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{aligned}{\dot {x}}(t)\in \{A_{1}x(t),A_{2}x(t)\}\end{aligned}}

The Data

The matrices $A_{1}\in R^{n\times n},A_{2}\in R^{n\times n}$ .

The LMI

{\begin{aligned}{\text{Find}}\;&P>0:\\A_{1}^{T}P+PA_{1}<0{\text{ and }}\\A_{2}^{T}P+PA_{2}<0\end{aligned}}

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

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.