Sample Efficient Certification of Discrete-Time Control Barrier Functions

2025-09-04

AI Summary

Verifying robust safety of discrete-time Control Barrier Functions (CBFs) in high-dimensional systems suffers from prohibitive computational complexity. Method: We propose a sample-efficient certification framework leveraging Lipschitz continuity. By reformulating the robust constraint verification as a scenario optimization problem and deriving tight bounds on CBF gradient variation via its Lipschitz constant, we drastically reduce the required number of sampling points. A lightweight verification algorithm is further designed to certify that the zero sublevel set satisfies robust forward invariance. Contribution/Results: Compared to conventional grid-based search or conservative approximations, our method achieves sublinear sample complexity while preserving rigorous theoretical safety guarantees. Numerical experiments on multiple nonlinear systems demonstrate that it attains accuracy comparable to full-state-space verification using less than 1% of the samples—significantly enhancing the deployability of CBFs in real-time safety-critical control.

Technology Category

• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Natural Language Processing: Safety and Robustness
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work

Abstract

Control Invariant (CI) sets are instrumental in certifying the safety of dynamical systems. Control Barrier Functions (CBFs) are effective tools to compute such sets, since the zero sublevel sets of CBFs are CI sets. However, computing CBFs generally involves addressing a complex robust optimization problem, which can be intractable. Scenario-based methods have been proposed to simplify this computation. Then, one needs to verify if the CBF actually satisfies the robust constraints. We present an approach to perform this verification that relies on Lipschitz arguments, and forms the basis of a certification algorithm designed for sample efficiency. Through a numerical example, we validated the efficiency of the proposed procedure.

Problem

• Certifying safety of dynamical systems efficiently
• Verifying Control Barrier Functions satisfy robust constraints
• Developing sample-efficient certification using Lipschitz arguments

Innovation

• Uses Lipschitz arguments for verification
• Implements sample-efficient certification algorithm
• Validates efficiency through numerical example