Prediction error identification of linear dynamic networks with rank-reduced noise

Dynamic networks are interconnected dynamic systems with measured node signals and dynamic modules reflecting the links between the nodes. We address the problem of identifying a dynamic network with known topology, on the basis of measured signals, for the situation of additive process noise on the node signals that is spatially correlated and that is allowed to have a spectral density that is singular. A prediction error approach is followed in which all node signals in the network are jointly predicted. The resulting joint-direct identification method, generalizes the classical direct method for closed-loop identification to handle situations of mutually correlated noise on inputs and outputs. When applied to general dynamic networks with rank-reduced noise, it appears that the natural identification criterion becomes a weighted LS criterion that is subject to a constraint. This constrained criterion is shown to lead to maximum likelihood estimates of the dynamic network and therefore to minimum variance properties, reaching the Cramér-Rao lower bound in the case of Gaussian noise. In order to reduce technical complexity, the analysis is restricted to dynamic networks with strictly proper modules.