Some Necessary and Sufficient Conditions for Synchronization of Second-Order Interconnected Networks

This paper presents some necessary and sufficient conditions for the synchronization of second-order interconnected networks, where fixed inner-linked connections with information communication exist. First, a novel derivation of the conditions for a second-order polynomial with complex coefficients to be Hurwitz is provided. Based on this, a sufficient and necessary condition is proposed for the synchronization of the coupled complex network. Next, the design method of the synchronization protocol is constructively given, where the upper and lower bounds of the control gains are obtained through the properties of the polynomial. In addition, we consider some special cases where the second-order model is a double integrator, or general multiagent model without fixed interactions. Finally, the simulation result is given to verify the theoretical analysis.