Bifurcation of limit cycles from a switched equilibrium in planar switched systems

Abstract

We consider a switched system of two subsystems that are activated as the trajectory enters the regions {(x,y):x>x¯} and {(x,y):x<−x¯}, respectively, where x¯ is a positive parameter. We prove that a regular asymptotically stable equilibrium of the associated Filippov equation of sliding motion (corresponding to x¯=0) yields an orbitally stable limit cycle for all x¯>0 sufficiently small. Such an equilibrium is called switched equilibrium in control theory, in which case considering x¯>0 refers to the effect of hysteresis. The fact that hysteresis perturbation of a switched equilibrium yields a limit cycle is known and is actively used in control. We not only prove this fact rigorously for the first time ever, but also offer a formula for the period of the limit cycle which turns out to be very sharp as our practical example demonstrates. Specifically, an application to a model of a dc–dc power converter concludes the paper.