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

${\displaystyle {\dot {x}}=Ax+Bu}$

${\displaystyle y=Cx}$

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

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

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

nonsingular, then

${\displaystyle CT^{-1}={\begin{bmatrix}I_{m}&0\end{bmatrix}}}$ .

Furthermore, let

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

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

Let

${\displaystyle Tx={\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}}$ , ${\displaystyle TB={\begin{bmatrix}B_{1}\\B_{2}\end{bmatrix}}}$ ,

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

${\displaystyle {\begin{bmatrix}{\dot {x}}_{1}\\{\dot {x}}_{2}\end{bmatrix}}={\begin{bmatrix}A_{11}&A_{12}\\A_{21}&A_{22}\end{bmatrix}}{\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}+{\begin{bmatrix}B_{1}\\B_{2}\end{bmatrix}}u}$ ,

${\displaystyle y=x_{1}}$

In the equivalent system, the substate vector ${\displaystyle x_{1}}$  is directly equal to the output ${\displaystyle y}$  of the original system. Thus to reconstruct the state of the original system, we suffice to get an estimate of the substate vector ${\displaystyle x_{2}}$ , namely, ${\displaystyle {\hat {x}}_{2}}$ , from the earlier equivalent system. Once an estimate ${\displaystyle {\hat {x}}_{2}}$  is obtained, an estimate of ${\displaystyle x(t)}$ , that is, the state vector of original system, can be obtained as

${\displaystyle {\hat {x}}(t)=T^{-1}{\begin{bmatrix}y(t)\\{\hat {x}}_{2}(t)\end{bmatrix}}}$ .

Problem Formulation

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

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

${\displaystyle {\hat {x}}_{2}=Mz+Ny}$

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

${\displaystyle {\text{lim}}_{t\rightarrow \infty }(x_{2}(t)-{\hat {x}}_{2}(t)=0}$ .

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

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

2. There exists a symmetric positive definite matrix P satisfying

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

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

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

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

where

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