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The Kalman filter is one of the best-known and most frequently used methods for dynamic state estimation. In addition to a measurement and state transition model, the Kalman filter requires knowledge about the covariance of the measurement and process noise. However, the noise covariances are mostly unknown and may vary during the application. Adaptive Kalman filters solve this problem by estimating the noise covariances online to improve the state estimation. Existing methods are often limited in their application because they are designed to adapt only the measurement noise or the process noise covariance and tend to diverge when both are unknown. Moreover, most methods provide no or only local convergence results, which implies that a poor initialization can adversely affect the estimation of the noise covariances, leading to a deteriorated state estimation. This paper introduces a novel adaptive Kalman filter based on additional Kalman smoothing and analytically derived covariance estimators. Firstly, the unbiased measurement and process noise covariance estimators are derived from the maximum a posteriori formulation of the Kalman smoother. Then, based on these estimators, which depend on the system formulation and the state estimates of the Kalman smoother, the adaptive Kalman filter algorithm is presented. The convergence of the derived estimators can be shown for time-invariant systems for one-dimensional measurement and process noise. For higher-dimensional problems, the convergence can be tested simulatively for the specific dynamical system. A detailed evaluation of various simulation scenarios is presented, demonstrating the accuracy and robustness of the proposed method.

Estimation of unknown noise covariances in a Kalman filter is a problem of significant practical interest in a wide array of applications. Although this problem has a long history, reliable algorithms for their estimation were scant, and necessary and sufficient conditions for identifiability of the covariances were in dispute until recently. Necessary and sufficient conditions for covariance estimation and a batch estimation algorithm were presented in our previous study. This paper presents stochastic gradient descent algorithms for noise covariance estimation in adaptive Kalman filters that are an order of magnitude faster than the batch method for similar or better root mean square error. More significantly, these algorithms are applicable to non-stationary systems where the noise covariances can occasionally jump up or down by an unknown magnitude. The computational efficiency of the new algorithms stems from adaptive thresholds for convergence, recursive fading memory estimation of the sample cross-correlations of the innovations, and accelerated stochastic gradient descent algorithms. The comparative evaluation of the proposed methods on a number of test cases demonstrates their computational efficiency and accuracy.

The Kalman filter requires knowledge of the noise statistics; however, the noise covariances are generally unknown. Although this problem has a long history, reliable algorithms for their estimation are scant, and necessary and sufficient conditions for identifiability of the covariances are in dispute. We address both of these issues in this paper. We first present the necessary and sufficient condition for unknown noise covariance estimation; these conditions are related to the rank of a matrix involving the auto and cross-covariances of a weighted sum of innovations, where the weights are the coefficients of the minimal polynomial of the closed-loop system transition matrix of a stable, but not necessarily optimal, Kalman filter. We present an optimization criterion and a novel six-step approach based on a successive approximation, coupled with a gradient algorithm with adaptive step sizes, to estimate the steady-state Kalman filter gain, the unknown noise covariance matrices, as well as the state prediction (and updated) error covariance matrix. Our approach enforces the structural assumptions on unknown noise covariances and ensures symmetry and positive definiteness of the estimated covariance matrices. We provide several approaches to estimate the unknown measurement noise covariance $R$ via post-fit residuals, an approach not yet exploited in the literature. The validation of the proposed method on five different test cases from the literature demonstrates that the proposed method significantly outperforms previous state-of-the-art methods. It also offers a number of novel machine learning motivated approaches, such as sequential (one sample at a time) and mini-batch-based methods, to speed up the computations.