Consecutive Synchronization of a Delayed Complex Dynamical Network via Distributed Adaptive Control Approach

  • Ali Kazemy
  • Jinde CaoEmail author
Regular Papers Control Theory and Applications


In this paper, a consecutive synchronization scheme is investigated to synchronize the nodes of a delayed complex dynamical network with an isolated node via an adaptive control approach. The specific feature of this scheme consists in the structure of the communication links: a communication connection is required between the isolated node and one selected node in the network, and further communication links exist between any node and one neighbor node. In this way, all nodes are connected together like a chain. Based on Lyapunov-Krasovskii theory, some conditions are obtained in the form of linear matrix inequalities to guarantee the consecutive synchronization by the designed distributed adaptive control. To make this synchronization scheme more practical, no constraints have been considered for coupling connection matrix such as being symmetric or zero row sum. Finally, a numerical example is presented to demonstrate the effectiveness of the proposed method.


Complex dynamical network distributed adaptive control Lyapunov-Krasovskii theory synchronization time-delay 


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  1. [1]
    A. Kazemy, “Global synchronization of neural networks with hybrid coupling: a delay interval segmentation approach,” Neural Computing and Applications, vol. 30, no. 2, pp. 627–637, 2018.Google Scholar
  2. [2]
    H.-B. Hu, K. Wang, L. Xu, and X.-F. Wang, “Analysis of online social networks based on complex network theory,” Complex Systems and Complexity Science, vol. 2, p. 1214, 2008.Google Scholar
  3. [3]
    Y.-W. Wang, T. Bian, J.-W. Xiao, and C. Wen, “Global synchronization of complex dynamical networks through digital communication with limited data rate,” IEEE Transactions on Neural Networks and Learning Systems, vol. 26, no. 10, pp. 2487–2499, 2015.MathSciNetGoogle Scholar
  4. [4]
    J. W. Simpson-Porco, F. Dörfler, and F. Bullo, “Synchronization and power sharing for droop-controlled inverters in islanded microgrids,” Automatica, vol. 49, no. 9, pp. 2603–2611, 2013.MathSciNetzbMATHGoogle Scholar
  5. [5]
    A. Sun, L. Lü, and C. Li, “Synchronization of an uncertain small-world neuronal network based on modified sliding mode control technique,” Nonlinear Dynamics, vol. 82, no. 4, pp. 1905–1912, 2015.MathSciNetzbMATHGoogle Scholar
  6. [6]
    J. Lu and D. Ho, “Stabilization of complex dynamical networks with noise disturbance under performance constraint,” Nonlinear Analysis: Real World Applications, vol. 12, no. 4, pp. 1974–1984, 2011.MathSciNetzbMATHGoogle Scholar
  7. [7]
    X. Mao and Z. Wang, “Stability, bifurcation, and synchronization of delay-coupled ring neural networks,” Nonlinear Dynamics, vol. 84, no. 2, pp. 1063–1078, 2016.MathSciNetzbMATHGoogle Scholar
  8. [8]
    Y. Wan, J. Cao, G. Chen, and W. Huang, “Distributed observer-based cyber-security control of complex dynamical networks,” IEEE Transactions on Circuits and Systems I, vol. 64, no. 11, pp. 2966–2975, 2017.MathSciNetGoogle Scholar
  9. [9]
    R. Li, J. Cao, A. Alsaedi, and F. Alsaadi, “Exponential and fixed-time synchronization of Cohen-Grossberg neural networks with time-varying delays and reaction-diffusion terms,” Applied Mathematics and Computation, vol. 313, pp. 37–51, 2017.MathSciNetGoogle Scholar
  10. [10]
    J. Cao and R. Li, “Fixed-time synchronization of delayed memristor-based recurrent neural networks,” Science China Information Sciences, vol. 60, no. 3, p. 032201, 2017.Google Scholar
  11. [11]
    R. Rakkiyappan, R. Sivasamy, and X. Li, “Synchronization of identical and nonidentical memristor-based chaotic systems via active backstepping control technique,” Circuits, Systems, and Signal Processing, vol. 34, no. 3, pp. 763–778, 2015.Google Scholar
  12. [12]
    W. Shen, Z. Zeng, and S. Wen, “Synchronization of complex dynamical network with piecewise constant argument of generalized type,” Neurocomputing, vol. 173, pp. 671–675, 2016.Google Scholar
  13. [13]
    X. Fang and W. Chen, “Synchronization of complex dynamical networks with time-varying inner coupling,” Nonlinear Dynamics, vol. 85, no. 1, pp. 13–21, 2016.MathSciNetzbMATHGoogle Scholar
  14. [14]
    A. Kazemy, “Synchronization criteria for complex dynamical networks with state and coupling time-delays,” Asian Journal of Control, vol. 19, no. 1, pp. 131–138, 2017.MathSciNetzbMATHGoogle Scholar
  15. [15]
    X. Li, R. Rakkiyappan, and N. Sakthivel, “Non-fragile synchronization control for markovian jumping complex dynamical networks with probabilistic time-varying coupling delay,” Asian Journal of Control, vol. 17, no. 5, pp. 1678–1695, 2015.MathSciNetzbMATHGoogle Scholar
  16. [16]
    E. Alzahrani, H. Akca, and X. Li, “New synchronization schemes for delayed chaotic neural networks with impulses,” Neural Computing and Applications, vol. 28, no. 9, pp. 2823–2837, 2017.Google Scholar
  17. [17]
    X. Li and X. Fu, “Lag synchronization of chaotic delayed neural networks via impulsive control,” IMA Journal of Mathematical Control and Information, vol. 29, pp. 133–145, 2012.MathSciNetzbMATHGoogle Scholar
  18. [18]
    X. Zhang, X. Lv, and X. Li, “Sampled-data based lag synchronization of chaotic delayed neural networks with impulsive control,” Nonlinear Dynamics, vol. 90, no. 3, pp. 2199–2207, 2017.MathSciNetzbMATHGoogle Scholar
  19. [19]
    N. Li and J. Cao, “Lag synchronization of memristor-based coupled neural networks via w-measure.” IEEE Transactions on Neural Networks and Learning Systems, vol. 27, no. 3, pp. 686–697, 2016.MathSciNetGoogle Scholar
  20. [20]
    J. Wang, H. Zhang, Z. Wang, and B. Wang, “Local exponential synchronization in complex dynamical networks with time-varying delay and hybrid coupling,” Applied Mathematics and Computation, vol. 225, pp. 16–32, 2013.MathSciNetzbMATHGoogle Scholar
  21. [21]
    Q. Ma and J. Lu, “Cluster synchronization for directed complex dynamical networks via pinning control,” Neurocomputing, vol. 101, pp. 354–360, 2013.Google Scholar
  22. [22]
    Y. Wang and J. Cao, “Cluster synchronization in nonlinearly coupled delayed networks of non-identical dynamic systems,” Nonlinear Analysis Real World Applications, vol. 14, no. 1, pp. 842–851, 2013.MathSciNetzbMATHGoogle Scholar
  23. [23]
    H. Du, “Function projective synchronization in complex dynamical networks with or without external disturbances via error feedback control,” Neurocomputing, vol. 173, pp. 1443–1449, 2016.Google Scholar
  24. [24]
    H. Bao and J. Cao, “Projective synchronization of fractional-order memristor-based neural networks.” Neural Networks the Official Journal of the International Neural Network Society, vol. 63, p. 1-9, 2015.Google Scholar
  25. [25]
    Y. Sun, Z. Ma, F. Liu, and J. Wu, “Theoretical analysis of synchronization in delayed complex dynamical networks with discontinuous coupling,” Nonlinear Dynamics, vol. 86, no. 1, pp. 489–499, 2016.MathSciNetzbMATHGoogle Scholar
  26. [26]
    D. Li, Z. Wang, and G. Ma, “Controlled synchronization for complex dynamical networks with random delayed information exchanges: a non-fragile approach,” Neurocomputing, vol. 171, pp. 1047–1052, 2016.Google Scholar
  27. [27]
    L. Zhang, Y. Wang, Y. Huang, and X. Chen, “Delaydependent synchronization for non-diffusively coupled time-varying complex dynamical networks,” Applied Mathematics and Computation, vol. 259, pp. 510–522, 2015.MathSciNetzbMATHGoogle Scholar
  28. [28]
    J.-A. Wang, “New synchronization stability criteria for general complex dynamical networks with interval timevarying delays,” Neural Computing and Applications, vol. 28, no. 2, pp. 805–815, 2017.Google Scholar
  29. [29]
    Z.Wu, D. Liu, and Q. Ye, “Pinning impulsive synchronization of complex-variable dynamical network,” Communications in Nonlinear Science and Numerical Simulation, vol. 20, no. 1, pp. 273–280, 2015.MathSciNetzbMATHGoogle Scholar
  30. [30]
    J. Lu, D. Ho, and L. Wu, “Exponential stabilization in switched stochastic dynamical networks,” Nonlinearity, vol. 22, no. 4, pp. 889–911, 2009.MathSciNetzbMATHGoogle Scholar
  31. [31]
    T. H. Lee, J. H. Park, H. Y. Jung, S. M. Lee, and O. M. Kwon, “Synchronization of a delayed complex dynamical network with free coupling matrix,” Nonlinear Dynamics, vol. 69, no. 3, pp. 1081–1090, 2012.MathSciNetzbMATHGoogle Scholar
  32. [32]
    H. Hou, Q. Zhang, and M. Zheng, “Cluster synchronization in nonlinear complex networks under sliding mode control,” Nonlinear Dynamics, vol. 83, no. 1–2, pp. 739–749, 2016.MathSciNetzbMATHGoogle Scholar
  33. [33]
    Y. Wang, Y. Xia, H. Shen, and P. Zhou, “SMC design for robust stabilization of nonlinear markovian jump singular systems,” IEEE Transactions on Automatic Control, vol. 63, no. 1, pp. 219–224, 2018.MathSciNetzbMATHGoogle Scholar
  34. [34]
    Y. Wang, H. Shen, H. R. Karimi, and D. Duan, “Dissipativity-based fuzzy integral sliding mode control of continuous-time T-S fuzzy systems,” IEEE Transactions on Fuzzy Systems, vol. 26, no. 3, pp. 1164–1176, 2018.Google Scholar
  35. [35]
    Y. Wang, Y. Gao, H. R. Karimi, H. Shen, and Z. Fang, “Sliding mode control of fuzzy singularly perturbed systems with application to electric circuit,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2017. DOI: 10.1109/TSMC.2017.2720968Google Scholar
  36. [36]
    N. Li, H. Sun, X. Jing, and Q. Zhang, “Exponential synchronisation of united complex dynamical networks with multi-links via adaptive periodically intermittent control,” IET Control Theory & Applications, vol. 7, no. 13, pp. 1725–1736, 2013.MathSciNetGoogle Scholar
  37. [37]
    X. Chen, J. Cao, J. Qiu, A. Alsaedi, and F. E. Alsaadi, “Adaptive control of multiple chaotic systems with unknown parameters in two different synchronization modes,” Advances in Difference Equations, vol. 2016, no. 1, p. 231, 2016.Google Scholar
  38. [38]
    L. Shi, H. Zhu, S. Zhong, K. Shi, and J. Cheng, “Cluster synchronization of linearly coupled complex networks via linear and adaptive feedback pinning controls,” Nonlinear Dynamics, vol. 88, no. 2, pp. 859–870, 2017.zbMATHGoogle Scholar
  39. [39]
    B. Niu, Y. Liu, G. Zong, Z. Han, and J. Fu, “Command filter-based adaptive neural tracking controller design for uncertain switched nonlinear output-constrained systems,” IEEE Transactions on Cybernetics, vol. 47, no. 10, pp. 3160–3171, 2017.Google Scholar
  40. [40]
    S. Cai, X. Li, Q. Jia, and Z. Liu, “Exponential cluster synchronization of hybrid-coupled impulsive delayed dynamical networks: average impulsive interval approach,” Nonlinear Dynamics, vol. 85, no. 4, pp. 2405–2423, 2016.zbMATHGoogle Scholar
  41. [41]
    J. Lu, C. Ding, J. Lou, and J. Cao, “Outer synchronization of partially coupled dynamical networks via pinning impulsive controllers,” Journal of the Franklin Institute, vol. 352, pp. 5024–5041, 2015.MathSciNetzbMATHGoogle Scholar
  42. [42]
    J. Lu, D. W. C. Ho, J. Cao, and J. Kurths, “Single impulsive controller for globally exponential synchronization of dynamical networks,” Nonlinear Analysis Real World Applications, vol. 14, no. 1, pp. 581–593, 2013.MathSciNetzbMATHGoogle Scholar
  43. [43]
    J. Lu, Z. Wang, J. Cao, D. Ho, and J. Kurths, “Pinning impulsive stabilization of nonlinear dynamical networks with time-varying delay,” International Journal of Bifurcation and Chaos, vol. 22, no. 7, p. 1250176, 2012.Google Scholar
  44. [44]
    T. H. Lee, J. H. Park, D. H. Ji, O. M. Kwon, and S.-M. Lee, “Guaranteed cost synchronization of a complex dynamical network via dynamic feedback control,” Applied Mathematics and Computation, vol. 218, no. 11, pp. 6469–6481, 2012.MathSciNetzbMATHGoogle Scholar
  45. [45]
    N. Li and J. Cao, “Intermittent control on switched networks via w-matrix measure method,” Nonlinear Dynamics, vol. 77, no. 4, pp. 1363–1375, 2014.MathSciNetzbMATHGoogle Scholar
  46. [46]
    B. Niu, C. K. Ahn, H. Li, and M. Liu, “Adaptive control for stochastic switched nonlower triangular nonlinear systems and its application to a one-link manipulator,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2017. DOI: 10.1109/TSMC.2017.2685638Google Scholar
  47. [47]
    S. Tong, Y. Li, and S. Sui, “Adaptive fuzzy tracking control design for SISO uncertain nonstrict feedback nonlinear systems,” IEEE Transactions on Fuzzy Systems, vol. 24, no. 6, pp. 1441–1454, 2016.Google Scholar
  48. [48]
    S. Liang, R. Wu, and L. Chen, “Adaptive pinning synchronization in fractional-order uncertain complex dynamical networks with delay,” Physica A: Statistical Mechanics and its Applications, vol. 444, pp. 49–62, 2016.MathSciNetzbMATHGoogle Scholar
  49. [49]
    Y. Li, S. Sui, and S. Tong, “Adaptive fuzzy control design for stochastic nonlinear switched systems with arbitrary switchings and unmodeled dynamics,” IEEE Transactions on Cybernetics, vol. 47, no. 2, pp. 403–414, 2017.Google Scholar
  50. [50]
    S. Bououden, M. Chadli, L. Zhang, and T. Yang, “Constrained model predictive control for time-varying delay systems: application to an active car suspension,” International Journal of Control, Automation and Systems, vol. 14, no. 1, pp. 51–58, 2016.Google Scholar
  51. [51]
    A. Chibani, M. Chadli, and N. B. Braiek, “A sum of squares approach for polynomial fuzzy observer design for polynomial fuzzy systems with unknown inputs,” International Journal of Control, Automation and Systems, vol. 14, no. 1, pp. 323–330, 2016.zbMATHGoogle Scholar
  52. [52]
    S. Marir, M. Chadli, and D. Bouagada, “A novel approach of admissibility for singular linear continuous-time fractional-order systems,” International Journal of Control, Automation and Systems, vol. 15, no. 2, pp. 959–964, 2017.zbMATHGoogle Scholar
  53. [53]
    Z. Wu and X. Fu, “Complex projective synchronization in drive-response networks coupled with complex-variable chaotic systems,” Nonlinear Dynamics, vol. 72, no. 1–2, pp. 9–15, 2013.MathSciNetzbMATHGoogle Scholar
  54. [54]
    C. Cai, Z. Wang, J. Xu, X. Liu, and F. E. Alsaadi, “An integrated approach to global synchronization and state estimation for nonlinear singularly perturbed complex networks,” IEEE Transactions on Cybernetics, vol. 45, no. 8, pp. 1597–1609, 2015.Google Scholar
  55. [55]
    S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, 1994.zbMATHGoogle Scholar

Copyright information

© Institute of Control, Robotics and Systems and The Korean Institute of Electrical Engineers and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Electrical EngineeringTafresh UniversityTafreshIran
  2. 2.School of MathematicsSoutheast UniversityNanjingChina
  3. 3.School of Electrical EngineeringNantong UniversityNantongChina
  4. 4.School of Mathematical SciencesShandong Normal UniversityJi’nanChina

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