Weak Lyapunov functions for hybrid dynamical systems: applications to electrical and mechanical systems

Abstract

In this work we propose a number of relevant engineering applications that exhibit both a continuous and a discrete evolution, and are therefore suitably described by a recent formalism for hybrid dynamical systems. More specifically, (i) we design observer schemes for a nonsmooth disturbance arising in AC/DC conversion, which we then cancel from a desirable signal; (ii) we show how reset actuation applied to nonlinear mechanical systems can at the same time sustain or damp oscillations; (iii) we study the feedback interconnection of a classical proportional-integral-derivative controller with a sliding mass under Coulomb friction through differential inclusions. In the context of dynamical systems, we analyze the properties of these applications in terms of asymptotic stability through Lyapunov functions tailored for hybrid systems. Instead of the standard Lyapunov conditions, we prove asymptotic stability through weaker, or relaxed, conditions that are compensated by additional (structural) properties that may be easier to verify.