Skip to main content
SpringerLink
Log in

Practical stabilization of third-order bilinear systems

  • Control Theory
  • Published:
Differential Equations Aims and scope Submit manuscript

Abstract

We consider the problem of practical stabilization of bilinear third-order dynamical systems by stationary state feedback. For bilinear systems in canonical form, we suggest to generalize the method of feedback linearization on the basis of a feedback of variable structure. This generalization is used for the derivation of sufficient conditions for the practical stabilization of the zero equilibrium of bilinear third-order systems in canonical form.

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. Leonov, G.A. and Shumafov, M.M., Vibrational Stabilization and the Brockett Problem, Differ. Uravn. Protsessy Upr., 2011, no. 4, http://www.math.spbu.ru/diffjournal.

  2. Emel’yanov, S.V. and Krishchenko, A.P., Stabilization of Irregular Systems, Differ. Uravn., 2012, vol. 48, no. 11, pp. 1515–1524.

    MathSciNet  Google Scholar 

  3. Emel’yanov, S.V. and Krishchenko, A.P., Stabilizability of Bilinear Systems of Canonical Form, Dokl. Akad. Nauk, 2012, vol. 445, no. 6, pp. 636–639.

    MathSciNet  Google Scholar 

  4. Emel’yanov, S.V., Korovin, S.K., and Shepit’ko, A.S., Stabilization of Bilinear Systems on the Plane by Constant and Relay Controls, Differ. Uravn., 2000, vol. 36, no. 8, pp. 1021–1028.

    MathSciNet  Google Scholar 

  5. Leonov, G.A., The Brockett Stabilization Problem, Avtomat. i Telemekh., 2001, no. 5, pp. 190–193.

    Google Scholar 

  6. Fomichev, V.V. and Shepit’ko, A.S., The Method of Rotating Lyapunov Functions in the Problem of Stabilization of Two-Dimensional Bilinear Systems, Differ. Uravn., 2000, vol. 36, no. 8, pp. 1136–1138.

    MathSciNet  Google Scholar 

  7. Bo, H., Zhai G., and Michel, A.N., Stabilization of Two-Dimensional Single-Input Bilinear Systems with a Finite Number of Constant Feedback Controllers, Proc. Amer. Control Conference, 2002, vol. 3, pp. 1874–1879.

    Google Scholar 

  8. Cong, S. and Yin, L.P., Switching Control Approach to Exponential Stabilization of Planar Bilinear Systems, Control Theory Appl. IET, 2012, vol. 6, no. 9, pp. 1313–1318.

    Article  MathSciNet  Google Scholar 

  9. Golubev, A.E., Stabilization of Third-Order Bilinear Systems Using Constant Controls, Science and Education. Electronic Scientific and Technical Journal, 2014, no. 7, http://technomag.bmstu.ru/doc/717640.html.

  10. Byrnes, C.I., On Brockett’s Necessary Condition for Stabilizability and the Topology of Liapunov Functions on ℝN, Commun. Inf. Syst., 2008, vol. 8, no. 4, pp. 333–352.

    MATH  MathSciNet  Google Scholar 

  11. Goncharov, O.I., Asymptotic Stabilization of a Class of Bilinear Systems Using Feedback of Variable Structure, Differ. Uravn., 2011, vol. 47, no. 11, pp. 1564–1572.

    MathSciNet  Google Scholar 

  12. Krasnoshchechenko, V.I. and Krishchenko, A.P., Nelineinye sistemy, geometricheskie metody analiza i sinteza (Nonlinear Systems: Geometric Methods of Analysis and Synthesis), Moscow, 2005.

    Google Scholar 

  13. Khalil, H.K., Nonlinear Systems, New York, 2002.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Bauman Moscow State Technical University, Moscow, Russia

    A. E. Golubev & A. P. Krishchenko

  2. Institute for Systems Analysis, Russian Academy of Sciences, Moscow, Russia

    A. E. Golubev & A. P. Krishchenko

Authors
  1. A. E. Golubev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. P. Krishchenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. E. Golubev.

Additional information

Original Russian Text © A.E. Golubev, A.P. Krishchenko, 2015, published in Differentsial’nye Uravneniya, 2015, Vol. 51, No. 8, pp. 1096–1100.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Golubev, A.E., Krishchenko, A.P. Practical stabilization of third-order bilinear systems. Diff Equat 51, 1092–1096 (2015). https://doi.org/10.1134/S0012266115080133

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0012266115080133

Keywords

Search

Navigation