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

The System


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

The Data
The matrices $$ A,B,M,N,Q,D_{12} $$.

The LMI: The Lyapunov Inequality


\begin{align} \text{Find} \; &P>0,Z:\\ \begin{bmatrix} AP+BZ+PA^T+Z^TB^T & PN^T+Z^TD_{12}^T \\ NP+D_{12}Z & 0\end{bmatrix} + \begin{bmatrix} M \Theta M^T & M \Theta Q^T \\ Q \Theta M^T & Q \Theta Q^T - \Theta \end{bmatrix} < 0\\ \end{align}$$

Conclusion:
$$ \begin{align} \text{The system is quadratically stable if and only if there exists some } \Theta \in P \Theta, P>0, \text{ and } Z \text{ such that the above LMI is feasible.}\\ \text{Furthermore, there exists a controller with } u(t) = Kx(t) \text{ where } K = ZP^{-1} \text{ is the quadratically stabilizing controller.} \end{align} $$

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

Optimal State-Feedback Controllers with Structured Norm-Bounded Uncertainty