Synchronization Control Of Complex Networks With Switching Topology

Complex networks,as the most challenging subject in complexity science,have attracted more and more attention of researchers.The research on complex networks has been widely used in physics,mathematics,biology,sociology and many other fields.As an important dynamic property of complex networks,synchronization has attracted many scholars' interest in research and produced a large number of results.From these results,researchers find that the topology of the network has a lot to do with the synchronization of complex networks.Because the change of the topology of networks usually cause the change of the property of synchronization.In the power grid,either the access of the new grid or the removal of the fault motor will lead to change in the network topology.Unlike the traditional time-varying network,the topology of networks jumps from one to the other.These 'jump' networks are called complex networks with switching topology,simple called switched networks.In recent years,some achievements have been made in the synchronization of switched networks,however,there still have a lot of problems to solve.In this article,the synchronization problem of complex networks with switching topology is studied by using the Lyapunov stability theory and theories of switched systems,the main contributions are list as follow:Firstly,we have studied the synchronization problem of complex networks with switching topology based on Lyapunov stability theory,the network has the following characteristics:(1)All the subnetworks in the switched networks are not self synchronous;(2)The nodes in networks have time delay;(3)The switching network can be directed,but there is no isolated node.Based on the Lyapunov stability theory,a set of switching signals based on time is designed for the networks mentioned above,and synchronize the complex network with the above features.Compared with the existing results,in this article,all the subnetworks in the switched network are not self synchronous.The delay of the nodes is added to make the switching network more realistic.The related signal can be solved by LMI toolbox in Matlab which is easy to implement.Secondly,we have studied the pinning synchronization problem of switched networks.First of all,the pinning synchronization problem under arbitrary switching is studied by using the common Lyapunov function method.Then the pinning controller and switching law are designed by using the single Lyapunov function method and the multi-Lyapunov function method,so that the synchronization of the switching network can be realized under the pinning controller and switching law.Due to the multiple Lyapunov function method,the design of the controller and switching law in this article only meets the relevant theory of Lyapunov stability in active time.Thus the pinning controller designed in this article is easier to implement.Finally,the work of this paper is summarized.Meanwhile,the shortcomings and the future research direction are pointed out.