On the identification of variances and adaptive Kalman filtering

A Kalman filter requires an exact knowledge of the process noise covariance matrix and the measurement noise covariance matrix . Here we consider the case in which the true values of and 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 and . This technique works only for the case in which the form of is known and the number of unknown elements in is less than where is the dimension of the state vector and is the dimension of the measurement vector. For other cases, the optimal steady-state gain Kopis obtained directly by an iterative procedure without identifying . As a corollary, it is shown that the steady-state optimal Kalman filter gain Kopdepends only on linear functionals of . 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.