# 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.
WIP, additional references to be added

## External Links

A list of references documenting and validating the LMI.
* - LMI in Control Systems Analysis, Design and Applications