A Single-pass Noise Covariance Estimation Algorithm in Multiple-model Adaptive Kalman Filtering

Estimation of unknown noise covariances in a Kalman filter is a problem of significant practical interest in a wide array of industrial applications. The reliable algorithms for their estimation are scarce, and the necessary and sufficient conditions for the identifiability of the covariances were in dispute until very recently. In this paper, we present a single-pass stochastic gradient descent (SGD) algorithm for noise covariance estimation for use in multiple-model adaptive Kalman filters. The multiple-model estimation algorithm assumes that the system obeys one of a finite number of models, and each model has its own dynamics. The overall estimate of the multiple-model approach is a convex combination of the estimates from the filters running in parallel based on the individual models, where the weights of the convex combination are the concomitant posterior model probabilities. Here, the identifiability conditions of the multiple-model approach depend on each candidate model since the corresponding Kalman filters are non-interacting. The proposed streaming algorithm reads measurement data exactly once and has an acceptable root mean square error (RMSE) of state estimates. The proposed estimating algorithm is implemented based on its single-pass nature, recursive fading memory estimation of the sample cross-correlations of the innovations, and the root mean square propagation (RMSprop) accelerated SGD algorithm. Our approach enforces the structural assumptions on unknown noise covariances, and ensures the symmetry and positive definiteness of the estimated covariance matrices. The proposed method is applicable to non-stationary systems where the noise covariances continuously change. The validation of the proposed method on several numerical examples demonstrates its accuracy and filter consistency.