LMIs in Control/Stability Analysis/Continuous Time/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

$\begin{array}{c}\dot{x}(t)\in \{A_{1}x(t),A_{2}x(t)\}\end{array}$

The Data

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

The LMI

$\begin{array}{cc}\text{Find}& P>0:\\ A_1^{T}P+P{A}_1<0\text{ and }\\ A_2^{T}P+P{A}_2<0\end{array}$

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

External links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.