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

Quadratic Polytopic Full State Feedback Optimal $$H_{\infty}$$ 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. $$H_{\infty}$$ methods formulate this task as an optimization problem and attempt to minimize the $$H_{\infty}$$ norm of the system.

The System
Consider System with following state-space representation.



\begin{align} \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{align}$$

where $$x\in\mathbb{R}^{m}$$, $$q\in\mathbb{R}^{n}$$ , $$w\in\mathbb{R}^{g}$$, $$A\in\mathbb{R}^{mxm}$$, $$B_1\in\mathbb{R}^{mxn}$$, $$B_2\in\mathbb{R}^{mxg}$$, $$p\in\mathbb{R}^{p}$$ , $$C_1\in\mathbb{R}^{pxm}$$, $$D_{11}\in\mathbb{R}^{pxn}$$,  $$D_{12}\in\mathbb{R}^{pxg}$$, $$z\in\mathbb{R}^{s}$$, $$C_2\in\mathbb{R}^{sxm}$$, $$D_{21}\in\mathbb{R}^{sxn}$$ ,  $$D_{22}\in\mathbb{R}^{sxg}$$ for any $$t\in\mathbb{R}$$.

Add uncertainty to system matrices

A, B_1, B_2, C_1, C_2, D_{11}, D_{12} $$

New state-space representation



\begin{align} \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{align}$$

The Optimization Problem:
Recall the closed-loop in state feedback is: $$ S(P,K) = $$

\begin{align} \begin{bmatrix} A + B_{2}F&&B_1\\C_1 + D_{12}F&& D_{11}\end{bmatrix}\\ \end{align}$$ This problem can be formulated as $$ 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{align} \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{align}$$

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

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

Related LMIs
Full State Feedback Optimal $H_{\infty}$ Controller