Adaptive Synchronization of Complex Dynamical Networks via Distributed Pinning Impulsive Control
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Adaptive Synchronization of Complex Dynamical Networks via Distributed Pinning Impulsive Control Dong Ding1 · Ze Tang1,2 · Yan Wang1 · Zhicheng Ji1 Accepted: 8 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, exponential synchronization of a class of nonlinearly coupled complex dynamical networks with time-varying delay is investigated. A novel distributed controller combined with pinning impulsive method is designed by selecting the systems with largest norms of errors to be controlled at every impulsive instant. By introducing the adaptive control protocol into the negative feedback controller, suitable control gains are obtained and therefore, the control cost is efficiently saved. Based on the Lyapunov stability theory and some mathematical techniques, some novel leader-following synchronization criteria are derived. Furthermore, with consideration of time-varying impulsive effects, the obtained results are extended to a more complicated situation. Finally, two numerical examples are performed to illustrate the effectiveness of the theoretical analysis. Keywords Exponential synchronization · Distributed impulsive control · Pinning adaptive control · Time-varying delay · Complex networks
1 Introduction Complex dynamical networks, as a kind of mathematical models, have been introduced into many practical applications, such as scheduling [1], traffic controlling [2] and managements [3]. In a complex network, every node is a dynamical system, while among different nodes, there exist some connections, which denote the information exchange among different nodes.
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Zhicheng Ji [email protected] Dong Ding [email protected] Ze Tang [email protected] Yan Wang [email protected]
1
Engineering Research Center of Internet of Things Technology Applications (Ministry of Education), Jiangnan University, Wuxi, China
2
Department of Electrical Engineering, Yeungnam University, Kyonsan, Republic of Korea
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D. Ding et al.
It is worth studying that how to force all of the nodes in the networks to behave uniformly. This phenomenon is called synchronization. With the deepening study on complex networks models, over the last few decades, a number of good results have been appeared, which focused on the leader-following problems [4,5], drive-response problems [6,7] and masterslave issues [8,9]. Since the complex networks may not be able to achieve synchronization only by adjusting coupling strengthes, different control schemes are effectively introduced to figure out such problems. For a complex network, how to achieve different types of synchronization phenomena and how to evaluate the convergence velocity have attracted a large number of attentions by the researchers from all over the world. In this regard, various methods on controller designing is currently one of the hottest topics in network science and control fields [10– 15]. Among different control protocols, impulsive control, as a kind of control methods who largely saves control cost
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