LMIs in Control/Click here to continue/LMIs in system and stability Theory/Parametric, Norm-Bounded Uncertain System Quadratic Stability

Given a system with matrices A,M,N,Q the quadratic stability of the system with parametric, norm-bounded uncertainty can be determined by the following LMI. The feasibility of the LMI tells if the system is quadratically stable or not.

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 \mathbf{\Delta} \; := \{\Delta \in \mathbb{R}^{n \times n} : \| \Delta \| \leq 1 \}\\ \end{align}$$

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

The LMI:


\begin{align} \text{Find} \; &P>0,\mu \geq 0:\\ \begin{bmatrix} AP+PA^T & PN^T \\ NP & 0\end{bmatrix} + \mu \begin{bmatrix} MM^T & MQ^T \\ QM^T & QQ^T - I \end{bmatrix} < 0\\ \end{align}$$

Conclusion:
The system above is quadratically stable if and only if there exists some mu >= 0 and P > 0 such that the LMI is feasible.

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

Related LMIs
Stability of Structured, Norm-Bounded Uncertainty

Stability under Arbitrary Switching

Quadratic Stability Margins