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Quadratic Polytopic Full State Feedback Optimal $$H_2$$ 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 $$H_2$$ 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.



\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 Data
The matrices necessary for this LMI are

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

\begin{align} \begin{bmatrix} A + B_{22}F&&B_1\\C_1 + D_{12}F&& D_{11}\end{bmatrix}\\ \end{align}$$ This problem can be formulated as $$ H_2 $$ optimal state-feedback, where K is a controller gain matrix.

The LMI: An LMI for Quadratic Polytopic $$ H_2 $$ Optimal
State-Feedback Control $$ ||S(P(\Delta),K(0,0,0,F))||_{H_2} \leq \gamma $$ $$ X > 0 $$

\begin{align} \begin{bmatrix} AX + B_2Z + XA^T + Z^TB^T_{2} &&B_1 \\B_1^T&&-I \end{bmatrix} +

\begin{bmatrix} A_iX + B_{2,i}Z + XA^T_i + Z^TB^T_{2,I} &&B_{1,i} \\B_{1,i}^T&&0 \end{bmatrix} < 0 \quad i = 1,......,k

\end{align}$$



\begin{align} \begin{bmatrix} X && (C_1X+D_{12}Z)^T \\C_1X+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{align}$$



\begin{align} \\TraceW < \gamma \end{align}$$

Conclusion:
The $$ H_2 $$ Optimal State-Feedback Controller is recovered by $$ F = ZX^{-1} $$

Implementation:
https://github.com/JalpeshBhadra/LMI/blob/master/H2_optimal_statefeedback_controller.m

Related LMIs
$ H_2 $ Optimal State-Feedback Controller