Team Optimal Decentralized State Estimation

We consider the problem of optimal decentralized estimation of a linear stochastic process by multiple agents. Each agent receives a noisy observation of the state of the process and delayed observations of its neighbors (according to a pre-specified, strongly connected, communication graph). Based on their observations, all agents generate a sequence of estimates of the state of the process. The objective is to minimize the total expected weighted mean square error between the state and the agents' estimates over a finite horizon. In centralized estimation with weighted mean square error criteria, the optimal estimator does not depend on the weight matrix in the cost function. We show that this is not the case when the information is decentralized. The optimal decentralized estimates depend on the weight matrix in the cost function. In particular, we show that the optimal estimate consists of two parts: a common estimate which is the conditional mean of the state given the common information and a correction term which is a linear function of the offset of the local information from the conditional expectation of the local information given the common information. The corresponding gain depends on the weight matrix as well as on the covariance between the offset of agents' local information from the conditional expectation of the local information given the common information. We show that the common estimate can be computed from single Kalman filter and derive recursive expressions for computing the offset covariances and the estimation gains.