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Structure selection of noise covariance matrices for linear Kalman filter design

In state reconstruction problems, the statistics of the noise affecting the state equations is often supposed to be known. Since an incorrect description of the model stochastic properties may have detrimental effects on the final filtering performance, many algorithms have been proposed to estimate the noise covariance matrices together with the unknown state. Due to the high computational load, a typical practical assumption is that the process noise covariance can be parameterized as a diagonal matrix. In this paper, we show by counterexamples that this is not always the best compromise between computational complexity and tracking accuracy. Furthermore, a combinatorial optimization algorithm originally employed for model structure selection in nonlinear identification applications is here adapted to the task of selecting the structure of the process noise covariance matrices. The effectiveness of the proposed approach is illustrated by means of some numerical examples.