# LMIs in Control/pages/Optimal Output Feedback Hinf LMI

## Optimal Output Feedback ${\displaystyle H_{\infty }}$ LMI

Optimal output feedback control is a problem which arises from not knowing all information about the output of the system. It correlates to the state feedback situation where the part of the state is unknown. This issue can arise in decentralized control problems, for example, and requires the use of an "observer-like" solution. One such method is the use of a Kalman Filter, a more classical technique. However, other methods exist that do not implement a Kalman Filter such as the one below which uses an LMI to preform the output feeback. The ${\displaystyle H_{\infty }}$ control methods form an optimization problem which attempts to minimize the ${\displaystyle H_{\infty }}$ norm of the system.

## The System

The system is represented using the 9-matrix notation shown below.

${\displaystyle {\begin{bmatrix}{\dot {x}}\\z\\y\end{bmatrix}}={\begin{bmatrix}A&B_{1}&B_{2}\\C_{1}&D_{11}&D_{12}\\C_{2}&D_{21}&D_{22}\end{bmatrix}}{\begin{bmatrix}x\\w\\u\end{bmatrix}}}$

where ${\displaystyle x(t)\in \mathbb {R} ^{n}}$ is the state, ${\displaystyle z(t)\in \mathbb {R} ^{p}}$ is the regulated output, ${\displaystyle y(t)\in \mathbb {R} ^{q}}$ is the sensed output, ${\displaystyle w(t)\in \mathbb {R} ^{r}}$ is the exogenous input, and ${\displaystyle u(t)\in \mathbb {R} ^{m}}$ is the actuator input, at any ${\displaystyle t\in \mathbb {R} }$.

## The Data

${\displaystyle A}$, ${\displaystyle B_{1}}$, ${\displaystyle B_{2}}$, ${\displaystyle C_{1}}$, ${\displaystyle C_{2}}$, ${\displaystyle D_{11}}$, ${\displaystyle D_{12}}$, ${\displaystyle D_{21}}$, ${\displaystyle D_{22}}$ are known.

## The LMI: Optimal Output Feedback ${\displaystyle H_{\infty }}$ Control LMI

The following are equivalent.

1) There exists a ${\displaystyle {\hat {K}}={\begin{bmatrix}A_{K}&B_{K}\\C_{K}&D_{K}\end{bmatrix}}}$ such that ${\displaystyle ||S(K,P)||_{H_{\infty }}<\gamma }$

2) There exists ${\displaystyle X_{1}}$, ${\displaystyle Y_{1}}$, ${\displaystyle Z}$, ${\displaystyle A_{n}}$, ${\displaystyle B_{n}}$, ${\displaystyle C_{n}}$, ${\displaystyle D_{n}}$ such that

${\displaystyle {\begin{bmatrix}X_{1}&I\\I&Y_{1}\end{bmatrix}}>0}$
${\displaystyle {\begin{bmatrix}AY_{1}+Y_{1}A^{\text{T}}+B_{2}C_{n}+C_{n}B_{2}^{\text{T}}&*^{\text{T}}&*^{\text{T}}&*^{\text{T}}\\A^{\text{T}}+A_{n}+(B_{2}D_{n}C_{2})^{\text{T}}&X_{1}A+A^{\text{T}}+B_{n}C_{2}+C_{2}^{\text{T}}B_{n}^{\text{T}}&*^{\text{T}}&*^{\text{T}}\\(B_{1}+B_{2}D_{n}D_{21})^{\text{T}}&(X_{1}B_{1}+B_{n}D_{21})^{\text{T}}&-\gamma I&*^{\text{T}}\\C_{1}Y_{1}+D_{12}C_{n}&C_{1}+D_{12}D_{n}C_{2}&D_{11}+D_{12}D_{n}D_{21}&-\gamma I\\\end{bmatrix}}<0}$

## Conclusion:

The above LMI determines the the upper bound ${\displaystyle \gamma }$ on the ${\displaystyle H_{\infty }}$ norm. In addition to this the controller ${\displaystyle {\hat {K}}(A_{K},B_{K},C_{K},D_{K})}$ can also be recovered.

${\displaystyle D_{K}=(I+D_{K2}D_{22})^{-1}D_{K2}}$
${\displaystyle B_{K}=B_{K2}(I+D_{22}D_{K})}$
${\displaystyle C_{K}=(I+D_{K}D_{22})C_{K2}}$
${\displaystyle A_{K}=A_{K2}-B_{K}(I+D_{22}D_{K})^{-1}D_{22}C_{K}}$

where,

${\displaystyle {\begin{bmatrix}A_{K2}&B_{K2}\\C_{K2}&D_{K2}\end{bmatrix}}={\begin{bmatrix}X_{2}&X_{1}B_{2}\\0&I\end{bmatrix}}^{-1}\left[{\begin{bmatrix}A_{n}&B_{n}\\C_{n}&D_{n}\end{bmatrix}}-{\begin{bmatrix}X_{1}AY_{1}&0\\0&0\end{bmatrix}}\right]{\begin{bmatrix}Y_{2}^{T}&0\\C_{2}Y_{1}&I\end{bmatrix}}^{-1}}$

for any full-rank ${\displaystyle X_{2}}$ and ${\displaystyle Y_{2}}$ such that

${\displaystyle {\begin{bmatrix}X_{1}&X_{2}\\X_{2}^{T}&X_{3}\end{bmatrix}}={\begin{bmatrix}Y_{1}&Y_{2}B_{2}\\Y_{2}^{T}&Y_{3}\end{bmatrix}}^{-1}}$.

## Implementation

This implementation requires Yalmip and Sedumi. https://github.com/eoskowro/LMI/blob/master/OF_Hinf.m