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Non-Smooth Lyapunov Functions for Stability Analysis of Hybrid Systems

Matteo Della Rossa 1
1 LAAS-MAC - Équipe Méthodes et Algorithmes en Commande
LAAS - Laboratoire d'analyse et d'architecture des systèmes
Abstract : Modeling of many phenomena in nature escape the rather common frameworks of continuous-time and discrete-time models. In fact, for many systems encountered in practice, these two paradigms need to be intrinsically related and connected, in order to reach a satisfactory level of description in modeling the considered physical/engineering process. These systems are often referred to as hybrid systems, and various possible formalisms have appeared in the literature over the past years. The aim of this thesis is to analyze the stability of particular classes of hybrid systems, by providing Lyapunov-based sufficient conditions for (asymptotic) stability. In particular, we will focus on non-differentiable locally Lipschitz candidate Lyapunov functions. The first chapters of this manuscript can be considered as a general introduction of this topic and the related concepts from non-smooth analysis. This will allow us to study a class of piecewise smooth maps as candidate Lyapunov functions, with particular attention to the continuity properties of the constrained differential inclusion comprising the studied hybrid systems. We propose ``relaxed'' Lyapunov conditions which require to be checked only on a dense set and discuss connections to other classes of locally Lipschitz or piecewise regular functions. Relaxing the continuity assumptions, we then investigate the notion of generalized derivatives when considering functions obtained as max-min combinations of smooth functions. This structure turns out to be particularly fruitful when considering the stability problem for differential inclusions arising from regularization of state-dependent switched systems. When the studied switched systems are composed of linear sub-dynamics, we refine our results, in order to propose algorithmically verifiable conditions. We further explore the utility of set-valued derivatives in establishing \emph{input-to-state} stability results, in the context of perturbed differential inclusions/switched systems, using locally Lipschitz candidate Lyapunov functions. These developments are then used in analyzing the stability problem for interconnections of differential inclusion, with an application in designing an observer-based controller for state-dependent switched systems.
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Matteo Della Rossa. Non-Smooth Lyapunov Functions for Stability Analysis of Hybrid Systems. Automatic Control Engineering. Institut National des Sciences Appliquées de Toulouse, 2020. English. ⟨NNT : 2020ISAT0004⟩. ⟨tel-03186040v2⟩

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