Matrix Control Barrier Functions

This paper generalizes the control barrier function framework by replacing scalar-valued functions with matrix-valued ones. Specifically, we develop barrier conditions for safe sets defined by matrix inequalities—both semidefinite and indefinite. Matrix inequalities provide a richer representation of safety constraints, generalizing multiple scalar CBFs and enabling nonsmooth safety constraints, including those arising from spectral conditions. The safety filters constructed from our proposed matrix control barrier functions via semidefinite programming (CBF-SDP) are shown to be continuous. Our matrix formulation naturally provides a continuous safety filter for Boolean-based control barrier functions, notably for disjunctions (OR), without relaxing the safe set. We illustrate the effectiveness of the proposed framework with applications in drone network connectivity maintenance—which relies on non-diagonal matrix-valued constraints—and nonsmooth obstacle avoidance, both in simulations and hardware experiments.