Skip to main content
SpringerLink
Log in

Pyragas Stabilizability of Unstable Equilibria by Nonstationary Time-Delayed Feedback

  • Topical Issue
  • Published:
Automation and Remote Control Aims and scope Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

The problem of Pyragas stabilizability of unstable equilibria of nonlinear systems is considered. Stabilization algorithm by periodic time-delayed feedback is constructed. An analytical criterion for stabilization is obtained. The proposed approach is based on the method of nonstationary stabilization.

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. Brockett, R., A Stabilization Problem/Open Problems in Mathematical Systems and Control Theory, London: Springer, 1999.

    Google Scholar 

  2. Leonov, G.A., The Brockett Stabilization Problem, Autom. Remote Control, 2001, vol. 62, no. 5, pp. 847–849.

    Article  MathSciNet  MATH  Google Scholar 

  3. Leonov, G.A., Brockett Problem in the Stability Theory of Linear Differential Equations, Algebra Analiz, 2001, vol. 13, no. 4, pp. 134–155.

    Google Scholar 

  4. Moreau, L. and Aeyels, D., Periodic Output Feedback Stabilization of Single-Input Single-Output Continuous-Time Systems with Odd Relative Degree, Syst. Control Lett., 2004, vol. 51, no. 5, pp. 395–406.

    Article  MathSciNet  MATH  Google Scholar 

  5. Leonov, G.A. and Shumafov, M.M., Stabilization of Linear Systems, Cambridge: Cambridge Scientific Publishers, 2012.

    MATH  Google Scholar 

  6. Pyragas, K., Continuous Control of Chaos by Self-Controlling Feedback, Phys. Lett. A, 1992, vol. 170, pp. 421–428.

    Article  Google Scholar 

  7. Pyragas, K., Control of Chaos via Extended Delay Feedback, Phys. Lett. A, 1995, vol. 206, pp. 323–330.

    Article  MathSciNet  MATH  Google Scholar 

  8. Namajūnas, A., Pyragas, K., and Tamaševičius, A., Stabilization of an Unstable Steady State in a Mackey–Glass System, Phys. Lett. A, 1995, vol. 204, pp. 255–262.

    Article  Google Scholar 

  9. Pyragas, K., Applications of Extended Delay Feedback to Control Chaos, J. Tech. Phys., 1996, vol. 37, pp. 409–412.

    Google Scholar 

  10. Pyragas, K., Control of Chaos via an Unstable Delayed Feedback Controller, Phys. Rev. Lett., 2001, vol. 86, pp. 2265–2268.

    Article  Google Scholar 

  11. Pyragas, K., Analytical Properties and Optimization of Time-Delayed Feedback Control, Phys. Rev. E, 2002, vol. 66, pp. 0262207–9.

    Article  Google Scholar 

  12. Pyragas, K., Pyragas, V., Kiss, I.Z., and Hudson, J.L., Stabilizing and Tracking Unknown Steady States of Dynamical Systems, Phys. Rev. Lett., 2002, vol. 89, pp. 244103–4.

    Article  Google Scholar 

  13. Pyragas, K., Control of Dynamical Systems via Time-Delayed Feedback and Unstable Controller, in Synchronization. Theory and Application, Dikovsky, A. and Maistrenko, Yu., Eds., Dordrecht: Kluwer, 2003, pp. 187–219.

    Google Scholar 

  14. Tamaševičius, A., Mykolaitis, G., Pyragas, V., and Pyragas, K., Delayed Feedback Control of Periodic Orbits Without Torsion in Nonautonomous Chaotic Systems. Theory and Experiment, Phys. Rev. E, 2007, vol. 76, pp. 026203–6.

    Article  Google Scholar 

  15. Novičenko, V. and Pyragas, K., Time-Delayed Feedback Control of Periodic Orbits with an Odd-number of Positive Unstable Floquet Multipliers, ENOC, Vienna, Austria, July 6–11, 2014.

    Google Scholar 

  16. Pyragas, K., A Twenty-Year Review of Time-Delay Feedback Control and Recent Developments, in Int. Sympos. on Nonlinear Theory and Its Applications, Spain, 2012, pp. 683–686.

    Google Scholar 

  17. Tian, Y., Zhu, J., and Chen G.J., A Survey on Delayed Feedback Control of Chaos, J. Control Theory Appl., 2005, no. 4, pp. 311–319.

    Article  MathSciNet  MATH  Google Scholar 

  18. Pyragas, K., Delayed Feedback Control of Chaos, Phil. Trans. Royal Soc. A, 2006, vol. 369, pp. 2039–2334.

    MathSciNet  MATH  Google Scholar 

  19. Kuznetsov, N.V., Leonov, G.A., and Shumafov, M.M., A Short Survey on Pyragas Time-Delay Feedback Stabilization and Odd Number Limitation, IFAC-PapersOnLine, 2015, vol. 48, no. 11, pp. 706–709.

    Article  Google Scholar 

  20. Fradkov, A.L., Application of Cybernetic Methods in Physics, Phys.-Usp., 2005, vol. 48, pp. 103–127.

    Article  Google Scholar 

  21. Fradkov, A.L., Kiberneticheskaya fizika. Printsipy i primery (Cybernetical Physics. Principles and Examples), Moscow: Nauka, 2003. Translated under the title Cybernetical Physics: From Control of Chaos to Quantum Control, Berlin: Springer, 2007.

    Google Scholar 

  22. Fradkov, A.L., Horizons of Cybernetical Physics, Philos. Trans. Royal Soc. A: Math. Phys. Eng. Sci., 2017, vol. 375, no. 2088, p. 20160223.

    Article  MathSciNet  Google Scholar 

  23. Ahlborn, A. and Parlitz, U., Stabilizing Unstable Steady States Using Multiple Delay Feedback Controls, Phys. Rev. Lett., 2004, vol. 93, p. 264101.

    Article  Google Scholar 

  24. Ahlborn, A. and Parlitz, U., Controlling Dynamical Systems Using Multiple Delay Feedback Controls, Phys. Rev. E, 2005, vol. 72, p. 016206.

    Article  MathSciNet  Google Scholar 

  25. Dahms, T., Hövel, Ph., and Schöll, E., Stabilization of Fixed Points by Extended Time-Delayed Feedback Control, Phys. Rev. E, 2007, vol. 76, p. 056213.

    Article  MathSciNet  Google Scholar 

  26. Hövel, Ph. and Schöll, E., Control of Unstable Steady States of Time-Delayed Feedback Methods, Phys. Rev. E, 2005, vol. 72, p. 046203.

    Article  Google Scholar 

  27. Huijberts, H., Michiels, W., and Nijmeijer, H., Stabilization via Time-Delayed Feedback. An Eigenvalues Optimization Approach, SIAM J. Appl. Dyn. Syst., 2009, vol. 8, no. 1, pp. 1–20.

    Article  MathSciNet  MATH  Google Scholar 

  28. Pyragas, K., Pyragas, V., Kiss, I.Z., and Hudson, J.L., Adaptive Control of Unknown Unstable Steady States of Dynamical Systems, Phys. Rev. E, 2004, vol. 70, p. 026215.

    Article  Google Scholar 

  29. Yanchuk, S., Wolfrum, M., Hövel, Ph., and Schöll, E., Control of Unstable Steady States by Long Delay Feedback, Phys. Rev. E, 2006, vol. 74, p. 026201.

    Article  MathSciNet  Google Scholar 

  30. Shumafov, M.M., On Stabilization of Two-Dimensional Linear Controlled Systems by a Time-Delayed Feedback, Vestn. Adygei. Gos. Univ., Ser. Estestvenno-Mat. Tekhn. Nauk., 2010, vol. 2 (61), pp. 40–52. http://vestnik.adygnet.ru

    Google Scholar 

  31. Shumafov, M.M., Stabilization of the Second Order Time-Invariant Control Systems by a Delay Feedback, Izv. Vyssh. Uchebn. Zaved., Mat., 2010, no. 12, pp. 87–90.

    MathSciNet  MATH  Google Scholar 

  32. Leonov, G.A., Shumafov, M.M., and Kuznetsov, N.V., Delayed Feedback Stabilization of Unstable Equilibria, IFAC Proc., 2014, vol. 19, pp. 6818–6825.

    Article  Google Scholar 

  33. Leonov, G.A., Shumafov, M.M., and Kuznetsov, N.N., Delayed Feedback Stabilization and the Huijberts-Michiels-Nijmeijer Problem, Differ. Equat., 2016, vol. 52, no. 13, pp. 1–25.

    MathSciNet  MATH  Google Scholar 

  34. Ushio, T., Limitation of Delayed Feedback Control in Nonlinear Discrete-Time Systems, IEEE Trans. Circuits Syst. I, 1996, vol. 43 (9), pp. 815–816.

    Article  Google Scholar 

  35. Nakajima, H., On Analytical Properties of Delayed Feedback Control of Chaos, Phys. Lett. A, 1997, vol. 232, pp. 207–210.

    Article  MathSciNet  MATH  Google Scholar 

  36. Nakajima, H. and Ueda, Y., Limitation of Generalized Delayed Feedback Control, Phys. D, 1998, vol. 111, pp. 143–150.

    Article  MathSciNet  MATH  Google Scholar 

  37. Just, W., Bernard, T., Ostheimer, M., Reibold, E., and Benner, H., Mechanism of Time-Delayed Feedback Control, Phys. Rev. Lett., 1997, vol. 78, pp. 203–206.

    Article  Google Scholar 

  38. Kokame, H., Hirata, K., Konishi, K., and Mori, T., Difference Feedback Can Stabilize Uncertain Steady States, IEEE Trans. Autom. Control, 2001, vol. 46(12), pp. 1908–1913.

    Article  MathSciNet  MATH  Google Scholar 

  39. Hino, T., Yamamoto, Sh., and Ushio, T., Stabilization of Unstable Periodic Orbits of Chaotic Discrete-Time Systems Using Prediction-Based Feedback Control, Int. J. Bifurcat. Chaos, 2002, vol. 12 (2), pp. 439–446.

    Article  MathSciNet  MATH  Google Scholar 

  40. Fiedler, B., Flunkert, V., Georgi, M., Hövel, Ph., and Schöll, E., Refuting the Odd-Number Limitation of Time-Delayed Feedback Control, Phys. Rev. Lett., 2007, vol. 98, p. 114101.

    Article  Google Scholar 

  41. Just, W., Fiedler, B., Georgi, M., Flunkert, V., Hövel, Ph., and Schöll, E., Beyond the Odd-Number Limitation. A Bifurcation Analysis of Time-Delayed Feedback Control, Phys. Rev. E, 2007, vol. 76, p. 7026210.

    Article  MathSciNet  Google Scholar 

  42. Hooton, E.W. and Amann, A., An Analitical Limitation for Time-Delayed Feedback Control in Autonomous Systems, 2012, arXiv:1109.1138v2[nlin.CD].

    Google Scholar 

  43. Hooton, E.W. and Amann, A., An Odd-Number Limitation of Extended Time-Delayed Feedback Control in Autonomous Systems, Phil. Trans. Royal Soc. A, 2013, 371, p. 20120463.

    Article  MathSciNet  MATH  Google Scholar 

  44. Pyragas, K. and Novičenko, V., Time-Delayed Feedback Control Design Beyond the Odd Number Limitation, Phys. Rev. E, 2013, vol. 88, no. 1, p. 012903.

    Article  Google Scholar 

  45. Pyragas, K., Act-and-Wait Time-Delayed Feedback Control of Nonautonomous Systems, Phys. Rev. E, 2016, vol. 94, p. 012201.

    Article  Google Scholar 

  46. Leonov, G.A., Pyragas Stabilizability via Delayed Feedback with Periodic Control Gain, Syst. Control Lett., 2014, vol. 69, pp. 34–37.

    Article  MathSciNet  MATH  Google Scholar 

  47. El’sgoltz, L.E. and Norkin, S.B., Introduction to the Theory and Application of Differential Equations with Deviations Arguments, New York: Academic, 1973.

    Google Scholar 

  48. Bellman, R.E. and Cooke, K.L., Differential-Difference Equations, New York: Academic, 1963, Translated under the title Differentsial’no-raznostnye uravneniya, Moscow: Mir, 1967.

    MATH  Google Scholar 

  49. Hale, J.K., Theory of Functional Differential Equations, New York: Springer, 1977.

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. St. Petersburg State University, St. Petersburg, Russia

    G. A. Leonov

  2. Adyghe State University, Maikop, Russia

    M. M. Shumafov

Authors
  1. G. A. Leonov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. M. Shumafov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. M. Shumafov.

Additional information

Original Russian Text © G.A. Leonov, M.M. Shumafov, 2018, published in Avtomatika i Telemekhanika, 2018, No. 6, pp. 87–98.

Deceased.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Leonov, G.A., Shumafov, M.M. Pyragas Stabilizability of Unstable Equilibria by Nonstationary Time-Delayed Feedback. Autom Remote Control 79, 1029–1039 (2018). https://doi.org/10.1134/S0005117918060048

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0005117918060048

Keywords

Search

Navigation