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On the Steady-State Distributionally Robust Kalman Filter

Minhyuk Jang, Astghik Hakobyan, Insoon Yang

TL;DR

The paper tackles state estimation under distributional mismatch by introducing a steady-state distributionally robust Kalman filter that uses Wasserstein ambiguity sets. The method relies on a single offline semidefinite program to compute a constant DR Kalman gain $K_{ss}^*$, and it establishes contraction-based convergence of the time-varying DR Riccati recursion to the steady-state solution under explicit radius bounds (notably with $\theta_v=0$). Theoretical results connect the DR filter to risk-sensitive filtering and provide practical convergence guarantees, while simulations demonstrate superior robustness and accuracy compared to baselines under Gaussian and non-Gaussian uncertainties. The work offers an efficient, scalable approach for robust state estimation in control and estimation tasks with limited or uncertain noise data, with clear pathways to extend to nonlinear settings and DR-controlled systems.

Abstract

State estimation in the presence of uncertain or data-driven noise distributions remains a critical challenge in control and robotics. Although the Kalman filter is the most popular choice, its performance degrades significantly when distributional mismatches occur, potentially leading to instability or divergence. To address this limitation, we introduce a novel steady-state distributionally robust (DR) Kalman filter that leverages Wasserstein ambiguity sets to explicitly account for uncertainties in both process and measurement noise distributions. Our filter achieves computational efficiency by requiring merely the offline solution of a single convex semidefinite program, which yields a constant DR Kalman gain for robust state estimation under distributional mismatches. Additionally, we derive explicit theoretical conditions on the ambiguity set radius that ensure the asymptotic convergence of the time-varying DR Kalman filter to the proposed steady-state solution. Numerical simulations demonstrate that our approach outperforms existing baseline filters in terms of robustness and accuracy across both Gaussian and non-Gaussian uncertainty scenarios, highlighting its significant potential for real-world control and estimation applications.

On the Steady-State Distributionally Robust Kalman Filter

TL;DR

The paper tackles state estimation under distributional mismatch by introducing a steady-state distributionally robust Kalman filter that uses Wasserstein ambiguity sets. The method relies on a single offline semidefinite program to compute a constant DR Kalman gain , and it establishes contraction-based convergence of the time-varying DR Riccati recursion to the steady-state solution under explicit radius bounds (notably with ). Theoretical results connect the DR filter to risk-sensitive filtering and provide practical convergence guarantees, while simulations demonstrate superior robustness and accuracy compared to baselines under Gaussian and non-Gaussian uncertainties. The work offers an efficient, scalable approach for robust state estimation in control and estimation tasks with limited or uncertain noise data, with clear pathways to extend to nonlinear settings and DR-controlled systems.

Abstract

State estimation in the presence of uncertain or data-driven noise distributions remains a critical challenge in control and robotics. Although the Kalman filter is the most popular choice, its performance degrades significantly when distributional mismatches occur, potentially leading to instability or divergence. To address this limitation, we introduce a novel steady-state distributionally robust (DR) Kalman filter that leverages Wasserstein ambiguity sets to explicitly account for uncertainties in both process and measurement noise distributions. Our filter achieves computational efficiency by requiring merely the offline solution of a single convex semidefinite program, which yields a constant DR Kalman gain for robust state estimation under distributional mismatches. Additionally, we derive explicit theoretical conditions on the ambiguity set radius that ensure the asymptotic convergence of the time-varying DR Kalman filter to the proposed steady-state solution. Numerical simulations demonstrate that our approach outperforms existing baseline filters in terms of robustness and accuracy across both Gaussian and non-Gaussian uncertainty scenarios, highlighting its significant potential for real-world control and estimation applications.

Paper Structure

This paper contains 18 sections, 5 theorems, 47 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Problem Setup
  5. DR State Estimation with Wasserstein Ambiguity Sets
  6. Distributionally Robust Kalman Filter
  7. Steady-State Distributionally Robust Kalman Filter
  8. Time-Invariant DR Kalman Filtering Algorithm
  9. Convergence of the DR Kalman filter
  10. Simulation Results
  11. Convergence of DR Kalman filter
  12. Randomly Generated Systems
  13. 2D System
  14. Performance of the Steady-State DR Kalman Filter
  15. Conclusions and Future Work
  16. ...and 3 more sections

Key Result

Lemma 1

Suppose ass:Gauss holds. Then, at any time stage $t\geq 0$, the DR-MMSE estimation problem eqn:DRMMSE given the nominal distributions $\hat{\mathbb{Q}}_{x,t}^- = \mathcal{N}(\bar{x}_t^-, \hat{\Sigma}_{x,t}^-)$ and $\hat{\mathbb{Q}}_{v,t} = \mathcal{N}(\hat{v}_t, \hat{\Sigma}_{v,t})$ is equivalent to Moreover, if $(\Sigma_{x,t}^{-,*},\Sigma_{v,t}^{*})$ is an optimal solution of eqn:DRMMSE_opt, then

Figures (4)

  • Figure 1: Logarithmic plot of the relative difference between $\mathop{\mathrm{Tr}}\nolimits[\Sigma_{x,t}]$ and $\mathop{\mathrm{Tr}}\nolimits[\Sigma_{x,ss}^*]$ over time.
  • Figure 2: Distributions of LQR cost and average MSE across 200 simulation runs for each filter under Gaussian noise distributions.
  • Figure 3: Distributions of LQR cost and average MSE across 200 simulation runs for each filter under U-Quadratic noise distributions.
  • Figure 4: 2D tracking performance under Gaussian noise for each filter across 200 runs. The black dashed line shows the desired trajectory, while the colored curve represents the mean trajectory. Shaded tubes indicate $\pm$1 standard deviation.

Theorems & Definitions (9)

  • Lemma 1: Wasserstein DR-MMSE Estimation Problem nguyen2023bridging
  • Theorem 1: DR Kalman Filter
  • Remark 1
  • Proposition 1
  • Theorem 2
  • Corollary 1
  • proof
  • proof
  • proof