A Spectral Perspective on Stochastic Control Barrier Functions

TL;DR

This paper introduces a spectral approach to stochastic control barrier functions, using the dominant eigenpair of a linear operator to accurately quantify and synthesize safety probabilities in stochastic systems.

Contribution

It proposes a novel spectral perspective that leverages the dominant eigenpair of a Koopman-like operator to synthesize valid safety-critical control barrier functions.

Findings

The dominant eigenfunction encodes the system's safety probability.

The dominant eigenvalue indicates the safety probability decay rate.

The power-policy iteration algorithm effectively computes the eigenpair and backup policy.

Abstract

Stochastic control barrier functions (SCBFs) provide a safety-critical control framework for systems subject to stochastic disturbances by bounding the probability of remaining within a safe set. However, synthesizing a valid SCBF that explicitly reflects the true safety probability of the system, which is the most natural measure of safety, remains a challenge. This paper addresses this issue by adopting a spectral perspective, utilizing the linear operator that governs the evolution of the closed-loop system's safety probability. We find that the dominant eigenpair of this Koopman-like operator encodes fundamental safety information of the stochastic system. The dominant eigenfunction is a natural and valid SCBF, with values that explicitly quantify the relative long-term safety of the state, while the dominant eigenvalue indicates the global rate at which the safety probability decays. A practical synthesis algorithm is proposed, termed power-policy iteration, which jointly computes the dominant eigenpair and an optimized backup policy. The method is validated using simulation experiments on safety-critical dynamics models.