User:Margav06/sandbox/Click here to continue/Optimal control systems/H2-Optimal Observer

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 $$ H_2$$ -optimal state estimation is to design an observer that minimizes the $$ H_2 $$ norm of the closed-loop transfer matrix from w to z. Kalman filter is a form of Optimal Observer.

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



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

The Data
The matrices needed as input are $$ A,B,C,D $$.

The Optimization Problem
The task is to design an observer of the following form:



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

The LMI: $$ H_2 $$ Optimal Observer
LMIs in the variables $$ P, G, Z, \nu $$ are given by:



\begin{align}

\begin{bmatrix} PA+ A^TP-GC_2-{C_2}^TG^T && PB_1-GD_{21}\\ \star && -1 \end{bmatrix} <0\\ trZ < \nu \end{align}$$

Conclusion:
The $$ H_2$$ -optimal observer gain is recovered by $$ L = P^{-1}G $$ and the $$ H_2 $$ norm of T(s) is $$ \mu = \sqrt {\nu} $$

Implementation
https://github.com/Ricky-10/coding107/blob/master/H2%20Optimal%20Observer

= External Links =

A list of references documenting and validating the LMI.


 * 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.
 * https://onlinelibrary.wiley.com/doi/abs/10.1002/rnc.1310