# LMIs in Control/pages/Mixed H2 HInf Optimal Observer

The goal of mixed $H_{2}-H\infty$ -optimal state estimation is to design an observer that minimizes the $H_{2}$ norm of the closed-loop transfer matrix from $w_{1}$ to $z_{1}$ , while ensuring that the $H\infty$ norm of the closed-loop transfer matrix from $w_{2}$ to $z_{2}$ is below a specified bound.

## The System

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

{\begin{aligned}{\dot {x}}&=Ax+B_{1,1}w_{1}+B_{1,2}w_{2},\\y&=C_{2}x+D_{21,1}w_{1}+D_{21,1}w_{2}\\\end{aligned}}

where it is assumed that $(A,C_{2})$  is detectable.

## 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 to minimize the $H_{2}$  norm of the closed-loop transfer matrix $T_{11}(s)$  from the exogenous input $w_{1}$  to the performance output $z_{1}$  while ensuring the $H\infty$  norm of the closed-loop transfer matrix $T_{22}(s)$  from the exogenous input $w_{2}$  to the performance output $z_{2}$  is less than $\gamma _{d}$ , where

{\begin{aligned}T_{11}(s)=C_{1,1}(s1-(A-LC_{2}))^{-1}(B_{1,1}-LD_{21,1})\\T_{22}(s)=C_{1,2}(s1-(A-LC_{2}))^{-1}(B_{1,2}-LD_{21,2})+D_{11,22}\end{aligned}}

is minimized. The form of the observer would be:

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

is to be designed, where $L\in \mathbb {R} ^{n_{x}\times n_{y}}$  is the observer gain.

## The LMI: $H_{\infty }$ Optimal Observer

The mixed $H_{2}-H\infty$ -optimal observer gain is synthesized by solving for $P\in \mathbb {S} ^{n_{x}},G\in \mathbb {R} ^{n_{x}\times n_{y}}$ , and $\nu \in \mathbb {R} _{>0}$  that minimize $\zeta (\nu )=\nu$  subject to $P>0,Z>0$ ,

{\begin{aligned}{\begin{bmatrix}PA+A^{T}P-GC_{2}-{C_{2}}^{T}G^{T}&&PB_{1}-GD_{21}\\\star &&-1\end{bmatrix}}<0\\{\begin{bmatrix}PA+A^{T}P-GC_{2}-{C_{2}}^{T}G^{T}&&PB_{1}-GD_{21}&&C_{1}\\\star &&-\gamma 1&&{D_{11}}^{T}\\\star &&\star &&-\gamma 1\end{bmatrix}}<0\\{\begin{bmatrix}P&&C{_{1,1}}^{T}\\\star &&Z\end{bmatrix}}>0\\trZ<\nu \end{aligned}}

## Conclusion:

The mixed $H_{2}-H_{\infty }$  -optimal observer gain is recovered by $L=P^{-1}G$  , the $H_{2}$  norm of $T_{11}(s)$  is less than $\mu ={\sqrt {\nu }}$  and the $H_{\infty }$  norm of T(s) is less than $\gamma _{d}$ .

## Implementation

Link to the MATLAB code designing $H_{2}-H\infty$ - Optimal Observer