Global synchronization under PI/PD controllers in general complex networks with time-delay

In complex networks, global synchronization problems have been widely studied over the past ten years due to its extensive applications. One of the challenge issue is how to design an available controller to make the complex networks reach global synchronization, especially with directed topologies and time-delay. As we all know, the classical proportional-integral-derivative (PID) controller is widely and successfully used in engineering systems with nonlinear dynamics, however, global synchronization under PI/PD controllers in general directed complex networks with time-delay is still unsolved. This paper proposes several sufficient conditions of global synchronization for directed networks with time-delay by designing appropriate PI/PD controllers, and gives some simple numerical examples to validate the main results.