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Recursive performance ranking of Kalman filter with mismatched noise covariances

The Kalman filter is a powerful recursive state estimator and has been widely used in many applications. To guarantee its optimality, the noise covariances should to be exactly known. In reality, however, for most practical applications, it is difficult or unrealistic to obtain the noise covariances. A typical practice is to use some pre-determined alternatives for unknown noise covariances. The main issue concerning this is how the pre-determined alternatives will affect the performance of the Kalman filter. In this study, the authors study recursive performance ranking of Kalman filter with mismatched noise covariances. For this purpose, three types of mean squared errors (MSEs) have been used, i.e., the ideal MSE (IMSE), the filter calculated MSE (FMSE), and the true MSE (TMSE). This study considers the recursive ranking of these three types of MSEs at each time step. It is found that for the case with positive semi-definite deviation from the truth, they have FMSE ≥ TMSE ≥ IMSE at each time step recursively. On the contrary, for the case with negative semi-definite deviation, they have TMSE ≥ IMSE ≥ FMSE at each time step recursively. Target tracking examples further verify these results.