Distributionally Robust Kalman Filtering over an Infinite-Horizon

This paper investigates distributionally robust filtering for state-space models subject to exogenous disturbances in state evolution and observation processes. The joint probability distribution of the disturbance process over an arbitrary horizon is unknown but is assumed to reside within a Wasserstein-2 ambiguity set centered around a nominal distribution. We seek to develop a causal estimator that minimizes the worst-case mean squared error (MSE) among all distributions in this ambiguity set. Unlike prior works, our framework accommodates disturbances with arbitrary temporal correlations. In the finite-horizon setting, this problem reduces to a semi-definite program (SDP) whose complexity scales with the time horizon. Consequently, we focus on the infinite-horizon case, where we derive the optimal linear time-invariant (LTI) filter using the Karush-Kuhn-Tucker (KKT) conditions and propose an efficient frequency-domain algorithm to compute it. While the optimal LTI filter generally has a non-rational transfer function, we provide a minimax rational approximation method to approximate the optimal non-rational filter, in the H<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</inf>-norm, with a near-optimal finite-order state-space estimator. This approach avoids the computational challenges associated with horizon-dependent scaling in the finite-horizon case. Numerical simulations demonstrate the effectiveness of the proposed method.