Skip to main content
Log in

Delayed feedback control based on the act-and-wait concept

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

Abstract

This paper proposes a delayed feedback control (DFC) based on the act-and-wait concept, which reduces the dynamics of DFC systems to that of discrete-time systems. Based on this concept, a delayed feedback controller is designed for a prototype two-dimensional oscillator using a simple systematic procedure. This control has two advantages: the feedback delay time can be large and it can obtain deadbeat behavior. A numerical example using a double-scroll circuit model demonstrates these theoretical results.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

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

    Google Scholar 

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

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  4. 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 

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

    Article  MATH  MathSciNet  Google Scholar 

  6. Just, W.: On the eigenvalue spectrum for time-delayed Floquet problems. Physica D 142, 153–165 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  7. 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 

  8. 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 

  9. 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  Google Scholar 

  10. Ding, Y., Jiang, W., Wang, H.: Delayed feedback control and bifurcation analysis of Rossler chaotic system. Nonlinear Dyn. (in press). doi:10.1007/s11071-010-9681-y

  11. 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 

  12. 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 

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

    Article  Google Scholar 

  14. 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 

  15. 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 

  16. 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  MathSciNet  Google Scholar 

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

  18. 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 

  19. 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 

  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. 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 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  24. 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 

  25. Michiels, W., Assche, V.V., Niculescu, S.-I.: Stabilization of time-delay systems with a controlled time-varying delay and applications. IEEE Trans. Autom. Control 50, 493–504 (2005)

    Article  Google Scholar 

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

    Article  Google Scholar 

  27. Konishi, K., Kokame, H., Hara, N.: Stability analysis and design of amplitude death induced by a time-varying delay connection. Phys. Lett. A 374, 733–738 (2010)

    Article  Google Scholar 

  28. Insperger, T.: Act-and-wait concept for continuous-time control systems with feedback delay. IEEE Trans. Control Syst. Technol. 14, 974–977 (2006)

    Article  Google Scholar 

  29. Stepan, G., Insperger, T.: Stability of time-periodic and delayed systems—A route to act-and-wait control. Ann. Rev. Control 30, 159–168 (2006)

    Article  Google Scholar 

  30. Insperger, T., Stepan, G.: Act-and-wait control concept for discrete-time systems with feedback delay. IET Control Theory Appl. 1, 553–557 (2007)

    Article  Google Scholar 

  31. Insperger, T., Wahi, P., Colombo, A., Stepan, G., Bernardo, M. di Hogan, J.S.: Full characterization of act-and-wait control for first order unstable lag processes. J. Vib. Control 16, 1209–1233 (2010)

    Article  Google Scholar 

  32. Insperger, T., Kovacs, L.L., Galambos, P., Stepan, G.: Increasing the accuracy of digital force control process using the act-and-wait concept. IEEE/ASME Trans. Mechatron. 15, 291–298 (2010)

    Article  Google Scholar 

  33. Gawthrop, P.: Act-and-wait and intermittent control: Some comments. IEEE Trans. Control Syst. Technol. (in press). doi:10.1109/TCST.2009.2034403

  34. Yanchuk, S., Wolfrum, M., Hövel, P., Schöll, E.: Control of unstable steady states by long delay feedback. Phys. Rev. E 74, 026201 (2006)

    Article  MathSciNet  Google Scholar 

  35. Ogata, K.: Discrete-time Control Systems. Pearson Education, Upper Saddle River (1994)

    Google Scholar 

  36. Matsumoto, T., Chua, L.O., Komuro, M.: The double scroll. IEEE Trans. Circuits Syst. 32, 797–818 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  37. Bayly, P.V., Virgin, L.N.: Practical considerations in the control of chaos. Phys. Rev. E 50, 604–607 (1994)

    Article  Google Scholar 

  38. Schuster, H.G., Stemmler, M.B.: Control of chaos by oscillating feedback. Phys. Rev. E 56, 6410–6417 (1997)

    Article  Google Scholar 

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

    Article  Google Scholar 

  40. Ueta, T., Toyosaki, Y., Tsuji, S., Kousaka, T.: Partial delayed feedback control and its DSP implementation. In: Proc. of IEEE International Midwest Symposium on Circuits and Systems, Hiroshima, pp. 629–632 (2004)

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Keiji Konishi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Konishi, K., Kokame, H. & Hara, N. Delayed feedback control based on the act-and-wait concept. Nonlinear Dyn 63, 513–519 (2011). https://doi.org/10.1007/s11071-010-9819-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-010-9819-y

Keywords

Navigation