Probabilistic Control Barrier Functions for Systems with State Estimation Uncertainty using Sub-Gaussian Concentration
This work addresses safety-critical control systems like spacecraft, offering incremental improvements in probabilistic safety guarantees for stochastic uncertainties.
The paper tackled the problem of providing formal safety guarantees for safety-critical control systems under stochastic uncertainties by developing a particle-based probabilistic Control Barrier Function framework that exploits sub-Gaussian structure. The result is a tractable optimization formulation with finite-sample safety certificates, demonstrated through numerical experiments to offer tight and provably valid probabilistic guarantees.
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.