Probabilistic Safety under Arbitrary Disturbance Distributions using Piecewise-Affine Control Barrier Functions

We propose a simple safety filter design for stochastic discrete-time systems based on piecewise affine probabilistic control barrier functions, providing an appealing balance between modeling flexibility and computational complexity. Exact evaluation of the safety filter consists of solving a mixed-integer quadratic program (MIQP) if the dynamics are control-affine (or a mixed-integer nonlinear program in general). We propose a heuristic search method that replaces this by a small number of small-scale quadratic programs (QPs), or nonlinear programs (NLPs) respectively. The proposed approach provides a flexible framework in which arbitrary (data-driven) quantile estimators can be used to bound the probability of safety violations. Through extensive numerical experiments, we demonstrate improvements in conservatism and computation time with respect to existing methods, and we illustrate the flexibility of the method for modeling complex safety sets. Supplementary material can be found at https://mathijssch.github.io/ecc26-supplementary/.