Optimal Kalman Filter Fusion with Singular Covariances of Filter Error Enbin Song Department of Mathematics Sichuan University Chengdu, Sichuan 610064, China Email: e.b.song@163.com Jie Xu Department of Mathematics Sichuan University Chengdu, Sichuan 610064, China Email: xujie − 2000@163.com Yunmin Zhu Department of Mathematics Sichuan University Chengdu, Sichuan 610064, China Email: ymzhu@scu.edu.cn Abstract—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. 1. I NTRODUCTION Multiple sensors estimation fusion has widespread appli- cations ranging from civil to military fields, for example, fault diagnosis, surveillance and monitoring, air traffic control, target tracking and localization, etc. An important practical problem in the above systems is to find an optimal state estimator based on the given observations. It is well known that Kalman filtering is one of the most popular recursive Mean Square Error (MSE) algorithm to optimally estimate the unknown state or process of a dynamic system. There are two basic fusion architectures [7]: centralized and decentralized/distributed (with respect to measurement fusion and tract fusion, respectively) depending on whether the raw measure are sent to the fusion center or not. For the case of centralized fusion, the centralized Kalman filtering can be implemented when the fusion center can obtain all raw measurements from the local sensors in time, and the corresponding state estimates are optimal in the the sense of MSE. Unfortunately, sending raw measurements needs more demanding in communication bandwidth and results a poor survivability of the system(particular, in a war situation), which leads to that the distributed fusion is more preferred. Consequently, for the case of distributed fusion, every local sensor has to implement Kalman filtering based on its own observations first for local requirement, and then sends the processed data–local state estimate to a fusion center. There- fore, the fusion center should fuse all received local estimates to yield an optimal state estimate in terms of MSE sense. Under the assumption that the estimation error covariance matrices are invertible, an optimal Kalman filtering fusion based on the information form was proposed in [1], [2], [3], [5], which was proved to be optimal in the sense that they are equivalent to the optimal centralized Kalman filtering using all sensor measurements. In this paper, we consider the Kalman filtering fusion for the case that the estimation error covariance matrices are singular because of the following reasons. First, we prove that our proposed fused state estimate is still equivalent to the centralized Kalman filtering using all sensor measurements. In addition, we also obtain the update formula of error covariance matrices. This shows that the centralized Kalman filtering can also be obtain by fusing the local estimate even the error covariance matrices are singular, which means even the local information matrices (error covariance matrices) is singular, there also exist some fusion formula, based on local estimate, that is equivalent to centralized Kalman filtering. To the best of our knowledge, this is new. Our result is different from that in [10], where they also consider the problem that estimation fusion based on local transformation of raw measurements (not based on the local estimate) when the the error covariance 1359