Dynamical Properties of Safety Filters for Linear Systems and Affine Control Barrier Functions

TL;DR

This paper analyzes the dynamical behavior of safety filters based on Control Barrier Functions for linear systems, providing conditions for stability, boundedness, and equilibria to ensure safe control.

Contribution

It offers a detailed characterization of the dynamical properties of CBF-based safety filters specifically for linear systems and affine constraints, addressing stability and equilibrium issues.

Findings

Conditions for global exponential stability of the origin.

Identification of scenarios with undesired equilibria.

Analysis of unbounded trajectories in the closed-loop system.

Abstract

This letter studies the dynamical properties of safety filters designed based on Control Barrier Functions (CBF). This mechanism, which is popular in safety-critical applications, takes a nominal controller and minimally modifies it to render it safe. Although CBF-based safety filters make the closed-loop system safe, characterizing their additional dynamical properties, such as stability, boundedness, or existence of spurious equilibria, remains a challenging problem. Here, we address this problem for the case of linear systems and an affine CBF constraint. We provide conditions under which the closed-loop system presents undesired equilibria, unbounded trajectories, or the origin is globally exponentially stable.