Analysis and Control of Multi-timescale Modular Directed Heterogeneous Networks

testWe examine the collective behavior of largescale networks of heterogeneous nonlinear systems with directed weighted interconnections, and containing a spanning tree. We consider networks that are composed of groups of densely interconnected nodes, called modules, that are in turn sparsely interconnected. Such networks are called modular. The modules represent sub-networks wherein consensus may be rapidly achieved, while synchronization among modules occurs at a lower pace. Furthermore, relying on the framework of [1], we identify an underlying dynamics that corresponds to a weighted average of the nodes' respective states. This average dynamics evolves on a yet slower timescale. Such triple timescale make modular networks are amenable to be analyzed via singular-perturbations theory. We show that if the nodes' dynamics are semi-passive and the average dynamics is globally asymptotically stable, so is the entire network. In the case that the average dynamics is not globally asymptotically stable we show how our main analysis statement can be used for network control, via the addition of control nodes, in order to globally asymptotically stabilize the network