Probabilistic Control Barrier Functions for Systems with State Estimation Uncertainty using Sub-Gaussian Concentration

TL;DR

This paper introduces a particle-based probabilistic Control Barrier Function framework that leverages sub-Gaussian properties to provide tight, computationally tractable safety guarantees for systems with stochastic uncertainties.

Contribution

It develops a novel particle-based approach exploiting sub-Gaussian structure to improve safety guarantees in stochastic control systems with state estimation uncertainties.

Findings

The framework provides explicit tail bounds for Gaussian uncertainties.

Finite-sample bounds on CVaR approximation errors are derived.

Numerical experiments demonstrate tight, valid safety guarantees.

Abstract

Safety-critical control systems, such as spacecraft performing proximity operations, must provide formal safety guarantees despite stochastic uncertainties from state estimation and unmodeled dynamics. Although Control Barrier Functions (CBFs) have been extended to stochastic systems, existing approaches typically face a trade-off between the tightness of probabilistic guarantees and computational tractability. This paper presents a particle-based probabilistic CBF framework that overcomes this limitation by exploiting the sub-Gaussian structure of the barrier function increment under Gaussian uncertainties. We establish that Gaussian uncertainties propagating through Lipschitz-continuous control-affine dynamics preserve sub-Gaussianity of the barrier function increment, with explicit tail bounds. Leveraging this structure, we derive finite-sample bounds on the approximation error between particle-based Conditional Value at Risk (CVaR) estimates and ground-truth probabilistic constraints; applying this yields a tractable optimization problem formulation with finite-sample safety certificates. We show through numerical experiments how the proposed approach provides tight yet provably valid probabilistic safety guarantees.