Pinning control and controllability of complex dynamical networks

  • Guanrong ChenEmail author


In this article, the notion of pinning control for directed networks of dynamical systems is introduced, where the nodes could be either single-input single-output (SISO) or multi-input multi-output (MIMO) dynamical systems, and could be non-identical and nonlinear in general but will be specified to be identical linear time-invariant (LTI) systems here in the study of network controllability. Both state and structural controllability problems will be discussed, illustrating how the network topology, node-system dynamics, external control inputs and inner dynamical interactions altogether affect the controllability of a general complex network of LTI systems, with necessary and sufficient conditions presented for both SISO and MIMO settings. To that end, the controllability of a special temporally switching directed network of linear time-varying (LTV) node systems will be addressed, leaving some more general networks and challenging issues to the end for research outlook.


Complex network pinning control controllability linear time-invariant (LTI) system temporally switching network graph theory 


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The author thanks Mario di Bernardo, Jian-Xi Gao, Bao- Yu Hou, Xiang Li, Yang-Yu Liu, LinWang, Xiao-FanWang, Lin-Ying Xiang and Gang Yan for their valuable comments and discussions.


  1. [1]
    D. J. Watts, S. H. Strogatz. Collective dynamics of ‘smallworld’ networks. Nature, vol. 393, no. 6684, pp. 440–442, 1998.CrossRefGoogle Scholar
  2. [2]
    A. L. Barabási, R. Albert. Emergence of scaling in random networks. Science, vol. 286, no. 5439, pp. 509–512, 1999.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    P. Erdos, A. Rényi. On the evolution of random graphs. Publication of the Mathematical Institute of the Hungarian Academy Sciences, vol. 5, pp. 17–60, 1960.MathSciNetzbMATHGoogle Scholar
  4. [4]
    C. K. Chui, G. R. Chen. Linear Systems and Optimal Control, New York, USA: Springer-Verlag, 1989.CrossRefzbMATHGoogle Scholar
  5. [5]
    G. R. Chen, Z. S. Duan. Network synchronizability analysis: A graph-theoretic approach. Chaos, vol. 18, Article number 037102, 2008.zbMATHGoogle Scholar
  6. [6]
    X. F. Wang, G. R. Chen. Pinning control of scale-free dynamical networks. Physica A: Statistical Mechanics and its Applications, vol. 310, no. 3–4, pp. 521–531, 2002.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    X. Li, X. F. Wang, G. R. Chen. Pinning a complex dynamical network to its equilibrium. IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 51, no. 10, pp. 2074–2087, 2004.MathSciNetCrossRefGoogle Scholar
  8. [8]
    G. R. Chen, X. F. Wang, X. Li. Introduction to Complex Networks: Models, Structures and Dynamics, 2nd ed., Beijing, China: Higher Education Press.Google Scholar
  9. [9]
    A. Cho. Scientific link-up yields ‘control panel’ for networks. Science, vol. 332, no. 6031, pp. 777, 2011.CrossRefGoogle Scholar
  10. [10]
    I. D. Couzin, J. Krause, N. R. Franks, S. A. Levin. Effective leadership and decision-making in animal groups on the move. Nature, vol. 433, no. 7025, pp. 513–516, 2005.CrossRefGoogle Scholar
  11. [11]
    G. R. Chen. Pinning control and synchronization on complex dynamical networks. International Journal of Control, Automation and Systems, vol. 12, no. 2, pp. 221–230, 2014.CrossRefGoogle Scholar
  12. [12]
    X. F.Wang, H. S. Su. Pinning control of complex networked systems: A decade after and beyond. Annual Reviews in Control, vol. 38, no. 1, pp. 103–111, 2014.CrossRefGoogle Scholar
  13. [13]
    F. F. Li. Pinning control design for the stabilization of Boolean networks. IEEE Transactions on Neural Networks and Learning Systems, vol. 27, no. 7, pp. 1585–1590, 2016.MathSciNetCrossRefGoogle Scholar
  14. [14]
    Y. Tang, H. J. Gao, J. Kurths, J. A. Fang. Evolutionary pinning control and its application in UAV coordination. IEEE Transactions on Industrial Informatics, vol. 8, no. 4, pp. 828–838, 2012.CrossRefGoogle Scholar
  15. [15]
    C. T. Lin. Structural controllability. IEEE Transactions on Automatic Control, vol. 19, no. 3, pp. 201–208, 1974.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    J. L. Willems. Structural controllability and observability. Systems & Control Letters, vol. 8, no. 1, pp. 5–12, 1986.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    H. Mayeda, T. Yamada. Strong structural controllability. SIAM Journal on Control and Optimization, vol. 17, no. 1, pp. 123–138, 1979.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    Y. Y. Liu, A. L. Barabási. Control principles of complex networks, [Online], Available:, 2015.Google Scholar
  19. [19]
    G. Yan, J. Ren, Y. C. Lai, C. H. Lai, B. W. Li. Controlling complex networks: How much energy is needed?. Physical Review Letters, vol. 108, no. 21, Article number 218703, 2012.CrossRefGoogle Scholar
  20. [20]
    W. X. Wang, X. Ni, Y. C. Lai, C. Grebogi. Optimizing controllability of complex networks by minimum structural perturbations. Physical Review E, vol. 85, no. 2, Article number 026115, 2012.CrossRefGoogle Scholar
  21. [21]
    T. Nepusz, T. Vicsek. Controlling edge dynamics in complex networks. Nature Physics, vol. 8, no. 7, pp. 568–573, 2012.CrossRefGoogle Scholar
  22. [22]
    T. Jia, Y. Y. Liu, E. Csóka, M. Pósfai, J. J. Slotine, A. L. Barabási. Emergence of bimodality in controlling complex networks. Nature Communications, vol. 4, Article number 2002, 2013.CrossRefGoogle Scholar
  23. [23]
    T. Jia, A. L. Barabási. Control capacity and a random sampling method in exploring controllability of complex networks. Scientific Reports, vol. 3, Article number 2354, 2013.Google Scholar
  24. [24]
    Z. Z. Yuan, C. Zhao, Z. R. Di, W. X. Wang, Y. C. Lai. Exact controllability of complex networks. Nature Communications, vol. 4, Article number 2447, 2013.Google Scholar
  25. [25]
    G. Menichetti, L. Dall’Asta, G. Bianconi. Network controllability is determined by the density of low in-degree and out-degree nodes. Physical Review Letters, vol. 113, no. 7, Article number 078701, 2014.CrossRefGoogle Scholar
  26. [26]
    J. X. Gao, Y. Y. Liu, R. M. D’Souza, A. L. Barabási. Target control of complex networks. Nature Communications, vol. 5, Article number 5415, 2014.CrossRefGoogle Scholar
  27. [27]
    A. E. Motter. Networkcontrology. Chaos, vol. 25, no. 9, Article number 097621, 2015.MathSciNetCrossRefGoogle Scholar
  28. [28]
    G. Yan, G. Tsekenis, B. Barzel, J. J. Slotine, Y. Y. Liu, A. L. Barabási. Spectrum of controlling and observing complex networks. Nature Physics, vol. 11, no. 9, pp. 779–786, 2015.CrossRefGoogle Scholar
  29. [29]
    A. J. Gates, L. M. Rocha. Control of complex networks requires both structure and dynamics. Scientific Reports, vol. 6, Article number 24456, 2016.CrossRefGoogle Scholar
  30. [30]
    T. H. Summers, F. L. Cortesi and J. Lygeros. On submodularity and controllability in complex dynamical networks. IEEE Transactions on Control of Network Systems, vol. 3, no. 1, pp. 91–101, 2016.MathSciNetCrossRefGoogle Scholar
  31. [31]
    B. Das, B. Subudhi, B. B. Pati. Cooperative formation control of autonomous underwater vehicles: An overview. International Journal of Automation and Computing, vol. 13, no. 3, pp. 199–225, 2016.CrossRefGoogle Scholar
  32. [32]
    F. Sorrentino, M. di Bernardo, F. Garofalo, G. R. Chen. Controllability of complex networks via pinning. Physical Review E, vol. 75, no. 4, Article number 046103, 2007.CrossRefGoogle Scholar
  33. [33]
    L. M. Pecora, T. L. Carroll. Master stability functions for synchronized coupled systems. Physical Review Letters, vol. 80, no. 10, pp. 2109–2112, 1998.CrossRefGoogle Scholar
  34. [34]
    M. Porfiri, M. di Bernardo. Criteria for global pinningcontrollability of complex networks. Automatica, vol. 44, no. 12, pp. 3100–3106, 2008.MathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    Y. L. Zou, G. R. Chen. Pinning controllability of asymmetrical weighted scale-free networks. Europhysics Letters, vol. 84, no. 5, Article number 58005, 2008.CrossRefGoogle Scholar
  36. [36]
    L. Y. Xiang, F. Chen, G. R. Chen. Pinning synchronization of networked multi-agent systems: Spectral analysis. Control Theory and Technology, vol. 13, no. 1, pp. 45–54, 2015.MathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    L. Lováz, M. D. Plummer. Matching Theory, New York: Elsevier, 1986.Google Scholar
  38. [38]
    Y. Y. Liu, J. J. Slotine, A. L. Barabási. Controllability of complex networks. Nature, vol. 473, no. 7346, pp. 167–173, 2011.CrossRefGoogle Scholar
  39. [39]
    L. Wang, X. F. Wang, G. Chen. Controllability of networked higher-dimensional systems with one-dimensional communication channels. Philosophical Transactions of the Royal Society A, to be published.Google Scholar
  40. [40]
    R. Shields, J. Pearson. Structural controllability of multiinput linear systems. IEEE Transactions on Automatic Control, vol. 21, no. 2, pp. 203–212, 1976.MathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    J. M. Dion, C. Commaulta, J. van der Woude. Generic properties and control of linear structured systems: A survey. Automatica, vol. 39, no. 7, pp. 1125–1144, 2003.MathSciNetCrossRefzbMATHGoogle Scholar
  42. [42]
    A. Lombardi, M. Hornquist. Controllability analysis of networks. Physical Review E, vol. 75, no. 5, Article number 056110, 2007.CrossRefGoogle Scholar
  43. [43]
    C. T. Lin. System structure and minimal structure controllability. IEEE Transactions on Automatic Control, vol. 22, no. 5, pp. 855–862, 1977.MathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    J. C. Jarczyk, F. Svaricek, B. Alt. Strong structural controllability of linear systems revisited. In Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference, IEEE, Orlando, USA, pp. 1213–1218, 2011.CrossRefGoogle Scholar
  45. [45]
    A. Chapman. Strong structural controllability of networked dynamics. Semi-Autonomous Networks, A. Chapman, Ed., New York: Springer, pp. 135–150, 2015.Google Scholar
  46. [46]
    H. G. Tanner. On the controllability of nearest neighbor interconnections. In Proceedings of the 43rd IEEE Conference on Decision and Control, IEEE, Nassau, Bahamas, 2004, vol. 3, pp. 2467–2472.Google Scholar
  47. [47]
    L. Y. Xiang, J. J. H. Zhu, F. Chen, G. R. Chen. Controllability of weighted and directed networks with nonidentical node dynamics. Mathematical Problems in Engineering, vol. 2013, Article number 405034, 2013.MathSciNetCrossRefzbMATHGoogle Scholar
  48. [48]
    T. Zhou. On the controllability and observability of networked dynamic systems. Automatica, vol. 52, pp. 63–75, 2015.MathSciNetCrossRefzbMATHGoogle Scholar
  49. [49]
    L. Wang, G. R. Chen, X. F. Wang, W. K. S. Tang. Controllability of networked MIMO systems. Automatica, vol. 69, pp. 405–409, 2016.MathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    L. Wang, G. R. Chen, X. F. Wang, W. K. S. Tang. Controllability of networked MIMO systems, [Online], Available:, 2015.zbMATHGoogle Scholar
  51. [51]
    B. Liu, T. G. Chu, L. Wang, G. M. Xie. Controllability of a leader-follower dynamic network with switching topology. IEEE Transactions on Automatic Control, vol. 53, no. 4, pp. 1009–1013, 2008.MathSciNetCrossRefGoogle Scholar
  52. [52]
    X. M. Liu, H. Lin, B. M. Chen. Graph-theoretic characterisations of structural controllability for multi-agent system with switching topology. International Journal of Control, vol. 86, no. 2, pp. 222–231, 2013.MathSciNetCrossRefzbMATHGoogle Scholar
  53. [53]
    X. M. Liu, H. Lin, B. M. Chen. Structural controllability of switched linear systems. Automatica, vol. 49, no. 12, pp. 3531–3537, 2013.MathSciNetCrossRefzbMATHGoogle Scholar
  54. [54]
    P. Holme, J. Saramäki. Temporal networks. Physics Reports, vol. 519, no. 3, pp. 97–125, 2012.CrossRefGoogle Scholar
  55. [55]
    X. Li, P. Yao, Y. J. Pan. Towards structural controllability of temporal complex networks. In Complex Systems and Networks: Dynamics, Controls and Applications, J. H. Lü, X. H. Yu, G. R. Chen, W. W. Yu, Eds., Berlin Heidelberg: Springer, pp. 341–371, 2015.Google Scholar
  56. [56]
    M. Pósfai, P. Hövel. Structural controllability of temporal networks. New Journal of Physics, vol. 16, no. 12, Article number 123055, 2014.MathSciNetCrossRefGoogle Scholar
  57. [57]
    G. Reissig, C. Hartung, F. Svaricek. Strong structural controllability and observability of linear time-varying systems. IEEE Transactions on Automatic Control, vol. 59, no. 11, pp. 3087–3092, 2014.MathSciNetCrossRefGoogle Scholar
  58. [58]
    Y. J. Pan, X. Li. Structural controllability and controlling centrality of temporal networks. PLoS One, vol. 9, no. 4, Article number 0094998, 2014.CrossRefGoogle Scholar
  59. [59]
    L. M. Silverman, H. E. Meadows. Controllability and observability in time-variable linear systems. SIAM Journal on Control, vol. 5, no. 1, pp. 64–73, 1967.MathSciNetCrossRefzbMATHGoogle Scholar
  60. [60]
    B. Y. Hou, X. Li, G. R. Chen. Structural controllability of temporally switching networks. IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 63, no. 10, pp. 1771–1781, 2016.MathSciNetCrossRefGoogle Scholar
  61. [61]
    Y. Y. Liu, J. J. Slotine, A. L. Barabási. Observability of complex systems. Proceedings of the National Academy of Sciences of the United States of America, vol. 110, no. 7, pp. 2460–2465, 2013.MathSciNetCrossRefzbMATHGoogle Scholar
  62. [62]
    B. B. Wang, L. Gao, Y. Gao, Y. Deng, Y. Wang. Controllability and observability analysis for vertex domination centrality in directed networks. Scientific Reports, vol. 4, Article number 5399, 2014.Google Scholar
  63. [63]
    A. M. Li, S. P. Cornelius, Y. Y. Liu, L. Wang, A. L. Barabási. The fundamental advantages of temporal networks, [Online], Available:, 2016.Google Scholar
  64. [64]
    S. Ghosh, J. Ruths. Structural control of single-input rank one bilinear systems. Automatica, vol. 64, pp. 8–17, 2016.MathSciNetCrossRefzbMATHGoogle Scholar
  65. [65]
    A. J. Gates, L. M. Rocha. Control of complex networks requires both structure and dynamics. Scientific Reports, vol. 6, Article number 24456, 2016.CrossRefGoogle Scholar

Copyright information

© Institute of Automation, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of Electronic EngineeringCity University of Hong KongHong KongChina

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