Bifurcation of limit cycles from a fold-fold singularity in planar switched systems

TL;DR

This paper studies how limit cycles bifurcate from a fold-fold singularity in planar switched systems, motivated by anti-lock braking systems, extending classical bifurcation theory to discontinuous systems.

Contribution

It introduces a novel bifurcation analysis for limit cycles emerging from fold-fold singularities in discontinuous planar systems, extending classical smooth bifurcation theory.

Findings

Limit cycles can bifurcate from fold-fold singularities as a parameter crosses zero.

The bifurcation analysis extends classical fold bifurcation theory to discontinuous systems.

The results are applied to the modeling of anti-lock braking systems.

Abstract

An anti-lock braking system (ABS) is the primary motivation for this research. The ABS controller switches the actions of charging and discharging valves in the hydraulic actuator of the brake cylinder based on the wheels' angular speed and acceleration. The controller is, therefore, modeled by discontinuous differential equations where two smooth vector fields are separated by a switching manifold S. The goal of the controller is to maximize the tire-road friction force during braking (and, in particular, to prevent the wheel lock-up). Since the optimal slip L of the wheel is known rather approximately, the actual goal of the controller is to achieve such a switching strategy that makes the dynamics converging to a limit cycle surrounding the region of prospective values of L. In this paper we show that the required limit cycle can be obtained as a bifurcation from a point x0 of S when a suitable parameter D crosses 0. The point x0 turns out to be a fold-fold singularity (the vector fields on the two sides of S are tangent one another at x0) and the parameter D measures the deviation of the switching manifold from a hyperplane. The proposed result is based on an extension of the classical fold bifurcation theory available for smooth maps. Construction of a suitable smooth map is a crucial step of the proof.