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

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 $R\in {ℝ}^{(n-m)\times n}$ be an arbitrarily chosen matrix which makes the matrix

$T=\left[\begin{array}{c}C\\ R\end{array}\right]$

nonsingular, then

$C{T}^{-1}=\left[\begin{array}{cc}{I}_m& 0\end{array}\right]$.

Furthermore, let

$TA{T}^{-1}=\left[\begin{array}{cc}{A}_{11}& A_{12}\\ A_{21}& A_{22}\end{array}\right]$, $A_{11}\in {ℝ}^{m\times m}$,

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

Let

$Tx=\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]$, $TB=\left[\begin{array}{c}{B}_1\\ B_2\end{array}\right]$,

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

$\left[\begin{array}{c}{\dot{x}}_1\\ {\dot{x}}_2\end{array}\right]=\left[\begin{array}{cc}{A}_{11}& A_{12}\\ A_{21}& A_{22}\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]+\left[\begin{array}{c}{B}_1\\ B_2\end{array}\right]u$,

$y=x_1$

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

$\hat{x}(t)=T^{-1}\left[\begin{array}{c}y(t)\\ {\hat{x}}_2(t)\end{array}\right]$.

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

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

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

$P{A}_{22}+A_{22}^{T}P+W{A}_{12}+A_{12}^T{W}^T<0$

2. There exists a symmetric positive definite matrix P satisfying

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

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

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

$H=B_2+L{B}_1,M=I,N=-L,$

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

External Links

A list of references documenting and validating the LMI.

• [1] - LMI in Control Systems Analysis, Design and Applications