A Nyquist criterion for synchronization in networks of heterogeneous linear systems *

We study the synchronization of a set of SISO subsystems interconnected via a time-invariant Laplacian matrix. By synchronization we mean that the outputs of all subsystems must be asymptotically equal to each other and behave in the same manner. We assume that the subsystems can be represented as the sum of a common nonzero transfer function plus a perturbation. A Nyquist-type criterion is established which ensures synchronization provided that the convex hull of the frequency responses of the subsystems does not intersect a certain region defined by the spectrum of the interconnection matrix. The result is applied to a variety of examples of different nature for which synchronization takes place. A counter example for which complete synchronization is impossible is also presented.