On structural behavioural controllability of linear discrete time systems with delays

In this paper we study the controllability of interconnected networks that are described by means of structured linear systems with state-like and control variables. We assume that the systems operate in discrete time with the set of integers as the time axis. Further, we assume that the state-like variables for their evolution only depend on recent values of their neighbours with, however, unknown weight factors. These recent values may be one step back in time, but also more steps. This yields a description of the systems by means of matrices containing fixed zeros and free parameters, together with a time lag structure. Knowing the dependency and lag structure, we represent (the structure of the) systems by means of weighted directed graphs and study questions concerning their structural controllability, where the latter has to be defined in an appropriate way, i.e., in behavioural sense. We provide a necessary and sufficient characterization of structural controllability of our systems using a graph representation. The obtained characterization makes use of well-known and efficient algorithms from graph theory. We prove that in this context finding the minimal number of driver (controller) nodes is an NP-hard problem. The concepts and results of the paper are illustrated on academic examples and on a gene regulatory network.