Sinkhorn Distributionally Robust State Estimation via System Level Synthesis

TL;DR

This paper introduces a novel Sinkhorn distributionally robust state estimation method using system level synthesis, which shapes errors under unknown continuous disturbances and offers probabilistic guarantees, improving robustness over existing approaches.

Contribution

It develops the first Sinkhorn ambiguity set with finite-sample guarantees, reformulates the robust estimation as a convex program, and proposes an efficient Frank-Wolfe algorithm with proven convergence.

Findings

Outperforms existing schemes in numerical case studies.

Provides finite-sample probabilistic guarantees.

Establishes connection with $\,\mathcal{H}_2$ and Wasserstein DRSE designs.

Abstract

In state estimation tasks, the usual assumption of exactly known disturbance distribution is often unrealistic and renders the estimator fragile in practice. The recently emerging Wasserstein distributionally robust state estimation (DRSE) design can partially mitigate this fragility; however, its worst-case distribution is provably discrete, which deviates from the inherent continuity of real-world distributions and results in over-pessimism. In this work, we develop a new Sinkhorn DRSE design within system level synthesis scheme with the aim of shaping the closed-loop errors under the unknown continuous disturbance distribution. For uncertainty description, we adopt the Sinkhorn ambiguity set that includes an entropic regularizer to penalize non-smooth and discrete distributions within a Wasserstein ball. We present the first result of finite-sample probabilistic guarantee of the Sinkhorn ambiguity set. Then we analyze the limiting properties of our Sinkhorn DRSE design, thereby highlighting its close connection with the generic $\mathcal{H}_2$ design and Wasserstein DRSE. To tackle the min-max optimization problem, we reformulate it as a finite-dimensional convex program through duality theory. By identifying a compact subset of the feasible set guaranteed to enclose the global optimum, we develop a tailored Frank-Wolfe solution algorithm and formally establish its convergence rate. The advantage of Sinkhorn DRSE over existing design schemes is verified through numerical case studies.