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Finite-time Convergent Control Barrier Functions with Feasibility Guarantees

Anni Li, Yingqing Chen, Christos G. Cassandras, Wei Xiao

Abstract

This paper studies the problem of finite-time convergence to a prescribed safe set for nonlinear systems whose initial states violate the safety constraints. Existing Control Lyapunov-Barrier Functions (CLBFs) can enforce recovery to the safe set but may suffer from the issue of chattering and they do not explicitly consider control bounds. To address these limitations, we propose a new Control Barrier Function (CBF) formulation that guarantees finite-time convergence to the safe set while ensuring feasibility under control constraints. Specifically, we strengthen the initially violated safety constraint by introducing a parameter which enables the exploitation of the asymptotic property of a CBF to converge to the safe set in finite time. Furthermore, the conditions for the existence of such a CBF under control bounds to achieve finite-time convergence are derived via reachability analysis and constraint comparison, providing a systematic approach for parameter design. A case study on 2D obstacle avoidance is presented to demonstrate the effectiveness and advantages of the proposed method.

Finite-time Convergent Control Barrier Functions with Feasibility Guarantees

Abstract

This paper studies the problem of finite-time convergence to a prescribed safe set for nonlinear systems whose initial states violate the safety constraints. Existing Control Lyapunov-Barrier Functions (CLBFs) can enforce recovery to the safe set but may suffer from the issue of chattering and they do not explicitly consider control bounds. To address these limitations, we propose a new Control Barrier Function (CBF) formulation that guarantees finite-time convergence to the safe set while ensuring feasibility under control constraints. Specifically, we strengthen the initially violated safety constraint by introducing a parameter which enables the exploitation of the asymptotic property of a CBF to converge to the safe set in finite time. Furthermore, the conditions for the existence of such a CBF under control bounds to achieve finite-time convergence are derived via reachability analysis and constraint comparison, providing a systematic approach for parameter design. A case study on 2D obstacle avoidance is presented to demonstrate the effectiveness and advantages of the proposed method.
Paper Structure (10 sections, 4 theorems, 46 equations, 4 figures)

This paper contains 10 sections, 4 theorems, 46 equations, 4 figures.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES
  3. Control Barrier Functions
  4. Control Lyapunov-Barrier Function
  5. PROBLEM FORMULATION AND APPROACH
  6. Feasibility-Guaranteed Finite-Time convergent CBFs
  7. Finite-Time convergent CBFs
  8. Feasibility Analysis of Finite-Time convergent CBFs
  9. CASE STUDY
  10. CONCLUSION & FUTURE WORK

Key Result

Theorem 1

(xiao2021high) Given a HOCBF $b(\bm x)$ from Def. def:hocbf with the safe sets $C_{i}, i\in\{1,\dots, m\}$ defined by (eqn:sets), if $\bm x(0) \in \cap_{i = 1}^m C_{i}$, then any Lipschitz continuous controller $\bm u(t)\in U$ that satisfies the HOCBF constraint in (eqn:constraint), $\forall t\geq0$

Figures (4)

  • Figure 1: Robot trajectories under different random initial positions with the proposed finite-time convergent CBF. The robot can reach the safety convergence set in specified time $t_f = 6s$ while guaranteeing safety (avoidance with respect to all obstacles).
  • Figure 2: Control and $h(\bm x)$ profiles corresponding to the 10 trajectories in Fig. \ref{['fig:traj']} with the proposed feasibility guaranteed finite-time CBF. The control bounds are strictly satisfied, and the finite-time convergence $h(\bm x(t_f)) \geq 0$ is satisfied at $t_f = 6s$.
  • Figure 3: Robot trajectories under the proposed Finite-time convergent CBF (FCCBF) and CLBF method. The safety constraint \ref{['eqn:safetycons-robot']} is violated as the blue trajectory (CLBF) enters the red circle, indicating a collision with the obstacle. The green curve (FCCBF) remains outside of the obstacle, ensuring conflict-free all the time.
  • Figure 4: Control and $b(\bm x)$ profiles under CLBF and the Finite-time convergent CBF (FCCBF) method. The control bounds are strictly satisfied. The safety constraint $b(x) \geq 0$ is not satisfied at around $t=5s$ under the CLBF method, whereas the proposed method remains nonnegative for all time.

Theorems & Definitions (16)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 1
  • Definition 5
  • Definition 6: Valid Finite-Time Convergent CBF
  • Theorem 2
  • proof
  • Remark 1: Existence of a finite-time convergent CBF
  • ...and 6 more