This article investigates the problem of data-driven state estimation for linear systems with both unknown system dynamics and noise covariances. We propose an Autocovariance Least-squares-based Data-driven Kalman Filter (ADKF), which provides a unified framework for simultaneous system identification and state estimation by utilizing pre-collected input-output trajectories and estimated initial states. Specifically, we design a SDP-based algorithm for estimating the noise covariances. We quantify the impact of model inaccuracy on noise covariances estimation using this identification algorithm, and introduce a feedback control mechanism for data collection to enhance the accuracy and stability of noise covariance estimation. The estimated noise covariances account for model inaccuracy, which are shown to be more suitable for state estimation. We also quantify the performance gap between the ADKF and the traditional Kalman filter with known system dynamics and noise covariances, showing that this gap decreases as the number and length of pre-collected trajectories increase. Finally, numerical simulations validate the robustness and effectiveness of the proposed ADKF.
Hu et al. (Sun,) studied this question.