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

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}$$

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_22F&&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 $$ F = ZY^{-1} $$

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

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