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

Samuel G. Gessow, Pio Ong, Aaron D. Ames, Brett T. Lopez

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.

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.

Paper Structure

This paper contains 16 sections, 6 theorems, 31 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Control Barrier Functions
  5. High-Order CBF
  6. Matrix Control Barrier Function
  7. High-Order Matrix CBFs
  8. Matrix Relative Degree Notions
  9. Scalar Case Revisited
  10. Matrix Constraint Well-Posedness
  11. Matrix Relative Degrees
  12. Optimal Decay HOMCBF
  13. Application in Safe Localization
  14. Problem
  15. Results
  16. ...and 1 more sections

Key Result

Theorem 1

Consider the control-affine system sys:ctrl_affine with sets $\{\mathcal{C}_i\}_{i\in[r]}$ defined as in eq:HO_safe_set. If $\mathbf{H}$ is a HOMCBF for sys:ctrl_affine, then any continuous feedback controller $\mathbf{k}:\mathbb{R}^n \rightarrow \mathbb{R}^m$ satisfying: for all $\mathbf{x}$ in an open neighborhood of $\cap_{i\in[r]}\mathcal{C}_i$, renders the intersection $\cap_{i\in[r]} \mathc

Figures (2)

  • Figure F1: (a): Trajectory using range-only measurements; the safe region is indicated as inside the red line and the desired state with the green $\times$. (b): The barrier value of $\lambda_{\min} (\mathbf{H})$
  • Figure F2: (a): Trajectory using heading-only measurements; the safe region is indicated as inside the red line and the desired state with the green $\times$. (b): The barrier value of $\lambda_{\min} (\mathbf{H})$.

Theorems & Definitions (21)

  • Definition 1
  • Definition 2: cf. ames2016control
  • Definition 3: cf. xiao2021high
  • Definition 4
  • Definition 5: cf. ong2025matrix
  • Definition 6
  • Theorem 1
  • proof
  • Definition 7
  • Theorem 2
  • ...and 11 more