Two-stage Kalman filtering via structured square-root

This paper considers the problem of estimating an unknown input (bias) by means of the augmented-state Kalman (AKF) filter. To reduce the computational complexity of the AKF, [12] recently developed an optimal two-stage Kalman filter (TS-AKF) that separates the bias estimation from the state estimation, and shows that his new two-stage estimator is equivalent to the standard AKF, but requires less computations per iteration. This paper focuses on the derivation of the optimal two-stage estimator for the square-root covariance implementation of the Kalman filter (TS-SRCKF), which is known to be numerically more robust than the standard covariance implementation. The new TS-SRCKF also estimates the state and the bias separately while at the same time it remains equivalent to the standard augmented-state SRCKF. It is experimentally shown in the paper that the new TS-SRCKF may require less flops per iteration for some problems than the Hsieh's TS-AKF [12]. Furthermore a second, even faster (single-stage) algorithm has been derived in the paper by exploiting the structure of the least-squares problem and the square-root covariance formulation of the AKF. The computational complexities of the two proposed methods have been analyzed and compared the those of other existing implementations of the AKF.