LMIs in Control/pages/reduced order state estimation

WIP, Description in progress

In this page, we investigate an LMI approach for the design of the problem of reduced-order observer design for the linear system.

System Setting
$$ \dot{x} = Ax + Bu $$

$$ y = Cx$$

where $$x \in \mathbb{R}^n, \ u \in \mathbb{R}^r, \text{ and } y \in \mathbb{R}^m$$ are the state vector, the input vector, and the output vector, respectively. Without loss of generality, it is assumed that rank$$(C) = m \leq n$$.

In the design of reduced-order state observers for linear systems, the following lemma performs a fundamental role.

Lemma
Given the linear system, and let $$R \in \mathbb{R}^{(n-m)\times n}$$ be an arbitrarily chosen matrix which makes the matrix

$$T=\begin{bmatrix}C \\ R \end{bmatrix} $$

nonsingular, then

$$CT^{-1}=\begin{bmatrix}I_m & 0 \end{bmatrix}$$.

Furthermore, let

$$TAT^{-1}=\begin{bmatrix}A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix}$$, $$A_{11} \in \mathbb{R}^{m\times m}$$,

then the matrix pair $$(A_{22}, A_{12})$$ is detectable if and only if $$(A, C)$$ is detectable.

Let

$$Tx=\begin{bmatrix}x_1 \\ x_2 \end{bmatrix} $$, $$TB=\begin{bmatrix}B_1 \\ B_2 \end{bmatrix} $$,

then it follows from the relations in previous 3 equations that system is equivalent to

$$\begin{bmatrix}\dot{x}_1 \\ \dot{x}_2 \end{bmatrix}=\begin{bmatrix}A_{11} & A_{12} \\ A_{21} & A_{22}\end{bmatrix}\begin{bmatrix}x_1 \\ x_2 \end{bmatrix} + \begin{bmatrix}B_1 \\ B_2 \end{bmatrix}u$$,

$$y=x_1$$

In the equivalent system, the substate vector $$x_1$$ is directly equal to the output $$y$$ of the original system. Thus to reconstruct the state of the original system, we suffice to get an estimate of the substate vector $$x_2$$, namely, $$\hat{x}_2$$, from the earlier equivalent system. Once an estimate $$\hat{x}_2$$ is obtained, an estimate of $$x(t)$$, that is, the state vector of original system, can be obtained as

$$\hat{x}(t)=T^{-1}\begin{bmatrix}y(t) \\ \hat{x}_2(t) \end{bmatrix} $$.

Problem Formulation
For the equivalent continuous-time linear system, design a reduced-order state observer in the form of

$$\dot{z} = Fz+Gy+Hu$$

$$\hat{x}_2=Mz+Ny$$

such that for arbitrary control input $$u(t)$$, and arbitrary initial system values $$x1(0),\ x2(0), \text{ and } z(0)$$, there holds

$$\text{lim}_{t\rightarrow \infty} (x_2(t) - \hat{x}_2(t) = 0$$.

Solution/Theorem
Problem has a solution if and only if one of the following two conditions holds:

1. There exist a symmetric positive definite matrix P and a matrix W satisfying

$$PA_{22}+A_{22}^TP+WA_{12}+A_{12}^TW^T < 0$$

2. There exists a symmetric positive definite matrix P satisfying

$$PA_{22}+A_{22}^TP-A_{12}A_{12}^T < 0$$

In this case, a reduced-order state observer can be obtained as in problem with

$$F=A_{22}+LA_{12}, \ G=(A_{21}+LA_{11})-(A_{22}+LA_{12})L,$$

$$H=B_{2}+LB_{1}, \ M=I, \ N=-L,$$

where

$$ L = P^{-1}W $$ with W and $$P >0$$ being a pair of feasible solutions to the first inequality condition or $$ L = -\frac{1}{2}P^{-1}A_{12}^T $$ with $$P>0$$ being a solution to the second inequality condition.

WIP, additional references to be added