Added a post on observability and detectability

_posts/2023-11-21-controllability-and-stabilisability.md

@@ -154,7 +154,8 @@ $$
 So this form of the system is very useful because it allows us to easily identify the dynamics of
 the system using the poles and zeros.
 
-Now consider the system with state feedback such that $\mathbf{u = Kx}$ \(note some resources use
+Now consider the system with state feedback such that $\mathbf{u = Kx}$ where
+$\mathbf{K} \in \mathbb{R}^{i \times k}$ for $i$ inputs and $k$ states \(note some resources use
 $\mathbf{u = -Kx}$ which is equivalent except that the signs of each element of $\mathbf{K}$ would
 be negated\). Then the system could be rewritten,
 

_posts/2023-12-06-observability-and-detectability.md

@@ -0,0 +1,180 @@
+---
+title: Observability and Detectability
+date: 2023-12-06
+categories: [Mechanics, Control]
+tags:
+ [
+ study,
+ mechanics,
+ dynamics,
+ mathematics,
+ linear systems,
+ control,
+ state space,
+ state feedback,
+ state estimators,
+ observability,
+ ]
+math: true
+toc: true
+---
+
+<!-- prettier-ignore -->
+* TOC
+{:toc}
+
+## State Estimators
+
+Previously I have discussed how [state feedback can be used to stabilise a
+system]({% post_url 2023-11-21-controllability-and-stabilisability %}) and [how to choose these
+parameters for the state feedback using LQR]({% post_url 2023-11-27-lqr %}). However, in both these
+cases it was assumed that we could measure the full state of the system. Unfortunately, this is
+rarely the case in practical implementations. As such, _state estimators_ are used to attempt to
+estimate the state of our system given some measurements.
+
+![A cart that moves left and right to balance a pendulum](/images/cart_pendulum.jpg)_A cart with a
+pendulum attached to it_
+
+For example, consider the cart and pendulum system above and assume the input is the force on the
+cart. Assume that we have encoders on the wheels of the cart and the rotational point of the
+pendulum so that we can measure their position. Despite this, the state of the system is also
+dependent on the _velocities_ of the cart and the pendulum which we are not directly measuring. In
+this case, we may attempt to differentiate the measured positions to determine the velocities
+however if the sensors are noisy then these estimates of velocity may not be accurate. We could
+attempt to add a low pass filter to the encoder measurements before differentiating it to try to
+reduce some of this noise but this will add delay to the measurements which may be unacceptable.
+
+## Observers
+
+### Open Loop Observer
+
+We might instead ask, is there some way that we can incorporate the knowledge about our system
+dynamics into the estimate for the state? To begin, we could create a simulation of the system and
+apply our inputs to it. This simulation would then give us both an estimate for the output
+$\mathbf{\hat{y}}$ and an internal estimate for the state $\mathbf{\hat{x}}$. This state estimate
+could then be used as the state feedback to produce appropriate inputs for the system.
+
+![A block diagram showing an input being fed into the real system and a simulation of the system](/images/open_loop_observer.jpg)_A
+block diagram of an open loop observer_
+
+This known as an open loop observer since we just feed in the input to both systems and then rely on
+our model to perfectly match the real system. Firstly we can define the error to be
+
+$$
+\mathbf{\tilde{x} = x - \hat{x}}
+$$
+
+Then we know,
+
+$$
+\begin{align*}
+ \mathbf{\dot{x} - \dot{\hat{x}}} &= \mathbf{Ax + Bu - A\hat{x} - Bu} \\
+ \implies \mathbf{\tilde{\dot{x}}} &= \mathbf{A\tilde{x}}
+\end{align*}
+$$
+
+From this we know that if our system is stable the error will converge to 0 over time but we can't
+increase the speed at which it will converge. Inaccuracies in the model for our system due to linear
+approximations or parameterisation errors may also lead to errors in the state estimate.
+
+### Closed Loop Observer with Output Feedback
+
+Instead, it would be better to try to incorporate the measurements from the real system to help get
+a better estimate of the state faster. To do this we will define the output feedback gain matrix
+$\mathbf{L} \in \mathbb{R}^{i \times p}$ where $i$ is the number of inputs to the system and $p$ is
+the number of outputs.
+
+![A block diagram for a closed loop observer (equation below)](/images/closed_loop_observer.jpg)_A
+closed loop observer block diagram_
+
+So now we have
+
+$$
+\begin{align*}
+ \mathbf{\dot{\hat{x}}} &= \mathbf{A\hat{x} + Bu + L(Cx - C\hat{x})} \\
+ \implies \mathbf{\dot{\tilde{x}}} &= \mathbf{(A - LC)\tilde{x}}
+\end{align*}
+$$
+
+Therefore, the eigenvalues of $\mathbf{(A - LH)}$ determine how fast the error will converge and
+these can be controlled by choosing an appropriate $\mathbf{L}$ matrix.
+
+## Observability and Detectability
+
+Another question it may be worth asking is can we even reconstruct the full state given our
+measurements? In this example, we know we could have by taking the derivatives of each of the
+measurements but what about a more complex system with coupled modes? For example, consider the
+rigid body with mounts below. This may represent a car's suspension system.
+
+![A rectangular prism rigid body attached to mounts at each corner](/images/rigid_body_mounts_3d.jpg)_A
+rigid body with mounts modelled as springs_
+
+![A view of the rigid body with mounts in one of the planes of perpendicular to the ground](/images/rigid_body_mounts_plane.jpg)_A
+planar view of the rigid body with transverse \(horizontal\) springs to model the transverse
+stiffness of the mounts_
+
+For the planar case, do we need to measure all of $x,\, y$ and $\theta$ to estimate the full state?
+The answer is actually no because $\theta$ and $x$ form coupled swaying and rocking modes \(try
+modelling this and finding the modes to check\). Therefore, if we have a model of the system we
+could just measure one and deduce the other.
+
+Observability and detectability are analogous to [controllability and
+stabilisability]({% post_url 2023-11-21-controllability-and-stabilisability %}) in that they
+determine whether a system's state can be fully reconstructed \(for observability\) or at least all
+of the unstable modes can be reconstructed \(for detectability\) from the measured outputs of the
+system. Just as there is a control canonical form and controllability matrix, there is also an
+observer canonical form \(denoted with subscript o\) and observability matrix \(denoted
+$\mathcal{O}$).
+
+$$
+\begin{align*}
+ \mathbf{F_o} &= \begin{pmatrix}
+ -a_1 & 1 & 0 & \dots & 0 \\
+ -a_2 & 0 & 1 & \dots & 0 \\
+ \vdots & \vdots & \vdots & \ddots & \vdots \\
+ -a_{n-1}& 0 & 0 & \dots & 1 \\
+ -a_n & 0 & 0 & \dots & 0
+ \end{pmatrix}
+ \\
+ \mathbf{B_o} &= \begin{pmatrix} b_1 \\ b_2 \\ \vdots \\ b_n \end{pmatrix},
+ \quad
+ \mathbf{C_o} = \begin{pmatrix} 1 & 0 & \dots & 0 \end{pmatrix}
+ \\
+ \mathcal{O} &= \begin{pmatrix} \mathbf{A} \\ \mathbf{CA} \\ \vdots \\ \mathbf{CA^{n-1}} \end{pmatrix}
+\end{align*}
+$$
+
+This is mathematically the same problem as choosing the state feedback which I showed
+[here]({% post_url 2023-11-21-controllability-and-stabilisability %}) and so these problems are
+known as dual problems. Specifically, the problems are the same if you replace $\mathbf{A}$ with
+$\mathbf{A^T}$, $\mathbf{B}$ with $\mathbf{C^T}$ and $\mathbf{C}$ with $\mathbf{B^T}$. This can be
+shown quite easily since the eigenvalues \(poles\) of a matrix and its transpose are the same.
+Therefore, finding the poles of $\mathbf{A - LC}$ is the same as finding the poles of
+$\mathbf{(A - LC)^T = A^T - C^T L^T}$ which has the same form as the control problem
+$\mathbf{A - BK}$ with the above substitutions. Similarly to the control problem, we can check if a
+system is observable or detectable by checking the rank of $\mathcal{O}$ or using the PBH test.
+
+Great! So if the system is observable we can choose the poles of the observer arbitrarily and make
+the error converge as fast as we like. What's more, the observer is implemented in simulation so
+there is no physical limitation on the gain. So we should just make the gain super large so that the
+system converges the fastest, right? Well, not quite.
+
+## The Noise Problem
+
+Unfortunately, in real systems there is noise. In general, we can categorise the noise as process
+noise, $\mathbf{w}$ which affects the state of the system and sensor noise, $\mathbf{v}$ which
+affects measurements of the output. Mathematically we can model these effects
+
+$$
+\begin{align*}
+ \mathbf{\dot{x}} &= \mathbf{Ax + Bu + B_n w} \\
+ \mathbf{y} &= \mathbf{Cx + v} \\
+ \implies \mathbf{\dot{\tilde{x}}} &= \mathbf{(A - LC) \tilde{x} - B_n w - Lv}
+\end{align*}
+$$
+
+This shows that if the gain of the $\mathbf{L}$ matrix is large then the sensor noise will cause
+large errors. On the other hand, if $\mathbf{L}$ has small gains then the sensor noise will not
+impact the error too much but the response to disturbances will be slow. Therefore, there is a
+tradeoff. So how can we choose an optimal state feedback? Well I'm going to save that for my next
+post!