# LMIs in Control/pages/MixedH2HinfinityOptimalobserver

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

## The System

Consider the continuous-time generalized plant ${\displaystyle P}$ with state-space realization

{\displaystyle {\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 ${\displaystyle (A,C_{2})}$ is detectable.

## The Data

The matrices needed as input are ${\displaystyle A,B_{1},B_{2},C_{2},D_{21},D_{11}}$.

## The Optimization Problem

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

{\displaystyle {\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:

{\displaystyle {\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 ${\displaystyle L\in \mathbb {R} ^{n_{x}\times n_{y}}}$ is the observer gain.

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

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

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

## Implementation

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