Along with the numerous highly interconnected nodes in real networks, a complex dynamical network shows the dynamic behaviors not only in the nodes but also in the links between nodes. This implies that the complex dynamical network can be considered to be composed of the two interconnected subsystems, the nodes subsystem (NS) and links subsystem (LS), and thus it has become evident that the dynamics of LS plays an important role in assisting NS to emerge the dynamic behavior. However, all the dynamic links are rarely considered to be the dynamic system as a whole in the existing literature. In this article, the outgoing links vector is introduced to describe the state of LS and the dynamics of LS is proposed and utilized to investigate the tracking task of NS. Here, the bounded uncertainties are considered in the dynamic model of NS and the nonlinearities are considered in the dynamic model of LS, which is rarely considered in the existing literature, by which the adaptive controller is synthesized to overcome the uncertainties and complete the tracking task of NS assisted with the constructed coupling term in LS. The results in this article show that when the controlled NS is tracking the given target, the LS is tracking the proposed auxiliary target. Finally, the effectiveness of the proposed theoretic results is verified via the example.
K E Y W O R D Sadaptive tracking control, complex dynamical network with uncertainties and nonlinearities, the links subsystem, the nodes subsystem, the outgoing links dynamics
INTRODUCTIONIn the last decades, the study of complex dynamical network mainly focuses on the structure and function of network, especially, how the links topology between nodes can effectively assist all the nodes to track the given reference signals, which has attracted extensive attention from many scholars. The complex dynamical network has been widely employed to describe many real networks such as the food webs, 1 the wireless sensor network, 2,3 the brain network, 4,5 the electrical power grids, 6,7 World-Wide Web, 8 and so on. It is remarkable that these studies as reviewed in References 9-15 only focus on the collective behavior of nodes such as synchronization, nevertheless, the all the links are regarded only as playing the auxiliary and secondary role for all the nodes tending synchronization. In order words, the dynamics of links in the complex dynamical network should be transformed into an independent dynamical subsystem and their dynamics should be considered as reviewed in References 16-22.
