Kalman filtering with no a priori information about noise--White noise case: Identification of covariances

Kalman filtering in the presence of white process and measurement noises having unknown means and covariances is considered. Only stationary linear discrete stochastic systems are considered. It is shown that the identification of noise covariances can be done without the knowledge of noise means. This means that the problem of identifying the noise statistics can be decomposed into two separate subproblems, namely, 1) identification of noise covariances and 2) identification of noise means, and that these two subproblems can be solved in that order. A procedure for identifying noise covariances is developed in this paper. It is a nontrivial extension of Mehra's results to the case where the process and measurement noises have unknown means, and are correlated with each other. This procedure, like Mehra's, can be used either in a nonrecursive mode, or in a batch-recursive mode. Solution of subproblem 2), and the standard Kalman filter algorithm are not discussed since they are well known in the literature.