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

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.