Journal of Electrical Engineering and Technology

The Korean Institute of Electrical Engineers (대한전기학회)
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A Nonlinear Synchronization Scheme for Hindmarsh-Rose Models

  • Kim, Jung-Su (Dept. of Control and Instrumentation Engineering, Seoul National University of Technology) ;
  • Allgower, Frank (Institute for Systems Theory and Automatic Control, University of Stuttgart)
  • Published : 2010.03.01
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Abstract

Multiple subsystems are required to behave synchronously or cooperatively in many areas. For example, synchronous behaviors are common in networks of (electro-) mechanical systems, cell biology, coupled neurons, and cooperating robots. This paper presents a feedback scheme for synchronization between Hindmarsh-Rose models which have polynomial vector fields. We show that the problem is equivalent to finding an asymptotically stabilizing control for error dynamics which is also a polynomial system. Then, an extension to a nonlinear observer-based scheme is presented, which reduces the amount of information exchange between models.

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References

  1. N. Motee, A. Jadbabaie and M. Barahona, "On the stability of the Kuramoto model of coupled nonlinear oscillators," Proceedings of American Control Conference, pp.4296-4301, 2004.
  2. M. K. Ali and J.-Q. Fangnon, "Synchronization of chaos and hyperchaos using linear and nonlinear feedback functions," Physical Review E, 55(5):5285-5290, 1997. https://doi.org/10.1103/PhysRevE.55.5285
  3. Murat Arcak and Petar Kokotovic, "Nonlinear observers: A circle criterion design and robustness analysis," Automatica, 37(12):1923-1930, 2001. https://doi.org/10.1016/S0005-1098(01)00160-1
  4. Vladimir N. Belykh, Grigory V. Osipov, Nina Kucklanderc, Bernd Blasiusc and Jurgen Kurths, "Automatic control of phase synchronization in coupled complex oscillators," Physica D, 200:81-104, 2005. https://doi.org/10.1016/j.physd.2004.10.008
  5. I. I. Blekhman, A. L. Fradkov, H. Nijmeijer and A. Y. Pogromsky, "On self-synchronization and controlled synchronization," Systems and Control Letters, 31:299-305, 1997. https://doi.org/10.1016/S0167-6911(97)00047-9
  6. S. Boyd, L. E. Ghaoui, E. Feron and V. Balakrishnan, "Linear Matrix Inequalities in System and Control Theory, SIAM, 1994.
  7. J. Buck, "Synchronous rhythmic flashing of fireies II," Quarterly Review of Biology, 63:265, 1988. https://doi.org/10.1086/415929
  8. N. Chopra and M. W. Spong, "On synchronization of Kuramoto oscillators" Proceedings of IEEE Control and Decision Conference, pp.3916-3922, 2005.
  9. Christian Ebenbauer, Polynomial Control Systems: Analysis and Design via Dissipation Inequalities and Sum of Squares, PhD thesis, University of Stuttgart, Germany, 2005.
  10. Christian Ebenbauer and Frank Allgower "Analysis and design of polynomial control systems using dissipation inequalities and sum of squares," Computers and Chemical Engineering, 30:1601-1614, 2006.
  11. Rong He and P. G. Vaidya, "Implementation of chaotic cryptography with chaotic synchronization," PHYSICAL REVIEW E, 57(2):1532-1535, 1998. https://doi.org/10.1103/PhysRevE.57.1532
  12. J. L. Hindmarsh and R. M. Rose, "A model of the nerve impulse using two first order differential equations," Nature, 296:162-164, 1982. https://doi.org/10.1038/296162a0
  13. J. L. Hindmarsh and R. M. Rose, "A model of neuronal bursting using three coupled first order differential equations," Proc. Roy. Soc. Lond. B, 221:87-102, 1984. https://doi.org/10.1098/rspb.1984.0024
  14. Debin Huang, "Simple adaptive-feedback controller for identical chaos synchronization," Physical Review E, 71:37203, 2005. https://doi.org/10.1103/PhysRevE.71.037203
  15. Alberto Isidori, Nonlinear Control Systems, 3rd Ed., Springer, 1995.
  16. A. Jadbabaie, J. Lin and A. S. Morse, "Coordination of groups of mobile autonomous agents using nearest neighbor rules," IEEE Trans. on Automatic Control, 48(6):988-1001, 2003. https://doi.org/10.1109/TAC.2003.812781
  17. Junan Lu, Xiaoqun Wu, Xiuping Han, and Jinhu Lu, "Adaptive feedback synchronization of a unified chaotic system," Physics Letters A, 329:327.333, 2004.
  18. Jung-Su Kim and F. Allgower, Analysis and Design of Nonlinear Control Systems. In Honor of Alberto Isidori, chapter Nonlinear synchronization of coupled oscillators: The polynomial case, pp.361-371, Springer, 2007.
  19. Jung-Su Kim and Frank Allgower, "A nonlinear synchronization scheme for polynomial systems" In the Proceedings of American Control Conference, 2007.
  20. Min-Chan Kim, Seung-Kyu Park, Tae-Won Kim and Ho-Gyun Ahn, "A study on the design of adaptive H1 controllerpolynomial approach," KIEE, 51(4):129-136, 2002.
  21. Seunghwan Kim, Hyungtae Kook, Sang Gui Lee and Myung-Han Park, "Synchronization and clustering in a network of three globally coupled neural oscillations," International Journal of Bifurcation and Chaos, 8(4):731-739, 1998. https://doi.org/10.1142/S0218127498000528
  22. Arthur J. Krener and Witold Respondek, "Nonlinear observers with linearizable error dynamics," SIAM J. Control and Optimization, 23(2):197-216, 1985. https://doi.org/10.1137/0323016
  23. Jinhu Lu and Junan Lu, "Controlling uncertain Lu system using linear feedback, Chaos Solutions and Fractals, 17:127-133, 2003. https://doi.org/10.1016/S0960-0779(02)00456-3
  24. Deepak Mishra, Abhishek Yadav, Sudipta Ray and Prem K. Kalra, "Controlling synchronization of modified Fitzhugh-Nagumo neurons under external electrical stimulation," NeuroQuantology, 4(1):50-67, 2006.
  25. T.C. Newell, P.M. Alsing, A. Gavrielides and V. Kovanis, "Synchronization of chaos using proportional feedback," Physical Review E, 49(1):313-318, 1994. https://doi.org/10.1103/PhysRevE.49.313
  26. H. Nijmeijer, "A dynamical control view on synchronization," Physica D, 154:219-228, 2001. https://doi.org/10.1016/S0167-2789(01)00251-2
  27. R. Olfati-Saber, "Flocking for multi-agent dynamic systems: Algorithms and theory," IEEE Trans. on Automatic Control, 51(3):401-420, 2006. https://doi.org/10.1109/TAC.2005.864190
  28. Ward T. Oud and Ivan Tyukin, "Sufficient conditions for synchronization in an ensemble of Hindmarsh and Rose neurons: Passivity-based approach," In IFAC Symposium on Nonlinear Control Systems, 2004.
  29. P. A. Parrilo, Structured Semidefnite Programs and Semialgebraic Geometry Methods in Robustness and Optimization. PhD thesis, California Institute of Technology, 2000.
  30. F. Pasemann, "Synchronous and asynchronous chaos in coupled neuromodules," International Journal of Bifurcation and Chaos, 9:1957-1968, 1999. https://doi.org/10.1142/S0218127499001425
  31. R. D. Pinto, P. Varona, A. R. Volkovskii, A. Szucs, Henry D. I. Abarbanel and M. I. Rabinovich, Synchronous behavior of two coupled electronic neurons," Physical Review E, 62(2):2644-2656, 2000. https://doi.org/10.1103/PhysRevE.62.2644
  32. Alexander Pogromsky, "Observer-based robust synchronization of dynamical systems," International Journal of Bifurcation and Chaos, 8(11):2243-2254, 1998. https://doi.org/10.1142/S0218127498001832
  33. Alexander Pogromsky and Henk Nijmeijer, "Cooperative oscillatory behavior of mutually coupled dynamical systems," IEEE Trans. on Circuit and Systems-I:Fundamental Theory and Applications, 48(2):152-162, 2001. https://doi.org/10.1109/81.904879
  34. S. Prajna, A. Papachristodoulou, P. Seiler and P.A. Parrilo, "SOSTOOLS and its control applications," In Positive Polynomials in Control, Lecture Notes in Control and Information Sciences, Springer, 312: 273-292, 2005. https://doi.org/10.1007/10997703_14
  35. Alejandro Rodriguez-Angeles and Henk Nijmeijer, "Mutual synchronization of robots via estimated state feedback: a cooperative approach," IEEE Trans. on Control Systems Technology, 12(4):542-534, 2004. https://doi.org/10.1109/TCST.2004.825065
  36. E.D. Sontag, "Comments on integral variants of ISS," Systems & Control Letters, 34:93-100, 1998. https://doi.org/10.1016/S0167-6911(98)00003-6
  37. G.-B. Stan and R. Sepulchre, "Analysis of interconnected oscillators by dissipativity theory," IEEE Trans. on Automatic Control, 52:256-270, 2007. https://doi.org/10.1109/TAC.2006.890471
  38. Steven H. Strogatz, "From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators," Physica D, 143:1-20, 2000. https://doi.org/10.1016/S0167-2789(00)00094-4
  39. W. Wang and J.J.E Slotine, "On partial contraction analysis for coupled nonlinear oscillators," Biological Cybernetics, 92(1), 2005.
  40. W. Wang and J.J.E Slotine, "A theoretical study of different leader roles in networks," IEEE Trans. on Automatic Control, 51(7):1156-1161, 2006. https://doi.org/10.1109/TAC.2006.878754
  41. Yanwu Wang, Zhi-Hong Guan and Hua O. Wang, "Feedback and adaptive control for the synchronization of Chen system via a single variable," Physics Letters A, 312:34-40, 2003. https://doi.org/10.1016/S0375-9601(03)00573-5
  42. Peter Wieland, Jung-Su Kim, Holger Scheu and Frank Allgower, "On consensus in multi-agent systems with linear highorder agents," In Proc. 17th IFAC World Congress, pp. 1541-1546, 2008.
  43. S.S. Yang and C. K. Duan, "A mathematical theory of synchronization of self-feedback," Journal of Physics A: Mathematical and General, 30:3273-3278, 1997. https://doi.org/10.1088/0305-4470/30/9/032
  44. Massahiko Yoshika, "Chaos synchronization in gapjunction-coupled neurons," Physical Review E, 71: 065203(R), 2005. https://doi.org/10.1103/PhysRevE.71.065203

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