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A Unified Stability Analysis of Safety-Critical Control using Multiple Control Barrier Functions

Matheus F. Reis, José P. Carvalho, A. Pedro Aguiar

TL;DR

This work tackles the problem of guaranteeing both safety and liveness for affine nonlinear systems with multiple safety constraints by providing a unified stability analysis for safety-filter QPs and CLF-CBF QPs. It introduces a generalized QP framework that subsumes both approaches and derives sufficient feasibility conditions when multiple CBFs are active, enabling safe operation under complex constraint sets. A central contribution is a systematic characterization of equilibrium points, including boundary equilibria at intersections of CBF boundaries, and a novel stability criterion based on the closed-loop Jacobian and active-CBF curvature to determine when such equilibria are destabilized or eliminated. The results extend prior work by connecting feasibility, equilibrium existence, and stability within a single analytic framework, with implications for robust safety filtering in autonomous and cyber-physical systems.

Abstract

Ensuring liveness and safety of autonomous and cyber-physical systems remains a fundamental challenge, particularly when multiple safety constraints are present. This letter advances the theoretical foundations of safety-filter Quadratic Programs (QP) and Control Lyapunov Function (CLF)-Control Barrier Function (CBF) controllers by establishing a unified analytical framework for studying their stability properties. We derive sufficient feasibility conditions for QPs with multiple CBFs and formally characterize the conditions leading to undesirable equilibrium points at possible intersecting safe set boundaries. Additionally, we introduce a stability criterion for equilibrium points, providing a systematic approach to identifying conditions under which they can be destabilized or eliminated. Our analysis extends prior theoretical results, deepening the understanding of the conditions of feasibility and stability of CBF-based safety filters and the CLF-CBF QP framework.

A Unified Stability Analysis of Safety-Critical Control using Multiple Control Barrier Functions

TL;DR

This work tackles the problem of guaranteeing both safety and liveness for affine nonlinear systems with multiple safety constraints by providing a unified stability analysis for safety-filter QPs and CLF-CBF QPs. It introduces a generalized QP framework that subsumes both approaches and derives sufficient feasibility conditions when multiple CBFs are active, enabling safe operation under complex constraint sets. A central contribution is a systematic characterization of equilibrium points, including boundary equilibria at intersections of CBF boundaries, and a novel stability criterion based on the closed-loop Jacobian and active-CBF curvature to determine when such equilibria are destabilized or eliminated. The results extend prior work by connecting feasibility, equilibrium existence, and stability within a single analytic framework, with implications for robust safety filtering in autonomous and cyber-physical systems.

Abstract

Ensuring liveness and safety of autonomous and cyber-physical systems remains a fundamental challenge, particularly when multiple safety constraints are present. This letter advances the theoretical foundations of safety-filter Quadratic Programs (QP) and Control Lyapunov Function (CLF)-Control Barrier Function (CBF) controllers by establishing a unified analytical framework for studying their stability properties. We derive sufficient feasibility conditions for QPs with multiple CBFs and formally characterize the conditions leading to undesirable equilibrium points at possible intersecting safe set boundaries. Additionally, we introduce a stability criterion for equilibrium points, providing a systematic approach to identifying conditions under which they can be destabilized or eliminated. Our analysis extends prior theoretical results, deepening the understanding of the conditions of feasibility and stability of CBF-based safety filters and the CLF-CBF QP framework.

Paper Structure

This paper contains 7 sections, 6 theorems, 33 equations, 1 figure.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES
  3. Control Lyapunov (CLFs) and Barrier Functions (CBFs)
  4. QP-Based Safety-Critical Controllers
  5. Closed-loop System Analysis
  6. Existence of Equilibrium Points
  7. Stability of Boundary Equilibrium Points

Key Result

Theorem 1

Under Assump. assumption:initial_state, consider the following assumptions: (i) There is only one CBF ($N=1$). (ii) System eq:affine_nonlinear is driftless: $f(x) = 0 \,\,\, \forall x \in \mathbb{R}^n$. (iii) Considering any number of CBFs ($N>1$), where $c \!=\! p^{-1} \!+\! \lVert\nabla V\rVert_{G}^2 > 0$, $G(x) \!=\! g(x) H(x)^{-1} g(x)^\mathsf{T}$, $f_{nom}(x) \!=\! f(x) \!+\! g(x) u_{nom}(x)

Figures (1)

  • Figure 1: Example of a asymptotically stable equilibrium point occurring at the intersection of two quadratic CBF boundaries.

Theorems & Definitions (18)

  • Definition II.1: CLFs
  • Definition II.2: Safety
  • Definition II.3: CBFs
  • Theorem 1
  • proof
  • Definition III.1: Equilibrium Manifold
  • Theorem 2: Existence of Equilibrium Points
  • proof
  • Lemma 1: Closed-Loop Jacobian
  • Remark III.1
  • ...and 8 more