Distributed estimation through randomized gossip Kalman filter

In this paper we consider the problem of estimating a random process from noisy measurements, collected by a sensor network. We analyze a distributed two-stage algorithm. The first stage is a Kalman-like estimate update, in which each agent makes use only of its own measurements. During the second phase agents communicate with their neighbors to improve their estimate. Estimate fusion is operated by running a consensus iteration. In literature it has been considered only the case of a fixed communication strategies, i.e. described by a fixed constant consensus matrix. However, in many practical cases this is just a rough model of communications in a sensor network, that usually happen according to a randomized strategy. This strategy is more properly modeled by assuming that the consensus matrices are drawn, according to a selection probability, from an alphabet of matrices compatible with the communication graph, at each time instant. This work deals therefore with randomized communication strategies and in particular with the symmetric gossip. A mean square performance analysis is carried out and an upper-bound for the trace of the estimation error variance is derived. The proposed upper-bound has to be considered the main technical contribution of the present paper, since it is based on a highly non-trivial inequality on matrix singular values, proved in the appendix. This upper-bound is a good performance assessment index and it is assumed therefore as a cost function to be minimized. We show moreover that problem of minimizing this cost function by choosing the Kalman gain and the selection probability is convex in each of the two variables separately although it is not jointly convex. Finally simulations are presented and the results discussed.