Kybernetika 54 no. 5, 1011-1032, 2018

Sliding-mode pinning control of complex networks

Oscar J. Suarez, Carlos J. Vega, Santiago Elvira-Ceja, Edgar N. Sanchez and David I. RodriguezDOI: 10.14736/kyb-2018-5-1011

Abstract:

In this paper, a novel approach for controlling complex networks is proposed; it applies sliding-mode pinning control for a complex network to achieve trajectory tracking. This control strategy does not require the network to have the same coupling strength on all edges; and for pinned nodes, the ones with the highest degree are selected. The illustrative example is composed of a network of 50 nodes; each node dynamics is a Chen chaotic attractor. Two cases are presented. For the first case the whole network tracks a reference for each one of the states; afterwards, the second case uses the backstepping technique to track a desired trajectory for only one state. Tracking performance and dynamical behavior of the controlled network are illustrated via simulations.

Keywords:

sliding mode, backstepping, pinning control, complex network, trajectory tracking

Classification:

05C82, 93D05, 93C10

paper.pdf

References:

  1. A. L. Barabási and R. Albert: Emergence of scaling in random networks. Science 286 (1999), 5439, 509-512.   DOI:10.1126/science.286.5439.509
  2. S. P. Bhat and D. U. Bernstein: Finite-time stability of continuous autonomous systems. SIAM J. Control Optim. 38 (2000), 3, 751-766.   DOI:10.1137/s0363012997321358
  3. S. Boccaletti, V. Latora, Y. Moreno, M. Chavez and D. U. Hwang: Complex networks: Structure and dynamics. Phys. Rep. 424 (2006), 4, 175-308.   DOI:10.1016/j.physrep.2005.10.009
  4. G. Chen and T. Ueta: Yet another chaotic attractor. Int. J. Bifurcation Chaos 9 (1999), 7, 1465-1466.   CrossRef
  5. G. Chen, X. Wang and X. Li: Fundamentals of Complex Networks: Models, Structures and Dynamics. John Wiley and Sons, Singapore 2014.   CrossRef
  6. T. Chen, X. Liu and W. Lu: Pinning complex networks by a single controller. IEEE Trans. Circuits Systems I: Regular Papers 54 (2007), 6, 1317-1326   CrossRef
  7. S. Drakunov, D. Izosimov, A. Lukyanov, V. A. Utkin and V. I. Utkin: The block control principle. 1. Automat. Remote Control 51 (1990), 5, 601-608.   CrossRef
  8. S. V. Emelyanov: Variable Structure Control Systems. Nouka, Moscow 1967.   CrossRef
  9. P. Erdos and A. Rényi: On the evolution of random graphs. Inst. Math. Hungar. Acad. Sci. 5 (1960), 1, 17-60.   CrossRef
  10. G. Hu and Z. Qu: Controlling spatiotemporal chaos in coupled map lattice systems. Phys. Rev. Lett. 72 (1994), 1, 68.   DOI:10.1103/physrevlett.72.68
  11. J. Guldner and V. I. Utkin: Tracking the gradient of artificial potential fields: Sliding mode control for mobile robots. Int. J. Control 63 (1996), 3, 417-432.   DOI:10.1080/00207179608921850
  12. J. Guldner and V. I. Utkin: The chattering problem in sliding mode systems. In: 14th Intenational Symposium of Mathematical Theory of Networks and Systems, (MTNS), Perpignan 2000.   CrossRef
  13. H. K. Khalil: Noninear Systems. Prentice-Hall, New Jersey 2002.   CrossRef
  14. A. Khanzadeh and M. Pourgholi: Fixed-time sliding mode controller design for synchronization of complex dynamical networks. Nonlinear Dynamics 88 (2017), 4, 2637-2649.   DOI:10.1007/s11071-017-3400-x
  15. M. Krstic, I. Kanellakopoulos and P. V. Kokotovic: Nonlinear and Adaptive Control Design. Wiley, 1995.   CrossRef
  16. H. Lee and V. I. Utkin: Chattering suppression methods in sliding mode control systems. Ann. Rev. Control 31 (2007), 2, 179-188.   DOI:10.1016/j.arcontrol.2007.08.001
  17. X. Li and G. Chen: Synchronization and desynchronization of complex dynamical networks: an engineering viewpoint. IEEE Trans. Circuits and Systems I: Fundamental Theory Appl. 50 (2003), 11, 1381-1390.   DOI:10.1109/tcsi.2003.818611
  18. X. Li, X. Wang and G. Chen: Pinning a complex dynamical network to its equilibrium. IEEE Trans. Circuits Systems I: Regular Papers 51 (2004), 10, 2074-2087.   DOI:10.1109/tcsi.2004.835655
  19. E. Lorenz: Deterministic nonperiodic flow. J. Atmospher. Sci. 20 (1963), 2, 130-141.   DOI:10.1175/1520-0469(1963)020<0130:dnf>2.0.co;2
  20. M. Pourmahmood, S. Khanmohammadi and G. Alizadeh: Finite-time synchronization of two different chaotic systems with unknown parameters via sliding mode technique. Appl. Math. Modell. 35 (2011), 3080-3091.   DOI:10.1016/j.apm.2010.12.020
  21. R. Roy, T. Murphy, T. Maier, Z. Gills and E. Hunt: Dynamical control of a chaotic laser: Experimental stabilization of a globally coupled system. Phys. Rev. Lett. 68 (1992), 9, 1259-1262.   DOI:10.1103/physrevlett.68.1259
  22. E. N. Sanchez and D. I. Rodriguez: Inverse optimal pinning control for complex networks of chaotic systems. Int. Bifurcation Chaos 25, (2015), 02, 1550031.   DOI:10.1142/s0218127415500315
  23. H. Sira-Ramírez: Sliding Mode Control: The Delta-Sigma Modulation Approach. Birkhauser, Basel 2015.   DOI:10.1007/978-3-319-17257-6
  24. J. Sun, Y. Shen, X. Wang and J. Chen: Finite-time combination-combination synchronization of four different chaotic systems with unknown parameters via sliding mode control. Nonlinear Dynamics 76 (2014), 1, 383-397.   DOI:10.1007/s11071-013-1133-z
  25. H. Su and W. Xiaofan: Pinning Control of Complex Networked Systems: Synchronization, Consensus and Flocking of Networked Systems via Pinning. Springer-Verlag, Berlin 2013.   CrossRef
  26. V. I. Utkin and H. Lee: Chattering problem in sliding mode control systems. In: Variable Structure Systems. VSS'06. International Workshop on Variable Structure Systems, (2006), pp. 346-350.   DOI:10.1109/vss.2006.1644542
  27. V. I. Utkin: Sliding Modes and their Application in Variable Structure Systems. Mir Publishers, Moscow 1978.   CrossRef
  28. V. I. Utkin: Sliding Modes in Control and Optimization. Springer-Verlag, Berlin 2013.   CrossRef
  29. D. Watts, J. Duncan and S. Strogatz: Collective dynamics of 'small-world' networks. Nature 393 (1998), 6684, 440-442.   DOI:10.1038/30918
  30. X. Wang and G. Cheng: Pinning control of scale-free dynamical networks. Physica A: Statist. Mech. Appl. 310 (2002), 3, 521-531.   DOI:10.1016/s0378-4371(02)00772-0
  31. S. Vaidyanathan and S. Sampath: Global chaos synchronization of hyperchaotic Lorenz systems by sliding mode control. In: Advances in Digital Image Processing and Information Technology, Springer 2011, pp. 156-164.   DOI:10.1007/978-3-642-24055-3\_16
  32. H. Yau: Design of adaptive sliding mode controller for chaos synchronization with uncertainties. Chaos, Solitons Fractals 22 (2004), 341-347.   DOI:10.1016/j.chaos.2004.02.004
  33. W. Yu, G. Chen and J. Lü: On pinning synchronization of complex dynamical networks. Automatica 45 (2009), 2, 429-435.   DOI:10.1016/j.automatica.2008.07.016
  34. M. Zhang, M. Xu and M. Han: Finite-time combination synchronization of uncertain complex networks by sliding mode control. Inform. Cybernet. Comput. Social Systems (ICCSS), (2017), 406-411.   DOI:10.1109/iccss.2017.8091448