On the existence of virtual exosystems for synchronized linear networks

When dealing with heterogeneous networks, where the agents are governed by non-identical models, interesting questions arise regarding the ability of the network to synchronize to a common non-trivial output trajectory, as well as the nature of such a trajectory. On this topic, Wieland, Allgöwer, and Sepulchre have recently derived results showing that for a class of heterogeneous networks of dynamically controlled linear agents, non-trivial output synchronization implies the existence of an observable virtual exosystem for which the regulator equations are solvable for each agent. Moreover, this virtual exosystem defines the output trajectories on the agreement manifold and is contained within each agent as an internal model. In this paper we shed further light on this topic by showing that, under a more general set of assumptions, non-trivial output synchronization can occur in the absence of such a virtual exosystem. We propose a modified result for this case that specifies the existence of a possibly \emph{unobservable} virtual exosystem for which the regulator equations are solvable, and for which the observable part defines the output trajectories on the agreement manifold. We also show that a variation of the virtual exosystem is contained within each agent as an internal model.