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Uniform Feasibility For Smoothed Backup Control Barrier Functions

Anil Alan, Bart De Schutter

TL;DR

The paper tackles the challenge of guaranteeing feasibility for CBF-based safety filters when the safe set is defined as a nonsmooth minimum of multiple constraints. It introduces a smooth inner approximation via the log-sum-exp (soft-min) operator and derives explicit uniform bounds on the smoothing parameter $\theta$ that certify the smoothed function $\tilde{h}_\theta$ as a CBF or eCBF, for both compact and unbounded safe sets. By applying converse safety theorems, the authors provide an a priori certificate of feasibility, eliminating the need for online certification in many scenarios. In the backup CBF framework, the work shows that safety of a compact backup set under a backup controller, together with a boundary-safety condition, suffices to certify the nonsmooth backup CBF, thus relaxing global compactness requirements. The results offer practical guidelines for implementing smooth inner-approximations of nonsmooth safe sets with guaranteed feasibility in a broad range of safety-critical control applications.

Abstract

We study feasibility guarantees for safety filters developed using Control Barrier Functions (CBFs) when a safe set is defined using the pointwise minimum of continuously differentiable functions, a construction that is common for the backup CBF method and typically nonsmooth. We replace the minimum by its log-sum-exp (soft-min) smoothing and show that, under a strict safety condition, the smooth function becomes a CBF (or extended CBF) for a range of the smoothing parameter. For compact safe sets, we derive an explicit lower bound on the smoothing parameter that makes the smooth function a CBF and hence renders the corresponding safety filter feasible. For unbounded sets, we introduce tail conditions under which the smooth function satisfies an extended CBF condition uniformly. Finally, we apply these results to backup CBFs. We show that safety of a compact (terminal) backup set under a backup controller, together with a condition ensuring safety of the backup trajectories on the relevant boundary of the safe set, is sufficient for feasibility for backup CBFs. These results provide a recipe for a priori feasibility guarantees for smooth inner approximations of nonsmooth safe sets without the need for additional online certification.

Uniform Feasibility For Smoothed Backup Control Barrier Functions

TL;DR

The paper tackles the challenge of guaranteeing feasibility for CBF-based safety filters when the safe set is defined as a nonsmooth minimum of multiple constraints. It introduces a smooth inner approximation via the log-sum-exp (soft-min) operator and derives explicit uniform bounds on the smoothing parameter that certify the smoothed function as a CBF or eCBF, for both compact and unbounded safe sets. By applying converse safety theorems, the authors provide an a priori certificate of feasibility, eliminating the need for online certification in many scenarios. In the backup CBF framework, the work shows that safety of a compact backup set under a backup controller, together with a boundary-safety condition, suffices to certify the nonsmooth backup CBF, thus relaxing global compactness requirements. The results offer practical guidelines for implementing smooth inner-approximations of nonsmooth safe sets with guaranteed feasibility in a broad range of safety-critical control applications.

Abstract

We study feasibility guarantees for safety filters developed using Control Barrier Functions (CBFs) when a safe set is defined using the pointwise minimum of continuously differentiable functions, a construction that is common for the backup CBF method and typically nonsmooth. We replace the minimum by its log-sum-exp (soft-min) smoothing and show that, under a strict safety condition, the smooth function becomes a CBF (or extended CBF) for a range of the smoothing parameter. For compact safe sets, we derive an explicit lower bound on the smoothing parameter that makes the smooth function a CBF and hence renders the corresponding safety filter feasible. For unbounded sets, we introduce tail conditions under which the smooth function satisfies an extended CBF condition uniformly. Finally, we apply these results to backup CBFs. We show that safety of a compact (terminal) backup set under a backup controller, together with a condition ensuring safety of the backup trajectories on the relevant boundary of the safe set, is sufficient for feasibility for backup CBFs. These results provide a recipe for a priori feasibility guarantees for smooth inner approximations of nonsmooth safe sets without the need for additional online certification.

Paper Structure

This paper contains 15 sections, 7 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Notation
  4. Control Barrier Functions and Converse Safety Theorems
  5. Backup Control Barrier Functions
  6. Main Results
  7. Problem Formulation
  8. Compact Sets
  9. General Case
  10. Connection to the Backup CBF Framework
  11. $\hat{\mathcal{B}}$ is compact
  12. $b_i\in C^1$ for all $i\in \mathbb{I}_N$
  13. MFCQ holds on $\partial \hat{\mathcal{B}}$
  14. (The backup system is strictly safe w.r.t. $\hat{\mathcal{B}}$
  15. Conclusion

Key Result

Theorem 1

Let $\mathcal{S}$ be a compact set defined with $h\in C^1$ as in eq:Sc and assume 0 is a regular value of $h$. If there exists a controller $k:\mathbb{R}^n\to\mathbb{U}$ such that $F(x)=f(x,k(x))$ is safe w.r.t. $\mathcal{S}$, then $h$ is a CBF.

Theorems & Definitions (19)

  • Definition 1: CBF, ames2019control
  • Definition 2: Extended CBF, mestres2024converse
  • Theorem 1: Converse safety for compact sets, ames2019control
  • Theorem 2: Converse safety for general case, mestres2024converse
  • Remark 1
  • Definition 3: Strict safety
  • Remark 2
  • Lemma 1
  • proof
  • Theorem 3: $\tilde{h}_\theta$ is a CBF for compact sets
  • ...and 9 more