Stability under dwell time constraints: Discretization revisited

Abstract

We investigate the stability of continuous-time linear switching systems with a guaranteed dwell time. It is known that the dwell time restrictions make the main methods for deciding stability such as Lyapunov functions and discretization inapplicable, at least in their standard forms. Our work focuses on adapting the discretization approach to address this limitation. The discretization is done merely by replacing arbitrary switching law with piecewise-constant functions with a fixed step size. We demonstrate that this classical method can be modified so that it not only becomes applicable under the dwell time constraints but also outperforms traditional methods for unconstrained systems. Namely, the discretization with the step size $h$ approximates the Lyapunov exponent with the precision $C{h}^2$, and the constant $C$ can be explicitly evaluated. This result is unexpected, as the approximation accuracy for systems without a guaranteed dwell time is linear in $h$. Our methods implementation is efficient in dimensions up to 10 for arbitrary systems and up to several hundreds for positive systems. The numerical results are provided.