Kalman Filter with Recursive Covariance Estimation Revisited with Technical Conditions Reduced

Abstract

Kalman filter algorithm is an optimal state estimator in the sense of minimum mean squared errors and maximum likelihood estimation. However, standard Kalman filter is highly restricted by strict a priori requirements on the information of dynamic model and statistical information of process noises. In practice, the covariance matrix of process noise is almost impossible to be directly determined a priori due to the intrinsic coupling of process noise and system state dynamics. Considering such background of Kalman filter algorithm, one algorithm, named as recursive covariance estimation (RCE), is introduced to estimate the covariance matrix when there is no information of process noise covariance matrix in the linear time-invariant (LTI) systems. Based on the framework of standard Kalman filter, a new algorithm named as Kalman filter with recursive covariance estimation (KF-RCE) is introduced to resolve this challenging problem of state estimation without the covariance matrix of process noise. Then the rigorous stability analysis is presented to show that the algorithm is optimal and the covariance matrix and state estimations are asymptotically consistent with the ideal Kalman filter using the exact covariance matrix of process noise. Finally, some simulation results of navigation system are given to examine theory analysis theorem and algorithm optimality.