A Joint Spectral Radius for \(\omega \) -Regular Language-Driven Switched Linear Systems

In this chapter, we introduce some tools to analyze stability properties of discrete-time switched linear systemsLinearsystems driven by switching signalsSwitchingsignal belonging to a given \(\omega \) -regular language-regular language. More precisely, we assume that the switching signalsSwitchingsignal are generated by a deterministic Büchi automatonDeterministic Büchi automaton whose alphabet coincides with the set of modes of the switched systemSwitched system. We present the notion Joint spectral radius-regular of -regular joint spectral radius-regular joint spectral radius ( \(\omega \) -RJSR), which intuitively describes the contraction of the state when the run associated with the switching signalSwitchingsignal visits an accepting state of the automaton. Then, we show how this quantity is related to the stability properties of such systems. Specifically, we show how this notion can characterize a class of stabilizing switching signalsSwitchingsignal for a switched systemSwitched system that is unstable for arbitrary switching. Though the introduced quantity is hard to compute, we present some methods to approximate it using Lyapunov and automata-theoretic techniques. More precisely, we show how upper bounds can be computed by solving a convex optimization problem. To validate the results of our work, we finally show a numerical example related to the synchronization of oscillators over a communication network.