An insight into noise covariance estimation for Kalman Filter design

In Kalman filtering applications, the variance of the estimation error is guaranteed to be minimized only if a complete description of the plant dynamics is available. While nowadays there are several established methods to find an accurate model of a physical system, the evaluation of the covariance matrices of the disturbances is still a hard task. Therefore, such matrices are usually parameterized as diagonal matrices and their entries are estimated from a preliminary set of measurements. In this paper, we analyze the properties of the Kalman filter for linear time-invariant systems and we show that, for some classes of systems, the number of design parameters can be significantly reduced. Moreover, we prove that, under some assumptions, the covariance of the process noise can be parameterized as a full matrix without increasing the complexity of the tuning procedure. The above theoretical results are tested on two numerical examples.