Kybernetika 47 no. 4, 612-629, 2011

Robust control of chaos in modified FitzHugh-Nagumo neuron model under external electrical stimulation based on internal model principle

Yuan Jiang and Jiyang Dai

Abstract:

This paper treats the question of robust control of chaos in modified FitzHugh-Nagumo neuron model under external electrical stimulation based on internal model principle. We first present the solution of the global robust output regulation problem for output feedback system with nonlinear exosystem. Then we show that the robust control problem for the modified FitzHugh-Nagumo neuron model can be formulated as the global robust output regulation problem and the solvability conditions for the output regulation problem for the modified FitzHugh-Nagumo neuron model are all satisfied. Then we apply the obtained output regulation results to the control problem for modified FitzHugh-Nagumo neuron model. Finally, an output feedback control law is designed for the modified FitzHugh-Nagumo neuron model to achieve global stability of the closed-loop system in the presence of uncertain parameters and external stimulus. An example is shown that the proposed algorithm can completely reject the external electrical stimulation generated from a Van der Pol circuit.

Keywords:

Van der Pol circuit, control theory, Lyapunov methods, internal model principle, modified FitzHugh-Nagumo model

Classification:

93E12, 62A10

References:

  1. M. Arcak and P. Kokotovic: Nonlinear observers: A circle criteria design and robustness analysis. Automatica 37 (2001), 1923-1930.   CrossRef
  2. C. I. Byrnes, F. D. Priscoli and A. Isidori: Structurally stable output regulation of nonlinear systems. Automatica 33 (1997), 369-385.   CrossRef
  3. C. Chen, Z. Ding and B. Lennox: Rejection of nonharmonic disturbances in nonlinear systems with semi-global stability. IEEE Trans. Circuits. Syst. II: Expr. Briefs 55 (2008), 1289-1293.   CrossRef
  4. Z. Chen and J. Huang: Global robust output regulation for output feedback systems. IEEE Trans. Automat. Control 50 (2005), 117-121.   CrossRef
  5. Z. Chen and J. Huang: Robust output regulation with nonlinear exosystems. Automatica 41 (2005), 1447-1454.   CrossRef
  6. E. J. Davison: The robust control of a servomechanism problem for linear time-invariant multivariable systems. IEEE Trans. Automat. Control 21 (1976), 25-34.   CrossRef
  7. C. A. Desoer and C. A. Lin: Tracking and disturbance rejection of MIMO nonlinear systems with PI controller. IEEE Trans. Automat. Control 30 (1985), 861-867.   CrossRef
  8. M. D. Di Benedetto: Synthesis of an internal model for nonlinear output regulation. Internat. J. Control {\mi 45} (1987), 1023-1034.   CrossRef
  9. Y. Q. Che, J. Wang, S. S. Zhou and B. Deng: Robust synchronization control of coupled chaotic neurons under external electrical stimulation. Chaos Solit. Fract. 40 (2009), 1333-1342.   CrossRef
  10. Z. Ding: Global output regulation of uncertain nonlinear systems with exogenous signals. Automatica 37 (2001), 113-119.   CrossRef
  11. Z. Ding: Output regulation of uncertain nonlinear systems with nonlinear exosystems. IEEE Trans. Automat. Control 51 (2006), 498-503.   CrossRef
  12. Z. Ding: Decentralized output regulation of large scale nonlinear systems with delay. Kybernetika. 45 (2009), 33-48.   CrossRef
  13. A. Isidori and C. I. Byrnes: Output regulation of nonlinear systems. IEEE Trans. Automat. Control {\mi 35} (1990), 131-140.   CrossRef
  14. D. A. Francis and W. M. Wonham: The internal model principle for linear multivariable regulators. Appl. Math. Optim. 2 (1975), 170-194.   CrossRef
  15. J. Huang and Z. Chen: A general framework for tackling the output regulation problem. IEEE Trans. Automat. Control 49 (2004), 2203-2218.   CrossRef
  16. T. Ideker, T. Galitski and L. Hood: A new approach to decoding life: Systems biology. Ann. Rev. Genom, Hum. Genet. 2 (2001), 343-372.   CrossRef
  17. Q. Gong and W. Lin: A note on global output regulation of nonlinear system in the output feedback form. IEEE Trans. Automat. Control 48 (2003), 1049-1054.   CrossRef
  18. J. Huang: Asymptotic tracking and disturbance rejection in uncertain nonlinear systems. IEEE Trans. Automat. Control 40 (1995), 1118-1122.   CrossRef
  19. J. Huang and C-F. Lin: On a robust nonlinear servomechanism problem. IEEE Trans. Automat. Control. 39 (1994), 1510-1513.   CrossRef
  20. J. Huang and W. J. Rugh: On a nonlinear multivariable servomechanism problem. Automatica 26 (1990), 963-972.   CrossRef
  21. A. Isidori and : Nonlinear Control Systems. 3rd eddition. Springer-Verlag, New York 1995.   CrossRef
  22. C. D. Johnson: Accommodation of external disturbances in linear regulator and servomechanism problems. IEEE Trans. Automat. Control. 16 (1971), 635-644.   CrossRef
  23. H. Kitano: Systems biology: A brief overview. Science 295 (2002), 1662-1664.   CrossRef
  24. K. E. Kürten and J. W. Clark: Chaos in neural systems. Phys. Lett. A, 114 (1986), 413-418.   CrossRef
  25. W. Lin and C. Qian: Adaptive control of nonlinearly parameterized systems: the smooth case. IEEE Trans. Automat. Control. 47 (2002), 1249-1266.   CrossRef
  26. S. Liu, Y. Jiang and P. Liu: Rejection of nonharmonic disturbances in nonlinear systems. Kybernetika 46 (2010), 758-798.   CrossRef
  27. R. Marino and P. Tomei: Nonlinear Control Design-Nonlinear, Robust and Adaptive. Prentice Hall, Englewood Cliffs, New York 1994.   CrossRef
  28. D. Mishra, A. Yadav, S. Ray and P. K. Kalra: Nonlinear Dynamical Analysis on Coupled Modified FitzHugh-Nagumo Neuron Model. Lecture Notes in Computer Science. Springer Berlin - Heidelberg. 3496 (2005), 95-101.   CrossRef
  29. D. Mishra, A. Yadav, S. Ray and P. K. Kalra: Controlling synchronization of modified FitzHugh-Nagumo neurons under external electrical stimulation. NeuroQuantology 1 (2006), 50-67.   CrossRef
  30. L. E. Ramos, S. \u{C}elikovský and V. Ku\u{c}era: Generalized output regulation problem for a class of nonlinear systems with nonautonomous exosystem. IEEE Trans. Automat. Control 49 (2004), 1737-1742.   CrossRef
  31. J. Rinzel: A formal classification of bursting mechanisms in excitable systems, in mathematical topics in population niology, morphogenesis and neurosciences. Lecture Notes in Biomath., Springer-Verlag, New York. 71 (1987), 267-281.   CrossRef
  32. B. Rehák, S. Čelikovský, J. Ruiz-León and J. Orozco-Mora: A comparison of two fem-based methods for the solution of the nonlinear output regulation problem. Kybernetika 45 (2009), 427-444.   CrossRef
  33. A. Serrani and A. Isidori: Global robust output regulation for a class of nonlinear systems. Syst. Control Lett. 39 (2000), 133-139.   CrossRef
  34. W. Sun and J. Huang: Output regulation for a class of uncertain nonlinear systems with nonlinear exosystems and its application. Science in China, Ser. F: Information Sciences 52 (2009), 2172-2179.   CrossRef
  35. K. V. Venkatesh, S. Bhartiya and A. Ruhela: Mulitple feedback loops are key to a robust dynamic performance of tryptophan regulation in Escherichia coli. FEBS Lett. 563, (2004), 234-240.   CrossRef
  36. J. Wang, T. Zhang and B. Deng: Synchronization of FitzHugh-Nagumo neurons in external electrical stimulation via nonlinear control. Chaos Solit. Fract. 31 (2007), 30-38.   CrossRef
  37. Z. Xi and Z. Ding: Global adaptive output regulation of a class of nonlinear systems with nonlinear exosystems. Automatica 43 (2007), 143-149.   CrossRef
  38. Z. Xi and Z. Ding: Global decentralised output regulation for a class of large-scale nonlinear systems with nonlinear exosystem. IET Control. Theory Appl. 1 (2007), 1504-1511.   CrossRef