Sequential Mini-Batch Noise Covariance Estimator

Abstract

Noise covariance estimation in an adaptive Kalman filter is a problem of significant practical interest in a wide array of industrial applications. Reliable algorithms for their estimation are scarce, and the necessary and sufficient conditions for identifiability of the covariances were in dispute until very recently. This chapter presents the necessary and sufficient conditions for the identifiability of noise covariances, and then develops sequential mini-batch stochastic optimization algorithms for estimating them. The optimization criterion involves the minimization of the sum of the normalized temporal cross-correlations of the innovations; this is based on the property that the innovations of an optimal Kalman filter are uncorrelated over time. Our approach enforces the structural constraints on noise covariances and ensures the symmetry and positive definiteness of the estimated covariance matrices. Our approach is applicable to non-stationary and multiple model systems, where the noise covariances can occasionally jump up or down by an unknown level. The validation of the proposed method on several test cases demonstrates its computational efficiency and accuracy.