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This paper investigates the state estimation problem for unknown linearsystems subject to both process and measurement noise. Based on a priorinput-output trajectory sampled at a higher frequency and a prior statetrajectory sampled at a lower frequency, we propose a novel robust data-drivenKalman filter (RDKF) that integrates model identification with state estimationfor the unknown system. Specifically, the state estimation problem isformulated as a non-convex maximum likelihood optimization problem. Then, weslightly modify the optimization problem to get a problem solvable with arecursive algorithm. Based on the optimal solution to this new problem, theRDKF is designed, which can estimate the state of a given but unknownstate-space model. The performance gap between the RDKF and the optimal Kalmanfilter based on known system matrices is quantified through a sample complexitybound. In particular, when the number of the pre-collected states tends toinfinity, this gap converges to zero. Finally, the effectiveness of thetheoretical results is illustrated by numerical simulations.

Robust Data-Driven Kalman Filtering for Unknown Linear Systems using Maximum Likelihood Optimization