Distributionally Robust Kalman Filter

TL;DR

This paper introduces a distributionally robust Kalman filter that accounts for ambiguity in noise distributions using Wasserstein sets, providing a computationally efficient, minimax optimal state estimation method with guaranteed stability.

Contribution

It proposes a novel noise-centric DRKF framework with SDP-based computation, convergence guarantees, and spectral bounds, advancing robust state estimation under distributional uncertainty.

Findings

The steady-state DRKF is obtained from a stationary SDP with constant gain.

The DR Riccati covariance iteration converges to the stationary SDP solution.

The steady-state DRKF is asymptotically minimax optimal for worst-case MSE.

Abstract

We study state estimation for discrete-time linear stochastic systems under distributional ambiguity in the initial state, process noise, and measurement noise. We propose a noise-centric distributionally robust Kalman filter (DRKF) based on Wasserstein ambiguity sets imposed directly on these distributions. This formulation excludes dynamically unreachable priors and yields a Kalman-type recursion driven by least-favorable covariances computed via semidefinite programs (SDP). In the time-invariant case, the steady-state DRKF is obtained from a single stationary SDP, producing a constant gain with Kalman-level online complexity. We establish the convergence of the DR Riccati covariance iteration to the stationary SDP solution, together with an explicit sufficient condition for a prescribed convergence rate. We further show that the proposed noise-centric model induces a priori spectral bounds on all feasible covariances and a Kalman filter sandwiching property for the DRKF covariances. Finally, we prove that the steady-state error dynamics are Schur stable, and the steady-state DRKF is asymptotically minimax optimal with respect to worst-case mean-square error.