Max-Min Lyapunov Functions for Switched Systems and Related Differential Inclusions

Matteo Della Rossa (1) , Aneel Tanwani (1) , Luca Zaccarian (2, 1)

1 LAAS-MAC - Équipe Méthodes et Algorithmes en Commande
2 Dipartimento di Ingegneria Industriale [Trento]

Starting from a finite family of continuously differentiable positive definite functions, we study conditions under which a function obtained by max-min combinations is a Lyapunov function, establishing stability for two kinds of nonlinear dynamical systems: a) Differential inclusions where the set-valued right-hand-side comprises the convex hull of a finite number of vector fields, and b) Autonomous switched systems with a state-dependent switching signal. We investigate generalized notions of directional derivatives for these max-min functions, and use them in deriving stability conditions with various degrees of conservatism, where more conservative conditions are numerically more tractable. The proposed constructions also provide nonconvex Lyapunov functions, which are shown to be useful for systems with state-dependent switching that do not admit a convex Lyapunov function. Several examples are included to illustrate the results.

Domaines: Systèmes dynamiques [math.DS]; Automatique / Robotique

Matteo Della Rossa, Aneel Tanwani, Luca Zaccarian. Max-Min Lyapunov Functions for Switched Systems and Related Differential Inclusions. Automatica, 2020, 120, pp.109123. ⟨10.1016/j.automatica.2020.109123⟩. ⟨hal-02886685⟩