Independence of Closed-Loop Equilibria and Stability from the Choice of Control Barrier Function for a Given Safe Set
This resolves a fundamental question in safe control for control-affine systems, showing that safety filter design is robust to CBF selection within an equivalence class.
The paper proves that for a given safe set, the choice of control barrier function (CBF) does not affect the number, location, or stability of equilibria in the interior of the safe set, and undesirable equilibria only appear on the boundary, independent of the CBF choice.
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 given, predefined ``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). Given that there exist many different CBFs associated with a given fixed safe set, 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 choice of 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 \textit{safety filters}, 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, provided that the CBFs are chosen from the same equivalence class.