Stability Margins of CBF-QP Safety Filters: Analysis and Synthesis

TL;DR

This paper analyzes the stability margins of control barrier function-based safety filters, providing exact characterizations and synthesis methods to ensure robustness in safety-critical control systems.

Contribution

It introduces a scalar loop representation for active-mode dynamics, enabling LMI-based certificates and synthesis conditions for robust stability guarantees.

Findings

Exact stability-margin characterizations for safety filters.

LMI-based certificates for controller robustness.

Numerical examples demonstrating margin enlargement.

Abstract

Control barrier function (CBF)-QP safety filters enforce safety by minimally modifying a nominal controller. While prior work has mainly addressed robustness of safety under uncertainty, robustness of the resulting closed-loop \emph{stability} is much less understood. This issue is important because once the safety filter becomes active, it modifies the nominal dynamics and can reduce stability margins or even destabilize the system, despite preserving safety. For linear systems with a single affine safety constraint, we show that the active-mode dynamics admit an exact scalar loop representation, leading to a classical robust-control interpretation in terms of gain, phase, and delay margins. This viewpoint yields exact stability-margin characterizations and tractable linear matrix inequality (LMI)-based certificates and synthesis conditions for controllers with certified robustness guarantees. Numerical examples illustrate the proposed analysis and the enlargement of certified stability margins for safety-filtered systems.