Determination of noise covariances for extended Kalman filter parameter estimators to account for modeling errors

The extended Kalman filter parameter estimation algorithm (EKFPE) is analyzed when modeling errors are present, focusing primarily on the parameter estimation error covariance. The recursive equations for the true error and the true linearized estimation error covariance for the EKFPE are derived. The true error covariance equations show that the additional terms resulting from mismodeling are analogous to the process and measurement noise covariance terms that appear in the EKFPE equations with perfect system and noise assumptions. A methodology for estimating appropriate process and measurement noise covariances to account for the additional terms is suggested. This method is based on the recursive minimization of the difference between the residual covariance matrix computed by the EKFPE and that calculated from time-averaging of the residual sequence generated by the filter at each time step. A Newton-type minimization algorithm is employed for this purpose. The covariance of a fictitious noise term that is sometimes assumed in the parameter variation model for the EKFPE, or pseudonoise, can also be determined by the method suggested in this work. The incorporation of the minimization algorithm to the standard EKFPE equations results in a modified EKFPE, whose improved performance in the presence of modeling errors is shown in various examples.