Optimal Kalman filter fusion with singular covariances of filter error

In this paper, we consider the optimal Kalman filtering fusion with singular estimation error covariance matrices. Here, our motivation comes from the following facts. First, the fused state estimate is still equivalent to the centralized Kalman filtering using all sensor measurements by our fusion formula. At the same time, we also obtain the update formula of error covariance matrices. Second, most of the existing fusion algorithms need inverse of estimation error covariance matrices, which can not be guaranteed to exist. A concrete application is state estimation for linear systems with equality constraints, which results the estimation error covariance matrices are singular. Third, from the viewpoint of theoretical perspective, we generalize the Kalman filtering fusion theorem to the case that error covariance matrices are singular. That is, our proposed formula can be used more extensively than the existing fusion formulas.