Sufficient Stability Conditions for a Class of Switched Systems With Multiple Steady States

In this letter, we present a novel approach to determine the stability of switched linear and nonlinear systems using Sum of Squares optimisation. Particularly, we use Sum of Squares optimisation to search for a Lyapunov function that defines an absorbing set that confines solution trajectories. For linear systems, we show that this also implies global asymptotic stability. Using this approach, we can study stability for a broader range of switched systems, particularly, we can search for a global attractor for switched nonlinear systems, whose dynamics are given by polynomial vector fields and which have multiple equilibria or limit cycles.

I. Introduction

Switched systems are used for modelling in many different fields [1]. Examples for their use range from the biological sciences [2], for example, the dynamics of human body thermoregulation during sleep abruptly changes when sleep transitions from non-rapid eye movement (REM) sleep to REM sleep [3], to mechanical engineering, where one example is the dynamics of an engine with shifting gears [4]. Switched systems can describe systems, whose dynamics are affected by instantaneous changes, by considering a set of continuous-time sub-systems and a rule governing the switching between them. Analysing switched systems is also important in the field of hybrid systems [1].