Relative closeness ranking of Kalman filtering with multiple mismatched measurement noise covariances

Due to the complexity of practical systems, the used models generally mismatch the practical ones. For the Kalman filtering with a single mismatched measurement noise covariance, the ranking of the ideal MSE (IMSE), the filter calculated MSE (FMSE), and the true MSE (TMSE) has been established. This study considers the ranking of the relative closenesses from the FMSE and TMSE to the IMSE when multiple mismatched measurement noise covariances are used. It is found that for the case with two deviations of the same signs, the larger the absolute deviation is, the farther away the FMSE or TMSE is from the IMSE. It is also found that for the case with two deviations of different signs, if the positive deviations are less than the absolute value of the negative deviation, then the FMSE or TMSE with positive deviation is relatively closer to the IMSE. Otherwise the ranking of the two corresponding relative closenesses depends on the parallel sum of two deviations with a non-zero threshold. Then the pair-wised ranking of the relative closenesses corresponding to two arbitrary deviations is further extended to the case with more than two deviations. Numerical examples are provided to validate the results.