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Adversarial Robustness for Matrix Control Barrier Functions in Sampled-Data Systems

James Usevitch

Abstract

This paper presents novel theoretical results to guarantee multi-agent set invariance using Matrix Control Barrier Functions in sampled-data systems. More specifically, the paper presents conditions under which heterogeneous control-affine agents applying zero-order-hold control inputs can compute control inputs to render safe sets defined by matrix inequalities forward invariant. It then introduces methods to guarantee set invariance while accounting for the presence of adversarial agents seeking to drive the system state to unsafe sets. Finally, the paper presents theoretical extensions of these set invariance results to systems having high relative degree with respect to the matrix-valued safe set function.

Adversarial Robustness for Matrix Control Barrier Functions in Sampled-Data Systems

Abstract

This paper presents novel theoretical results to guarantee multi-agent set invariance using Matrix Control Barrier Functions in sampled-data systems. More specifically, the paper presents conditions under which heterogeneous control-affine agents applying zero-order-hold control inputs can compute control inputs to render safe sets defined by matrix inequalities forward invariant. It then introduces methods to guarantee set invariance while accounting for the presence of adversarial agents seeking to drive the system state to unsafe sets. Finally, the paper presents theoretical extensions of these set invariance results to systems having high relative degree with respect to the matrix-valued safe set function.
Paper Structure (10 sections, 8 theorems, 39 equations)

This paper contains 10 sections, 8 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Notation and Problem Formulation
  3. Problem Formulation
  4. Preliminaries and Overview of Matrix Control Barrier Functions
  5. Main Results
  6. Zero-Order Hold Exponential Matrix Control Barrier Functions
  7. Multi-Agent Exponential Matrix Control Barrier Functions
  8. Adversarial Robustness for Exponential Matrix Control Barrier Functions
  9. Systems with High Relative Degree
  10. Conclusion

Key Result

Proposition 1

Consider the autonomous system with control affine dynamics $\dot{\vec{x}} = \widehat{F}(\vec{x})$ with a continuous vector field $\widehat{F}$ Let $S$ be defined as in eq:safe_set. If the following barrier condition holds with a positive constant $c_\alpha > 0$, for all $\vec{x}$ in an open neighborhood $\mathcal{E} \supset S$, then set $S$ is forward invariant for the system.

Theorems & Definitions (20)

  • Definition 1
  • Proposition 1: Adapted from ong2025matrix
  • Proposition 2: Weyl's Inequality horn2012matrix
  • Corollary 1
  • proof
  • Remark 1
  • Lemma 1
  • proof
  • Definition 2
  • Theorem 1
  • ...and 10 more