User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Stability under Arbitrary Switching

LMIs in Control/Stability Analysis/Continuous Time/Stability under Arbitrary Switching

Using the LMI below, find a P matrix that fits the constraints. If there exists one, then the system can switch between subsystems $$ A_1 $$ and $$ A_2 $$ arbitrarily and remain stable.

The System


\begin{align} \dot x(t) \in \{ A_1x(t),A_2x(t) \} \end{align}$$

The Data
The matrices $$ A_1 \in R^{n \times n} ,A_2 \in R^{n \times n} $$.

The LMI


\begin{align} \text{Find} \; &P>0:\\ A_1^TP+PA_1 < 0 \text{ and }\\ A_2^TP+PA_2 < 0 \end{align}$$

Conclusion
The switched system is stable under arbitrary switching if there exists some P > 0 such that the LMIs hold.

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

Related LMIs
Parametric, Norm-Bounded Uncertain System Quadratic Stability

Stability of Structured, Norm-Bounded Uncertainty

Quadratic Stability Margins