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

Matisse Teuwen, Mathijs Schuurmans, Panagiotis Patrinos

TL;DR

This work develops probabilistic safety filters for stochastic discrete-time systems using piecewise-affine control barrier functions (PW-CBFs). It derives tractable sufficient conditions for enforcing N-step safety via quantile-based constraints and proposes a fast approximate safety-filter algorithm based on index selection, avoiding expensive mixed-integer programs. The paper extends the framework to data-driven settings, providing high-confidence guarantees when disturbance distributions are unknown, and demonstrates reduced conservatism and strong performance in quadruped navigation and obstacle-rich path planning. Overall, the approach combines modeling flexibility for complex safe sets with computational practicality for real-time deployment in uncertain environments.

Abstract

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/.

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

TL;DR

This work develops probabilistic safety filters for stochastic discrete-time systems using piecewise-affine control barrier functions (PW-CBFs). It derives tractable sufficient conditions for enforcing N-step safety via quantile-based constraints and proposes a fast approximate safety-filter algorithm based on index selection, avoiding expensive mixed-integer programs. The paper extends the framework to data-driven settings, providing high-confidence guarantees when disturbance distributions are unknown, and demonstrates reduced conservatism and strong performance in quadruped navigation and obstacle-rich path planning. Overall, the approach combines modeling flexibility for complex safe sets with computational practicality for real-time deployment in uncertain environments.

Abstract

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/.

Paper Structure

This paper contains 17 sections, 10 theorems, 40 equations, 4 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Verification of piecewise affine CBFs
  4. Sufficient conditions
  5. Safety filter design
  6. Risk allocation
  7. Search over index $j$
  8. Data-driven extensions
  9. General case
  10. Special case: g(x,xi)=xi
  11. Numerical Examples
  12. Quadruped navigation through a narrow corridor
  13. Known disturbance distribution
  14. Unknown disturbance distribution
  15. General path planning
  16. ...and 2 more sections

Key Result

Proposition II.1

Fix $N \in \N$, and let $h$ be a $\delta$-PCBF with Then, for any policy $\pi = (\kappa_k)_{k\in [N]}$, satisfying eq:p-cbf with $u_k = \kappa_k(x_k)$, for all $k \in [N]$, $P_{N}(x, \pi) \leq \varepsilon$ for all $x \in S$.

Figures (4)

  • Figure 1: Example of the search heuristic described in \ref{['sec:index-search']}. At the first iteration, $\bm j = (4, 2)$ would be selected, as shown in the figure, since this combination provides the least violation in the current state. In the case of infeasibility, the next candidates are $(1, 2), (3,2), (2,2), (4,3), (4, 3), (1, 3), \dots, (2, 4).$
  • Figure 2: Combinations of the noise standard deviation $\sigma$ and the tolerated 20-step exit probability $\varepsilon$ for which the different methods are feasible.
  • Figure 3: Empirical exit probabilities $\hat{P}_N$ in the experiment of \ref{['sec:unknown-dist']}.
  • Figure 4: State trajectories in the path planning example of \ref{['sec:path-planning']}.

Theorems & Definitions (24)

  • Definition 1: $N$-step exit probability
  • Definition 2: Probabilistic CBF mestres_ProbabilisticControlBarrier_2025
  • Remark 1
  • Proposition II.1: mestres_ProbabilisticControlBarrier_2025
  • Lemma III.1
  • Lemma III.2
  • Proposition III.1
  • proof
  • Remark 2
  • Corollary III.1
  • ...and 14 more