Sinkhorn Ambiguity Sets for Distributionally Robust Control: Convexity, Weak Compactness, and Tractability

TL;DR

This paper introduces a novel distributionally robust control framework using Sinkhorn divergence, demonstrating convexity, weak compactness, and tractability, especially with limited data, and validates it through a trajectory planning example.

Contribution

It establishes the convexity and weak compactness of Sinkhorn ambiguity sets and develops a convex programming approach for robust control with safety constraints.

Findings

Sinkhorn ambiguity sets are convex and weakly compact.

The control problem can be solved via convex programming.

The approach is effective in a trajectory planning example.

Abstract

Classical stochastic control assumes perfect knowledge of the uncertainty affecting the plant. In practice, however, such information is often incomplete. To address this limitation, we consider a distributionally robust control (DRC) problem with ambiguity sets defined via the Sinkhorn discrepancy. Compared to other discrepancy measures based on optimal transport, such as the popular Wasserstein distance, the Sinkhorn divergence does not constrain the worst-case distribution to be discrete, and allows combining observed data with prior knowledge in the form of a reference distribution, making this choice particularly suitable when only few noise samples are available for control design. We first study the properties of Sinkhorn ambiguity sets, establishing convexity and weak compactness under standard assumptions. We then leverage these results to prove that, the Sinkhorn DR linear quadratic control problem over linear policies can be solved through convex programming-even in the presence of DR safety constraints. Finally, we validate our theoretical findings and demonstrate the effectiveness of the proposed approach on a trajectory planning example.