Finite-Sample-Based Reachability for Safe Control with Gaussian Process Dynamics

2025-05-12

AI Summary

Gaussian process model predictive control (GP-MPC) for safety-critical control of unknown nonlinear systems often suffers from excessive conservatism or lacks rigorous theoretical guarantees. Method: This paper proposes a finite-sample-based stochastic reachable set construction framework. It establishes, for the first time, a complexity theory for posterior GP sampling, enabling high-probability recursive feasibility and closed-loop safety and stability. Instead of relying on deterministic confidence bands, the approach propagates epistemic uncertainty in a data-efficient manner. By integrating stochastic reachable set analysis with probabilistic stability theory, the proposed sampling-based GP-MPC avoids over-conservatism while preserving formal guarantees. Results: Numerical experiments demonstrate a significant improvement in safety rate, strict satisfaction of state constraints, and computationally tractable online optimization—achieving both enhanced performance and theoretical rigor.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Calibration & Uncertainty Quantification
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

Gaussian Process (GP) regression is shown to be effective for learning unknown dynamics, enabling efficient and safety-aware control strategies across diverse applications. However, existing GP-based model predictive control (GP-MPC) methods either rely on approximations, thus lacking guarantees, or are overly conservative, which limits their practical utility. To close this gap, we present a sampling-based framework that efficiently propagates the model's epistemic uncertainty while avoiding conservatism. We establish a novel sample complexity result that enables the construction of a reachable set using a finite number of dynamics functions sampled from the GP posterior. Building on this, we design a sampling-based GP-MPC scheme that is recursively feasible and guarantees closed-loop safety and stability with high probability. Finally, we showcase the effectiveness of our method on two numerical examples, highlighting accurate reachable set over-approximation and safe closed-loop performance.

Problem

• Ensuring safe control with finite-sample-based reachability in GP dynamics
• Overcoming conservatism and approximation issues in GP-MPC methods
• Guaranteeing recursive feasibility and closed-loop safety in control schemes

Innovation

• Sampling-based framework for uncertainty propagation
• Finite-sample reachable set construction from GP
• Recursively feasible and safe GP-MPC scheme