Conditions for Synchronizability in Arrays of Coupled Linear Systems

Synchronization in arrays of coupled continuous-time linear systems is studied. Sufficiency of certain conditions for the existence of a synchronizing feedback law are analyzed. It is shown that, for neutrally stable systems that are detectable from their outputs, a linear feedback law exists under which any number of coupled systems synchronize provided that the (directed, weighted) graph describing the interconnection is fixed and connected. An algorithm generating one such feedback law as well as the trajectory that the solutions converge to are presented. It is also shown that, for critically unstable systems, detectability is not sufficient, whereas full-state coupling is, for the existence of a linear feedback law that is synchronizing for all coupling configurations described by a connected graph.