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Recurrent Control Barrier Functions: A Path Towards Nonparametric Safety Verification

Jixian Liu, Enrique Mallada

TL;DR

The paper addresses safety verification for high-dimensional dynamical systems by relaxing the invariance requirement of traditional reachability and barrier-function methods to a finite-time recurrence framework. It introduces Recurrent Control Barrier Functions (RCBFs), showing that the signed distance to a $\tau$-recurrent safe set is a valid RCBF under sector containment, enabling nonparametric, data-driven synthesis of safe sets. A GPU-friendly, sampling-based verification algorithm certifies the RCBF conditions on trajectory neighborhoods, balancing conservativeness and computational cost through adaptive cell sampling. Numerical experiments on a 3D evasion task demonstrate provable safety guarantees with scalable performance, outperforming conventional HJ reachability in computation time while preserving safety. This work provides a practical pathway to scalable, interpretable safety verification for complex dynamical systems.

Abstract

Ensuring the safety of complex dynamical systems often relies on Hamilton-Jacobi (HJ) Reachability Analysis or Control Barrier Functions (CBFs). Both methods require computing a function that characterizes a safe set that can be made (control) invariant. However, the computational burden of solving high-dimensional partial differential equations (for HJ Reachability) or large-scale semidefinite programs (for CBFs) makes finding such functions challenging. In this paper, we introduce the notion of Recurrent Control Barrier Functions (RCBFs), a novel class of CBFs that leverages a recurrent property of the trajectories, i.e., coming back to a safe set, for safety verification. Under mild assumptions, we show that the RCBF condition holds for the signed-distance function, turning function design into set identification. Notably, the resulting set need not be invariant to certify safety. We further propose a data-driven nonparametric method to compute safe sets that is massively parallelizable and trades off conservativeness against computational cost.

Recurrent Control Barrier Functions: A Path Towards Nonparametric Safety Verification

TL;DR

The paper addresses safety verification for high-dimensional dynamical systems by relaxing the invariance requirement of traditional reachability and barrier-function methods to a finite-time recurrence framework. It introduces Recurrent Control Barrier Functions (RCBFs), showing that the signed distance to a -recurrent safe set is a valid RCBF under sector containment, enabling nonparametric, data-driven synthesis of safe sets. A GPU-friendly, sampling-based verification algorithm certifies the RCBF conditions on trajectory neighborhoods, balancing conservativeness and computational cost through adaptive cell sampling. Numerical experiments on a 3D evasion task demonstrate provable safety guarantees with scalable performance, outperforming conventional HJ reachability in computation time while preserving safety. This work provides a practical pathway to scalable, interpretable safety verification for complex dynamical systems.

Abstract

Ensuring the safety of complex dynamical systems often relies on Hamilton-Jacobi (HJ) Reachability Analysis or Control Barrier Functions (CBFs). Both methods require computing a function that characterizes a safe set that can be made (control) invariant. However, the computational burden of solving high-dimensional partial differential equations (for HJ Reachability) or large-scale semidefinite programs (for CBFs) makes finding such functions challenging. In this paper, we introduce the notion of Recurrent Control Barrier Functions (RCBFs), a novel class of CBFs that leverages a recurrent property of the trajectories, i.e., coming back to a safe set, for safety verification. Under mild assumptions, we show that the RCBF condition holds for the signed-distance function, turning function design into set identification. Notably, the resulting set need not be invariant to certify safety. We further propose a data-driven nonparametric method to compute safe sets that is massively parallelizable and trades off conservativeness against computational cost.

Paper Structure

This paper contains 17 sections, 6 theorems, 25 equations, 3 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries and Related Work
  3. Problem Statement
  4. Safety Assessment
  5. Reachability Analysis
  6. Control Barrier Functions
  7. Recurrent Control Barrier Function
  8. Control Recurrent Sets
  9. Recurrent Control Barrier Function
  10. Signed Distance Function: a Valid RCBF
  11. Safety Enforcement Using Recurrence
  12. Verification of a Cell
  13. Numerical Methods
  14. Numerical Simulations
  15. Results Comparison
  16. ...and 2 more sections

Key Result

Theorem 1

An immediate consequence of Definition def:CBF is that any Lipschitz-continuous controller $k(x)$ satisfying renders the set $h_{\ge 0}:=\{x:h(x)\ge0\}$ invariant. Thus, $h_{\ge0}$ is, by definition, control invariant.

Figures (3)

  • Figure 1: Illustration of Recurrent Sets and Recurrent Trajectories
  • Figure 2: Contour Plot of the Boundary of the Unsafe Region with Different Precision and Different Methods when $x_3 = \pi$
  • Figure 3: Volume gap and computation time versus $\tau$ (and $\alpha$); $n_{\mathrm{s}}{=}3000$, $r_{\min}{=}0.370$.

Theorems & Definitions (19)

  • Definition 1: Safe State
  • Definition 2: Control Invariant Set
  • Definition 3: Backward Reachable Tube
  • Definition 4: Extended Class $\mathcal{K}$ Function
  • Definition 5: Control Barrier Function acenst2019ecc
  • Theorem 1: acenst2019ecc
  • Definition 6: Control Recurrent Sets
  • Definition 7: Recurrent Control Barrier Function
  • Theorem 2: Safety Assessment via RCBFs
  • proof
  • ...and 9 more