From closed-loop identification to dynamic networks: Generalization of the direct method

Identification methods for identifying (modules in) dynamic cyclic networks, are typically based on the standard methods that are available for identification of dynamic systems in closed-loop. The commonly used direct method for closed-loop prediction error identification is one of the available tools. In this paper we are going to show the consequences when the direct method is used under conditions that are more general than the classical closed-loop case. We will do so by focusing on a simple two-node (feedback) network where we add additional disturbances, excitation signals and sensor noise. The direct method loses consistency when correlated disturbances are present on node signals, or when sensor noises are present. A generalization of the direct method, the joint-direct method, is explored, that is based on a vector predictor and includes a conditioning on external excitation signals. It is shown to be able to cope with the above situations, and to retain consistency of the module estimates.