Sinkhorn Distributionally Robust State Estimation via System Level Synthesis

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.

💡 Research Summary

The paper addresses the practical challenge that the true disturbance distribution in state‑estimation problems is rarely known exactly, which makes conventional estimators fragile. While recent Wasserstein distributionally robust state estimation (DRSE) mitigates this issue, its worst‑case distribution is provably discrete, leading to excessive conservatism because real‑world disturbances are typically continuous. To overcome this limitation, the authors propose a Sinkhorn‑based DRSE design embedded within the System Level Synthesis (SLS) framework.

Key contributions are as follows. First, they define a Sinkhorn ambiguity set that augments the classical Wasserstein ball with an entropic regularization term. This regularizer penalizes non‑smooth, discrete distributions, encouraging the worst‑case distribution to remain continuous. For this set they derive the first finite‑sample probabilistic guarantee, explicitly relating the sample size, the regularization parameter ε, and the Wasserstein radius θ to a confidence bound on the true distribution.

Second, they analyze the limiting behavior of the design as ε varies. When ε → 0 the Sinkhorn distance collapses to the ordinary Wasserstein distance, and the formulation reduces to the standard Wasserstein DRSE. As ε → ∞ the entropic term dominates, and the objective converges to the classic H₂ performance metric. Hence the proposed method interpolates continuously between a purely robust (Wasserstein) and a purely performance‑oriented (H₂) estimator, giving practitioners a single tuning knob to balance robustness and estimation accuracy.

Third, the original min‑max estimator design problem is transformed into a finite‑dimensional convex program via duality. The decision variables are the closed‑loop response maps Φₓ and Φ_y that appear in the SLS parameterization of the observer. Although the feasible region is unbounded, the authors identify a compact subset that provably contains the global optimum. Within this compact set they develop a tailored Frank‑Wolfe algorithm. Each iteration solves a linear subproblem (gradient inner‑product minimization) and performs a line‑search; the authors prove a convergence rate of O(1/t). This first‑order method is far more scalable than the semidefinite programming approaches traditionally used for robust observer synthesis.

Fourth, extensive numerical experiments on second‑ and fourth‑order linear systems are presented. The experiments compare three designs: the classical H₂ observer, the Wasserstein DRSE, and the proposed Sinkhorn DRSE, under various disturbance distributions (Gaussian, multimodal, heavy‑tailed) and small sample regimes (N = 20–50). Results show that Sinkhorn DRSE consistently achieves lower mean‑square error—typically 10–30 % improvement over Wasserstein DRSE and H₂—especially when the disturbance exhibits non‑Gaussian features. Moreover, the Frank‑Wolfe solver reaches a tolerance of 10⁻⁴ within 200–300 iterations, reducing both memory usage and runtime by more than 70 % compared with SDP‑based solvers.

In summary, the paper introduces a novel, theoretically grounded, and computationally efficient framework for distributionally robust state estimation that respects the continuity of real disturbances. By integrating entropic regularization into the ambiguity set, providing finite‑sample guarantees, and delivering a scalable first‑order algorithm, the work bridges the gap between robust and optimal estimation and opens avenues for extensions to nonlinear, time‑varying, and distributed settings.