Regions of Attraction of CBF-Based Safety Filters for Stable Planar Linear Systems

In this paper we study the dynamical properties induced by safety filters based on control barrier functions (CBFs). These are obtained by minimally modifying a nominal controller with the goal of rendering it safe. We consider the simple setting of a planar linear plant with a stable linear nominal controller and an affine constraint. We show that even in this simple setting, the filtered system can oscillate or diverge while remaining safe. We fully characterize the parameters for which the origin is globally (exponentially or asymptotically) stable. For all other parameters, the filter creates a saddle on the constraint boundary, and we characterize the region of attraction of the origin. For underdamped nominal dynamics, its boundary is an unstable periodic orbit, a homoclinic loop, or the saddle's stable manifold, depending on whether the CBF gain is below, at, or above a homoclinic transition value. A vehicle sideslip example and two three-dimensional examples illustrate the results and what changes beyond planar dynamics.