# LMIs in Control/pages/Discrete-Time Mixed H2 HInf Optimal Observer

LMIs in Control/pages/Discrete-Time Mixed H2 HInf Optimal Observer

In many applications, perhaps even most, the state of the system cannot be directly known. In this case, you will need to strategically to measure key system outputs that will make the system states indirectly observable. Observers need to converge much faster than the system dynamics in order for their estimations to be accurate. Optimal observer synthesis is therefore advantageous. In this LMI, we seek to optimize both H2 and Hinf norms, to minimize both the average and the maximum error of the observer.

## The System

{\displaystyle {\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 ${\displaystyle x\in R^{n}}$  and is the state vector, ${\displaystyle A\in R^{n*n}}$  and is the state matrix, ${\displaystyle B\in R^{n*r}}$  and is the input matrix, ${\displaystyle w\in R^{r}}$  and is the exogenous input, ${\displaystyle C\in R^{m*n}}$  and is the output matrix, ${\displaystyle D\in R^{m*r}}$  and is the feedthrough matrix, ${\displaystyle y\in R^{m}}$  and is the output, and it is assumed that ${\displaystyle (A_{d},C_{d2})}$  is detectable.

${\displaystyle A\in R^{n*n}}$


## The Data

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

## The Optimization Problem

An observer of the form:

{\displaystyle {\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 ${\displaystyle L_{d}\in R^{n_{x}*n_{y}}}$  is the observer gain.

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

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

and the performance output is defined as

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

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

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

{\displaystyle {\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

where ${\displaystyle tr}$  refers to the trace of a matrix.

## Conclusion:

The mixed-${\displaystyle H_{2}H_{inf}}$ -optimal observer gain is recovered by ${\displaystyle L_{d}=P^{-1}G_{d}}$ , the ${\displaystyle H_{2}}$  norm of ${\displaystyle T_{11}(z)}$  is less than ${\displaystyle \mu ={\sqrt {v}}}$ , and the ${\displaystyle H_{inf}}$  norm of ${\displaystyle T_{22}(z)}$  is less than ${\displaystyle \gamma _{d}}$ . This result gives us a matrix of observer gains ${\displaystyle L_{d}}$  that allow us to optimally observe the states of the system indirectly as:

{\displaystyle {\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}}}

## Implementation

This implementation requires Yalmip and Sedumi.