An optimal adaptive Kalman filter

In a robustly adaptive Kalman filter, the key problem is to construct an adaptive factor to balance the contributions of the kinematic model information and the measurements on the state vector estimates, and the corresponding learning statistic for identifying the kinematic model biases. What we pursue in this paper are some optimal adaptive factors under the particular conditions that the state vector can or cannot be estimated by measurements. Two optimal adaptive factors are derived, one of which is deduced by requiring that the estimated covariance matrix of the predicted residual vector equals the corresponding theoretical one. The other is obtained by requiring that the estimated covariance matrix of the predicted state vector equals its theoretical one. The two related optimal adaptive factors are given. These are analyzed and compared in theory and in an actual example. This shows, through the actual computations, that the filtering results obtained by optimal adaptive factors are superior to those obtained by adaptive factors based on experience.