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

TL;DR

This paper introduces certificates that verify the compatibility of multiple control barrier function constraints over convex hulls, enabling efficient safety filtering in control systems.

Contribution

It presents novel feasibility conditions and computational methods for certifying control constraint compatibility from sampled vertices to their convex hull.

Findings

Feasibility conditions for control constraints over convex hulls.

Methods based on interval intersections and linear programming.

Conditions for affine quadratic-program safety filters.

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.