LMIs in Control/LMI for Polytopic Uncertainity/Quadratic Polytopic Hinf- Optimal State Feedback Control

The System:
Consider System with following state-space representation.

{\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}}

Add uncertainty to system matrices

A,B_{1},B_{2},C_{1},C_{2},D_{11},D_{12}

New state-space representation

{\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:
$S(P,K)=$

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

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

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

{\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 $H\infty$ Optimal State-Feedback Controller is recovered by $F=ZY^{-1}$

Implementation: https://github.com/JalpeshBhadra/LMI/tree/master

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