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

TL;DR

This paper introduces a sampling-based Gaussian Process control method that guarantees safety and stability with finite samples, improving over conservative or approximate existing approaches.

Contribution

We develop a novel finite-sample-based framework for reachability analysis in GP-driven control, ensuring safety and stability with high probability.

Findings

Accurate reachable set over-approximation demonstrated

Guarantees of safety and stability in closed-loop control

Efficient sampling method reduces conservatism

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.