A Spectral Perspective on Stochastic Control Barrier Functions

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.

Figures (7)

• Figure 1: Visualizations of $\psi$ during the course of convergence of power-policy iteration in the omnidirectional case (Case 1) of the double integrator example (Section \ref{['sec: double integrator']}). In all subfigures, the red boxes represent the safety constraint $C = [-1, 1] \times [-2, 2]$ where the horizontal and vertical axes represent the position and the velocity, respectively. Regardless of what we take as the initial guess, power-policy iteration converges to the same dominant eigenfunction.
• Figure 2: (Top) The error between consecutive iterates $\psi$ during the course of convergence. The error is with respect to the Banach space norm. (Bottom) The convergence of $\gamma^\pi$ value estimates. It can be seen that the iteration converges to the same value $\gamma^\pi = 1.2424$, regardless of the initial guess.
• Figure 3: (a) The closed-loop trajectories for the omnidirectional $\sigma$ case (Case 1) of the double integrator example (Section \ref{['sec: double integrator']}). (b) Empirical safety probability and S. P. = Safety Probability.
• Figure 4: The converged eigenfunctions for cases 2, 3, and 4 in the double integrator example (Section \ref{['sec: double integrator']}).
• Figure 5: The $0.1$- and $0.9$-level sets of the converged eigenfunction in the WIG example (Section \ref{['sec: aircraft']}). The red skeletons denote the safety constraints \ref{['eq: aircraft constraints']}. The computed $\gamma^\pi$ value is $1.2 \times 10^{-4} \; \mathrm{/s}$. FPA = Flight path angle.

Theorems & Definitions (34)

• Definition 1: Control Barrier Function ames2019control
• Definition 2: Infinitesimal Generator of a Stochastic Process
• Lemma 1: Itô's Lemma ito1944stochastic
• Corollary 1
• Definition 3: Stochastic CBF
• Theorem 1: Safety Probability Lower Bound
• proof
• Theorem 2: Properties of $T_t^\pi$
• proof
• Theorem 3