Interacting Kalman Filters for Linear Systems with Coupling Based on Empirical Covariances

The problem of designing estimators for stochastic linear systems with distributed observations is considered. Each observation process is associated to a node in an undirected graph, which is used to compute a local estimate at the node. The injection gain used in the filter at each node is obtained from the empirical covariance of all the estimates available at that node. This way the underlying idea comes from the theory of ensemble filtering and we analyze the evolution of the coupled covariances over the entire graph. After providing a detailed derivation of the evolution of covariance matrix, we observe that the drift term in the differential equation for the coupled covariances has some inherent stability structure in case of regular graphs, which leads to the fluctuations around the steady state. We provide an illustration of our algorithm on an academic example while comparing it with centralized and ensemble Kalman filters.