Small-gain results for discrete-time networks of systems with delay

In this paper input-to-state stability for large-scale discrete-time networks of systems with delay is considered. Firstly, interconnection delays between systems, which can arise due to the propagation of signals over large distances, are treated. It is established that such delays cannot cause the instability of the network if a delay-independent small-gain condition holds. Secondly, local delays in each system, which can arise due to inherent delays in each local dynamical process, are considered. For this set-up, using a small-gain condition, the stability of the network of systems is established via the Razumikhin method and the Krasovskii approach, respectively. By combining the results for networks of systems with communication and local delays, respectively, a framework for ISS analysis for general networks with delay is obtained.