LMIs in Control/pages/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 ${\displaystyle H_{2}}$ -optimal state estimation is to design an observer that minimizes the ${\displaystyle H_{2}}$ 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 ${\displaystyle P}$ with state-space realization

{\displaystyle {\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 ${\displaystyle A,B,C,D}$.

The Optimization Problem

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

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

The LMI: ${\displaystyle H_{2}}$ Optimal Observer

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

{\displaystyle {\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 ${\displaystyle H_{2}}$ -optimal observer gain is recovered by ${\displaystyle L=P^{-1}G}$ and the ${\displaystyle H_{2}}$ norm of T(s) is ${\displaystyle \mu ={\sqrt {\nu }}}$