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A complex dynamical network (CDN) can be considered as the composition system with the nodes subsystem (NS) and the links subsystem (LS), and both subsystems are coupled with each other. In this paper, two vector differential equations (VDE) are used to describe the dynamical behaviours of NS and LS, respectively, in which the dynamical behaviour of NS is considered as the VDE with the second derivative term (SDT). This paper mainly focuses on the dynamics of LS, which is represented as VDE with the intuitive topologic feature of outgoing links, and investigates the design of the tracking controller for NS and the auxiliary tracking objectives (ATO) for LS. Firstly, the dynamical models of NS and LS in CDN are proposed, and the corresponding assumptions are given. Secondly, based on Lyapunov stability theory, the controller of NS and the ATO of LS are designed so that the state of NS can asymptotically track the given reference signal. Finally, the effectiveness of the proposed control strategy in this paper is verified by the numerical simulation example with N two-links robots. Abbreviations: ATO: auxiliary tracking objectives; CDN: complex dynamical network; LS: links subsystem; MDE: matrix differential equation; NS: nodes subsystem; SDT:second derivative term; VDE: vector differential equation; [ABSTRACT FROM AUTHOR]

Summary: This article formulates and solves a problem of model following adaptive control (MFAC) of nodes in the complex dynamic network (CDN) with the dynamic matrix‐relation links. From the perspective of composite systems, the CDN with the dynamic matrix‐relation links is considered as an interconnected system composed of intercoupling node subsystem (NS) and link subsystem (LS). Based on this idea, employing the Riccati matrix differential equation as the dynamic equation of LS, and the vector differential equation with the second derivative term as the dynamic equation of NS. Then, to remove the restriction that the state of LS is unavailable, an adaptive control scheme is proposed for NS with the help of constructed dynamic auxiliary tracking target for LS, such that the problem of MFAC of nodes is solved. Besides, it is proved mathematically that the state of NS is asymptotically tracking the state of the given reference model when the state of LS is asymptotically tracking the constructed dynamic auxiliary tracking target. Finally, the numerical simulation is given to verify the effectiveness of the theoretical results in this article. [ABSTRACT FROM AUTHOR]

For the complex dynamical network (CDN) with uncertainties, this paper defines the links synchronization and synthesizes the adaptive control scheme to realize it. Generally speaking, a CDN can be considered as the composition system with the two coupled dynamic subsystems: the nodes subsystem (NS) and the links subsystem (LS). We observed that there are many results in the existing literature on the NS synchronization, which implies that the state at each node tends to be the same when the NS synchronization happens. However, the LS synchronization is rarely discussed in the existing literature due to its unknown meaning and engineering practice background. Inspired by the ego-networks, this paper employs the time-varying outgoing links at each node to geometrically describe the changing of network topologic structure and regards the values-weighted of outgoing links as the state variables of LS, by which the LS synchronization is introduced, the corresponding control scheme for its implementation is synthesized under the condition that the state of NS is available and the state of LS is unavailable. The control scheme includes the adaptive controller for NS and the coupling strategy for LS. Finally, the effectiveness of proposed control scheme in this paper is verified by a numerical example. [ABSTRACT FROM AUTHOR]

This paper formulates and solves a new problem of the double model following adaptive control (MFAC) of nodes and links in a complex dynamical network (CDN). This is different from most existing studies on CDN and MFAC. Inspired by the concept of composite systems, the CDN with dynamic links is regarded as an interconnected system composed of an interconnected node group (NG) and link group (LG). Guided by the above-mentioned new idea of viewing a CDN from the perspective of composite systems, by means of Lyapunov theory and proposed related mathematical preliminaries, a new adaptive control scheme is proposed for NG. In addition, to remove the restriction that the states of links in a CDN are unavailable due to physical constraints, technical restraints, and expensive measurement costs, we synthesize the coupling term in LG with the proposed adaptive control scheme for NG, such that the problem of double MFAC of nodes and links in CDN is solved. Finally, a simulation example is presented to verify the theoretical results. [ABSTRACT FROM AUTHOR]