**Stability of constrained switched systems driven by ω-regular languages**

by Georges Aazan

Friday 27 October 2023 at 2pm

ENS Paris-Saclay, Room 1Z14 and Zoom

**Abstract.** Switched systems are dynamical systems with several operating modes, each mode being described by a differential (continuous time) or difference (discrete time) equation.
At all times, the active operating mode is determined by a switching signal. Switched systems are very useful in practice for accurately describing the execution of control algorithms on distributed computing infrastructures and thus for taking into account the constraints linked to the use of shared computing and communication resources. Furthermore, switched systems have unexpected properties (unstable behavior can for example result from switching between stable operating modes) that justify the development of specific theoretical tools for their study.

Early work on stability of switched systems has focused on stability for switching signals that are arbitrary or that satisfy some (minimum or average) dwell-time condition. More recently, several works have considered the problem of proving stability for subsets of switching signals. In general, such switching signals are assumed to be generated by some finite state automaton and stability is characterized either in term of constrained joint spectral radius or using Lyapunov functions. However, there are some subsets of switching signals that cannot be specified using classical finite state automata. Examples are switching signals belonging to some omega-regular languages e.g. defined by Linear Temporal Logic (LTL) formulas, which are often used to specify scheduling and communication protocols. A representative example of omega-regular language is the set of shuffled switching signals: a switching signal is shuffled if and only if all the modes are activated infinitely often. In a preliminary study, the stability of switched systems under shuffled switching signals was characterized by means of Lyapunov functions.

This thesis aims at developing theoretical and numerical tools to analyze the stability of switched systems under shuffled switching signals and more generally under constraints given by an omega-regular language. We define a notion of shuffled joint spectral radius that allows us to quantify the speed of convergence of the switched system under shuffled switching signals. We develop numerical algorithms based on Linear Matrix Inequalities (LMIs) and automata theoretic techniques to compute approximations of the shuffled joint spectral radius. In the second part of the thesis, we extend these results to more general classes of switching signals such as those specified by omega-regular languages. These languages can always be characterized by Büchi automata. Finally, we will present an observer design for switched systems based on the Büchi automata and reconstructible sequences, i.e. sequences allowing to estimate the state of the system. This design consists of an application of our theoretical results.

**Jury:**

- Jamal Daafouz, Université de Lorraine, CRAN (Reviewer)
- Raphael Jungërs, Université catholique de Louvain, ICTEAM (Reviewer)
- Carolina Albea Sanchez, Universitad de Sevilla, (Examiner)
- Aneel Tanwani, CNRS, LAAS Toulouse, (Examiner)
- Elena Panteley, CNRS, L2S (Examiner)
- Antoine Girard, CNRS, L2S (Supervisor)
- Laurent Fribourg, CNRS, LMF (Supervisor)
- Luca Greco, Université Paris-Saclay, L2S (Supervisor)
- Paolo Mason, CNRS, L2S (Guest)