Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Control Barrier Function based Quadratic Programs Introduce Undesirable Asymptotically Stable Equilibria

Matheus F. Reis, A. Pedro Aguiar, Paulo Tabuada

TL;DR

The paper addresses undesired equilibria arising in the quadratic-programming framework that combines Control Lyapunov Functions and Control Barrier Functions for safety-critical control. It shows these equilibria can be asymptotically stable and reside on the boundary of the safe set, and derives a condition characterizing their stability. To mitigate this, it introduces Lyapunov shaping via a rotated CLF $V(x,Q)$ and a proximity barrier $h_{\\mathcal{D}}(x,Q)$, and presents a modified QP that prevents boundary equilibria while preserving safety and stability. Simulation results on integrator and nonlinear systems demonstrate the approach eliminates the undesired equilibria and yields collision-free, stabilizing behavior. The work advances safety-critical control by providing a theoretically grounded method to avoid boundary artifacts in CLF-CBF QP controllers, with potential impact on multi-robot collision avoidance and other safety-critical applications.

Abstract

Control Lyapunov functions (CLFs) and control barrier functions (CBFs) have been used to develop provably safe controllers by means of quadratic programs (QPs), guaranteeing safety in the form of trajectory invariance with respect to a given set. In this manuscript, we show that this framework can introduce equilibrium points (particularly at the boundary of the unsafe set) other than the minimum of the Lyapunov function into the closed-loop system. We derive explicit conditions under which these undesired equilibria (which can even appear in the simple case of linear systems with just one convex unsafe set) are asymptotically stable. To address this issue, we propose an extension to the QP-based controller unifying CLFs and CBFs that explicitly avoids undesirable equilibria on the boundary of the safe set. The solution is illustrated in the design of a collision-free controller.

Control Barrier Function based Quadratic Programs Introduce Undesirable Asymptotically Stable Equilibria

TL;DR

The paper addresses undesired equilibria arising in the quadratic-programming framework that combines Control Lyapunov Functions and Control Barrier Functions for safety-critical control. It shows these equilibria can be asymptotically stable and reside on the boundary of the safe set, and derives a condition characterizing their stability. To mitigate this, it introduces Lyapunov shaping via a rotated CLF and a proximity barrier , and presents a modified QP that prevents boundary equilibria while preserving safety and stability. Simulation results on integrator and nonlinear systems demonstrate the approach eliminates the undesired equilibria and yields collision-free, stabilizing behavior. The work advances safety-critical control by providing a theoretically grounded method to avoid boundary artifacts in CLF-CBF QP controllers, with potential impact on multi-robot collision avoidance and other safety-critical applications.

Abstract

Control Lyapunov functions (CLFs) and control barrier functions (CBFs) have been used to develop provably safe controllers by means of quadratic programs (QPs), guaranteeing safety in the form of trajectory invariance with respect to a given set. In this manuscript, we show that this framework can introduce equilibrium points (particularly at the boundary of the unsafe set) other than the minimum of the Lyapunov function into the closed-loop system. We derive explicit conditions under which these undesired equilibria (which can even appear in the simple case of linear systems with just one convex unsafe set) are asymptotically stable. To address this issue, we propose an extension to the QP-based controller unifying CLFs and CBFs that explicitly avoids undesirable equilibria on the boundary of the safe set. The solution is illustrated in the design of a collision-free controller.

Paper Structure

This paper contains 13 sections, 3 theorems, 37 equations, 4 figures.

Table of Contents

  1. INTRODUCTION
  2. Quadratic Programs for Safety Critical Systems
  3. Quadratic Program Formulation
  4. Analysis of the closed-loop system with QP-based controller
  5. Existence of equilibrium points
  6. Stability of equilibrium points
  7. Lyapunov Shaping for QP-based Controllers
  8. Simulation Results
  9. CONCLUSIONS
  10. APPENDIX
  11. Proof of Theorem \ref{['theorem1']}
  12. Proof of Theorem \ref{['theorem2']}
  13. Proof of Theorem \ref{['theorem3']}

Key Result

Theorem 1

The set $\mathcal{E}$ of equilibrium points of the closed-loop system resulting from the application of the control law eq:QP_control into eq:nonlinear_system is given by where $0 \in \mathbb{R}^{n}$ is the origin of the state space and where $G(x) = g(x) g(x)^\mathsf{T}$, $\mathcal{E}_{\text{int}}$ is the set of interior equilibria and $\mathcal{E}_{\partial \mathcal{C}}$ is the set of bounda

Figures (4)

  • Figure 1: System trajectories for the QP-based control of the integrator with a circular obstacle. The undesirable asymptotically stable equilibrium is shown in red.
  • Figure 2: System trajectories for the closed-loop system with the proposed QP-based control for the integrator.
  • Figure 3: System trajectories for the closed-loop system with nominal and proposed QP-based controller for $f(x) = f_1(x)$ and $g(x) = I_n$ with a circular obstacle.
  • Figure 4: System trajectories for the closed-loop system with nominal and proposed QP-based controller $f(x) = f_2(x)$ and $g(x) = I_n$ with a circular obstacle.

Theorems & Definitions (9)

  • Definition II.1: CLFs
  • Definition II.2: Safety
  • Definition II.3: CBFs
  • Theorem 1
  • Remark III.1
  • Theorem 2
  • Remark IV.1
  • Remark IV.2
  • Theorem 3