Distributionally Robust Kalman Filter

TL;DR

The paper addresses robust state estimation for linear systems when noise statistics are uncertain, introducing a noise-centric distributionally robust Kalman filter (DRKF) that places Wasserstein ambiguity sets on the initial state, process noise, and measurement noise.It preserves the structural form of the classical Kalman filter while providing a priori spectral bounds on feasible covariances and enabling tractable offline and online computation through convex SDPs.A steady-state DRKF is obtained from a single stationary SDP, yielding a constant-gain filter with Kalman-level online complexity and strong robustness guarantees, including spectral boundedness, convergence, stability, and asymptotic minimax optimality.Numerical experiments demonstrate improved estimation accuracy and more reliable uncertainty quantification under unknown or uncertain noise models, with favorable performance in trajectory tracking and computational efficiency relative to existing robust and distributionally robust filters.

Abstract

In this work, we propose a noise-centric formulation of the distributionally robust Kalman filter (DRKF) for discrete-time linear stochastic systems with uncertain noise statistics. By placing Wasserstein ambiguity sets directly on the process and measurement noise distributions, the proposed DRKF preserves the analytical structure of the classical Kalman filter while providing a priori spectral bounds on all feasible covariances. In the time-invariant setting, we derive a steady-state DRKF from a single stationary semidefinite program, yielding a constant-gain estimator with the same per-step computational complexity as the standard Kalman filter. We establish conditions guaranteeing the existence, uniqueness, and convergence of this steady-state solution, and we prove its asymptotic minimax optimality with respect to the worst-case mean-square error. Numerical experiments validate the theory and demonstrate that the proposed DRKF improves estimation accuracy under unknown or uncertain noise models while offering computational advantages over existing robust and distributionally robust filters.

Figures (9)

• Figure 1: Effect of $\theta$ on the average regret MSE under a) Gaussian and b) U-Quadratic noise distributions, averaged over 20 simulation runs.
• Figure 2: Effect of the number of samples used to construct the nominal distributions on the average MSE under Gaussian noise, averaged over 10 runs.
• Figure 3: Average regret MSE of the proposed steady-state DRKF under Gaussian noise, as a function of $\theta_w$ and $\theta_v$, averaged over 100 runs.
• Figure 4: Effect of $\theta$ on the average MPC cost under a) Gaussian and b) nonzero-mean U-Quadratic noise, averaged over 20 simulation runs.
• Figure 5: 2D trajectories averaged over 200 simulations under Gaussian noise. Each filter uses its optimal parameter $\theta \in [10^{-2}, 10^{1}]$ minimizing the MPC cost. The black dashed line denotes the desired trajectory; colored curves show the mean, and shaded tubes indicate $\pm 1$ standard deviation.

Theorems & Definitions (32)

• Remark 1
• Lemma 1
• Remark 2
• Remark 3
• Corollary 1
• Theorem 1: DR Kalman Filter
• Lemma 2
• Proposition 1
• Corollary 2
• Theorem 2