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Data-Driven Permissible Safe Control with Barrier Certificates

Rayan Mazouz, John Skovbekk, Frederik Baymler Mathiesen, Eric Frew, Luca Laurenti, Morteza Lahijanian

TL;DR

The paper targets safety for stochastic systems with unknown dynamics and continuous control by learning the dynamics with Gaussian processes and certifying safety via piecewise stochastic barrier functions. It constructs a maximal set of permissible strategies that guarantee probabilistic safety, with guarantees derived from learned transition-kernel bounds and PW-SBF-based invariance theorems. The approach combines GP-based learning error bounds, partitioned state-control spaces, and LP-based synthesis to produce a safe, data-driven action set, validated on linear and nonlinear systems where more data expands the permissible strategy set and tightens safety guarantees. This enables safe exploration and data collection in learning-enabled systems, providing theoretical guarantees while handling continuous control and nonconvex safe sets. Practical impact includes scalable, provable safety for unknown systems and a framework for data-efficient safe policy exploration.

Abstract

This paper introduces a method of identifying a maximal set of safe strategies from data for stochastic systems with unknown dynamics using barrier certificates. The first step is learning the dynamics of the system via Gaussian process (GP) regression and obtaining probabilistic errors for this estimate. Then, we develop an algorithm for constructing piecewise stochastic barrier functions to find a maximal permissible strategy set using the learned GP model, which is based on sequentially pruning the worst controls until a maximal set is identified. The permissible strategies are guaranteed to maintain probabilistic safety for the true system. This is especially important for learning-enabled systems, because a rich strategy space enables additional data collection and complex behaviors while remaining safe. Case studies on linear and nonlinear systems demonstrate that increasing the size of the dataset for learning the system grows the permissible strategy set.

Data-Driven Permissible Safe Control with Barrier Certificates

TL;DR

The paper targets safety for stochastic systems with unknown dynamics and continuous control by learning the dynamics with Gaussian processes and certifying safety via piecewise stochastic barrier functions. It constructs a maximal set of permissible strategies that guarantee probabilistic safety, with guarantees derived from learned transition-kernel bounds and PW-SBF-based invariance theorems. The approach combines GP-based learning error bounds, partitioned state-control spaces, and LP-based synthesis to produce a safe, data-driven action set, validated on linear and nonlinear systems where more data expands the permissible strategy set and tightens safety guarantees. This enables safe exploration and data collection in learning-enabled systems, providing theoretical guarantees while handling continuous control and nonconvex safe sets. Practical impact includes scalable, provable safety for unknown systems and a framework for data-efficient safe policy exploration.

Abstract

This paper introduces a method of identifying a maximal set of safe strategies from data for stochastic systems with unknown dynamics using barrier certificates. The first step is learning the dynamics of the system via Gaussian process (GP) regression and obtaining probabilistic errors for this estimate. Then, we develop an algorithm for constructing piecewise stochastic barrier functions to find a maximal permissible strategy set using the learned GP model, which is based on sequentially pruning the worst controls until a maximal set is identified. The permissible strategies are guaranteed to maintain probabilistic safety for the true system. This is especially important for learning-enabled systems, because a rich strategy space enables additional data collection and complex behaviors while remaining safe. Case studies on linear and nonlinear systems demonstrate that increasing the size of the dataset for learning the system grows the permissible strategy set.
Paper Structure (15 sections, 4 theorems, 21 equations, 2 figures, 1 algorithm)

This paper contains 15 sections, 4 theorems, 21 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Problem Formulation
  4. Preliminaries
  5. Gaussian Process Regression
  6. Piecewise Constant Stochastic Barrier Functions
  7. Piecewise Stochastic Barriers for Invariance via GP Regression
  8. Learning Transition Kernel Bounds
  9. Learning $f$
  10. Bounding the Transition Kernel
  11. Permissible Strategy Set Synthesis
  12. Case Studies
  13. Linear System
  14. Nonlinear System
  15. Conclusion

Key Result

Theorem 1

PWC function $B(z)$ in eq:pwc is a stochastic barrier function for process $\mathbf{z}$ with safe sets $Z \subset \mathbb{R}^{n_z}$ and initial set $Z_0 \subseteq Z$, if $\forall i \in \{1,\ldots,K_z\}$, there exist scalars $\beta,\eta \geq 0$ such that Then, for a $N\in \mathbb{N}_{\geq 0}$, it follows that

Figures (2)

  • Figure 1: Results for the linear system case studies using the known system and two datasets. Regions with # of removed control regions and simulated trajectories over 100 steps are shown in each plot. Adversarial trajectories show permissible sets maintain safety.
  • Figure 2: Results for the nonlinear system case studies using various sizes of datasets. Regions with # of removed control regions and simulated trajectories over 100 steps are shown in each plot. Adversarial trajectories show permissible control sets maintain safety.

Theorems & Definitions (9)

  • Definition 1: Probabilistic Safety
  • Definition 2: Permissible strategy
  • Remark 1
  • Theorem 1: mazouz2024piecewise
  • Proposition 1: skovbekk2023formal
  • Remark 2
  • Theorem 2: PWC-SBF for Permissible Strategies
  • Corollary 1
  • Remark 3