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From Vertices to Convex Hulls: Certifying Set-Wise Compatibility for CBF Constraints

Shima Sadat Mousavi, Xiao Tan, Aaron D. Ames

TL;DR

This work addresses ensuring safety for systems subject to multiple CBF constraints under input bounds over a region. It introduces Compatibility Propagation Certificates (CPCs) that leverage vertex feasibility to certify hull-wide feasibility under concavity and curvature-alignment assumptions, using offline interval intersections (CPC--Interval) or small linear programs (CPC--Common), and, when needed, convex blending of vertex inputs (CPC--Blend). An additional result shows that, with affine data, the CBF-QP safety filter can be evaluated as an affine interpolation of vertex solutions, enabling explicit, online-safe controllers without real-time optimization. The approach provides a hierarchy of lightweight, region-wise certificates that complement pointwise CBF-QP methods and reduce online computation, with numerical case studies illustrating practical applicability. Together, these contributions enable robust, explicit safety controllers across convex hulls of operating states by performing offline certifications and leveraging affine interpolations when possible.

Abstract

This paper develops certificates that propagate compatibility of multiple control barrier function (CBF) constraints from sampled vertices to their convex hull. Under mild concavity and affinity assumptions, we present three sufficient feasibility conditions under which feasible inputs over the convex hull can be obtained per coordinate, with a common input, or via convex blending. We also describe the associated computational methods, based on interval intersections or an offline linear program (LP). Beyond certifying compatibility, we give conditions under which the quadratic-program (QP) safety filter is affine in the state. This enables explicit implementations via convex combinations of vertex-feasible inputs. Case studies illustrate the results.

From Vertices to Convex Hulls: Certifying Set-Wise Compatibility for CBF Constraints

TL;DR

This work addresses ensuring safety for systems subject to multiple CBF constraints under input bounds over a region. It introduces Compatibility Propagation Certificates (CPCs) that leverage vertex feasibility to certify hull-wide feasibility under concavity and curvature-alignment assumptions, using offline interval intersections (CPC--Interval) or small linear programs (CPC--Common), and, when needed, convex blending of vertex inputs (CPC--Blend). An additional result shows that, with affine data, the CBF-QP safety filter can be evaluated as an affine interpolation of vertex solutions, enabling explicit, online-safe controllers without real-time optimization. The approach provides a hierarchy of lightweight, region-wise certificates that complement pointwise CBF-QP methods and reduce online computation, with numerical case studies illustrating practical applicability. Together, these contributions enable robust, explicit safety controllers across convex hulls of operating states by performing offline certifications and leveraging affine interpolations when possible.

Abstract

This paper develops certificates that propagate compatibility of multiple control barrier function (CBF) constraints from sampled vertices to their convex hull. Under mild concavity and affinity assumptions, we present three sufficient feasibility conditions under which feasible inputs over the convex hull can be obtained per coordinate, with a common input, or via convex blending. We also describe the associated computational methods, based on interval intersections or an offline linear program (LP). Beyond certifying compatibility, we give conditions under which the quadratic-program (QP) safety filter is affine in the state. This enables explicit implementations via convex combinations of vertex-feasible inputs. Case studies illustrate the results.
Paper Structure (9 sections, 8 theorems, 24 equations, 3 figures, 1 table)

This paper contains 9 sections, 8 theorems, 24 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Compatibility over Convex Hulls
  4. Certificate I: CPC--Interval
  5. Certificate II: CPC--Common
  6. Certificate III: CPC--Blend
  7. Affine Interpolation of the Safety Filter
  8. Numerical Examples
  9. Conclusion

Key Result

Lemma 1

Under (A3), for any $u\in\mathcal{S}(\Psi)$ the map $\phi(x)\triangleq\Psi(x)u$ is concave on $H$.

Figures (3)

  • Figure 1: Three-room temperature trajectories using different controllers.
  • Figure 2: Partition of the polytope $H$ into critical regions: green region $H_1$ (both constraints inactive), pink region $H_2$ (second constraint active), and cyan region $H_3$ (first constraint active).
  • Figure 3: Trajectories and inputs under affine interpolation of the safety filters. Stars mark random initial states.

Theorems & Definitions (25)

  • Definition 1
  • Definition 2
  • Definition 3
  • Remark 1
  • Lemma 1
  • proof
  • Remark 2
  • Example 1
  • Theorem 1
  • proof
  • ...and 15 more