Counterexample-Guided Synthesis of Robust Discrete-Time Control Barrier Functions

TL;DR

This paper introduces a counterexample-guided method for synthesizing robust discrete-time control barrier functions that ensure safety across entire regions, addressing conservatism in learning-based approaches.

Contribution

It proposes a novel counterexample-guided synthesis approach for robust discrete-time control barrier functions applicable to systems with input constraints and disturbances.

Findings

Effective synthesis of R-DTCBFs demonstrated on numerical case studies

Reduces conservatism compared to existing methods

Ensures safety over entire regions, not just sampled states

Abstract

Learning-based methods have gained popularity for training candidate Control Barrier Functions (CBFs) to satisfy the CBF conditions on a finite set of sampled states. However, since the CBF is unknown a priori, it is unclear which sampled states belong to its zero-superlevel set and must satisfy the CBF conditions, and which ones lie outside it. Existing approaches define a set in which all sampled states are required to satisfy the CBF conditions, thus introducing conservatism. In this paper, we address this issue for robust discrete-time CBFs (R-DTCBFs). Furthermore, we propose a class of R-DTCBFs that can be used in an online optimization problem to synthesize safe controllers for general discrete-time systems with input constraints and bounded disturbances. To train such an R-DTCBF that is valid not only on sampled states but also across the entire region, we employ a verification algorithm iteratively in a counterexample-guided approach. We apply the proposed method to numerical case studies.