Control Barrier Function-Based Safety Filters: Characterization of Undesired Equilibria, Unbounded Trajectories, and Limit Cycles

TL;DR

This paper analyzes the dynamical behaviors of control barrier function-based safety filters, identifying conditions for boundedness, absence of limit cycles, and characterizing undesired equilibria, with explicit formulas for planar systems.

Contribution

It provides new conditions for bounded trajectories, characterizes undesired equilibria via algebraic equations, and offers explicit formulas for planar systems, advancing understanding of safety filter dynamics.

Findings

Undesired equilibria can be characterized algebraically.

Conditions for bounded trajectories and absence of limit cycles are established.

Explicit formulas for equilibria in planar systems are derived.

Abstract

This paper focuses on safety filters designed based on Control Barrier Functions (CBFs): these are modifications of a nominal stabilizing controller typically utilized in safety-critical control applications to render a given subset of states forward invariant. The paper investigates the dynamical properties of the closed-loop systems, with a focus on characterizing undesirable behaviors that may emerge due to the use of CBF-based filters. These undesirable behaviors include unbounded trajectories, limit cycles, and undesired equilibria, which can be locally stable and even form a continuum. Our analysis offer the following contributions: (i) conditions under which trajectories remain bounded and (ii) conditions under which limit cycles do not exist; (iii) we show that undesired equilibria can be characterized by solving an algebraic equation, and (iv) we provide examples that show that asymptotically stable undesired equilibria can exist for a large class of nominal controllers and design parameters of the safety filter (even for convex safe sets). Further, for the specific class of planar systems, (v) we provide explicit formulas for the total number of undesired equilibria and the proportion of saddle points and asymptotically stable equilibria, and (vi) in the case of linear planar systems, we present an exhaustive analysis of their global stability properties. Examples throughout the paper illustrate the results.