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Complexity analysis of network-based dynamical systems

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Abstract

This paper investigates the nonlinear dynamics of network-based dynamical systems where network communication channels of finite data rates are inserted into the closed loops of the control systems. The authors analyze the bifurcation and chaotic behavior of the non-smooth dynamical systems. The authors first prove that for almost all system parameters there are no periodic orbits. This result distinguishes this type of non-smooth dynamical systems from many others exhibiting border-collision bifurcations. Next, the authors show analytically that the chaotic sets are separated from the region containing the line segment of all fixed points with a finite distance. Finally, the authors employ a simple model to highlight that both the number of clients sharing a common network channel and fluctuations in the available network bandwidth have significant influence on the performance of such dynamical systems.

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References

  1. P. Octanez, J. Monyne, and D. Tilbury, Using deadbands to reduce communication in networked control systems, Proc. American Control Conference, 2002.

  2. G. Walsh, O. Beldiman, and L. Bushnell, Error encoding algorithms for networked control systems, IEEE Conf. Decision and Control, 1999.

  3. G. Walsh and H. Ye, Scheduling of networked control systems, IEEE Control Systems Magazine, 1002, 21(1): 57–65.

    Article  Google Scholar 

  4. G. Walsh, O. Beldiman, and L. Bushnell, Asymptotic behavior of nonlinear networked control systems, IEEE, Trans. Auto. Contr., 2001, 46(7): 1093–1097.

    Article  MathSciNet  MATH  Google Scholar 

  5. G. Walsh, Y. Hong, and L. Bushnell, Stability analysis of networked control systems, IEEE Trans. Contr. Syst. Techno., 2002, 10(3): 438–446.

    Article  Google Scholar 

  6. G. Walsh, Y. Hong, and L. Bushnell, Error encoding algorithms for networked control systems, Automatica, 2002, 38(2): 261–267.

    Article  MATH  Google Scholar 

  7. J. K. Yook, D. M. Tilbury, and N. R. Soparkar, Trading computation for bandwidth reducing communication indistributed control systems using state estimators, IEEE Trans. Contr. Sys. Techno., 2002, 10(4): 503–518.

    Article  Google Scholar 

  8. G. Zhang and T. Chen, Networked control systems: A perspective from chaos, Int. J. of Bifurcation and Chaos, 2005, 15(10): 3075–3101.

    Article  MATH  Google Scholar 

  9. G. Zhang, X. Chen, and T. Chen, A model predictive control approach to networked systems, The 46th Conference on Decision and Control, New Orleans, December 12–14, 2007.

  10. L. Peterson and B. Davie, Computer Networks: A Systems Approach, 2nd Ed., Morgan Kaufmann Publishers, 2000.

  11. J. Walrand and P. Varaiya, High Performance Communication Networks, Morgan Kaufmann Publishers, 1996.

  12. R. Krtolica, Ü. Özgüner, D. Chan, et al., Stability of linear feedback systems with random communication delays, Int. J. Control, 1994, 59(4): 925–953.

    Article  MATH  Google Scholar 

  13. U. Özgüner, H. Goktag, and H. Chan, Automative suspension control through a computer communication network, IEEE, Conf. Control Applications, 1992.

  14. A. Ray, Performance evaluation of medium access control protocols for distributed digital avionics, ASME Journal of Dynamic Systems, Measurement and Control, 1987, 109(4): 370–377.

    Article  Google Scholar 

  15. J. Sparks, Low-cost technologies for aerospace applications, Microprocessors and Microsystems, 1997, 20(8): 449–454.

    Article  Google Scholar 

  16. A. Malinowshi, T. Booth, S. Grady, et al., Real time control of a robotic manipulator via unreliable internet connection, IEEE Conf. Indus. Elect. Society, IECON’01, 2001.

  17. R. Safaric, K. Jezernik, D. Calkin, et al., Telerobot control via internet, Proc. IEEE Sympo. Indus. Elect., 1999, 1: 298–303.

    Google Scholar 

  18. G. Zhang, G. Chen, T. Chen, et al., Dynamical analysis of a networked control system, Int. J. of Bifurcation and Chaos, 2007, 17(1): 61–83.

    Article  MATH  Google Scholar 

  19. A. Ray, Introduction to networking for integrated controls Systems, IEEE Control Systems Magazine, 1989, 9(1): 76–79.

    Article  Google Scholar 

  20. J. Richard, Time-delay systems: An overview of some recent advances and open problems, Automatica, 2003, 39(10): 1667–1694.

    Article  MathSciNet  MATH  Google Scholar 

  21. D. Nesic and A. R. Teel, Input output stability properties of networked control systems, IEEE Trans. Automat. Contr., 2004, 49(10): 1650–1667.

    Article  MathSciNet  Google Scholar 

  22. G. Zhang, G. Chen, T. Chen, et al., Analysis of a type of nonsmooth dynamical systems, Chaos, Solitons & Fractals, 2006, 30(5): 1153–1164.

    Article  MathSciNet  MATH  Google Scholar 

  23. R. D. Eyres, P. T. Piiroinen, A. R. Champneys, et al., Grazing bifurcations and chaos in the dynamics of a hydraulic damper with relief valves, SIAM J. Applied and Dynamical Systems, 2005, 4(4): 1076–1106.

    Article  MathSciNet  MATH  Google Scholar 

  24. S. Banerjee, J. A. Yorke, and C. Grebogi, Robust chaos, Phys. Rev. Lett., 1998, 80(14): 3049–2052.

    Article  MATH  Google Scholar 

  25. F. Casas, W. Chin, C. Grebogi, et al., Universal grazing bifurcations in impact oscillators, Phys. Rev. E, 1996, 53(1): 134–139.

    Article  MathSciNet  Google Scholar 

  26. W. Chin, E. Ott, H. Nusse, et al., Grazing bifurcations in impact oscillators, Phys. Rev. E, 1994, 50(6): 4427–4444.

    Article  MathSciNet  Google Scholar 

  27. W. Chin, E. Ott, H. Nusse, et al., Universal behavior of impact oscillators near grazing-incidence, Phys. Lett. A, 1995, 201(2–3): 197–204.

    Article  MathSciNet  MATH  Google Scholar 

  28. M. di Bernardo, C. Budd, and A. Champneys, Grazing and border-collision in piecewise-smooth systems: A unified analytical framework, Phys. Rev. Lett., 2001, 86: 2553–2556.

    Article  Google Scholar 

  29. M. di Bernardo, C. Budd, and A. Champneys, Corner collision implies border-collision bifurcation, Physica D, 2001, 154(3–4): 171–194.

    Article  MathSciNet  MATH  Google Scholar 

  30. P. Kowalczyk, Robust chaos and border-collision bifurcations in non-invertible piecewise linear maps, Nonlinearity, 2005, 18(2): 485–504.

    Article  MathSciNet  MATH  Google Scholar 

  31. H. Nusse, E. Ott, and J. A. Yorke, Border-collision bifurcations: An explanation for observed bifurcation phenomena, Phys. Rev. E, 1994, 49(2): 1073–1076.

    Article  MathSciNet  Google Scholar 

  32. H. Nusse, E. Ott, and J. A. Yorke, Border-collision bifurcations including “period two to period three” for piecewise smooth systems, Physica D, 1992, 57(1–2): 39–57.

    Article  MathSciNet  MATH  Google Scholar 

  33. C. Robinson, An Introduction to Dynamical Systems: Continuous and Discrete, Pearson Prentice Hall, 2004.

  34. I. Sushko, L. Gardini, and T. Puu, Tongues of periodicity in a family of two-dimensional discontinuous maps of real Mobius type, Chaos, Solitons & Fractals, 2004, 21(2): 403–412.

    Article  MathSciNet  MATH  Google Scholar 

  35. G. Zhang, Y. Shi, H. Xu, et al., A further study of a nonsmooth dynamical economic model, The 4th Asia-pacific Workshop on Chaos Control and Synchronization, August 24–26, Harbin, China, 2007.

  36. G. Goodwin, H. Haimovich, D. Quevedo, et al., A moving horizon approach to networked control system design, IEEE Trans. Auto. Contr., 2004, 49(9): 1427–1445.

    Article  MathSciNet  Google Scholar 

  37. W. Wong and R. Brockett, Systems with finite communication bandwidth constraints, part II: Stabilization with limited information feedback, IEEE Trans. Automat. Contr., 1999, 44(5): 1049–1053.

    Article  MathSciNet  MATH  Google Scholar 

  38. S. Tatikonda and S. Mitter, Control under communication constraints, IEEE Trans. Auto. Contr., 2004, 49(7): 1056–1068.

    Article  MathSciNet  Google Scholar 

  39. N. Elia, When Bode meets Shannon: Control-oriented feedback communication schemes, IEEE Trans. Auto. Contr., 2004, 49(9): 1477–1488.

    Article  MathSciNet  Google Scholar 

  40. G. Zhang, X. Chen, and T. Chen, A mixed-integer programming approach to networked control systems, International Journal of Numerical Analysis and Modeling, 2008, 5(4): 590–611.

    MathSciNet  MATH  Google Scholar 

  41. F. Lian, J. Moyne, and D. Tilbury, Performance evaluation of control networks, IEEE Control Systems Magazine, 2001, 21(1): 66–83.

    Article  Google Scholar 

  42. F. Lian, J. Moyne, and D. Tilbury, Networked design consideration for distributed control systems, IEEE Trans. Contr. Syst. Techno., 2002, 10(2): 297–307.

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu, 610054, China

    Guofeng Zhang

  2. Intelligent Control Laboratory, Center for Systems and Control, College of Engineering, Peking University, Beijing, 100871, China

    Long Wang

  3. Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Alberta, T6G 2V 4, Canada

    Tongwen Chen

Authors
  1. Guofeng Zhang
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  2. Long Wang
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  3. Tongwen Chen
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Corresponding author

Correspondence to Guofeng Zhang.

Additional information

This research of the first author is partially supported by an the National Natural Science Foundation of China under Grant No. 60804015, and an NSERC grant to the third author.

This paper was recommended for publication by Editor Jing HAN.

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Zhang, G., Wang, L. & Chen, T. Complexity analysis of network-based dynamical systems. J Syst Sci Complex 24, 413–432 (2011). https://doi.org/10.1007/s11424-011-8173-6

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  • DOI: https://doi.org/10.1007/s11424-011-8173-6

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