Covariance correction filter with unknown disturbance associated to system state

This paper is motivated by the fact that the unknown disturbances (UDs) in discrete-time stochastic systems may be associated with state, such as the perturbation and the model error. In such case, the UD takes on the first two moment (FTM) at least, i.e. the property of both mean and covariance. If use the UD-FTM to correct the state estimation and its covariance simultaneously, it should result in a better accuracy than the classical methods including the augmentation, robust filter and interacting multiple model (IMM), which only consider the first moment (mean) property of UD to correct the state estimation, regardless of the second moment (covariance) of UD. In this paper, a two-stage expectation maximization (EM) algorithm is proposed to jointly identify the UD-FTM. The first EM is for joint state estimation and UD's pseudo measurement (UD-PM) identification, while the second EM is for Gaussian mixture (GM), which uses the identified UD-PM from the first EM to fit out the UD-FTM. Further we can improve the state estimation accuracy by using the fitted UD-FTM with an open-loop correction. Finally, simulation results illustrate the effectiveness of the proposed method.