A new generic bifurcation from a fold-fold singularity of a discontinuous system
Oleg Makarenkov
Mathematical Sciences, University of Texas - Dallas
Date and time:
Monday, June 16, 2014 - 11:00am
Location:
ECCR 257
Abstract:
An anti-lock braking system (ABS) is the primary motivation for this talk. 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, modelled 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 converge to a limit cycle surrounding the region of prospective values of ??L.
In this talk I show that the required limit cycle can be obtained as a bifurcation from an equilibrium x0 of S when a suitable parameter D crosses 0. The respective equilibrium 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. This required development of a new theory because the available studies examine the response of a fold-fold singularity to perturbations of the vector field as opposed to perturbations of the switching manifold. The new theory is not a consequence of the available results because the switching manifold of our model is discontinuous.