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

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

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:

${\begin{aligned}{\dot {\hat {x}}}=A{\hat {x}}+L(y-{\hat {y}}),\\{\hat {y}}=C_{2}{\hat {x}}\\\end{aligned}}$

The LMI: $H_{2}$ Optimal Observer

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

${\begin{aligned}{\begin{bmatrix}PA+A^{T}P-GC_{2}-{C_{2}}^{T}G^{T}&&PB_{1}-GD_{21}\\\star &&-1\end{bmatrix}}<0\\trZ<\nu \end{aligned}}$

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