Modelling and identification of diffusively coupled linear networks with additional directed links

Abstract

Dynamic networks consist of interconnected dynamical systems. The subsystems can be viewed as transformations of input signals into output signals, where signals flow from one system into another through interconnections. The signal flows represent directions of information flow, thus a dynamic network can be visualised by a directed graph. In contrast, natural and physical laws only impose relations between systems variables, while variables are shared among systems via interconnections. Sharing is independent of direction, and therefore a dynamic network originating from physics can be visualised by an undirected graph. Typically, dynamic networks are considered to have either directed or undirected interconnections. For both situations, network models, analytic tools, and identification algorithms have been developed. However, dynamic networks can also have both directed and undirected interconnections, for example, in physical networks equipped with digital controllers. In this work, we present mixed linear dynamic networks that contain both undirected and directed interconnections, where the nature of the interconnecting dynamics needs to be incorporated into the modelling framework, identifiability analysis, and identification procedure. For these mixed networks, we derive dynamic network models; formulate conditions for consistent identification of all dynamics in the network; and develop a tractable identification algorithm that delivers consistent estimates.