Global Pinning Synchronization with PI Controller in General Complex Directed Networks

Global pinning synchronization in complex networks has been extensively studied in the last decades due to its potential applications. The question is how to design an available controller to achieve global synchronization when the network is directed. As we all know, the classical PI controller is most widely and successfully used in engineering systems with nonlinear dynamics. However, the global synchronization with PI controller in general complex directed networks still remain unresolved. In this paper we will theoretically show that the global synchronization of some directed networks can also be ensured by designing an appropriate PI controller and pioneering some selected nodes, then a simple numerical example will also be given to verify our main results.