LMIs in Control/H2Optimalobserver

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

$\begin{array}{cc}\dot{x}& =Ax+B_{1}w(t),\\ y& =C_{2}x+D_{21}w\\ \end{array}$

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:

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

The LMI: $H_2$ Optimal Observer

LMIs in the variables $P,G,Z,\nu$ are given by:

$\begin{array}{c}\left[\begin{array}{ccc}PA+A^{T}P-G{C}_2-{C_2}^T{G}^T& & P{B}_1-G{D}_{21}\\ \star & & -1\end{array}\right]<0\\ trZ<\nu \end{array}$

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