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

This paper investigates the state estimation problem for unknown linear systems subject to both process and measurement noise. Based on a prior input-output trajectory sampled at a higher frequency and a prior state trajectory sampled at a lower frequency, we propose a novel robust data-driven Kalman filter (RDKF) that integrates model identification with state estimation for the unknown system. Specifically, the state estimation problem is formulated as a non-convex maximum likelihood optimization problem. Then, we slightly modify the optimization problem to get a problem solvable with a recursive algorithm. Based on the optimal solution to this new problem, the RDKF is designed, which can estimate the state of a given but unknown state-space model. The performance gap between the RDKF and the optimal Kalman filter based on known system matrices is quantified through a sample complexity bound. In particular, when the number of the pre-collected states tends to infinity, this gap converges to zero. Finally, the effectiveness of the theoretical results is illustrated by numerical simulations.