Skip to main content
SpringerLink
Log in

Design and experimental verification of multiple delay feedback control for time-delay nonlinear oscillators

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

This study aims to show that a multiple delay feedback control method can stabilize unstable fixed points of time-delay nonlinear oscillators. The boundary curves of stability in a control parameter space are derived using linear stability analysis. A simple procedure for designing a feedback gain is provided. The main advantage of this procedure is that the designed controller can stabilize a system even if the controller delay times are long. These analytical results are experimentally verified using electronic circuits.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Chen, G., Dong, X.: From Chaos to Order. World Scientific, Singapore (1998)

    MATH  Google Scholar 

  2. Schuster, H.G.: Handbook of Chaos Control. Wiley, New York (1999)

    Book  MATH  Google Scholar 

  3. Andrievskii, B.R., Fradkov, A.L.: Control of chaos: Methods and applications—I. Methods. Autom. Remote Control 64, 673–713 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  4. Andrievskii, B.R., Fradkov, A.L.: Control of chaos: Methods and applications—II. Applications. Autom. Remote Control 65, 505–533 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  5. Schöll, E., Schuster, H.G.: Handbook of Chaos Control. Wiley, New York (2008)

    MATH  Google Scholar 

  6. Pyragas, K.: Continuous control of chaos by self-controlling feedback. Phys. Lett. A 170, 421–428 (1992)

    Article  Google Scholar 

  7. Pyragas, K.: Delayed feedback control of chaos. Philos. Trans. R. Soc. Lond. A 364, 2309–2334 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  8. Gu, K., Niculescu, S.I.: Survey on recent results in the stability and control of time-delay systems. J. Dyn. Syst. Meas. Control 125, 158–165 (2003)

    Article  Google Scholar 

  9. Chang, A., Bienfang, J.C., Hall, G.M., Gardner, J.R., Gauthier, D.J.: Stabilizing unstable steady states using extended time-delay autosynchronization. Chaos 8, 782–790 (1998)

    Article  MATH  Google Scholar 

  10. Hövel, P., Schöll, E.: Control of unstable steady states by time-delayed feedback methods. Phys. Rev. E 72, 046203 (2005)

    Article  Google Scholar 

  11. Dahms, T., Hövel, P., Schöll, E.: Control of unstable steady states by extended time-delayed feedback. Phys. Rev. E 76, 056201 (2007)

    Article  MathSciNet  Google Scholar 

  12. Kokame, H., Hirata, K., Konishi, K., Mori, T.: State difference feedback for stabilizing uncertain steady states of non-linear systems. Int. J. Control 74, 537–546 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  13. Kokame, H., Hirata, K., Konishi, K., Mori, T.: Difference feedback can stabilize uncertain steady states. IEEE Trans. Autom. Control 46, 1908–1913 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  14. Hirata, K., Kokame, H., Konishi, K., Fujita, H.: Observer-based delayed-feedback control of continuous-time systems. In: Proc. of American Control Conference, pp. 25–27 (2001)

    Google Scholar 

  15. Tronciu, V.Z., Wünsche, H.-J., Wolfrum, M., Radziunas, M.: Semiconductor laser under resonant feedback from a Fabry–Perot resonator: Stability of continuous-wave operation. Phys. Rev. E 73, 046205 (2006)

    Article  Google Scholar 

  16. Ahlborn, A., Parlitz, U.: Stabilizing unstable steady states using multiple delay feedback control. Phys. Rev. Lett. 93, 264101 (2004)

    Article  Google Scholar 

  17. Ahlborn, A., Parlitz, U.: Controlling dynamical systems using multiple delay feedback control. Phys. Rev. E 72, 016206 (2005)

    Article  MathSciNet  Google Scholar 

  18. Ahlborn, A., Parlitz, U.: Laser stabilization with multiple-delay feedback control. Opt. Lett. 31, 465–467 (2006)

    Article  Google Scholar 

  19. Hikihara, T., Kawagoshi, T.: An experimental study on stabilization of unstable periodic motion in magneto-elastic chaos. Phys. Lett. A 211, 29–36 (1996)

    Article  Google Scholar 

  20. Kittel, A., Parisi, J., Pyragas, K., Richter, R.: Delayed feedback control of chaos in an electronic double-scroll oscillator. Z. Naturforsch. 49a, 843–846 (1994)

    Google Scholar 

  21. Konishi, K., Kokame, H.: Odd-number property of multiple delayed feedback control. In: Proc. of the 15th International IEEE Workshop on Nonlinear Dynamics of Electronic Systems, pp. 249–252 (2007)

    Google Scholar 

  22. Lu, J., Ma, Z., Li, L.: Double delayed feedback control for the stabilization of unstable steady states in chaotic systems. Commun. Nonlinear Sci. Numer. Simul. 14, 3037–3045 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  23. Farmer, J.D.: Chaotic attractors of an infinite-dimensional dynamical system. Physica D 4, 366–393 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  24. MacDonald, N.: Biological Delay Systems: Linear Stability Theory. Cambridge University Press, Cambridge (1989)

    MATH  Google Scholar 

  25. Voss, H.U.: Anticipating chaotic synchronization. Phys. Rev. E 61, 5115–5119 (2000)

    Article  Google Scholar 

  26. Bunner, M., Just, W.: Synchronization of time-delay systems. Phys. Rev. E 58, 4072–4075 (1998)

    Article  Google Scholar 

  27. Shahverdiev, E.M., Shore, K.A.: Generalized synchronization in time-delayed systems. Phys. Rev. E 71, 016201 (2005)

    Article  Google Scholar 

  28. Sano, S., Uchida, A., Yoshimori, S., Roy, R.: Dual synchronization of chaos in Mackey–Glass electronic circuits with time-delayed feedback. Phys. Rev. E 75, 016207 (2007)

    Article  Google Scholar 

  29. Kye, W.H., Choi, M., Kim, C.M.: Encryption with synchronized time-delayed systems. Phys. Rev. E 71, 045202 (2005)

    Article  MathSciNet  Google Scholar 

  30. Suzuki, M., Sakamoto, N.: Controlling ideal turbulence in time-delayed Chua’s circuit: Stabilization and synchronization. Int. J. Bifurc. Chaos Appl. Sci. Eng. 20, 1351–1363 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  31. Feng, C.F.: Projective synchronization between two different time-delayed chaotic systems using active control approach. Nonlinear Dyn. 62, 453–459 (2010)

    Article  MATH  Google Scholar 

  32. Moon, F.C.: Dynamics and Manufacturing Processes. Wiley, New York (1998)

    Google Scholar 

  33. Radons, G., Neugebauer, R.: Nonlinear Dynamics of Production Systems. Wiley, New York (2004)

    Book  Google Scholar 

  34. Sowa, N., Kondou, T., Mori, H., Choi, M.S.: Method of preventing unstable vibration caused by time delays in contact rotating systems: Application of new stability analysis. JSME Int. J. Ser. C, Dyn. Control Robot. Des. Manuf. 49, 973–982 (2006)

    Article  Google Scholar 

  35. Namajūnas, A., Pyragas, K., Tamaševičius, A.: Stabilization of an unstable steady state in a Mackey–Glass system. Phys. Lett. A 204, 255–262 (1995)

    Article  Google Scholar 

  36. Guan, X., Feng, G., Chen, C., Chen, G.: A full delayed feedback controller design method for time-delay chaotic systems. Physica D 227, 36–42 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  37. Gjurchinovski, A., Urumov, V.: Variable-delay feedback control of unstable steady states in retarded time-delayed systems. Phys. Rev. E 81, 016209 (2010)

    Article  Google Scholar 

  38. Hale, J.K., Lunel, S.M.V.: Introduction to Functional Differential Equations. Springer, New York (1993)

    MATH  Google Scholar 

  39. Michiels, W., Niculescu, S.I.: Stability and Stabilization of Time-Delay Systems: An Eigenvalue-Based Approach. SIAM, Philadelphia (2007)

    Book  MATH  Google Scholar 

  40. Ishii, M., Konishi, K., Kokame, H.: Robust stability of extended delayed-feedback control in one-dimensional chaotic systems. Phys. Lett. A 235, 603–609 (1997)

    Article  Google Scholar 

  41. Konishi, K., Senda, K., Kokame, H.: Amplitude death in time-delay chaotic oscillators coupled by diffusive connections. Phys. Rev. E 78, 056216 (2008)

    Article  MathSciNet  Google Scholar 

  42. Blyuss, K.B., Kyrychko, Y.N., Hövel, P., Schöll, E.: Control of unstable steady states in neutral time-delayed systems. Eur. Phys. J. B 65, 571–576 (2008)

    Article  MATH  Google Scholar 

  43. Xu, S., Lam, J., Zou, Y.: Delay-dependent approach to stabilization of time-delay chaotic systems via standard and delayed feedback controllers. Int. J. Bifurc. Chaos Appl. Sci. Eng. 15, 1455–1465 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  44. Rezaie, B., Motlagh, M.R.J., Analoui, M., Khorsandi, S.: Stabilizing fixed points of time-delay systems close to the Hopf bifurcation using a dynamic delayed feedback control method. J. Phys. A, Math. Theor. 42, 395102 (2009)

    Article  Google Scholar 

  45. Vasegh, N., Sedigh, A.K.: Delayed feedback control of time-delayed chaotic systems: Analytical approach at Hopf bifurcation. Phys. Lett. A 372, 5110–5114 (2008)

    Article  MATH  Google Scholar 

  46. Vasegh, N., Sedigh, A.K.: Chaos control via TDFC in time-delayed systems: The harmonic balance approach. Phys. Lett. A 373, 354–358 (2009)

    Article  MATH  Google Scholar 

  47. Konishi, K., Kokame, H., Hara, N.: Stabilization of a steady state in network oscillators by using diffusive connections with two long time delays. Phys. Rev. E 81, 016201 (2010)

    Article  Google Scholar 

  48. Ushio, T.: Limitation of delayed feedback control in nonlinear discrete-time systems. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 43, 815–816 (1996)

    Article  Google Scholar 

  49. Nakajima, H.: On analytical properties of delayed feedback control of chaos. Phys. Lett. A 232, 207–210 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  50. Just, W., Reibold, E., Bennerb, H., Kacperskic, K., Fronczakc, P., Holyst, J.: Limits of time-delayed feedback control. Phys. Lett. A 254, 158–164 (1999)

    Article  Google Scholar 

  51. Fiedler, B., Flunkert, V., Georgi, M., Hövel, P., Schöll, E.: Refuting the odd number limitation of time-delayed feedback control. Phys. Rev. Lett. 98, 114101 (2007)

    Article  Google Scholar 

  52. Just, W., Fiedler, B., Georgi, M., Flunkert, V., Hövel, P., Schöll, E.: Beyond the odd number limitation: A bifurcation analysis of time-delayed feedback control. Phys. Rev. E 76, 026210 (2007)

    Article  MathSciNet  Google Scholar 

  53. Postlethwaite, C.M., Silber, M.: Stabilizing unstable periodic orbits in the Lorenz equations using time-delayed feedback control. Phys. Rev. E 76, 056214 (2007)

    Article  MathSciNet  Google Scholar 

  54. Konishi, K., Kokame, H.: Observer-based delayed-feedback control for time-discrete chaotic systems. Phys. Lett. A 248, 359–368 (1998)

    Article  Google Scholar 

  55. Yamamoto, S., Hino, T., Ushio, T.: Dynamic delayed feedback controllers for chaotic discrete-time systems. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 48, 785–789 (2001)

    Article  MATH  Google Scholar 

  56. Pyragas, K., Pyragas, V., Kiss, I.Z., Hudson, J.L.: Stabilizing and tracking unknown steady states of dynamical systems. Phys. Rev. Lett. 89, 244103 (2002)

    Article  Google Scholar 

  57. Pyragas, K., Pyragas, V., Kiss, I.Z., Hudson, J.L.: Adaptive control of unknown unstable steady states of dynamical systems. Phys. Rev. E 70, 026215 (2004)

    Article  Google Scholar 

  58. Gjurchinovski, A., Urumov, V.: Stabilization of unstable steady states by variable-delay feedback control. Europhys. Lett. 84, 40013 (2008)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electrical and Information Systems, Osaka Prefecture University, 1–1 Gakuen-cho, Naka-ku, Sakai, Osaka, 599-8531, Japan

    Luan Ba Le, Keiji Konishi & Naoyuki Hara

Authors
  1. Luan Ba Le
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Keiji Konishi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Naoyuki Hara
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Keiji Konishi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Le, L.B., Konishi, K. & Hara, N. Design and experimental verification of multiple delay feedback control for time-delay nonlinear oscillators. Nonlinear Dyn 67, 1407–1418 (2012). https://doi.org/10.1007/s11071-011-0077-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-011-0077-4

Keywords

Search

Navigation