# LMIs in Control/pages/Full-State Feedback Optimal Control Hinf LMI

## Full State Feedback Optimal $H_{\infty }$ Control

Full State Feedback is a control technique that attempts to place the system's closed loop system poles in specified locations based off of performance specifications given. $H_{\infty }$ methods formulate this task as an optimization problem and attempt to minimize the $H_{\infty }$ norm of the system. In a single-input single-output (SISO) system this norm represents the maximum gain on a magnitude Bode plot. In the case of multi-input multi-output (MIMO) systems it can be interpreted as maximum response to a perturbation introduced to the system. In either, by minimizing the $H_{\infty }$ we are minimizing the worst case effect of a disturbance to the system, whether it is noise or another perturbation.

## The System

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

${\begin{bmatrix}{\dot {x}}\\z\\y\end{bmatrix}}={\begin{bmatrix}A&B_{1}&B_{2}\\C_{1}&D_{11}&D_{12}\\C_{2}&D_{21}&D_{22}\end{bmatrix}}{\begin{bmatrix}x\\w\\u\end{bmatrix}}$ where $x(t)\in \mathbb {R} ^{n}$ is the state, $z(t)\in \mathbb {R} ^{p}$ is the regulated output, $y(t)\in \mathbb {R} ^{q}$ is the sensed output, $w(t)\in \mathbb {R} ^{r}$ is the exogenous input, and $u(t)\in \mathbb {R} ^{m}$ is the actuator input, at any $t\in \mathbb {R}$ .

The lower linear fractional transformation (LFT) is used to implement a controller $K$ into the system. The lower LFT is denoted as ${\underline {S}}(P,K)$ and is formed by ${\underline {S}}(P,K)=P_{11}+P_{12}(I-KP_{22})^{-1}KP_{21}$ with ${\begin{bmatrix}z\\y\end{bmatrix}}={\begin{bmatrix}P_{11}&P_{12}\\P_{21}&P_{22}\end{bmatrix}}{\begin{bmatrix}w\\u\end{bmatrix}}$ . For full-state feedback we consider a controller of the form $u(t)=Fx(t)$ . This is a special case where $y(t)=x(t)$ and results in a controller of the form $K={\begin{bmatrix}0&0\\0&F\end{bmatrix}}$ .

## The Data

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

## The LMI:Full State Feedback Optimal $H_{\infty }$ Control LMI

The following are equivalent.

1) There exists a $F$ such that $||{\underline {S}}(P,K(0,0,0,F)||_{H_{\infty }}\leq \gamma$ 2) There exists $Y>0$ and $Z$ such that

${\begin{bmatrix}YA^{T}+AY+Z^{T}B_{2}^{T}+B_{2}Z&B_{1}&YC_{1}^{T}+Z^{T}D_{12}^{T}\\B1_{1}^{T}&-\gamma I&D_{11}^{T}\\C_{1}Y+D_{12}Z&D_{11}&-\gamma I\end{bmatrix}}<0$ .

Then $F=ZY^{-1}$ .

## Conclusion:

The above LMI, if feasible, will determine the bound $\gamma$ on the $H_{\infty }$ norm of the system. In addition to this $F$ is also determined allowing the closed loop system to be determined using the controller ${\hat {K}}(0,0,0,F)$ found during the optimization.

## Implementation

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