LMIs in Control/pages/HInf Optimal Observer

$$ H\infin $$-Optimal observers yield  robust  estimates  of  some  or  all  internal  plant states  by  processing  measurement  data. Robust observers are increasingly  demanded in industry  as they  may provide state  and  parameter  estimates  for  monitoring  and  diagnosis purposes even in the presence of large disturbances such as noise etc. It is there where Kalman filters may tend to fail. State observer is a system that provides estimates of internal states of a given real system, from measurements of the inputs and outputs of the real system. The goal of $$ H_\infin$$ -optimal state estimation is to design an observer that minimizes the $$ H_\infin $$ norm of the closed-loop transfer matrix from w to z.

The System
Consider the continuous-time generalized plant $$ P $$ with state-space realization



\begin{align} \dot x&=Ax+B_1w,\\ y&=C_2x+D_{21}w\\ \end{align}$$

The Data
The matrices needed as input are $$ A,B_1,B_2,C_2,D_{21},D_{11} $$.

The Optimization Problem
The observer gain $$ L $$ is to be designed such that the $$ H\infin $$ of the transfer matrix from w to z, given by

\begin{align} T(s) = C_1(s1-(A-LC_2))^{-1}(B_1-LD_{21})+D_{11}\\ \end{align} $$ is minimized. The form of the observer would be:



\begin{align} \dot{\hat{x}}=A\hat{x} + L(y - \hat{y}),\\ \hat{y} = C_2\hat{x}\\ \end{align}$$

The LMI: $$ H_\infin $$ Optimal Observer
The $$H\infin $$-optimal observer gain is synthesized by solving for $$ P \in \mathbb{S} ^{n_x}, G \in \mathbb{R} ^{n_x \times n_y} $$, and $$ \gamma \in \mathbb{R}_{>0} $$ that minimize $$ \zeta(\gamma) = \gamma $$ subject to $$ P>0 $$ and



\begin{align}

\begin{bmatrix} PA+ A^TP-GC_2-{C_2}^TG^T && PB_1-GD_{21} && C_1^T\\ \star && -\gamma1 && {D_{11}}^T\\ \star && \star && -\gamma1 \end{bmatrix} <0\\ \end{align}$$

Conclusion:
The $$ H_\infin$$ -optimal observer gain is recovered by $$ L = P^{-1}G $$ and the $$ H_\infin $$ norm of T(s) is $$ \gamma$$.

Implementation
Link to the MATLAB code designing $$ H\infin $$- Optimal Observer

https://github.com/Ricky-10/coding107/blob/master/HinfinityOptimalobserver

= External Links =


 * LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
 * LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
 * LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
 * 
 *