Identifiability of dynamic networks with part of the nodes noise-free

In dynamic network identification a major goal is to uniquely identify the topology and dynamic links between the measured node variables. It is common practice to assume that process noises affect every output in multivariable system identification, and every node in dynamic networks with a full rank noise process. For many practical situations this assumption might be overly strong. This leads to the question of how to handle situations where the process noise is not full rank, i.e. when the number of white noise processes driving the network is strictly smaller than the number of nodes. In this paper a first step towards answering this question is taken by addressing the case of a dynamic network where some nodes are noise-free, and others are disturbed with a (correlated) process noise. In this situation the predictor filters that generate the one-step-ahead prediction of the node signals are non-unique, and the appropriate identification criterion leads to a constrained optimization problem. It is assessed when it is possible to distinguish between models on the basis of this criterion, leading to new notions of network identifiability. It appears that a sufficient condition for network identifiability is that every node signal in the network is excited by an external excitation signal or a process noise signal that is uncorrelated with other node excitations.