LMIs in Control/LMIs for Controller Synthesis/Optimal State-Feedback Controllers with Structured Norm-Bounded Uncertainty

The System


\begin{align} \dot x(t)&=Ax(t)+Bu(t)+Mp(t)+B_2w(t),&&p(t) = \Delta (t)q(t),\\ q(t) &= Nx(t)+D_{12}u(t),&&\Delta \in \bf{\Delta} \;, || \Delta || \leq 1 \\ y(t) &= Cx(t)+D_{22}u(t)\\ \end{align}$$

The Data
The matrices $$ A,B,M,B_2,N,D_{12},C,D_{22} $$.

The Optimization Problem
$$ \text{Minimize } \gamma $$

The LMI: The Lyapunov Inequality


\begin{align} \text{Find} \; &P>0,Z:\\ \begin{bmatrix} AP+BZ+PA^T+Z^TB^T+B_2B_2^T+M \Theta M^T & (CP+D_{22}Z)^T & PN^T+Z^TD_{12}^T \\ CP+D_{22}Z & -\gamma ^2 I & 0 \\ NP+D_{12}Z & 0 & -\Theta\end{bmatrix} < 0\\ \end{align}$$

Conclusion:
$$ \begin{align} \text{If there exists some } \Theta \in P \Theta, P>0, \text{ and } Z \text{ such that the above LMI is feasible, then the system satisfies } ||y||_{L_2} \leq \gamma ||u||_{L_2}. \\ \text{and there exists a controller with } u(t) = Kx(t) \text{ where } K=ZP^{-1} \text{ is the corresponding controller.} \end{align} $$

Implementation
https://github.com/mcavorsi/LMI

Related LMIs
Quadratically Stabilizing Controllers with Parametric Norm-Bounded Uncertainty

H-infinity Optimal Quadratically Stabilizing Controllers with Parametric Norm-Bounded Uncertainty

Stabilizing State-Feedback Controllers with Structured Norm-Bounded Uncertainty