Distributionally Robust Kalman Filter
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arXiv
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| Format: | Preprint |
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2025
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| author | Jang, Minhyuk Hakobyan, Astghik Yang, Insoon |
| author_facet | Jang, Minhyuk Hakobyan, Astghik Yang, Insoon |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_06286 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Distributionally Robust Kalman Filter Jang, Minhyuk Hakobyan, Astghik Yang, Insoon Systems and Control Optimization and Control 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. |
| title | Distributionally Robust Kalman Filter |
| topic | Systems and Control Optimization and Control |
| url | https://arxiv.org/abs/2512.06286 |