Performance Analysis of the Kalman Filter With Mismatched Noise Covariances

The Kalman filter is a powerful state estimator and has been successfully applied in many fields. To guarantee the optimality of the Kalman filter, the noise covariances need to be exactly known. However, this is not necessarily true in many practical applications. Usually, they are either completely unknown or at most partially known. In this technical note, we study performance of the Kalman filter with mismatched process and measurement noise covariances. For this purpose, three mean squared errors (MSEs) are used, namely the ideal MSE (IMSE), the filter calculated MSE (FMSE), and the true MSE (TMSE). The main contribution of this work is that the relationships between the three MSEs are disclosed from two points of views. The first view is about their ordering and the second view is about the relative closeness from the FMSE and TMSE to the IMSE. Using the first view, it is found that for the case with positive (definite) deviation from the truth, the FMSE is the worst and the IMSE is the best. And for the case with negative (definite) deviation, the TMSE is the worst and the best is the FMSE. Using the second view, it is found that the TMSE is relatively closer to the IMSE than the FMSE if the deviation is larger than certain threshold, and the TMSE will be farther away otherwise. Numerical examples further verify these conclusions.