High-Order Matrix Control Barrier Functions: Well-Posedness and Feasibility via Matrix Relative Degree

Abstract

Control barrier functions (CBFs) provide an effective framework for enforcing safety in dynamical systems with scalar constraints. However, many safety constraints are more naturally expressed as matrix-valued conditions, such as positive definiteness or eigenvalue bounds - scalar formulations introduce potential nonsmoothness that complicates analysis. Matrix control barrier functions (MCBFs) address this limitation by directly enforcing matrix-valued safety constraints. Yet for constraints where the control input does not appear in the first derivative, high-order formulations are required. While such extensions are well understood in the scalar case, they remain largely unexplored in the matrix case. This paper develops high-order matrix control barrier functions (HOMCBFs) and establishes conditions ensuring well-posedness and feasibility of the associated constraints, enabling enforcement of matrix-valued safety constraints for systems with high-order dynamics. We further show that, using an optimal-decay HOMCBF formulation, forward invariance can be ensured while requiring control only over the minimum eigenspace. The framework is demonstrated on a localization safety problem by enforcing positive definiteness of the information matrix for a double integrator system with a nonlinear measurement model.