# LMIs in Control/pages/discrete time mixed h2 hinf optimal observer

LMIs in Control/pages/discrete time mixed h2 hinf optimal observer

## The System

{\begin{aligned}x_{k+1}&=A_{d}x_{k}+B_{d1,1}w_{1,k}+B_{d1,2}w_{2,k},\\y_{k}&=C_{c2}x_{k}+D_{d21,1}w_{1,k}+D_{d21,2}w_{2,k}\\\end{aligned}} Where it is assumed that :$(A_{d},C_{d2})$ is detectable.

## The Data

The matrices $A_{d},B_{d1},C_{cd2},C_{cd1},D_{d21}$ .

## The Optimization Problem

An observer of the form:

{\begin{aligned}{\hat {x}}_{k+1}&=A_{d}{\hat {x}}_{k}+L_{d}(y_{k}-{\hat {y}}_{k}),\\{\hat {y}}_{k}&=C_{d2}{\hat {x}}_{k}\\\end{aligned}} is to be designed, where $L_{d}\in R^{n_{x}*n_{y}}$ is the observer gain.

Defining the error state $e_{k}=x_{k}-{\hat {x}}_{k}$ , the error dynamics are found to be

$e_{k+1}=(A_{d}-L_{d}C_{d2})e_{k}+(B_{d1,1}-L_{d}D_{d21,1})w_{1,k}+(B_{d1,2}-L_{d}D_{d21,2})w_{2,k}$ ,

and the performance output is defined as

${\begin{bmatrix}Z_{1,k}\\Z_{2,k}\end{bmatrix}}={\begin{bmatrix}C_{d1,1}\\C_{d1,2}\end{bmatrix}}e_{k}+{\begin{bmatrix}0&D_{d11,12}\\D_{d11,21}&D_{d11,22}\end{bmatrix}}{\begin{bmatrix}w_{1,k}\\w_{2,k}\end{bmatrix}}$ .

The observer gain $L_{d}$ is to be designed to minimize the $H_{2}$ norm of the closed loop transfer matrix $T_{11}(z)$ from the exogenous input $w_{2,k}$ to the performance output $z_{2,k}$ is less than $\gamma _{d}$ , where

{\begin{aligned}T_{11}(z)&=C_{d1,1}(z1-(A_{d}-L_{d}C_{d2}))^{-1}(B_{d1,1}-L_{d}D_{d21,1}),\\T_{22}(z)&=C_{d1,2}(z1-(A_{d}-L_{d}C_{d2}))^{-1}(B_{d1,2}-L_{d}D_{d21,2})+D_{d11,22}\end{aligned}} ## The LMI: Discrete-Time Mixed H2-Hinf-Optimal Observer

The discrete-time mixed-$H_{2}H_{inf}$ -optimal observer gain is synthesized by solving for $P\in S^{n_{x}}$ , $Z\in S^{n_{z}}$ , $G_{d}\in R^{n_{x}*n_{y}}$ , and $v\in R_{>0}$ that minimize J$(v)=v$ subject to $P>0,Z>0$ ,

{\begin{aligned}{\begin{bmatrix}P&PA_{d}-G_{d}C_{d2}&PB_{d1,1}-G_{d}D_{d21,1}\\*&P&0\\*&*&1\end{bmatrix}}&>0,\\{\begin{bmatrix}P&PA_{d}-G_{d}C_{d2}&PB_{d1,2}-G_{d}D_{d21,2}&0\\*&P&0&C_{d1,2}^{T}\\*&*&\gamma _{d}1&D_{d11,22}^{T}\\*&*&*&\gamma _{d}1\end{bmatrix}}&>0,\\{\begin{bmatrix}Z&PC_{d1,1}\\*&P\end{bmatrix}}&>0,\\trZ ## Conclusion:

The mixed-$H_{2}H_{inf}$ -optimal observer gain is recovered by $L_{d}=P^{-1}G_{d}$ , the $H_{2}$ norm of $T_{11}(z)$ is less than $\mu ={\sqrt {v}}$ , and the $H_{inf}$ norm of $T_{22}(z)$ is less than $\gamma _{d}$ .

## Implementation

WIP! NEEDS TO BE UPDATED: THE CURRENTLY LINKED CODE IS FOR DISCRETE TIME H2 OPTIMAL OBSERVER: https://github.com/rezajamesahmed/LMImatlabcode/blob/master/Discrete_Time_H2_Optimal_Observer_LMIs_Wikibook_Example.m

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