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

Yiting Chen, Pol Mestres, Jorge Cortes, Emiliano Dall'Anese

TL;DR

This paper analyzes how the choice of a Control Barrier Function (CBF) affects the equilibria and dynamics of optimization-based safe controllers for control-affine systems. Across general CBF-based controllers, including CLF-CBF QP and safety filters, it proves that interior equilibria are invariant to the CBF, and undesirable equilibria appear only on the boundary with their number and location independent of the CBF; for CLF-CBF QP and safety filters, it also shows that stability properties of these equilibria are invariant within CBF equivalence classes. The authors derive explicit Jacobian expressions at boundary undesired equilibria to quantify how dynamics depend on Leray-type terms and CBF-related parameters, while showing that equivalent CBFs yield identical spectral characteristics up to a simple factor. Numerical simulations corroborate the theoretical findings, illustrating invariance of boundary equilibria across equivalent CBFs and revealing how the region of attraction can depend on the chosen CBF pair. Overall, the work provides design-insensitive guidance for selecting CBFs and clarifies when the CBF choice truly matters for closed-loop safety and stability.

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.

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

TL;DR

This paper analyzes how the choice of a Control Barrier Function (CBF) affects the equilibria and dynamics of optimization-based safe controllers for control-affine systems. Across general CBF-based controllers, including CLF-CBF QP and safety filters, it proves that interior equilibria are invariant to the CBF, and undesirable equilibria appear only on the boundary with their number and location independent of the CBF; for CLF-CBF QP and safety filters, it also shows that stability properties of these equilibria are invariant within CBF equivalence classes. The authors derive explicit Jacobian expressions at boundary undesired equilibria to quantify how dynamics depend on Leray-type terms and CBF-related parameters, while showing that equivalent CBFs yield identical spectral characteristics up to a simple factor. Numerical simulations corroborate the theoretical findings, illustrating invariance of boundary equilibria across equivalent CBFs and revealing how the region of attraction can depend on the chosen CBF pair. Overall, the work provides design-insensitive guidance for selecting CBFs and clarifies when the CBF choice truly matters for closed-loop safety and stability.

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.
Paper Structure (12 sections, 15 theorems, 68 equations, 3 figures)

This paper contains 12 sections, 15 theorems, 68 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Control Barrier Functions
  4. Control Lyapunov Functions
  5. Problem Statement
  6. Impact of CBF on Dynamics and Equilibria
  7. Impact of CBF Selection in CBF-CLF Controllers
  8. Characterization of Undesirable Equilibria
  9. Stability of Undesirable Equilibria
  10. Impact of CBF Selection in Safety Filters
  11. Numerical Simulation
  12. Conclusions

Key Result

Lemma 2.2

(Relation between Gradients of CBFs): Let $h_1,h_2:\mathbb{R}^n\to\mathbb{R}$ be two CBFs of ${\mathcal{S}}$. Then $\nabla h_2({\mathbf{x}})=\zeta({\mathbf{x}}) \nabla h_1({\mathbf{x}})$ with $\zeta :\partial \mathcal{S}\mapsto \mathbb{R}_{>0}$ a function that is unique.

Figures (3)

  • Figure 1: Closed-loop system that is the subject of the paper. We consider a general formulation for the safe optimization-based controllers that subsumes existing safety filters and CLF-CBF QP approaches.
  • Figure 2: Examples of undesirable equilibria of an LTI planar system with a CLF-CBF-QP controller for a circular obstacle. The simulation setup and the CLF are as in MFR-APA-PT:21. Plots (a) and (b) show the origin (which is a desirable equilibrium), the undesirable (i.e., spurious) equilibria, and the vector field of the closed-loop system for two different choices of the CBF and of the associated extended class-${\mathcal{K}}$ function (details are provided in Section ). The plots show that the number, location, and stability properties of both the desired equilibrium and the undesired equilibria do not change with the CBF pair.
  • Figure 3: Examples of trajectories of an LTI planar system with a safety filter for a circular obstacle; the figures show the vector fields, the undesired equilibria, the stable sets (stable manifolds) of the saddle equilibria, and the desired equilibrium (which is the origin). The plots are generated with different CBF pairs: (a): with CBF pair $(h_1,\alpha_1)$; (b): with CBF pair $(h_2,\alpha_1)$; (c): with CBF pair $(h_1,\alpha_2)$; (d) with CBF pair $(h_2,\alpha_2)$.

Theorems & Definitions (25)

  • Definition 2.1
  • Lemma 2.2
  • Example 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Corollary 2.6
  • Definition 2.7
  • Remark 3.1
  • Example 3.3
  • Example 3.4
  • ...and 15 more