On the identification of variances and adaptive Kalman filtering

Abstract

A Kalman filter requires an exact knowledge of the process noise covariance matrixQand the measurement noise covariance matrixR. Here we consider the case in which the true values ofQandRare 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 ofQandR. This technique works only for the case in which the form ofQis known and the number of unknown elements inQis less thann \times rwherenis the dimension of the state vector andris the dimension of the measurement vector. For other cases, the optimal steady-state gain Kopis obtained directly by an iterative procedure without identifyingQ. As a corollary, it is shown that the steady-state optimal Kalman filter gain Kopdepends only onn \times rlinear functionals ofQ. 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.