# LMIs in Control/pages/quadratic polytopic h2 optimal state feedback control

LMIs in Control/pages/quadratic polytopic h2 optimal state feedback control

## Quadratic Polytopic Full State Feedback Optimal ${\displaystyle H_{2}}$ 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 on performance specifications given, such as requiring stability or bounding the overshoot of the output. By minimizing the ${\displaystyle H_{2}}$ norm of this system we are minimizing the effect noise has on the system as part of the performance specifications.

## 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 Data

The matrices necessary for this LMI are

## The Optimization Problem:

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

{\displaystyle {\begin{aligned}{\begin{bmatrix}A+B_{22}F&&B_{1}\\C_{1}+D_{12}F&&D_{11}\end{bmatrix}}\\\end{aligned}}}

This problem can be formulated as ${\displaystyle H_{2}}$ optimal state-feedback, where K is a controller gain matrix.

## The LMI: An LMI for Quadratic Polytopic ${\displaystyle H_{2}}$ Optimal

State-Feedback Control
${\displaystyle ||S(P(\Delta ),K(0,0,0,F))||_{H_{2}}\leq \gamma }$
${\displaystyle X>0}$

{\displaystyle {\begin{aligned}{\begin{bmatrix}AX+B_{2}Z+XA^{T}+Z^{T}B_{2}^{T}&&B_{1}\\B_{1}^{T}&&-I\end{bmatrix}}+{\begin{bmatrix}A_{i}X+B_{2,i}Z+XA_{i}^{T}+Z^{T}B_{2,I}^{T}&&B_{1,i}\\B_{1,i}^{T}&&0\end{bmatrix}}<0\quad i=1,......,k\end{aligned}}}
{\displaystyle {\begin{aligned}{\begin{bmatrix}X&&(C_{1}X+D_{12}Z)^{T}\\C_{1}X+D_{12}Z&&W\end{bmatrix}}+{\begin{bmatrix}0&&(C_{1,i}X+D_{12,i}Z)^{T}\\C_{1,i}X+D_{12,i}Z&&0\end{bmatrix}}>0\quad i=1,......,k\end{aligned}}}
{\displaystyle {\begin{aligned}\\TraceW<\gamma \end{aligned}}}

## Conclusion:

The ${\displaystyle H_{2}}$ Optimal State-Feedback Controller is recovered by ${\displaystyle F=ZX^{-1}}$