# LMIs in Control/pages/Quadratic Polytopic Hinf- Optimal State Feedback Control

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

For a system having polytopic uncertainties, 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. ${\displaystyle H_{\infty }}$  methods formulate this task as an optimization problem and attempt to minimize the ${\displaystyle H_{\infty }}$  norm of the system.

## The System

Consider System with following state-space representation.

{\displaystyle {\begin{aligned}{\dot {x}}(t)&=Ax(t)+B_{1}q(t)+B_{2}w(t)\\p(t)&=C_{1}x(t)+D_{11}q(t)+D_{12}w(t)\\z(t)&=C_{2}x(t)+D_{21}q(t)+D_{22}w(t)\\\end{aligned}}}

where ${\displaystyle x\in \mathbb {R} ^{m}}$  , ${\displaystyle q\in \mathbb {R} ^{n}}$  , ${\displaystyle w\in \mathbb {R} ^{g}}$ , ${\displaystyle A\in \mathbb {R} ^{mxm}}$ , ${\displaystyle B_{1}\in \mathbb {R} ^{mxn}}$ , ${\displaystyle B_{2}\in \mathbb {R} ^{mxg}}$ , ${\displaystyle p\in \mathbb {R} ^{p}}$  , ${\displaystyle C_{1}\in \mathbb {R} ^{pxm}}$ , ${\displaystyle D_{11}\in \mathbb {R} ^{pxn}}$ , ${\displaystyle D_{12}\in \mathbb {R} ^{pxg}}$ , ${\displaystyle z\in \mathbb {R} ^{s}}$ , ${\displaystyle C_{2}\in \mathbb {R} ^{sxm}}$ , ${\displaystyle D_{21}\in \mathbb {R} ^{sxn}}$  , ${\displaystyle D_{22}\in \mathbb {R} ^{sxg}}$  for any ${\displaystyle t\in \mathbb {R} }$ .

${\displaystyle A,B_{1},B_{2},C_{1},C_{2},D_{11},D_{12}}$

New state-space representation

{\displaystyle {\begin{aligned}{\dot {x}}(t)&=(A+A_{i})x(t)+(B_{1}+B_{i})q(t)+(B_{2}+B_{i})w(t)\\p(t)&=(C_{1}+C_{i})x(t)+(D_{11}+D_{i})q(t)+(D_{12}+D_{i})w(t)\\z(t)&=C_{2}x(t)+D_{21}q(t)+D_{22}w(t)\\\end{aligned}}}

## The Optimization Problem:

Recall the closed-loop in state feedback is:
${\displaystyle S(P,K)=}$

{\displaystyle {\begin{aligned}{\begin{bmatrix}A+B_{2}F&&B_{1}\\C_{1}+D_{12}F&&D_{11}\end{bmatrix}}\\\end{aligned}}}

This problem can be formulated as ${\displaystyle H\infty }$  optimal state-feedback, where K is a controller gain matrix.

## The LMI:

An LMI for Quadratic Polytopic ${\displaystyle H\infty }$  Optimal State-Feedback Control ${\displaystyle ||S(P(\Delta ),K(0,0,0,F))||_{H\infty }\leq \gamma }$
${\displaystyle Y>0}$

{\displaystyle {\begin{aligned}{\begin{bmatrix}Y(A+A_{i})^{T}+(A+A_{i})Y+Z^{T}(B_{2}+B_{1,i})^{T}+(B_{2}+B_{1,i})Z&&*^{T}&&*^{T}\\(B_{1}+B_{1,i})^{T}&&-\gamma I&&*^{T}\\(C_{1}+C_{1,i})Y+(D_{12}+D_{12,i})Z&&(D_{11}+D_{11,i})&&-\gamma I\end{bmatrix}}<0\end{aligned}}}

## Conclusion:

The ${\displaystyle H\infty }$  Optimal State-Feedback Controller is recovered by ${\displaystyle F=ZY^{-1}}$
Controller will determine the bound ${\displaystyle \gamma }$  on the ${\displaystyle H_{\infty }}$  norm of the system.