Equilibria and Their Stability Do Not Depend on the Control Barrier Function in Safe Optimization-Based Control

TL;DR

This paper demonstrates that the choice of control barrier function (CBF) does not influence the number, location, or stability of equilibria in safe optimization-based control systems, especially within the safe set boundary.

Contribution

It provides a theoretical analysis showing that CBF selection does not affect equilibrium properties, clarifying the role of CBFs in safe control design.

Findings

Equilibria inside the safe set are unaffected by CBF choice.

Undesirable equilibria only occur on the safe set boundary.

Stability of equilibria is independent of the CBF used.

Abstract

Control barrier functions (CBFs) play a critical role in the design of safe optimization-based controllers for control-affine systems. Given a CBF associated with a desired ``safe'' set, the typical approach consists in embedding CBF-based constraints into the optimization problem defining the control law to enforce forward invariance of the safe set. While this approach effectively guarantees safety for a given CBF, the CBF-based control law can introduce undesirable equilibrium points (i.e., points that are not equilibria of the original system); open questions remain on how the choice of CBF influences the number and locations of undesirable equilibria and, in general, the dynamics of the closed-loop system. This paper investigates how the choice of CBF impacts the dynamics of the closed-loop system and shows that: (i) The CBF does not affect the number, location, and (local) stability properties of the equilibria in the interior of the safe set; (ii) undesirable equilibria only appear on the boundary of the safe set; and, (iii) the number and location of undesirable equilibria for the closed-loop system do not depend of the choice of the CBF. Additionally, for the well-established safety filters and controllers based on both CBF and control Lyapunov functions (CLFs), we show that the stability properties of equilibria of the closed-loop system are independent of the choice of the CBF and of the associated extended class-K function.