A note on stability conditions for planar switched systems

TL;DR

This paper establishes simple, coordinate-invariant criteria for the asymptotic stability of planar switched systems with Hurwitz matrices, unifying and simplifying previous conditions by analyzing key matrix functions.

Contribution

It provides new, unified, and easier-to-verify stability conditions for planar switched systems that improve upon prior results by reducing the number of cases to consider.

Findings

Derived necessary and sufficient stability conditions for all switching functions.

Unified previous disparate conditions into a simpler framework.

Reduced analysis complexity by focusing on four key cases.

Abstract

This paper is concerned with the stability problem for the planar linear switched system $\dot x(t)=u(t)A_1x(t)+(1-u(t))A_2x(t)$, where the real matrices $A_1,A_2\in \R^{2\times 2}$ are Hurwitz and $u(\cdot) [0,\infty[\to\{0,1\}$ is a measurable function. We give coordinate-invariant necessary and sufficient conditions on $A_1$ and $A_2$ under which the system is asymptotically stable for arbitrary switching functions $u(\cdot)$. The new conditions unify those given in previous papers and are simpler to be verified since we are reduced to study 4 cases instead of 20. Most of the cases are analyzed in terms of the function $\Gamma(A_1,A_2)={1/2}(\tr(A_1) \tr(A_2)- \tr(A_1A_2))$.