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

$PA_{22}+A_{22}^{T}P+WA_{12}+A_{12}^{T}W^{T}<0$

2. There exists a symmetric positive definite matrix P satisfying

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

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.