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Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 578539, 7 pages
http://dx.doi.org/10.1155/2015/578539
Research Article

Adaptive Second-Order Synchronization of Two Heterogeneous Nonlinear Coupled Networks

College of Science, North China University of Technology, Beijing 100144, China

Received 12 June 2015; Accepted 16 July 2015

Academic Editor: Michael Z. Q. Chen

Copyright © 2015 Bo Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper investigates the second-order synchronization of two heterogeneous nonlinear coupled networks by introducing controller and adaptive laws. Based on Lyapunov stability properties and LaSalle invariance principle, it is proved that the position and the velocity of two heterogeneous nonlinear coupled networks are asymptotically stable. Finally, some numerical simulations are presented to verify the analytical results.

1. Introduction

In recent years, people have paid more attention to the synchronization problem of complex networks due to their broad applications, such as biology, physics, communication, computer, and the traffic [13].

Various models and algorithms about complex networks have been investigated based on different tasks or interests. To achieve the synchronization of complex networks, the adaptive strategy is one of the most interesting topics on the synchronization problem of complex networks. In [4], the authors introduced an adaptive synchronization scheme in complex networks which was linked through nonlinearly coupling. [5] considered the consensus problems of multiagent systems with second-order nonlinear dynamics by introducing the distributed control gains. In [6], the agents of second-order multiagent systems were governed by both position and velocity consensus protocol with time-varying velocity. In [7], the authors studied the group-consensus problem of second-order nonlinear multiagent systems. There have been many studies about the second-order networks [811]. However, due to the limit of outside influences and communication conditions, the dynamics of the coupling nodes can be different; so the heterogeneous networks models were proposed in [1215]. In [16], the authors investigated the consensus problem of heterogeneous multiagent systems. In [17], the authors discussed the adaptive consensus of second-order multiagent systems with heterogeneous nonlinear dynamics and time-varying delays. In [18], the authors studied the finite-time consensus problem of heterogeneous multiagent systems consisting of first-order and second-order integrator agents. In [19], the authors investigated the containment control problem of heterogeneous multiagent systems. The recent papers focus on the synchronization of single network [20, 21]. However, the synchronization can also occur in two or more networks [22, 23], such as the inside doors and the outside doors of city subways. In [24], the authors investigated the synchronization between two coupled complex networks. In [25], the authors further solved the synchronization problem of two nonlinear coupled networks. In [26], the number of nodes, dynamics, and topological structures of the two complex networks were different. However, the second-order synchronization of two heterogeneous nonlinear coupled networks has not been investigated.

Motivated by this, in this paper, we focus on the problem of adaptive second-order synchronization of two heterogeneous nonlinear coupled networks. The main contributions of this paper are threefold: (1) the nonlinear intrinsic dynamics of each node is heterogeneous; (2) the synchronization occurs in the two heterogeneous nonlinear coupled networks; (3) controller and adaptive laws are introduced to solve the second-order synchronization of the two heterogeneous nonlinear coupled networks. Particularly, even if the topological structure is unknown, the two heterogeneous nonlinear networks can achieve synchronization by introducing the suitable controller and adaptive laws.

This paper is organized as follows. In Section 2, the second-order models of two heterogeneous nonlinear coupled networks are given. Moreover, some preliminaries are introduced to solve the adaptive synchronization. Section 3 presents the main results and the theoretical analysis of the second-order synchronization of two heterogeneous nonlinear coupled networks. Some numerical simulations of the theoretical results are given in Section 4. Finally, the conclusion is made in Section 5.

2. Preliminaries and Problem Statement

Consider the second-order models of two heterogeneous nonlinear coupled networks consisting of identical nodes described bywhere , () describe the position vectors of networks (1) and (2), respectively, and , are their velocity vectors, respectively. and are continuous functions. are the position and velocity coupling strengths in two networks, respectively. and denote the coupling configurations of the two networks, respectively. If there exists communication channel between node and node , then ,  ; otherwise, , , and the diagonal elements are defined as , . is the neighbor set of node . describe the nonlinear coupling parameter of both networks. is the controller of network.

Define the position error and velocity error of the th node as

Differentiating and , then

Denoting and , we can have

The controller is designed as

Combining (6), system (4) can be rewritten as

In the following, the necessary definition, assumption, and lemma will be presented for discussing the second-order synchronization of two heterogeneous nonlinear coupled networks.

Definition 1. Networks (1) and (2) are said to achieve second-order synchronization if that is, for .

Assumption 2. For every of network (1) and of network (2) (), there exist the constants , such that

Lemma 3 (see [17]). For any vectors and positive definite matrix, the following matrix inequality holds:

3. Main Results

In this section, we will investigate the second-order synchronization of two heterogeneous nonlinear coupled networks and provide the detailed analysis.

Theorem 4. Consider networks (1) and (2) steered by (6) under Assumption 2, then the position and velocity of each node can asymptotically synchronize.

Proof. Constructing the Lyapunov function, whereand where is a positive constant.
Differentiating , then where .
Differentiating , we get Differentiating , then Combining , , and , then we can have if . Based on LaSalle invariance principle, we can know that, for any initial states, the error solution will tend to zero, which implies that networks (1) and (2) can asymptotically synchronize with controller (6).

Remark 5. When the topology structure is unknown, the two heterogeneous nonlinear coupled networks also can be asymptotically synchronized by controller (6).

Corollary 6. If the coupled networks (1) and (2) have the identical dynamics, the networks can asymptotically synchronize through the following controller: where are the same as Theorem 4.

4. Simulations

In this section, several numerical simulations are given to illustrate the analytical results.

We choose Lorentz system of two different parameters as nonlinear dynamics of networks (1) and (2). The Lorentz system is described as the following:where , , and are the parameters. For network (1), let , , and ; and for network (2), let , , and .

The coupling matrixes of networks (1) and (2) with four nodes, respectively, are described by the following matrixes:

With controller (6), it can be found that the velocity and position of networks (1) and (2) can synchronize to the synchronous state, described as Figures 1-2, respectively. However, if networks (1) and (2) without the controller are in the same conditions, we can find that velocity and position of the networks cannot achieve synchronization, when . The simulations are shown in Figures 3 and 4.

Figure 1: The error trajectories for velocity with controllers.
Figure 2: The error trajectories for position with controllers.
Figure 3: The error trajectories for velocity without controllers.
Figure 4: The error trajectories for position without controllers.

5. Conclusion

In this paper, we have considered the adaptive second-order synchronization of two heterogeneous nonlinear coupled networks. By constructing a valid Lyapunov function, we have proved that the networks can achieve asymptotically synchronization with the given controller and adaptive laws. Particularly, even if the topological structure is unknown, the networks also can be synchronized by the given controller and adaptive laws.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was partly supported by the National Natural Science Foundation of China under Grant (no. 61304049), “The-Great-Wall-Scholar” Candidate Training-Plan of North China University of Technology, Construction Plan for Innovative Research Team of North China University of Technology, and the Plan training project of excellent young teacher of North China University of Technology, the special project of North China University of Technology (no. XN085).

References

  1. H. Su, G. Jia, and M. Z. Q. Chen, “Semi-global containment control of multi-agent systems with input saturation,” IET Control Theory & Applications, vol. 8, no. 18, pp. 2229–2237, 2014. View at Publisher · View at Google Scholar · View at Scopus
  2. H. Su and M. Z. Chen, “Multi-agent containment control with input saturation on switching topologies,” IET Control Theory & Applications, vol. 9, no. 3, pp. 399–409, 2015. View at Publisher · View at Google Scholar
  3. B. Liu, W.-N. Hu, J. Zhang, and H.-S. Su, “Controllability of discrete-time multi-agent systems with multiple leaders on fixed networks,” Communications in Theoretical Physics, vol. 58, no. 6, pp. 856–862, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. Y. Liang and X. Wang, “Adaptive synchronization in complex networks through nonlinearly coupling,” Computer Engineering and Applications, vol. 48, no. 10, pp. 25–28, 2012. View at Google Scholar
  5. W. Yu, W. Ren, W. X. Zheng, G. Chen, and J. Lü, “Distributed control gains design for consensus in multi-agent systems with second-order nonlinear dynamics,” Automatica, vol. 49, no. 7, pp. 2107–2115, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. W. Yu, G. Chen, M. Cao, and J. Kurths, “Second-Order consensus for multiagent systems with directed topologies and nonlinear dynamics,” IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, vol. 40, no. 3, pp. 881–891, 2010. View at Publisher · View at Google Scholar · View at Scopus
  7. Q. Ma, Z. Wang, and G. Miao, “Second-order group consensus for multi-agent systems via pinning leader-following approach,” Journal of the Franklin Institute, vol. 351, no. 3, pp. 1288–1300, 2014. View at Publisher · View at Google Scholar · View at Scopus
  8. B. Liu, S. Li, and L. Wang, “Adaptive synchronization of two time-varying delay nonlinear coupled networks,” in Proceedings of the 33rd Chinese Control Conference (CCC '14), pp. 3800–3804, July 2014. View at Publisher · View at Google Scholar · View at Scopus
  9. P. Lin and Y. Jia, “Consensus of second-order discrete-time multi-agent systems with nonuniform time-delays and dynamically changing topologies,” Automatica, vol. 45, no. 9, pp. 2154–2158, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. Y. Zheng, Y. Zhu, and L. Wang, “Finite-time consensus of multiple second-order dynamic agents without velocity measurements,” International Journal of Systems Science, vol. 45, no. 3, pp. 579–588, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. Z.-J. Tang, T.-Z. Huang, J.-L. Shao, and J.-P. Hu, “Consensus of second-order multi-agent systems with nonuniform time-varying delays,” Neurocomputing, vol. 97, pp. 410–414, 2012. View at Publisher · View at Google Scholar · View at Scopus
  12. H. Su, Z. Rong, M. Z. Q. Chen, X. Wang, G. Chen, and H. Wang, “Decentralized adaptive pinning control for cluster synchronization of complex dynamical networks,” IEEE Transactions on Cybernetics, vol. 43, no. 1, pp. 394–399, 2013. View at Publisher · View at Google Scholar · View at Scopus
  13. B. Liu, H. Su, R. Li, D. Sun, and W. Hu, “Switching controllability of discrete-time multi-agent systems with multiple leaders and time-delays,” Applied Mathematics and Computation, vol. 228, pp. 571–588, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. H. Su, M. Z. Chen, J. Lam, and Z. Lin, “Semi-global leader-following consensus of linear multi-agent systems with input saturation via low gain feedback,” IEEE Transactions on Circuits and Systems. I. Regular Papers, vol. 60, no. 7, pp. 1881–1889, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. H. Su, N. Zhang, M. Z. Chen, H. Wang, and X. Wang, “Adaptive flocking with a virtual leader of multiple agents governed by locally Lipschitz nonlinearity,” Nonlinear Analysis: Real World Applications, vol. 14, no. 1, pp. 798–806, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. Y. Zheng, Y. Zhu, and L. Wang, “Consensus of heterogeneous multi-agent systems,” IET Control Theory & Applications, vol. 5, no. 16, pp. 1881–1888, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. B. Liu, X. Wang, H. Su, Y. Gao, and L. Wang, “Adaptive second-order consensus of multi-agent systems with heterogeneous nonlinear dynamics and time-varying delays,” Neurocomputing, vol. 118, pp. 289–300, 2013. View at Publisher · View at Google Scholar · View at Scopus
  18. Y. Zheng and L. Wang, “Finite-time consensus of heterogeneous multi-agent systems with and without velocity measurements,” Systems & Control Letters, vol. 61, no. 8, pp. 871–878, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. Y. Zheng and L. Wang, “Containment control of heterogeneous multi-agent systems,” International Journal of Control, vol. 87, no. 1, pp. 1–8, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. H. Su, M. Z. Q. Chen, X. Wang, and J. Lam, “Semiglobal observer-based leader-following consensus with input saturation,” IEEE Transactions on Industrial Electronics, vol. 61, no. 6, pp. 2842–2850, 2014. View at Publisher · View at Google Scholar · View at Scopus
  21. B. Liu, X. Wang, H. Su, H. Zhou, Y. Shi, and R. Li, “Adaptive synchronization of complex dynamical networks with time-varying delays,” Circuits, Systems, and Signal Processing, vol. 33, no. 4, pp. 1173–1188, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. J.-A. Wang, “Adaptive generalized synchronization between two different complex networks with time-varying delay coupling,” Acta Physica Sinica, vol. 61, no. 2, Article ID 020509, 2012. View at Google Scholar · View at Scopus
  23. H. Tang, L. Chen, J.-A. Lu, and C. K. Tse, “Adaptive synchronization between two complex networks with nonidentical topological structures,” Physica A: Statistical Mechanics and its Applications, vol. 387, no. 22, pp. 5623–5630, 2008. View at Publisher · View at Google Scholar · View at Scopus
  24. C. Li, W. Sun, and J. Kurths, “Synchronization between two coupled complex networks,” Physical Review E, vol. 76, no. 4, Article ID 046204, 2007. View at Publisher · View at Google Scholar · View at Scopus
  25. Y. Du, W. Sun, and C. Li, “Adaptive synchronization between two nonlinear complex networks,” Communications in Applied Mathematics and Computational Science, vol. 23, no. 2, pp. 87–94, 2009. View at Google Scholar
  26. J. Chen, L. Jiao, J. Wu, and X. Wang, “Adaptive synchronization between two different complex networks with time-varying delay coupling,” Chinese Physics Letters, vol. 26, no. 6, Article ID 060505, 2009. View at Publisher · View at Google Scholar · View at Scopus