prediction steps Hp\n 1 control steps Hc\n 1 slack variable ϵ (control constraints)\n 1 manipulated inputs u (0 integrating states)\n 4 estimated states x̂\n 2 measured outputs ym (2 integrating states)\n 0 unmeasured outputs yu\n 0 measured disturbances d\n\nExtended Help\n\ndetails: Extended Help\nManipulated inputs setpoints mathbfr_u are not common but they can be interesting for over-actuated systems, when nu > ny (e.g. prioritize solutions with lower economical costs). The default Lwt value implies that this feature is disabled by default.The objective function follows this nomenclature:VARIABLE DESCRIPTION SIZE\nH_p prediction horizon (integer) ()\nH_c control horizon (integer) ()\nmathbfΔU manipulated input increments over H_c (nu*Hc,)\nmathbfY predicted outputs over H_p (ny*Hp,)\nmathbfU manipulated inputs over H_p (nu*Hp,)\nmathbfR_y predicted output setpoints over H_p (ny*Hp,)\nmathbfR_u predicted manipulated input setpoints over H_p (nu*Hp,)\nmathbfM_H_p output setpoint tracking weights over H_p (ny*Hp, ny*Hp)\nmathbfN_H_c manipulated input increment weights over H_c (nu*Hc, nu*Hc)\nmathbfL_H_p manipulated input setpoint tracking weights over H_p (nu*Hp, nu*Hp)\nC slack variable weight ()\nϵ slack variable for constraint softening ()\n\n\n\n\n\nLinMPC(estim::StateEstimator; )\n\nUse custom state estimator estim to construct LinMPC.\n\nestim.model must be a LinModel. Else, a NonLinMPC is required. \n\nExamples\n\njulia> estim = KalmanFilter(LinModel([tf(3, [30, 1]); tf(-2, [5, 1])], 4), i_ym=[2]);\n\njulia> mpc = LinMPC(estim, Mwt=[0, 1], Nwt=[0.5], Hp=30, Hc=1)\nLinMPC controller with a sample time Ts = 4.0 s, OSQP optimizer, KalmanFilter estimator and:\n 30 prediction steps Hp\n 1 control steps Hc\n 1 slack variable ϵ (control constraints)\n 1 manipulated inputs u (0 integrating states)\n 3 estimated states x̂\n 1 measured outputs ym (1 integrating states)\n 1 unmeasured outputs yu\n 0 measured disturbances d\n\n\n\n\n\n","category":"type"},{"location":"public/predictive_control/#ExplicitMPC","page":"Predictive Controllers","title":"ExplicitMPC","text":"","category":"section"},{"location":"public/predictive_control/","page":"Predictive Controllers","title":"Predictive Controllers","text":"ExplicitMPC","category":"page"},{"location":"public/predictive_control/#ModelPredictiveControl.ExplicitMPC","page":"Predictive Controllers","title":"ModelPredictiveControl.ExplicitMPC","text":"ExplicitMPC(model::LinModel; )\n\nConstruct an explicit linear predictive controller based on LinModel model.\n\nThe controller minimizes the following objective function at each discrete time k:\n\nbeginaligned\nmin_mathbfΔU mathbf(R_y - Y) mathbfM_H_p mathbf(R_y - Y) \n + mathbf(ΔU) mathbfN_H_c mathbf(ΔU) \n + mathbf(R_u - U) mathbfL_H_p mathbf(R_u - U) \nendaligned\n\nSee LinMPC for the variable definitions. This controller does not support constraints but the computational costs are extremely low (array division), therefore suitable for applications that require small sample times. The keyword arguments are identical to LinMPC, except for Cwt and optim which are not supported. \n\nThis method uses the default state estimator, a SteadyKalmanFilter with default arguments.\n\nExamples\n\njulia> model = LinModel([tf(3, [30, 1]); tf(-2, [5, 1])], 4);\n\njulia> mpc = ExplicitMPC(model, Mwt=[0, 1], Nwt=[0.5], Hp=30, Hc=1)\nExplicitMPC controller with a sample time Ts = 4.0 s, SteadyKalmanFilter estimator and:\n 30 prediction steps Hp\n 1 control steps Hc\n 1 manipulated inputs u (0 integrating states)\n 4 estimated states x̂\n 2 measured outputs ym (2 integrating states)\n 0 unmeasured outputs yu\n 0 measured disturbances d\n\n\n\n\n\nExplicitMPC(estim::StateEstimator; )\n\nUse custom state estimator estim to construct ExplicitMPC.\n\nestim.model must be a LinModel. Else, a NonLinMPC is required.\n\n\n\n\n\n","category":"type"},{"location":"public/predictive_control/#NonLinMPC","page":"Predictive Controllers","title":"NonLinMPC","text":"","category":"section"},{"location":"public/predictive_control/","page":"Predictive Controllers","title":"Predictive Controllers","text":"NonLinMPC","category":"page"},{"location":"public/predictive_control/#ModelPredictiveControl.NonLinMPC","page":"Predictive Controllers","title":"ModelPredictiveControl.NonLinMPC","text":"NonLinMPC(model::SimModel; )\n\nConstruct a nonlinear predictive controller based on SimModel model.\n\nBoth NonLinModel and LinModel are supported (see Extended Help). The controller minimizes the following objective function at each discrete time k:\n\nbeginaligned\nmin_mathbfΔU ϵ mathbf(R_y - Y) mathbfM_H_p mathbf(R_y - Y) \n + mathbf(ΔU) mathbfN_H_c mathbf(ΔU) \n + mathbf(R_u - U) mathbfL_H_p mathbf(R_u - U) \n + C ϵ^2 \n + E J_E(mathbfU_E mathbfY_E mathbfD_E mathbfp)\nendaligned\n\nSee LinMPC for the variable definitions. The custom economic function J_E can penalizes solutions with high economic costs. Setting all the weights to 0 except E creates a pure economic model predictive controller (EMPC). The arguments of J_E include the manipulated inputs, the predicted outputs and measured disturbances from k to k+H_p inclusively:\n\n mathbfU_E = beginbmatrix mathbfU mathbfu(k+H_p-1) endbmatrix quad\n mathbfY_E = beginbmatrix mathbfy(k) mathbfY endbmatrix quad\n mathbfD_E = beginbmatrix mathbfd(k) mathbfD endbmatrix\n\nsince H_c H_p implies that mathbfΔu(k+H_p) = mathbf0 or mathbfu(k+H_p)= mathbfu(k+H_p-1). The vector mathbfD includes the predicted measured disturbance over H_p. The argument mathbfp is a custom parameter object of any type but use a mutable one if you want to modify it later e.g.: a vector.\n\ntip: Tip\nReplace any of the 4 arguments with _ if not needed (see JE default value below).\n\nThis method uses the default state estimator :\n\nif model is a LinModel, a SteadyKalmanFilter with default arguments;\nelse, an UnscentedKalmanFilter with default arguments. \n\nwarning: Warning\nSee Extended Help if you get an error like: MethodError: no method matching Float64(::ForwardDiff.Dual).\n\nArguments\n\nmodel::SimModel : model used for controller predictions and state estimations.\nHp=nothing: prediction horizon H_p, must be specified for NonLinModel.\nHc=2 : control horizon H_c.\nMwt=fill(1.0,model.ny) : main diagonal of mathbfM weight matrix (vector).\nNwt=fill(0.1,model.nu) : main diagonal of mathbfN weight matrix (vector).\nLwt=fill(0.0,model.nu) : main diagonal of mathbfL weight matrix (vector).\nM_Hp=diagm(repeat(Mwt,Hp)) : positive semidefinite symmetric matrix mathbfM_H_p.\nN_Hc=diagm(repeat(Nwt,Hc)) : positive semidefinite symmetric matrix mathbfN_H_c.\nL_Hp=diagm(repeat(Lwt,Hp)) : positive semidefinite symmetric matrix mathbfL_H_p.\nCwt=1e5 : slack variable weight C (scalar), use Cwt=Inf for hard constraints only.\nEwt=0.0 : economic costs weight E (scalar). \nJE=(_,_,_,_)->0.0 : economic function J_E(mathbfU_E mathbfY_E mathbfD_E mathbfp).\np=model.p : J_E function parameter mathbfp (any type).\noptim=JuMP.Model(Ipopt.Optimizer) : nonlinear optimizer used in the predictive controller, provided as a JuMP.Model (default to Ipopt optimizer).\nadditional keyword arguments are passed to UnscentedKalmanFilter constructor (or SteadyKalmanFilter, for LinModel).\n\nExamples\n\njulia> model = NonLinModel((x,u,_,_)->0.5x+u, (x,_,_)->2x, 10.0, 1, 1, 1, solver=nothing);\n\njulia> mpc = NonLinMPC(model, Hp=20, Hc=1, Cwt=1e6)\nNonLinMPC controller with a sample time Ts = 10.0 s, Ipopt optimizer, UnscentedKalmanFilter estimator and:\n 20 prediction steps Hp\n 1 control steps Hc\n 1 slack variable ϵ (control constraints)\n 1 manipulated inputs u (0 integrating states)\n 2 estimated states x̂\n 1 measured outputs ym (1 integrating states)\n 0 unmeasured outputs yu\n 0 measured disturbances d\n\nExtended Help\n\ndetails: Extended Help\nNonLinMPC controllers based on LinModel compute the predictions with matrix algebra instead of a for loop. This feature can accelerate the optimization, especially for the constraint handling, and is not available in any other package, to my knowledge.The optimization relies on JuMP automatic differentiation (AD) to compute the objective and constraint derivatives. Optimizers generally benefit from exact derivatives like AD. However, the NonLinModel state-space functions must be compatible with this feature. See Automatic differentiation for common mistakes when writing these functions.Note that if Cwt≠Inf, the attribute nlp_scaling_max_gradient of Ipopt is set to 10/Cwt (if not already set), to scale the small values of ϵ.\n\n\n\n\n\nNonLinMPC(estim::StateEstimator; )\n\nUse custom state estimator estim to construct NonLinMPC.\n\nExamples\n\njulia> model = NonLinModel((x,u,_,_)->0.5x+u, (x,_,_)->2x, 10.0, 1, 1, 1, solver=nothing);\n\njulia> estim = UnscentedKalmanFilter(model, σQint_ym=[0.05]);\n\njulia> mpc = NonLinMPC(estim, Hp=20, Hc=1, Cwt=1e6)\nNonLinMPC controller with a sample time Ts = 10.0 s, Ipopt optimizer, UnscentedKalmanFilter estimator and:\n 20 prediction steps Hp\n 1 control steps Hc\n 1 slack variable ϵ (control constraints)\n 1 manipulated inputs u (0 integrating states)\n 2 estimated states x̂\n 1 measured outputs ym (1 integrating states)\n 0 unmeasured outputs yu\n 0 measured disturbances d\n\n\n\n\n\n","category":"type"},{"location":"public/predictive_control/#Move-Manipulated-Input-u","page":"Predictive Controllers","title":"Move Manipulated Input u","text":"","category":"section"},{"location":"public/predictive_control/","page":"Predictive Controllers","title":"Predictive Controllers","text":"moveinput!","category":"page"},{"location":"public/predictive_control/#ModelPredictiveControl.moveinput!","page":"Predictive Controllers","title":"ModelPredictiveControl.moveinput!","text":"moveinput!(mpc::PredictiveController, ry=mpc.estim.model.yop, d=[]; ) -> u\n\nCompute the optimal manipulated input value u for the current control period.\n\nSolve the optimization problem of mpc PredictiveController and return the results mathbfu(k). Following the receding horizon principle, the algorithm discards the optimal future manipulated inputs mathbfu(k+1) mathbfu(k+2) Note that the method mutates mpc internal data but it does not modifies mpc.estim states. Call preparestate!(mpc, ym, d) before moveinput!, and updatestate!(mpc, u, ym, d) after, to update mpc state estimates.\n\nCalling a PredictiveController object calls this method.\n\nSee also LinMPC, ExplicitMPC, NonLinMPC.\n\nArguments\n\ninfo: Info\nKeyword arguments with emphasis are non-Unicode alternatives.\n\nmpc::PredictiveController : solve optimization problem of mpc.\nry=mpc.estim.model.yop : current output setpoints mathbfr_y(k).\nd=[] : current measured disturbances mathbfd(k).\nD̂=repeat(d, mpc.Hp) or Dhat : predicted measured disturbances mathbfD, constant in the future by default or mathbfd(k+j)=mathbfd(k) for j=1 to H_p.\nR̂y=repeat(ry, mpc.Hp) or Rhaty : predicted output setpoints mathbfR_y, constant in the future by default or mathbfr_y(k+j)=mathbfr_y(k) for j=1 to H_p.\nR̂u=mpc.Uop or Rhatu : predicted manipulated input setpoints, constant in the future by default or mathbfr_u(k+j)=mathbfu_op for j=0 to H_p-1. \n\nExamples\n\njulia> mpc = LinMPC(LinModel(tf(5, [2, 1]), 3), Nwt=[0], Hp=1000, Hc=1);\n\njulia> preparestate!(mpc, [0]); ry = [5];\n\njulia> u = moveinput!(mpc, ry); round.(u, digits=3)\n1-element Vector{Float64}:\n 1.0\n\n\n\n\n\n","category":"function"},{"location":"public/generic_func/#Functions:-Generic-Functions","page":"Generic Functions","title":"Functions: Generic Functions","text":"","category":"section"},{"location":"public/generic_func/","page":"Generic Functions","title":"Generic Functions","text":"Pages = [\"generic_func.md\"]","category":"page"},{"location":"public/generic_func/","page":"Generic Functions","title":"Generic Functions","text":"This page contains the documentation of functions that are generic to SimModel, StateEstimator and PredictiveController types.","category":"page"},{"location":"public/generic_func/#Set-Constraint","page":"Generic Functions","title":"Set Constraint","text":"","category":"section"},{"location":"public/generic_func/","page":"Generic Functions","title":"Generic Functions","text":"setconstraint!","category":"page"},{"location":"public/generic_func/#ModelPredictiveControl.setconstraint!","page":"Generic Functions","title":"ModelPredictiveControl.setconstraint!","text":"setconstraint!(estim::MovingHorizonEstimator; ) -> estim\n\nSet the constraint parameters of the MovingHorizonEstimator estim.\n\nIt supports both soft and hard constraints on the estimated state mathbfx, process noise mathbfw and sensor noise mathbfv:\n\nbeginalignat*3\n mathbfx_min - c_x_min ϵ mathbfx_k(k-j+p) mathbfx_max + c_x_max ϵ qquad j = N_k N_k - 1 0 \n mathbfw_min - c_w_min ϵ mathbfw(k-j+p) mathbfw_max + c_w_max ϵ qquad j = N_k N_k - 1 1 \n mathbfv_min - c_v_min ϵ mathbfv(k-j+1) mathbfv_max + c_v_max ϵ qquad j = N_k N_k - 1 1\nendalignat*\n\nand also ϵ 0. All the constraint parameters are vector. Use ±Inf values when there is no bound. The constraint softness parameters mathbfc, also called equal concern for relaxation, are non-negative values that specify the softness of the associated bound. Use 0.0 values for hard constraints (default for all of them). Notice that constraining the estimated sensor noises is equivalent to bounding the innovation term, since mathbfv(k) = mathbfy^m(k) - mathbfy^m(k). See Extended Help for details on the constant p, on model augmentation and on time-varying constraints.\n\nArguments\n\ninfo: Info\nAll the keyword arguments have non-Unicode alternatives e.g. xhatmin or Vhatmax. The default constraints are mentioned here for clarity but omitting a keyword argument will not re-assign to its default value (defaults are set at construction only).\n\nestim::MovingHorizonEstimator : moving horizon estimator to set constraints\nx̂min=fill(-Inf,nx̂) / x̂max=fill(+Inf,nx̂) : estimated state bound mathbfx_minmax\nŵmin=fill(-Inf,nx̂) / ŵmax=fill(+Inf,nx̂) : estimated process noise bound mathbfw_minmax\nv̂min=fill(-Inf,nym) / v̂max=fill(+Inf,nym) : estimated sensor noise bound mathbfv_minmax\nc_x̂min=fill(0.0,nx̂) / c_x̂max=fill(0.0,nx̂) : x̂min / x̂max softness weight mathbfc_x_minmax\nc_ŵmin=fill(0.0,nx̂) / c_ŵmax=fill(0.0,nx̂) : ŵmin / ŵmax softness weight mathbfc_w_minmax\nc_v̂min=fill(0.0,nym) / c_v̂max=fill(0.0,nym) : v̂min / v̂max softness weight mathbfc_v_minmax\nall the keyword arguments above but with a first capital letter, e.g. X̂max or C_ŵmax: for time-varying constraints (see Extended Help)\n\nExamples\n\njulia> estim = MovingHorizonEstimator(LinModel(ss(0.5,1,1,0,1)), He=3);\n\njulia> estim = setconstraint!(estim, x̂min=[-50, -50], x̂max=[50, 50])\nMovingHorizonEstimator estimator with a sample time Ts = 1.0 s, OSQP optimizer, LinModel and:\n 3 estimation steps He\n 0 slack variable ϵ (estimation constraints)\n 1 manipulated inputs u (0 integrating states)\n 2 estimated states x̂\n 1 measured outputs ym (1 integrating states)\n 0 unmeasured outputs yu\n 0 measured disturbances d\n\nExtended Help\n\ndetails: Extended Help\nThe constant p=0 if estim.direct==true (current form), else p=1 (prediction form). Note that the state mathbfx and process noise mathbfw constraints are applied on the augmented model, detailed in SteadyKalmanFilter Extended Help. For variable constraints, the bounds can be modified after calling updatestate!, that is, at runtime, except for ±Inf bounds. Time-varying constraints over the estimation horizon H_e are also possible, mathematically defined as:beginalignat*3\n mathbfX_min - C_x_min ϵ mathbfX mathbfX_max + C_x_max ϵ \n mathbfW_min - C_w_min ϵ mathbfW mathbfW_max + C_w_max ϵ \n mathbfV_min - C_v_min ϵ mathbfV mathbfV_max + C_v_max ϵ\nendalignat*For this, use the same keyword arguments as above but with a first capital letter:X̂min / X̂max / C_x̂min / C_x̂max : mathbfX constraints (nx̂*(He+1),).\nŴmin / Ŵmax / C_ŵmin / C_ŵmax : mathbfW constraints (nx̂*He,).\nV̂min / V̂max / C_v̂min / C_v̂max : mathbfV constraints (nym*He,).\n\n\n\n\n\nsetconstraint!(mpc::PredictiveController; ) -> mpc\n\nSet the constraint parameters of the PredictiveController mpc.\n\nThe predictive controllers support both soft and hard constraints, defined by:\n\nbeginalignat*3\n mathbfu_min - c_u_min ϵ mathbfu(k+j) mathbfu_max + c_u_max ϵ qquad j = 0 1 H_p - 1 \n mathbfΔu_min - c_Δu_min ϵ mathbfΔu(k+j) mathbfΔu_max + c_Δu_max ϵ qquad j = 0 1 H_c - 1 \n mathbfy_min - c_y_min ϵ mathbfy(k+j) mathbfy_max + c_y_max ϵ qquad j = 1 2 H_p \n mathbfx_min - c_x_min ϵ mathbfx_i(k+j) mathbfx_max + c_x_max ϵ qquad j = H_p\nendalignat*\n\nand also ϵ 0. The last line is the terminal constraints applied on the states at the end of the horizon (see Extended Help). See MovingHorizonEstimator constraints for details on bounds and softness parameters mathbfc. The output and terminal constraints are all soft by default. See Extended Help for time-varying constraints.\n\nArguments\n\ninfo: Info\nThe keyword arguments Δumin, Δumax, c_Δumin, c_Δumax, x̂min, x̂max, c_x̂min, c_x̂max and their capital letter versions have non-Unicode alternatives e.g. Deltaumin, xhatmax and C_DeltauminThe default constraints are mentioned here for clarity but omitting a keyword argument will not re-assign to its default value (defaults are set at construction only).\n\nmpc::PredictiveController : predictive controller to set constraints\numin=fill(-Inf,nu) / umax=fill(+Inf,nu) : manipulated input bound mathbfu_minmax\nΔumin=fill(-Inf,nu) / Δumax=fill(+Inf,nu) : manipulated input increment bound mathbfΔu_minmax\nymin=fill(-Inf,ny) / ymax=fill(+Inf,ny) : predicted output bound mathbfy_minmax\nx̂min=fill(-Inf,nx̂) / x̂max=fill(+Inf,nx̂) : terminal constraint bound mathbfx_minmax\nc_umin=fill(0.0,nu) / c_umax=fill(0.0,nu) : umin / umax softness weight mathbfc_u_minmax\nc_Δumin=fill(0.0,nu) / c_Δumax=fill(0.0,nu) : Δumin / Δumax softness weight mathbfc_Δu_minmax\nc_ymin=fill(1.0,ny) / c_ymax=fill(1.0,ny) : ymin / ymax softness weight mathbfc_y_minmax\nc_x̂min=fill(1.0,nx̂) / c_x̂max=fill(1.0,nx̂) : x̂min / x̂max softness weight mathbfc_x_minmax\nall the keyword arguments above but with a first capital letter, except for the terminal constraints, e.g. Ymax or C_Δumin: for time-varying constraints (see Extended Help)\n\nExamples\n\njulia> mpc = LinMPC(setop!(LinModel(tf(3, [30, 1]), 4), uop=[50], yop=[25]));\n\njulia> mpc = setconstraint!(mpc, umin=[0], umax=[100], Δumin=[-10], Δumax=[+10])\nLinMPC controller with a sample time Ts = 4.0 s, OSQP optimizer, SteadyKalmanFilter estimator and:\n 10 prediction steps Hp\n 2 control steps Hc\n 1 slack variable ϵ (control constraints)\n 1 manipulated inputs u (0 integrating states)\n 2 estimated states x̂\n 1 measured outputs ym (1 integrating states)\n 0 unmeasured outputs yu\n 0 measured disturbances d\n\nExtended Help\n\ndetails: Extended Help\nTerminal constraints provide closed-loop stability guarantees on the nominal plant model. They can render an unfeasible problem however. In practice, a sufficiently large prediction horizon H_p without terminal constraints is typically enough for stability. If mpc.estim.direct==true, the estimator computes the states at i = k (the current time step), otherwise at i = k - 1. Note that terminal constraints are applied on the augmented state vector mathbfx (see SteadyKalmanFilter for details on augmentation).For variable constraints, the bounds can be modified after calling moveinput!, that is, at runtime, but not the softness parameters mathbfc. It is not possible to modify ±Inf bounds at runtime.tip: Tip\nTo keep a variable unconstrained while maintaining the ability to add a constraint later at runtime, set the bound to an absolute value sufficiently large when you create the controller (but different than ±Inf).It is also possible to specify time-varying constraints over H_p and H_c horizons. In such a case, they are defined by:beginalignat*3\n mathbfU_min - C_u_min ϵ mathbfU mathbfU_max + C_u_max ϵ \n mathbfΔU_min - C_Δu_min ϵ mathbfΔU mathbfΔU_max + C_Δu_max ϵ \n mathbfY_min - C_y_min ϵ mathbfY mathbfY_max + C_y_max ϵ\nendalignat*For this, use the same keyword arguments as above but with a first capital letter:Umin / Umax / C_umin / C_umax : mathbfU constraints (nu*Hp,).\nΔUmin / ΔUmax / C_Δumin / C_Δumax : mathbfΔU constraints (nu*Hc,).\nYmin / Ymax / C_ymin / C_ymax : mathbfY constraints (ny*Hp,).\n\n\n\n\n\n","category":"function"},{"location":"public/generic_func/#Evaluate-Output-y","page":"Generic Functions","title":"Evaluate Output y","text":"","category":"section"},{"location":"public/generic_func/","page":"Generic Functions","title":"Generic Functions","text":"evaloutput","category":"page"},{"location":"public/generic_func/#ModelPredictiveControl.evaloutput","page":"Generic Functions","title":"ModelPredictiveControl.evaloutput","text":"evaloutput(model::SimModel, d=[]) -> y\n\nEvaluate SimModel outputs y from model.x0 states and measured disturbances d.\n\nIt returns model output at the current time step mathbfy(k). Calling a SimModel object calls this evaloutput method.\n\nExamples\n\njulia> model = setop!(LinModel(tf(2, [10, 1]), 5.0), yop=[20]);\n\njulia> y = evaloutput(model)\n1-element Vector{Float64}:\n 20.0\n\n\n\n\n\nevaloutput(estim::StateEstimator, d=[]) -> ŷ\n\nEvaluate StateEstimator outputs ŷ from estim.x̂0 states and disturbances d.\n\nIt returns estim output at the current time step mathbfy(k). If estim.direct is true, the method preparestate! should be called beforehand to correct the state estimate. \n\nCalling a StateEstimator object calls this evaloutput method.\n\nExamples\n\njulia> kf = SteadyKalmanFilter(setop!(LinModel(tf(2, [10, 1]), 5), yop=[20]), direct=false);\n\njulia> ŷ = evaloutput(kf)\n1-element Vector{Float64}:\n 20.0\n\n\n\n\n\n","category":"function"},{"location":"public/generic_func/#Change-State-x","page":"Generic Functions","title":"Change State x","text":"","category":"section"},{"location":"public/generic_func/#Prepare-State-x","page":"Generic Functions","title":"Prepare State x","text":"","category":"section"},{"location":"public/generic_func/","page":"Generic Functions","title":"Generic Functions","text":"preparestate!","category":"page"},{"location":"public/generic_func/#ModelPredictiveControl.preparestate!","page":"Generic Functions","title":"ModelPredictiveControl.preparestate!","text":"preparestate!(model::SimModel, _ , _ ) -> x\n\nDo nothing for SimModel and return the current model state mathbfx(k). \n\n\n\n\n\npreparestate!(estim::StateEstimator, ym, d=estim.model.dop) -> x̂\n\nPrepare estim.x̂0 estimate with meas. outputs ym and dist. d for the current time step.\n\nThis function should be called at the beginning of each discrete time step. Its behavior depends if estim is a StateEstimator in the current/filter (1.) or delayed/predictor (2.) formulation:\n\nIf estim.direct is true, it removes the operating points with remove_op!, calls correct_estimate!, and returns the corrected state estimate mathbfx_k(k).\nElse, it does nothing and returns the current best estimate mathbfx_k-1(k).\n\nExamples\n\njulia> estim2 = SteadyKalmanFilter(LinModel(ss(0.1, 0.5, 1, 0, 4)), nint_ym=0, direct=true);\n\njulia> x̂ = round.(preparestate!(estim2, [1]), digits=3)\n1-element Vector{Float64}:\n 0.01\n\njulia> estim1 = SteadyKalmanFilter(LinModel(ss(0.1, 0.5, 1, 0, 4)), nint_ym=0, direct=false);\n\njulia> x̂ = preparestate!(estim1, [1])\n1-element Vector{Float64}:\n 0.0\n\n\n\n\n\npreparestate!(mpc::PredictiveController, ym, d=[]) -> x̂\n\nCall preparestate! on mpc.estim StateEstimator.\n\n\n\n\n\n","category":"function"},{"location":"public/generic_func/#Update-State-x","page":"Generic Functions","title":"Update State x","text":"","category":"section"},{"location":"public/generic_func/","page":"Generic Functions","title":"Generic Functions","text":"updatestate!","category":"page"},{"location":"public/generic_func/#ModelPredictiveControl.updatestate!","page":"Generic Functions","title":"ModelPredictiveControl.updatestate!","text":"updatestate!(model::SimModel, u, d=[]) -> xnext\n\nUpdate model.x0 states with current inputs u and meas. dist. d for the next time step.\n\nThe method computes and returns the model state for the next time step mathbfx(k+1).\n\nExamples\n\njulia> model = LinModel(ss(1.0, 1.0, 1.0, 0, 1.0));\n\njulia> x = updatestate!(model, [1])\n1-element Vector{Float64}:\n 1.0\n\n\n\n\n\nupdatestate!(estim::StateEstimator, u, ym, d=[]) -> x̂next\n\nUpdate estim.x̂0 estimate with current inputs u, measured outputs ym and dist. d. \n\nThis function should be called at the end of each discrete time step. It removes the operating points with remove_op!, calls update_estimate! and returns the state estimate for the next time step mathbfx_k(k+1). The method preparestate! should be called prior to this one to correct the estimate when applicable (if estim.direct == true). Note that the MovingHorizonEstimator with the default direct=true option is not able to estimate mathbfx_k(k+1), the returned value is therefore the current corrected state mathbfx_k(k).\n\nExamples\n\njulia> kf = SteadyKalmanFilter(LinModel(ss(0.1, 0.5, 1, 0, 4.0))); u = [1]; ym = [0];\n\njulia> preparestate!(kf, ym);\n\njulia> x̂ = updatestate!(kf, u, ym) # x̂[2] is the integrator state (nint_ym argument)\n2-element Vector{Float64}:\n 0.5\n 0.0\n\n\n\n\n\nupdatestate!(mpc::PredictiveController, u, ym, d=[]) -> x̂next\n\nCall updatestate! on mpc.estim StateEstimator.\n\n\n\n\n\n","category":"function"},{"location":"public/generic_func/#Init-State-x","page":"Generic Functions","title":"Init State x","text":"","category":"section"},{"location":"public/generic_func/","page":"Generic Functions","title":"Generic Functions","text":"initstate!","category":"page"},{"location":"public/generic_func/#ModelPredictiveControl.initstate!","page":"Generic Functions","title":"ModelPredictiveControl.initstate!","text":"initstate!(model::SimModel, u, d=[]) -> x\n\nInit model.x0 with manipulated inputs u and meas. dist. d steady-state and reset time.\n\nThe method tries to initialize the model state mathbfx at steady-state. It removes the operating points on u and d and calls steadystate!:\n\nIf model is a LinModel, the method computes the steady-state of current inputs u and measured disturbances d.\nElse, model.x0 is left unchanged. Use setstate! to manually modify it.\n\nExamples\n\njulia> model = LinModel(tf(6, [10, 1]), 2.0);\n\njulia> u = [1]; x = initstate!(model, u); y = round.(evaloutput(model), digits=3)\n1-element Vector{Float64}:\n 6.0\n \njulia> x ≈ updatestate!(model, u)\ntrue\n\n\n\n\n\ninitstate!(estim::StateEstimator, u, ym, d=[]) -> x̂\n\nInit estim.x̂0 states from current inputs u, measured outputs ym and disturbances d.\n\nThe method tries to find a good steady-state for the initial estimate mathbfx. It stores u - estim.model.uop at estim.lastu0 and removes the operating points with remove_op!, and call init_estimate!:\n\nIf estim.model is a LinModel, it finds the steady-state of the augmented model using u and d arguments, and uses the ym argument to enforce that mathbfy^m(0) = mathbfy^m(0). For control applications, this solution produces a bumpless manual to automatic transfer. See init_estimate! for details.\nElse, estim.x̂0 is left unchanged. Use setstate! to manually modify it.\n\nIf applicable, it also sets the error covariance estim.P̂ to estim.P̂_0.\n\nExamples\n\njulia> estim = SteadyKalmanFilter(LinModel(tf(3, [10, 1]), 0.5), nint_ym=[2], direct=false);\n\njulia> u = [1]; y = [3 - 0.1]; x̂ = round.(initstate!(estim, u, y), digits=3)\n3-element Vector{Float64}:\n 5.0\n 0.0\n -0.1\n\njulia> x̂ ≈ updatestate!(estim, u, y)\ntrue\n\njulia> evaloutput(estim) ≈ y\ntrue\n\n\n\n\n\ninitstate!(mpc::PredictiveController, u, ym, d=[]) -> x̂\n\nInit the states of mpc.estim StateEstimator and warm start mpc.ΔŨ at zero.\n\n\n\n\n\n","category":"function"},{"location":"public/generic_func/#Set-State-x","page":"Generic Functions","title":"Set State x","text":"","category":"section"},{"location":"public/generic_func/","page":"Generic Functions","title":"Generic Functions","text":"setstate!","category":"page"},{"location":"public/generic_func/#ModelPredictiveControl.setstate!","page":"Generic Functions","title":"ModelPredictiveControl.setstate!","text":"setstate!(model::SimModel, x) -> model\n\nSet model.x0 to x - model.xop from the argument x. \n\n\n\n\n\nsetstate!(estim::StateEstimator, x̂) -> estim\n\nSet estim.x̂0 to x̂ - estim.x̂op from the argument x̂. \n\n\n\n\n\nsetstate!(mpc::PredictiveController, x̂) -> mpc\n\nSet mpc.estim.x̂0 to x̂ - estim.x̂op from the argument x̂.\n\n\n\n\n\n","category":"function"},{"location":"public/generic_func/#Set-Model-and-Weights","page":"Generic Functions","title":"Set Model and Weights","text":"","category":"section"},{"location":"public/generic_func/","page":"Generic Functions","title":"Generic Functions","text":"setmodel!","category":"page"},{"location":"public/generic_func/#ModelPredictiveControl.setmodel!","page":"Generic Functions","title":"ModelPredictiveControl.setmodel!","text":"setmodel!(estim::StateEstimator, model=estim.model; ) -> estim\n\nSet model and covariance matrices of estim StateEstimator.\n\nAllows model adaptation of estimators based on LinModel at runtime. Modification of NonLinModel state-space functions is not supported. New covariance matrices can be specified with the keyword arguments (see SteadyKalmanFilter documentation for the nomenclature). Not supported by Luenberger and SteadyKalmanFilter, use the time-varying KalmanFilter instead. The MovingHorizonEstimator model is kept constant over the estimation horizon H_e. The matrix dimensions and sample time must stay the same. Note that the observability and controllability of the new augmented model is not verified (see Extended Help for more info).\n\nArguments\n\ninfo: Info\nKeyword arguments with emphasis are non-Unicode alternatives.\n\nestim::StateEstimator : estimator to set model and covariances.\nmodel=estim.model : new plant model (not supported by NonLinModel).\nQ̂=nothing or Qhat : new augmented model mathbfQ covariance matrix.\nR̂=nothing or Rhat : new augmented model mathbfR covariance matrix.\n\nExamples\n\njulia> kf = KalmanFilter(LinModel(ss(0.1, 0.5, 1, 0, 4.0)), σQ=[√4.0], σQint_ym=[√0.25]);\n\njulia> kf.model.A[], kf.Q̂[1, 1], kf.Q̂[2, 2] \n(0.1, 4.0, 0.25)\n\njulia> setmodel!(kf, LinModel(ss(0.42, 0.5, 1, 0, 4.0)), Q̂=[1 0;0 0.5]);\n\njulia> kf.model.A[], kf.Q̂[1, 1], kf.Q̂[2, 2] \n(0.42, 1.0, 0.5)\n\nExtended Help\n\ndetails: Extended Help\nUsing the default model augmentation computed by the default_nint method, switching from a non-integrating plant model to an integrating one will produce an augmented model that is not observable. Moving the unmeasured disturbances at the model input (nint_u parameter) can fix this issue.\n\n\n\n\n\nsetmodel!(mpc::PredictiveController, model=mpc.estim.model; ) -> mpc\n\nSet model and objective function weights of mpc PredictiveController.\n\nAllows model adaptation of controllers based on LinModel at runtime. Modification of NonLinModel state-space functions is not supported. New weight matrices in the objective function can be specified with the keyword arguments (see LinMPC for the nomenclature). If Cwt ≠ Inf, the augmented move suppression weight is mathbfN_H_c = mathrmdiag(mathbfN_H_c C), else mathbfN_H_c = mathbfN_H_c. The StateEstimator mpc.estim cannot be a Luenberger observer or a SteadyKalmanFilter (the default estimator). Construct the mpc object with a time-varying KalmanFilter instead. Note that the model is constant over the prediction horizon H_p.\n\nArguments\n\ninfo: Info\nKeyword arguments with emphasis are non-Unicode alternatives.\n\nmpc::PredictiveController : controller to set model and weights.\nmodel=mpc.estim.model : new plant model (not supported by NonLinModel).\nMwt=nothing : new main diagonal in mathbfM weight matrix (vector).\nNwt=nothing : new main diagonal in mathbfN weight matrix (vector).\nLwt=nothing : new main diagonal in mathbfL weight matrix (vector).\nM_Hp=nothing : new mathbfM_H_p weight matrix.\nÑ_Hc=nothing or Ntilde_Hc : new mathbfN_H_c weight matrix (see def. above).\nL_Hp=nothing : new mathbfL_H_p weight matrix.\nadditional keyword arguments are passed to setmodel!(mpc.estim).\n\nExamples\n\njulia> mpc = LinMPC(KalmanFilter(LinModel(ss(0.1, 0.5, 1, 0, 4.0)), σR=[√25]), Hp=1, Hc=1);\n\njulia> mpc.estim.model.A[1], mpc.estim.R̂[1], mpc.M_Hp[1], mpc.Ñ_Hc[1]\n(0.1, 25.0, 1.0, 0.1)\n\njulia> setmodel!(mpc, LinModel(ss(0.42, 0.5, 1, 0, 4.0)); R̂=[9], M_Hp=[10], Nwt=[0.666]);\n\njulia> mpc.estim.model.A[1], mpc.estim.R̂[1], mpc.M_Hp[1], mpc.Ñ_Hc[1]\n(0.42, 9.0, 10.0, 0.666)\n\n\n\n\n\n","category":"function"},{"location":"public/generic_func/#Get-Additional-Information","page":"Generic Functions","title":"Get Additional Information","text":"","category":"section"},{"location":"public/generic_func/","page":"Generic Functions","title":"Generic Functions","text":"getinfo","category":"page"},{"location":"public/generic_func/#ModelPredictiveControl.getinfo","page":"Generic Functions","title":"ModelPredictiveControl.getinfo","text":"getinfo(estim::MovingHorizonEstimator) -> info\n\nGet additional info on estim MovingHorizonEstimator optimum for troubleshooting.\n\nIf estim.direct==true, the function should be called after calling preparestate!. Otherwise, call it after updatestate!. It returns the dictionary info with the following fields:\n\ninfo: Info\nFields with emphasis are non-Unicode alternatives.\n\n:Ŵ or :What : optimal estimated process noise over N_k, mathbfW\n:ϵ or :epsilon : optimal slack variable, ϵ\n:X̂ or :Xhat : optimal estimated states over N_k+1, mathbfX\n:x̂ or :xhat : optimal estimated state for the next time step, mathbfx_k(k+1)\n:V̂ or :Vhat : optimal estimated sensor noise over N_k, mathbfV\n:P̄ or :Pbar : estimation error covariance at arrival, mathbfP\n:x̄ or :xbar : optimal estimation error at arrival, mathbfx\n:Ŷ or :Yhat : optimal estimated outputs over N_k, mathbfY\n:Ŷm or :Yhatm : optimal estimated measured outputs over N_k, mathbfY^m\n:x̂arr or :xhatarr : optimal estimated state at arrival, mathbfx_k(k-N_k+p)\n:J : objective value optimum, J\n:Ym : measured outputs over N_k, mathbfY^m\n:U : manipulated inputs over N_k, mathbfU\n:D : measured disturbances over N_k, mathbfD\n:sol : solution summary of the optimizer for printing\n\nExamples\n\njulia> model = LinModel(ss(1.0, 1.0, 1.0, 0, 5.0));\n\njulia> estim = MovingHorizonEstimator(model, He=1, nint_ym=0, direct=false);\n\njulia> updatestate!(estim, [0], [1]);\n\njulia> round.(getinfo(estim)[:Ŷ], digits=3)\n1-element Vector{Float64}:\n 0.5\n\n\n\n\n\ngetinfo(mpc::PredictiveController) -> info\n\nGet additional info about mpc PredictiveController optimum for troubleshooting.\n\nThe function should be called after calling moveinput!. It returns the dictionary info with the following fields:\n\ninfo: Info\nFields with emphasis are non-Unicode alternatives.\n\n:ΔU or :DeltaU : optimal manipulated input increments over H_c, mathbfΔU\n:ϵ or :epsilon : optimal slack variable, ϵ\n:D̂ or :Dhat : predicted measured disturbances over H_p, mathbfD\n:ŷ or :yhat : current estimated output, mathbfy(k)\n:Ŷ or :Yhat : optimal predicted outputs over H_p, mathbfY\n:Ŷs or :Yhats : predicted stochastic output over H_p of InternalModel, mathbfY_s\n:R̂y or :Rhaty : predicted output setpoint over H_p, mathbfR_y\n:R̂u or :Rhatu : predicted manipulated input setpoint over H_p, mathbfR_u\n:x̂end or :xhatend : optimal terminal states, mathbfx_i(k+H_p)\n:J : objective value optimum, J\n:U : optimal manipulated inputs over H_p, mathbfU\n:u : current optimal manipulated input, mathbfu(k)\n:d : current measured disturbance, mathbfd(k)\n\nFor LinMPC and NonLinMPC, the field :sol also contains the optimizer solution summary that can be printed. Lastly, the optimal economic cost :JE is also available for NonLinMPC.\n\nExamples\n\njulia> mpc = LinMPC(LinModel(tf(5, [2, 1]), 3), Nwt=[0], Hp=1, Hc=1);\n\njulia> preparestate!(mpc, [0]); u = moveinput!(mpc, [10]);\n\njulia> round.(getinfo(mpc)[:Ŷ], digits=3)\n1-element Vector{Float64}:\n 10.0\n\n\n\n\n\n","category":"function"},{"location":"public/generic_func/#Real-Time-Simulate-and-Control","page":"Generic Functions","title":"Real-Time Simulate and Control","text":"","category":"section"},{"location":"public/generic_func/","page":"Generic Functions","title":"Generic Functions","text":"danger: Disclaimer\nThese utilities are for soft real-time applications. They are not suitable for hard real-time environnement like safety-critical processes.","category":"page"},{"location":"public/generic_func/#Save-current-time-t","page":"Generic Functions","title":"Save current time t","text":"","category":"section"},{"location":"public/generic_func/","page":"Generic Functions","title":"Generic Functions","text":"savetime!","category":"page"},{"location":"public/generic_func/#ModelPredictiveControl.savetime!","page":"Generic Functions","title":"ModelPredictiveControl.savetime!","text":"savetime!(model::SimModel) -> t\n\nSet model.t to time() and return the value.\n\nUsed in conjunction with periodsleep for simple soft real-time simulations. Call this function before any other in the simulation loop.\n\n\n\n\n\nsavetime!(estim::StateEstimator) -> t\n\nCall savetime!(estim.model) and return the time t.\n\n\n\n\n\nsavetime!(mpc::PredictiveController) -> t\n\nCall savetime!(mpc.estim.model) and return the time t.\n\n\n\n\n\n","category":"function"},{"location":"public/generic_func/#Period-Sleep","page":"Generic Functions","title":"Period Sleep","text":"","category":"section"},{"location":"public/generic_func/","page":"Generic Functions","title":"Generic Functions","text":"periodsleep","category":"page"},{"location":"public/generic_func/#ModelPredictiveControl.periodsleep","page":"Generic Functions","title":"ModelPredictiveControl.periodsleep","text":"periodsleep(model::SimModel, busywait=false) -> nothing\n\nSleep for model.Ts s minus the time elapsed since the last call to savetime!.\n\nIt calls sleep if busywait is false. Else, a simple while loop implements busy-waiting. As a rule-of-thumb, busy-waiting should be used if model.Ts < 0.1 s, since the accuracy of sleep is around 1 ms. Can be used to implement simple soft real-time simulations, see the example below.\n\nwarning: Warning\nThe allocations in Julia are garbage-collected (GC) automatically. This can affect the timing. In such cases, you can temporarily stop the GC with GC.enable(false), and restart it at a convenient time e.g.: just before calling periodsleep.\n\nExamples\n\njulia> model = LinModel(tf(2, [0.3, 1]), 0.1);\n\njulia> function sim_realtime!(model)\n t_0 = time()\n for i=1:3\n t = savetime!(model) # first function called\n println(round(t - t_0, digits=3))\n updatestate!(model, [1])\n periodsleep(model, true) # last function called\n end\n end;\n\njulia> sim_realtime!(model)\n0.0\n0.1\n0.2\n\n\n\n\n\nperiodsleep(estim::StateEstimator) -> nothing\n\nCall periodsleep(estim.model).\n\n\n\n\n\nperiodsleep(mpc::PredictiveController) -> nothing\n\nCall periodsleep(mpc.estim.model).\n\n\n\n\n\n","category":"function"},{"location":"#ModelPredictiveControl.jl","page":"Home","title":"ModelPredictiveControl.jl","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"An open source model predictive control package for Julia.","category":"page"},{"location":"","page":"Home","title":"Home","text":"The package depends on ControlSystemsBase.jl for the linear systems and JuMP.jl for the solving.","category":"page"},{"location":"","page":"Home","title":"Home","text":"The objective is to provide a simple, clear and modular framework to quickly design model predictive controllers (MPCs) in Julia, while preserving the flexibility for advanced real-time optimization. Modern MPCs based on closed-loop state estimators are the main focus of the package, but classical approaches that rely on internal models are also possible. The JuMP.jl interface allows the user to test different solvers easily if the performance of the default settings is not satisfactory.","category":"page"},{"location":"","page":"Home","title":"Home","text":"The documentation is divided in two parts:","category":"page"},{"location":"","page":"Home","title":"Home","text":"Manual — This section includes step-by-step guides to design predictive controllers on multiple case studies.\nFunctions — Documentation of methods and types exported by the package. The \"Internals\" section provides implementation details of functions that are not exported.","category":"page"},{"location":"#Manual","page":"Home","title":"Manual","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Depth = 2\nPages = [\n \"manual/installation.md\",\n \"manual/linmpc.md\",\n \"manual/nonlinmpc.md\",\n \"manual/mtk.md\"\n]","category":"page"},{"location":"#Functions:-Public","page":"Home","title":"Functions: Public","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Depth = 2\nPages = [\n \"public/sim_model.md\",\n \"public/state_estim.md\",\n \"public/predictive_control.md\",\n \"public/generic_func.md\",\n \"public/plot_sim.md\",\n]","category":"page"},{"location":"#Functions:-Internals","page":"Home","title":"Functions: Internals","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Depth = 1\nPages = [\n \"internals/sim_model.md\",\n \"internals/state_estim.md\",\n \"internals/predictive_control.md\",\n]","category":"page"},{"location":"internals/sim_model/#Functions:-SimModel-Internals","page":"Plant Models","title":"Functions: SimModel Internals","text":"","category":"section"},{"location":"internals/sim_model/","page":"Plant Models","title":"Plant Models","text":"Pages = [\"sim_model.md\"]","category":"page"},{"location":"internals/sim_model/#State-Space-Functions","page":"Plant Models","title":"State-Space Functions","text":"","category":"section"},{"location":"internals/sim_model/","page":"Plant Models","title":"Plant Models","text":"ModelPredictiveControl.f!\nModelPredictiveControl.h!","category":"page"},{"location":"internals/sim_model/#ModelPredictiveControl.f!","page":"Plant Models","title":"ModelPredictiveControl.f!","text":"f!(xnext0, model::LinModel, x0, u0, d0, p) -> nothing\n\nEvaluate xnext0 = A*x0 + Bu*u0 + Bd*d0 in-place when model is a LinModel.\n\n\n\n\n\nCall model.f!(xnext0, x0, u0, d0, p) for NonLinModel.\n\n\n\n\n\n","category":"function"},{"location":"internals/sim_model/#ModelPredictiveControl.h!","page":"Plant Models","title":"ModelPredictiveControl.h!","text":"h!(y0, model::LinModel, x0, d0, p) -> nothing\n\nEvaluate y0 = C*x0 + Dd*d0 in-place when model is a LinModel.\n\n\n\n\n\nCall model.h!(y0, x0, d0, p) for NonLinModel.\n\n\n\n\n\n","category":"function"},{"location":"internals/sim_model/#Steady-State-Calculation","page":"Plant Models","title":"Steady-State Calculation","text":"","category":"section"},{"location":"internals/sim_model/","page":"Plant Models","title":"Plant Models","text":"ModelPredictiveControl.steadystate!","category":"page"},{"location":"internals/sim_model/#ModelPredictiveControl.steadystate!","page":"Plant Models","title":"ModelPredictiveControl.steadystate!","text":"steadystate!(model::LinModel, u0, d0)\n\nSet model.x0 to u0 and d0 steady-state if model is a LinModel.\n\nFollowing setop! notation, the method evaluates the equilibrium from:\n\n mathbfx_0 = mathbf(I - A)^-1(B_u u_0 + B_d d_0 + f_op - x_op)\n\nwith constant manipulated inputs mathbfu_0 = u - u_op and measured disturbances mathbfd_0 = d - d_op. The Moore-Penrose pseudo-inverse computes mathbf(I - A)^-1 to support integrating model (integrator states will be 0).\n\n\n\n\n\nDo nothing if model is a NonLinModel.\n\n\n\n\n\n","category":"function"}] +} diff --git a/previews/PR114/siteinfo.js b/previews/PR114/siteinfo.js new file mode 100644 index 00000000..6a7c0baf --- /dev/null +++ b/previews/PR114/siteinfo.js @@ -0,0 +1 @@ +var DOCUMENTER_CURRENT_VERSION = "previews/PR114";