A Spectral Perspective on Stochastic Control Barrier Functions
This work addresses safety-critical control for stochastic systems, offering a method to explicitly quantify long-term safety probabilities, though it appears incremental as it builds on existing SCBF frameworks.
The paper tackled the challenge of synthesizing stochastic control barrier functions (SCBFs) that accurately reflect true safety probabilities in systems with stochastic disturbances, by introducing a spectral perspective using a Koopman-like operator to derive a dominant eigenfunction as a valid SCBF and a synthesis algorithm called power-policy iteration.
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.