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

TL;DR

This work addresses safe control under unknown GP-modeled dynamics by introducing a finite-sample approach to propagate epistemic uncertainty. By sampling from the GP posterior a finite number of dynamics functions, it constructs a probabilistic reachable set that contains the true trajectory with high probability, avoiding the conservatism of worst-case propagation. Building on this, the authors design a recursive-feasible sampling-based GP-MPC that enforces constraints with high probability and provides safety and stability guarantees, with formal sample-complexity and stability proofs. The approach is demonstrated on car and pendulum examples, showing accurate reachable-set over-approximations and safe closed-loop performance. The results offer a scalable, less-conservative alternative to robust GP-MPC for safety-critical applications while maintaining tractable computation through finite sampling and online sample management.

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.

Figures (4)

• Figure 1: Illustration of the proposed sampling-based reachable set (blue region) based on a finite number of dynamics samples (green lines). Using sampling, we treat the epistemic uncertainty to an arbitrarily small tolerance $\epsilon$ (i.e., at least one sample is $\epsilon$-close to the unknown dynamics). We sequentially propagate the residual $\epsilon$-epistemic and aleatoric uncertainty, represented by black balls. As a result, the true (unknown) system response (black dashed line) is contained within a tube (green region) around the $\epsilon$-close sample.
• Figure 2: Comparison of uncertainty propagation for a given input sequence $\bm{u}$ in the car example. The dashed line represents the true trajectory of the unknown system, navigating while satisfying the constraints (solid parallel black lines). On the left, \ref{['fig:samplingMPC_car']} shows sampling-based reachable sets using $N = 2\times 10^{2}$ (red), $2\times 10^{4}$ (blue) and $2\times 10^{7}$ (green) samples. On the right, \ref{['fig:RobustMPC_car']} shows the reachable set over-approximation (red ellipsoids) obtained by the robust GP-MPC approach koller_learning-based_2018, which grows exponentially in a small horizon of 14. We observe that a tighter approximation of the reachable set achieved via the sampling-based method enables the car to navigate while satisfying the track constraints in a lane-changing maneuver, given enough samples.
• Figure 3: Illustration of sample complexity rate for the car example, showing the increase in the required number of samples as the tolerance $\epsilon$ decreases.
• Figure 4: Demonstration of safe closed-loop control using the proposed sampling-based GP-MPC with pendulum dynamics. The blue dot indicates the current state of the pendulum, the cyan region shows the computed reachable set, and the blue line depicts the predicted trajectory under the mean dynamics, all at time step $k=3$. The sampling-based GPMPC ensures that the reachable set satisfies the angular velocity constraint (red dashed line) and enters the terminal set (green ellipse). The dashed black line represents the resulting closed-loop trajectory until stabilization at the upright position.

Theorems & Definitions (29)

• Definition 1: van2011information
• Lemma 1: van2008reproducing
• Lemma 2
• Theorem 1
• proof
• Remark 1: Extension to vector-valued $g^{\star}$
• Corollary 1: Sample complexity rates
• proof
• Theorem 2: Reachable set
• proof