An Online Exploratory Maximum Likelihood Estimation Approach to Adaptive Kalman Filtering

Over the past few decades, numerous adaptive Kalman filters (AKFs) have been proposed. However, achieving online estimation with both high estimation accuracy and fast convergence speed is challenging, especially when both the process noise and measurement noise covariance matrices are relatively inaccurate. Maximum likelihood estimation (MLE) possesses the potential to achieve this goal, since its theoretical accuracy is guaranteed by asymptotic optimality and the convergence speed is fast due to weak dependence on accurate state estimation. Unfortunately, the maximum likelihood cost function is so intricate that the existing MLE methods can only simply ignore all historical measurement information to achieve online estimation, which cannot adequately realize the potential of MLE. In order to design online MLE-based AKFs with high estimation accuracy and fast convergence speed, an online exploratory MLE approach is proposed, based on which a mini-batch coordinate descent noise covariance matrix estimation framework is developed. In this framework, the maximum likelihood cost function is simplified for online estimation with fewer and simpler terms which are selected in a mini-batch and calculated with a backtracking method. This maximum likelihood cost function is sidestepped and solved by exploring possible estimated noise covariance matrices adaptively while the historical measurement information is adequately utilized. Furthermore, four specific algorithms are derived under this framework to meet different practical requirements in terms of convergence speed, estimation accuracy, and calculation load. Abundant simulations and experiments are carried out to verify the validity and superiority of the proposed algorithms as compared with existing state-of-the-art AKFs.