Synchronization of linear time-invariant systems on rooted graphs

This paper studies the synchronization properties of networks of identical, linear time-invariant systems. It is shown that networks of passive systems synchronize for any directed and rooted communication topology, thus generalizing a well known result on synchronization in networks of single integrators. If the passivity property is relaxed to output-feedback-passivity, synchronization is proven for arbitrarily large subsets of directed and rooted communication graphs, by imposing a sufficiently high feedback gain. Finally, it is shown that, in networks of single-input single-output systems, passivity is not only sufficient but it is also, to a certain extent, necessary for the above property to hold.