Gain Scheduled PID Equation 2.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from scipy.integrate import odeint\n",
    "from matplotlib import pyplot as plt\n",
    "\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Equation\n",
    "===\n",
    "\n",
    "The system is governed by the following equation\n",
    "\n",
    "$$ \\dddot{y} + 2.14\\ddot{y} + 9.276\\dot{y} + 4.228y = 4.228x $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def model(y_cap, t, params):\n",
    "    \"\"\"\n",
    "        Given a vector <y, y1, y2> returns <y1, y2, y3> according to equation\n",
    "        above.\n",
    "    \"\"\"\n",
    "    y, y1, y2 = y_cap # y, dy/dt, d2y/dt2\n",
    "    x = params\n",
    "\n",
    "    y3 = -2.14*y2 - 9.276*y1 - 4.228*y + 4.228*x\n",
    "    dy_cap = y1, y2, y3\n",
    "\n",
    "    return dy_cap"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f0e203b27d0>]"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "t = np.linspace(0, 1000, num=1000)\n",
    "params = 2 # let x = 2\n",
    "y_cap = odeint(model, (0, 0, 0), t, args=(params, ))\n",
    "\n",
    "plt.plot(y_cap[0]) # Constant input system does not produce anything"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "class PIDController:\n",
    "    \"\"\"\n",
    "        v = Final output of plant\n",
    "        u = Input signal to the plant\n",
    "    \"\"\"\n",
    "    def __init__(self, Ki, Kp, Kd, dt, initial_control_var, set_point):\n",
    "        self.params = (Ki, Kp, Kd)\n",
    "\n",
    "        self.sum_error = 0\n",
    "        self.dt = dt\n",
    "\n",
    "        self.prev_v = initial_control_var\n",
    "        self.set_point = set_point\n",
    "\n",
    "    def set_set_point(self, sp):\n",
    "        self.set_point = sp\n",
    "\n",
    "    def set_k_constants(Ki, Kp, Kd):\n",
    "        self.params = (Ki, Kp, Kd)\n",
    "\n",
    "    def get_control_signal(self, v):\n",
    "        \"\"\"\n",
    "            Given the latest measurement of v(t) we obtain new u(t)\n",
    "        \"\"\"\n",
    "        Ki, Kp, Kd = self.params\n",
    "        dt = self.dt\n",
    "        \n",
    "        error = self.set_point - v\n",
    "        self.sum_error += error*dt\n",
    "        derror = (v - self.prev_v)/dt\n",
    "        \n",
    "        u = Kp*error + Ki*self.sum_error + Kd*derror\n",
    "\n",
    "        self.prev_v = v\n",
    "        \n",
    "        return u"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Simulation\n",
    "---"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f0e20469c50>]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "t = np.linspace(0, 1000, num=10000)\n",
    "delta_t = t[1]-t[0]\n",
    "\n",
    "SP = np.ones(t.shape[0])\n",
    "SP[500: 1000] = 0\n",
    "SP[1000: 1500] = 15\n",
    "SP[1500: 2000] = 20\n",
    "SP[2000: ] = 10\n",
    "\n",
    "plt.plot(SP)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Gain schedule\n",
    "---"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "controller = PIDController(Ki=0.1, Kp=1.5, Kd=0, dt=delta_t, initial_control_var=0, set_point=SP[0])\n",
    "\n",
    "y_caps = [(0, 0, 0)] # initialize with initial values of y_cap\n",
    "u = np.zeros_like(t)\n",
    "\n",
    "for i in range(len(t)-1):\n",
    "    cur_y = y_caps[-1][0]\n",
    "\n",
    "    u[i+1] = controller.get_control_signal(cur_y) # Get next u from current y\n",
    "    y_cap_solved = odeint(model, y_caps[i], t[i:i+2], args=(u[i+1], ))[-1]\n",
    "\n",
    "    y_caps.append(y_cap_solved)\n",
    "\n",
    "    controller.set_set_point(SP[i])\n",
    "    controller.set_k_constants()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "y_caps = np.array(y_caps)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7f0e20427a50>"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(SP, linestyle='dashed', label=\"Set Point\")\n",
    "plt.plot(y_caps[:, 0], label=\"Response\")\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7f0e1f4d9710>"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(SP-y_caps[:, 0], label=\"Error\")\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.6"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}