# LMIs in Control/pages/Optimal Output Feedback H2 LMI

## Optimal Output Feedback ${\displaystyle H_{2}}$ LMI

Similar to state feedback, output feedback is necessary when information about the output is not known. Often techniques such as Kalman filtering are implemented to tackle this problem. The method below, however, does not use a filtering technique and instead uses a combination of LMI constraints to perform the output feedback as well as find the minimal bound on the ${\displaystyle H_{2}}$  norm of the system. ${\displaystyle H_{2}}$  is often done using more classical tools such as Riccati equations. More recently LMI techniques have been created to solve problems such as full state feedback or output feedback as seen below.

## 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 ${\displaystyle H_{2}}$ 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 ${\displaystyle ||S(K,P)||_{H_{2}}<\gamma }$

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

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

## Conclusion:

The above LMI determines the the upper bound ${\displaystyle \gamma }$  on the H2 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_H2.m