Decision-directed adaptive recursive estimators: Divergence prevention

In recent years, minimum-variance recursive estimators, such as the Kalman filter, have been used successfully in many practical applications. However, a common problem, known in the literature as the divergence phenomenon, is often encountered in these applications. Divergence is said to occur when the error covariance calculated by the estimator becomes inconsistent with the actual error covariance. Previous methods of dealing with this problem have involved limited memory filtering or simultaneous estimation of random process statistics. The method presented here is different in that the state model and statistics are accepted as given; the form of the optimal estimator is used, but a constant check is made on the consistency of the calculated and actual error covariances. The method is independent of the source of error, whether it be inaccuracies in the system model, incorrect values of the a priori and random process statistics, approximations required in the case of nonlinear systems, or computational roundoff. A test for inconsistency and an adaptive decision-directed procedure for adjusting the calculated covariance, shown to be optimal in a certain sense, are discussed. Several simulated examples, in which inconsistencies in the calculated and actual error covariances exist, show a significant improvement in the performance of the estimator when the given procedure is appended.