Bifurcation of limit cycles from a switched equilibrium in planar switched systems and its application to power converters

TL;DR

This paper demonstrates how a stable equilibrium in a switched system can bifurcate into a stable limit cycle when a parameter is varied, with applications to power converter design.

Contribution

It provides a theoretical analysis of limit cycle bifurcation in planar switched systems and applies it to improve power converter stability.

Findings

Stable equilibrium leads to a limit cycle for small parameter changes.

Application to power converters avoids sliding motions.

Theoretical results validated through practical example.

Abstract

We consider a switched system of two subsystems that are activated as the trajectory enters the regions $\{(x,y):x>\bar x\}$ and $\{(x,y):x<-\bar x\}$ respectively, where $\bar x$ is a positive parameter. We prove that a regular asymptotically stable equilibrium of the associated Filippov equation of sliding motion (corresponding to $\bar x=0$) yields an orbitally stable limit cycle for all $\bar x>0$ sufficiently small. The research is motivated by an application to a dc-dc power converter, where $\bar x>0$ is used in place of $\bar x=0$ to avoid sliding motions.