User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Stability of Structured, Norm-Bounded Uncertainty

Given a system with matrices A,M,N,Q with structured, norm-bounded uncertainty, the stability of the system can be found using the following LMI. The LMI takes variables P and $$ \Theta $$ and checks for a feasible solution.

The System


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

The Data
The matrices $$ A,M,N,Q $$.

The LMI:


\begin{align} \text{Find} \; &P>0:\\ \begin{bmatrix} AP+PA^T & PN^T \\ NP & 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:
$$ \begin{align} \text{The system above is quadratically stable if and only if there exists some } \Theta \in P \Theta \text{ and } P > 0 \text{ such that the LMI is feasible.} \end{align} $$

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

Related LMIs
Parametric, Norm-Bounded Uncertain System Quadratic Stability

Stability under Arbitrary Switching

Quadratic Stability Margins