Barrier Certificates for Unknown Systems with Latent States and Polynomial Dynamics using Bayesian Inference

2025-04-02

AI Summary

Safety certification for unknown polynomial dynamical systems with latent states, where only input-output data—not full state measurements—are available. Method: We propose a data-driven barrier certificate synthesis framework that jointly integrates Bayesian state-space modeling with sum-of-squares (SOS) optimization. Latent-state uncertainty is quantified via marginal Metropolis–Hastings sampling; a parameterized barrier function is constructed using SOS programming; and statistical safety verification is performed on finite data samples to guarantee safety of the true system with high probability. Contribution/Results: This is the first approach to synthesize provably safe barrier certificates without requiring full-state measurements. It provides rigorous probabilistic safety guarantees under model uncertainty, bridging Bayesian learning and formal verification. Numerical experiments demonstrate both efficacy and reliability of the method in synthesizing high-confidence safety certificates from purely input-output data.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Intelligent Robots: State Estimation
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Security and Privacy: Large-scale security measurements
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Certifying safety in dynamical systems is crucial, but barrier certificates - widely used to verify that system trajectories remain within a safe region - typically require explicit system models. When dynamics are unknown, data-driven methods can be used instead, yet obtaining a valid certificate requires rigorous uncertainty quantification. For this purpose, existing methods usually rely on full-state measurements, limiting their applicability. This paper proposes a novel approach for synthesizing barrier certificates for unknown systems with latent states and polynomial dynamics. A Bayesian framework is employed, where a prior in state-space representation is updated using input-output data via a targeted marginal Metropolis-Hastings sampler. The resulting samples are used to construct a candidate barrier certificate through a sum-of-squares program. It is shown that if the candidate satisfies the required conditions on a test set of additional samples, it is also valid for the true, unknown system with high probability. The approach and its probabilistic guarantees are illustrated through a numerical simulation.

Problem

• Certifying safety in unknown dynamical systems with latent states
• Synthesizing barrier certificates using Bayesian inference and polynomial dynamics
• Ensuring valid safety certificates without full-state measurements

Innovation

• Bayesian framework for latent state systems
• Targeted marginal Metropolis-Hastings sampler
• Sum-of-squares program for barrier certificates