Abstract
This paper discusses the distributed adaptive control schemes for pinning synchronization in complex dynamical network with non-delayed and delayed coupling. An effective distributed adaptive strategy to adjust simultaneously coupling strength and feedback gains is designed based on information of the non-delayed network’s configuration. For a special case where the information of delay is available, a distributed adaptive scheme to tune the coupling weights of non-delayed coupling network is proposed by using the delayed feedback controllers. Based on the small-world network and scalefree network, simulation examples are given to demonstrate the effectiveness of theoretical analysis.
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References
D. Watts and S. Strogatz, “Collective dynamics of ‘small-world’ networks,” Nature, vol. 393, pp. 440–442, June 1998.
A. Barabási and R. Albert, “Emergence of scaling in random networks,” Science, vol. 286, pp. 509–512, September 1999.
M. Newman, A. Barabási, and D. Watts, The Structure and Dynamics of Networks, Princeton University Press, 2006.
L. Pecora and T. Carroll, “Synchronization in chaotic systems,” Physical Review Letters, vol. 64, no. 8, pp. 821–824, February 1990.
X. Wang and G. Chen, “Synchronization in smallworld dynamical networks,” International Journal of Bifurcation and Chaos, vol. 12, no. 1, pp. 187–192, January 2002.
C. Li and G. Chen, “Synchronization in general complex dynamical networks with coupling delays,” Physica A: Statistical Mechanics and its Applications, vol. 343, pp. 263–278, November 2004.
H. Gao, J. Lam, and G. Chen, “New criteria for synchronization stability of general complex dynamical networks with coupling delays,” Physics Letters A, vol. 360, no. 2, pp. 263–273, December 2006.
J. Cao, P. Li, and W. Wang, “Global synchronization in arrays of delayed neural networks with constant and delayed coupling,” Physics Letters A, vol. 353, no. 4, pp. 318–325, May 2006.
W. Yu and J. Cao, “Synchronization control of stochastic delayed neural networks,” Physica A: Statistical Mechanics and its Applications, vol. 373, pp. 252–260, January 2007.
W. Yu, G. Chen, and J. Lü, “On pinning synchronization of complex dynamical networks,” Automatica, vol. 45, no. 2, pp. 429–435, February 2009.
J. Lu, D. Ho, and J. Cao, “A unified synchronization criterion for impulsive dynamical networks,” Automatica, vol. 46, no. 7, pp. 1215–1221, July 2010.
C. Zheng and J. Cao, “Robust synchronization of dynamical network with impulsive disturbances and uncertain parameters,” International Journal of Control, Automation and Systems, vol. 11, no. 4, pp. 657–665, August 2013.
J. Lu and J. Cao, “Adaptive complete synchronization of two identical or different chaotic (hyperchaotic) systems with fully unknown parameters,” Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 15, no. 4, 043901, 2005.
S. Zheng, Q. Bi, and G. Cai, “Adaptive projective synchronization in complex networks with timevarying coupling delay,” Physics Letters A, vol. 373, no. 17, pp. 1553–1559, April 2009.
W. He and J. Cao, “Generalized synchronization of chaotic systems: an auxiliary system approach via matrix measure,” Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 19, no. 1, 013118, 2009.
W. Yu and J. Cao, “Adaptive synchronization and lag synchronization of uncertain dynamical system with time delay based on parameter identification,” Physica A: Statistical Mechanics and its Applications, vol. 375, no. 2, pp. 467–482, March 2007.
Y. Sun and J. Cao, “Adaptive lag synchronization of unknown chaotic delayed neural networks with noise perturbation,” Physics Letters A, vol. 364, no. 3, pp. 277–285, April 2007.
X. Wang and G. Chen, “Pinning control of scalefree dynamical networks,” Physica A: Statistical Mechanics and its Applications, vol. 310, no. 3, pp. 521–531, July 2002.
X. Li, X. Wang, and G. Chen, “Pinning a complex dynamical network to its equilibrium,” IEEE Trans. on Circuits and Systems I: Regular Papers, vol. 51, no. 10, pp. 2074–2087, October 2004.
J. Zhou, J.-A. Lu, and J. Lü, “Pinning adaptive synchronization of a general complex dynamical network,” Automatica, vol. 44, no. 4, pp. 996–1003, April 2008.
W. Xia and J. Cao, “Pinning synchronization of delayed dynamical networks via periodically intermittent control,” Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 19, no. 1, 013120, 2009.
X. Yang and J. Cao, “Adaptive pinning synchronization of complex networks with stochastic perturbations,” Discrete Dynamics in Nature and Society, vol. 2010, 416182, 2010.
G. Chen, “Pinning control and synchronization on complex dynamical networks,” International Journal of Control, Automation and Systems, vol. 12, no. 2, pp. 221–230, April 2014.
W. Lu, T. Chen, and G. Chen, “Synchronization analysis of linearly coupled systems described by differential equations with a coupling delay,” Physica D: Nonlinear Phenomena, vol. 221, no. 2, pp. 118–134, September 2006.
P. Li and Z. Yi, “Synchronization analysis of delayed complex networks with time-varying couplings,” Physica A: Statistical Mechanics and its Applications, vol. 387, no. 14, pp. 3729–3737, June 2008.
M. Park, O. Kwon, J. H. Park, S. Lee, and E. Cha, “Synchronization criteria of fuzzy complex dynamical networks with interval time-varying delays,” Applied Mathematics and Computation, vol. 218, no. 23, pp. 11634–11647, August 2012.
M. Park, O. Kwon, J. Park, S. M. Lee, and E. J. Lee, “On synchronization criterion for coupled discrete-time neural networks with interval timevarying delays,” Neurocomputing, vol. 99, no. 1, pp. 188–196, January 2013.
S. Wen, S. Chen, and W. Guo, “Adaptive global synchronization of a general complex dynamical network with non-delayed and delayed coupling,” Physics Letters A, vol. 372, no. 42, pp. 6340–6346, October 2008.
W. Guo, F. Austin, and S. Chen, “Pinning synchronization of the complex networks with non-delayed and delayed coupling,” Physics Letters A, vol. 373, no. 17, pp. 1565–1572, April 2009.
W. Guo, S. Chen, and W. Sun, “Topology identification of the complex networks with non-delayed and delayed coupling,” Physics Letters A, vol. 373, no. 41, pp. 3724–3729, October 2009.
Y. Wu, C. Li, and A. Yang, “Pinning adaptive antisynchronization between two general complex dynamical networks with non-delayed and delayed coupling,” Applied Mathematics and Computation, vol. 218, no. 14, pp. 7445–7452, March 2012.
Z. Wang, L. Huang, and Y. Wang, “Synchronization analysis of networks with both delayed and nondelayed couplings via adaptive pinning control method,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 12, pp. 4202–4208, December 2010.
W. Yu, P. DeLellis, and G. Chen, “Distributed adaptive control of synchronization in complex networks,” IEEE Trans. on Automatic Control, vol. 57, no. 8, pp. 2153–2158, August 2012.
W. Yu, W. Ren, and W. Zheng, “Distributed control gains design for consensus in multi-agent systems with second-order nonlinear dynamics,” Automatica, vol. 49, no. 7, pp. 2107–2115, July 2013.
T. Chen, X. Liu, and W. Lu, “Pinning complex networks by a single controller,” IEEE Trans. on Circuits and Systems I: Regular Papers, vol. 54, no. 6, pp. 1317–1326, June 2007.
S. Boyd, L. Ghaoui, and E. Feron, Linear Matrix Inequalities in System and Control Theory, Society for Industrial and Applied Mathematics, Philadelphia, 1994.
R. Horn and C. Johnson, Topics in Matrix Analysis, Cambridge University Press, 1994.
W. Lu and T. Chen, “New approach to synchronization analysis of linearly coupled ordinary differential systems,” Physica D: Nonlinear Phenomena, vol. 213, no. 2, pp. 214–230, January 2006.
H. Khalil and W. Grizzle, Nonlinear Systems, 3rd edition, Prentice Hall, 2002.
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This work was supported by the National Natural Science Foundation of China (Grant Nos. 61272530, and 11072059), the Natural Science Foundation of Jiangsu Province of China (Grant No. BK2012741). The authors are grateful to the anonymous reviewers for their resourceful comments, and constructive suggestions.
Shaolin Li received his B.S. and M.S. degrees from Yunnan University, Kunming, China, in 1999 and 2006, respectively. He is currently a Visiting Scholar in Department of Mathematics, Southeast University, Nanjing, China. He is an Associate Professor with the Department of Mathematics, Honghe University, Yunan, China. His current research interests include collective behavior in complex dynamical networks and neural networks.
Jinde Cao received his B.S. degree from Anhui Normal University, Wuhu, China, an M.S. degree from Yunnan University, Kunming, China, and a Ph.D. degree from Sichuan University, Chengdu, China, all in Mathematics/Applied Mathematics, in 1986, 1989, and 1998, respectively. From March 1989 to May 2000, he was with the Yunnan University. In May 2000, he joined the Department of Mathematics, Southeast University, Nanjing, China. From July 2001 to June 2002, he was a Postdoctoral Research Fellow at the Department of Automation, and Computer-Aided Engineering, Chinese University of Hong Kong, Hong Kong. In the period from 2006 to 2008, he was a Visiting Research Fellow, and a Visiting Professor at the School of Information Systems, Computing and Mathematics, Brunel University, UK. On August 2014, he was a Visiting Professor at the School of Electrical and Computer Engineering, RMIT University, Australia. Currently, he is a distinguished professor and doctoral advisor at the Southeast University, prior to which he was a Professor at Yunnan University from 1996 to 2000. He is the author or coauthor of more than 400 journal papers and five edited books. His research interests include nonlinear systems, neural networks, complex systems, complex networks and applied mathematics. Dr. Cao was an Associate Editor of the IEEE Transactions on Neural Networks, Journal of the Franklin Institute and Neurocomputing. He is the dean of Department of Mathematics at Southeast University, Nanjing, China and also an Associate Editor of the IEEE Transactions on Cybernetics, Differential Equations and Dynamical Systems, Mathematics and Computers in Simulation and Neural Networks. Dr. Cao is a Reviewer of Mathematical Reviews and Zentralblatt-Math. He is a ISI Highly-Cited Researcher in Mathematics and Engineering listed by Thomson Reuters.
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Li, S., Cao, J. Distributed adaptive control of pinning synchronization in complex dynamical networks with non-delayed and delayed coupling. Int. J. Control Autom. Syst. 13, 1076–1085 (2015). https://doi.org/10.1007/s12555-014-0177-2
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DOI: https://doi.org/10.1007/s12555-014-0177-2