Optimal Decay Matrix Control Barrier Functions: Soundness and Completeness for Matrix Safety Constraints

Abstract

Many safety requirements, such as keeping a manipulator away from kinematic singularities, a network connected, or a system observable, amount to keeping a state-dependent matrix positive definite. Matrix control barrier functions (MCBFs) enforce such requirements with a semidefinite program that minimally modifies a nominal control input. The MCBF constraint bounds how fast the safety margin, the smallest eigenvalue of the matrix, may decay. Yet a fixed decay rule is conservative, rejecting safe controls that drive the state quickly toward, but never across, the boundary of the safe set. Scalar optimal decay (SOD) addresses this issue by scaling the decay term by a scalar weight that is optimized together with the control. For scalar barrier functions, SOD then rejects a control at the boundary only if the control would decrease the margin. For MCBFs, however, SOD can reject controls that leave the margin unchanged, and near the safety boundary SOD accepts such controls only at a high cost. We introduce matrix optimal decay (MOD), which replaces the scalar weight with a matrix multiplier while keeping the semidefinite program convex. MOD is provably sound and complete, certifying at the boundary exactly the controls that do not decrease the margin. If some control can strictly increase the margin at every boundary point, the MOD-MCBF filtered controller is continuous and renders the safe set forward invariant. We also add an optional recovery condition that drives the system toward the safe set after a violation without affecting soundness or completeness. On a planar two-link arm avoiding singularities, MOD-MCBF certifies safe motions that both a fixed decay rule and SOD reject, and tracks an end-effector target more closely than SOD.