Feedback Control of Border Collision Bifurcations

Abstract

The feedback control of border collision bifurcations is considered. These bifurcations can occur when a fixed point of a piecewise smooth system crosses the border between two regions of smooth operation. The goal of the control effort in this work is to modify the bifurcation so that the bifurcated steady state is locally attracting. In this way, the system’s local behavior is ensured to remain stable and close to the original operating condition. Linear and piecewise linear feedbacks are used since the system linearization on the two sides of the border generically determines the type and stability properties of any border collision bifurcation. A two-dimensional example on quenching of heart arrhythmia is used to illustrate the ideas.