On the identification of variances and adaptive Kalman filtering

A Kalman filter requires an exact knowledge of the process noise covariance matrix Q and the measurement noise covariance matrix R. Here we consider the case in which the true values of Q and R are unknown. The system is assumed to be constant, and the random inputs are stationary. First, a correlation test is given which checks whether a particular Kalman filter is working optimally or not. If the filter is suboptimal, a technique is given to obtain asymptotically normal, unbiased, and consistent estimates of Q and R. This technique works only for the case in which the form of Q is known and the number of unknown elements in Q is less than n r where n is the dimension of the state vector and r is the dimension of the measurement vector. For other cases, the optimal steady-state gain K op is obtained directly by an iterative procedure without identifying Q. As a corollary, it is shown that the steady-state optimal Kalman filter gain K op depends only on n r linear functionals of Q. The results are first derived for discrete systems. They are then extended to continuous systems. A numerical example is given to show the usefulness of the approach.