LMIs in Control/pages/Quadratic polytopic stabilization

A Quadratic Polytopic Stabilization Controller Synthesis can be done using this LMI, requiring the information about $$A$$,$$\Delta_{A(t)} $$ ,$$B$$ and $$\Delta_{B(t)} $$ matrices.

The System


\begin{align} \dot x(t)&=Ax(t)+Bu(t),\\ x(0)&=x_0, \end{align}$$ where $$x(t)\in \R^n$$, $$u(t)\in \R^m$$, at any $$t\in \R$$. The system consist of uncertainties of the following form

\begin{align} \Delta_{A(t)} = A_1 \delta_1 (t)+....+A_k \delta_k (t)\\ \Delta_{B(t)} = B_1 \delta_1 (t)+....+B_k \delta_k (t)\\ \end{align}$$

where $$x\in\mathbb{R}^{m}$$,$$u\in\mathbb{R}^{n}$$,$$A\in\mathbb{R}^{mxm}$$ and $$B\in\mathbb{R}^{mxn}$$

The Data
The matrices necessary for this LMI are $$A$$,$$\Delta_{A(t)} \, ie \, A_i $$ ,$$B$$ and $$\Delta_{B(t)} \, ie \, B_i $$

The Optimization and LMI:LMI for Controller Synthesis using the theorem of Polytopic Quadratic Stability
There exists a K such that

\begin{align} \dot x(t)&=(A+\Delta_A+(B+\Delta_B)K)x(t)\\ \end{align}$$ is quadratically stable for $$ (\Delta_A,\Delta_B) \in C_0 ((A_1,B_2),...,(A_k,B_k)) $$ if and only if there exists some P>0 and Z such that



\begin{align} (A+A_i)P + P(A+A_i)^T + (B+B_i)Z+Z^T(B+B_i)^T < 0 \quad for \quad i = 1,...k. \end{align}$$

Conclusion:
The Controller gain matrix is extracted as $$ K = ZP^{-1} $$ Note that here the controller doesn't depend on $$ \Delta $$
 * If you want K to depend on $$ \Delta $$, the problem is harder.
 * But this would require sensing $$ \Delta $$ in real-time.

Implementation
This implementation requires Yalmip and Sedumi. https://github.com/JalpeshBhadra/LMI/blob/master/quadraticpolytopicstabilization.m

Related LMIs
Quadratic Polytopic $H_{\infty}$ Controller Quadratic Polytopic $H_{2}$ Controller