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

TL;DR

This paper extends control barrier functions to matrix-valued safety constraints, introducing high-order formulations that ensure well-posedness and feasibility for systems with high-order dynamics.

Contribution

It develops high-order matrix control barrier functions (HOMCBFs) and establishes conditions for their well-posedness and feasibility, enabling matrix safety enforcement in complex systems.

Findings

HOMCBFs ensure safety constraints are well-posed and feasible.

Optimal-decay HOMCBFs can enforce invariance by controlling only the minimum eigenspace.

Application demonstrated on a localization safety problem with positive definite information matrix.

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.