Abstract
In this chapter, we introduce some tools to analyze stability properties of discrete-time switched linear systems driven by switching signals belonging to a given \(\omega \)-regular language. More precisely, we assume that the switching signals are generated by a deterministic Büchi automaton whose alphabet coincides with the set of modes of the switched system. We present the notion of \(\omega \)-regular joint spectral radius (\(\omega \)-RJSR), which intuitively describes the contraction of the state when the run associated with the switching signal 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 signals for a switched 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.
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Notes
- 1.
We identify a switching signal \(\theta :\mathbb N \rightarrow \varSigma \) with the infinite word \(\theta (0)\theta (1) \dots \) and use the notation \(\theta \in Lang (\mathcal B)\) for \(\theta (0)\theta (1) \dots \in Lang(\mathcal B)\)
- 2.
The MATLAB® scripts of this numerical example are available at the following repository: https://github.com/georgesaazan/w-regular-oscillators.
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Acknowledgements
The authors would like to thank Laurent Fribourg for useful discussions. This work was supported in part by the “Agence Nationale de la Recherche” (ANR) under Grant HANDY ANR-18-CE40-0010.
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Aazan, G., Girard, A., Mason, P., Greco, L. (2024). A Joint Spectral Radius for \(\omega \)-Regular Language-Driven Switched Linear Systems. In: Postoyan, R., Frasca, P., Panteley, E., Zaccarian, L. (eds) Hybrid and Networked Dynamical Systems. Lecture Notes in Control and Information Sciences, vol 493. Springer, Cham. https://doi.org/10.1007/978-3-031-49555-7_7
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