Tracking control for the nonlinear complex dynamical network assisted with outgoing links dynamics

Abstract

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.

Model following adaptive control for nodes in complex dynamical network via the state observer of links

“…Different from the processing method of NS in the Refs. [24,25,32], this paper introduces the matrix form in Equation (3) to describe the state of NS for MFAC research. This helps simplify mathematical analysis and synthesizing controller.…”

“…The constant matrix in this research is unknown, which is different from the researches on synchronization and tracking control of CDN in the Refs. [34,35,25]. (c).…”

“…On the other hand, although literature [17] studied the MFAC problem of NS in CDN, it did not consider the important dynamic auxiliary role played by links, which should not be ignored. Because the dynamic change of links can promote the controlled nodes to emerge some DCs, for example, the synchronization and tracking 24,25 . Accordingly, inspired by the above discussions, this paper proposes the MFAC problem of NS in CDN via the links dynamics, which can be used to fill the insufficiencies above mentioned.…”

“…As far as we know, in order to solve the problem that the state information of the links is unavailable in the CDN, two processing methods have emerged in the existing literature. The first method is to construct the dynamic auxiliary tracking targets of links firstly, then use the state information of auxiliary targets for links to design an appropriate control scheme for NS, then promote the nodes to emerge the required DCs through mutual coupling 25,32 . Another method is to design an asymptotical state observer for links 33 , and use the state information of observer to achieve required control target.…”

Partial‐node‐based resilient set‐membership filtering for output‐coupled complex networks with event‐triggered mechanisms

“…Complex networks represent the intricate connections among elements in many real-world systems and constitute an efficacious formalism, 1,2 such as the Internet, biological networks, 3,4 communication networks, 5,6 social and transport networks, 7,8 power networks, 9,10 and so on. It is a scientific research method that complex networks are modeled as graphs, where the elements are abstracted as nodes and the interactions among these elements are abstracted as edges.…”