Global stability analysis of on/off systems

Abstract

This paper considers quadratic surface Lyapunov functions in the study of global stability analysis of on/off systems (OFS), including those OFS with unstable nonlinearity sectors. In previous work, quadratic surface Lyapunov functions were successfully applied to prove global asymptotic stability of limit cycles of relay feedback systems. In this work, we show that these ideas can be used to prove global asymptotic stability of equilibrium points of piecewise linear systems (PLS). We present conditions in the form of LMI that, when satisfied, guarantee global asymptotic stability of an equilibrium point. A large number of examples was successfully proven globally stable. These include systems with an unstable affine linear subsystem, systems of relative degree larger than one and of high dimension, and systems with unstable nonlinearity sectors, for which all classical fail to analyze. In fact, existence of an example with a globally stable equilibrium point that could not be successfully analyzed with this new methodology is still an open problem. This work opens the door to the possibility that more general PLS can be systematically globally analyzed using quadratic surface Lyapunov functions.