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In this paper, the optimal distributed Kalman filtering fusion with linear equality constraint (LEC) is proposed. When the Kalman filter subject to LEC is applied in state estimation of distributed linear dynamic system, all local error covariance matrices are singular, which is a challenge for most of the existing distributed fusion algorithms. However, the new fusion algorithm is rigorously derived with matrix analysis technique on the Moore-Penrose inverse. Updating state estimate and error covariance matrix, the new fusion algorithm only uses local estimates and their corresponding error covariances. We have proved theoretically that the distributed Kalman filtering with LEC (DKF-LEC) just using all local measurements is equivalent to the centralized Kalman filtering with LEC (CKF-LEC), which means it is optimal. Furthermore, we also prove the optimality of the proposed fusion formula for Kalman filtering with LEC and feedback. In addition, the simulation results show that the proposed fusion formula for Kalman filtering with LEC have the same performance of the centralized Kalman filtering with LEC whether considering feedback or not. It is consistent with the theoretical result that the DKF-LEC is equivalent to CKF-LEC as the estimate of CKF-LEC is disassembled into local estimates in distributed system. And the performance of the new fusion formula for Kalman filtering with LEC and feedback is obviously better than those without feedback.

Multi-sensor distributed fusion estimation algorithms based on machine learning are proposed in this paper. Firstly, using local estimations as inputs and estimations of three classic distributed fusion (weighted by matrices, by diagonal matrices and by scalars) as the training sets, three distributed fusion algorithms based on BP network (BP net-based fusion weighted by matrices, by diagonal matrices and by scalar) are proposed and the selection basis of the number of nodes in hidden layer is given. Furthermore, by using local estimations as inputs and centralized fusion estimation as training set, another recurrent net-based distributed fusion algorithm is proposed, in the case that neither true states nor cross-covariance matrices is available. This method is not limited to the linear minimum variance (LMV) criterion, so its accuracy is higher than the classical three distributed fusion algorithms. A radar tracking simulation verifies the effectiveness of the proposed fusion networks.