Abstract
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.
Highlights
T HE Kalman filter (KF) [8] is the minimum mean square error (MMSE) state estimator for discrete-time linear dynamic systems under Gaussian white noise with known mean and covariance parameters
We explore enhanced covariance estimation methods based on sequential mini-batch stochastic gradient descent (SGD) algorithms with adaptive step sizes and iteration-dependent convergence thresholds
CONTRIBUTION AND ORGANIZATION OF THE PAPER In this paper, we develop sequential mini-batch estimation methods with adaptive step size rules to improve the computational efficiency of the algorithm in [21]
Summary
T HE Kalman filter (KF) [8] is the minimum mean square error (MMSE) state estimator for discrete-time linear dynamic systems under Gaussian white noise with known mean and covariance parameters. It is the best linear estimation algorithm when the noise processes are nonGaussian with known first and second-order statistics, i.e., mean and covariance. It has found successful applications in numerous fields, such as navigation, weather forecasting, signal processing, econometrics and structural health monitoring, to name a few [2]. We explore enhanced covariance estimation methods based on sequential mini-batch stochastic gradient descent (SGD) algorithms with adaptive step sizes and iteration-dependent convergence thresholds
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