Skip to main content
SpringerLink
Log in

Local and semiglobal stabilization in a cascade with delay

  • Determinate Systems
  • Published:
Automation and Remote Control Aims and scope Submit manuscript

Abstract

Consideration is given to the problem of stability and stabilization for a system of differential equations with delay of a “cascade” structure. Sufficient conditions for local asymptotic stability as well as “semiglobal” stabilization for a cascade with a linear subsystem are obtained.

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. Arcak, M., Sepulchre, R., and Teel, A.R., Trading the Stability of Finite Zeros for Global Stabilization of Nonlinear Cascade Systems, Proc. 40th IEEE Conf. on Decision and Control, Orlando, 2001, pp. 3025–3029.

  2. Jankovic, M., Sepulchre, R., and Kokotovic, P., Global Adaptive Stabilization of Cascaded Nonlinear Systems, Automatica, 1997, vol. 33, no. 2, pp. 363–368.

    Article  MathSciNet  Google Scholar 

  3. Lozano, R., Brogliato, B., and Landau, I.D., Passivity and Global Stabilization of Cascaded Nonlinear System, IEEE Trans. Automat. Control, 1992, AC-37, pp. 1386–1389.

    Article  MathSciNet  Google Scholar 

  4. Panteley, E. and Loria, A., Global Uniform Asymptotic Stability of Cascaded Non-autonomous Nonlinear Systems, Proc. 4th Eur. Control Conf., 1997, paper no. 259.

  5. Krasovskii, N.N., Nekotorye zadachi teorii ustoichivosti dvizheniya (Some Porblems of the Theory of Motion Stability), Moscow: Fizmatgiz, 1959.

    Google Scholar 

  6. Hou, C., Gao, F., and Qian, J., Stability Criterion for Linear Systems with Nonlinear Delayed Perturbations, J. Math. Anal. Appl., 1999, vol. 237, pp. 573–582.

    Article  MATH  MathSciNet  Google Scholar 

  7. Kato, J., Asymptotic Behavior in Functional Differential Equations with Infinite Delay, Lecture Notes Math., 1982, vol. 1017, pp. 300–312.

    Article  Google Scholar 

  8. Michiels, W., Sepulchre, R., and Roose, D., Robustness of Nonlinear Delay Equations w.r.t. Bounded Input Perturbations, Proc. 14th Int. Sympos. Math. Theory Networks Syst. (MTNS2000), 2000, pp. 1–5.

  9. Michiels, W., Sepulchre, R., and Roose, D., Stability of Perturbed Delay Differential Equations and Stabilization of Nonlinear Cascade Systems, TW Report 298, Department Comput. Sci., Katholieke Univ., Leuven, Belgium, 2000.

    Google Scholar 

  10. Murakami, S., Perturbation Theorem for Functional Differential Equations with Infinite Delay via Limiting Equations, J. Diff. Eq., 1985, vol. 59, pp. 314–335.

    Article  MATH  MathSciNet  Google Scholar 

  11. Krasovskii, N.N., Asymptotic Stability of Systems with Aftereffect, Prikl. Mat. Mekh., 1956, vol. 20, no. 4, pp. 513–518.

    MathSciNet  Google Scholar 

  12. Andreev, A.S., Stability of the Non-autonomous Functional-Differential Equation, Dokl. Akad. Nauk, 1997, vol. 356, no. 2, pp. 151–153.

    MathSciNet  Google Scholar 

  13. Razumikhin, B.S., Ustoichivost’ ereditarnykh sistem (Stability of Ereditary Systems), Moscow: Nauka, 1988.

    Google Scholar 

  14. Sedova, N.O., Degenerate Functions in Studying Asymptotic Stability of Solutions to Functional-Differential Equations, Mat. Zametki, 2005, vol. 78, no. 3, pp. 468–472.

    MathSciNet  Google Scholar 

  15. Andreev, A.S. and Pavlikov, S.V., Non-sign-definite Functionals in the Problem of Stability of Functional-Differential Equations with Finite Delay, Mekh. Tverdogo Tela, 2004, no. 34, pp. 112–118.

  16. Hale, J.K., Theory of Functional Differential Equations, New York: Springer-Verlag, 1977. Translated under the title Teoriya funktsional’no-differentsial’nykh uravnenii, Moscow: Mir, 1984.

    MATH  Google Scholar 

  17. Kim, A.V., Pryamoi metod Lyapunova v teorii ustoichivosti sistem s posledeistviem (A Direct Lyapunov Method in the Theory of Stability of Systems with Aftereffect), Yekaterinburg: Ural. Univ., 1992.

    Google Scholar 

  18. Khusainov, D.Ya. and Shatyrko, A.V., Metod funktsii Lyapunova v issledovanii ustoichivosti differentsial’no-funktsional’nykh sistem (The Method of Lyapunov Functions in Studying Sability of Differential Functional Systems), Kiev: Kiev. Univ., 1997.

    Google Scholar 

  19. Fowler, A.C. and Mackey, M.C., Relaxation Oscilations in a Class of Delay Differential Equations, SIAM J. Appl. Math., 2002, vol. 63, no. 1, pp. 299–323.

    Article  MATH  MathSciNet  Google Scholar 

  20. Xiao, Y. and Chen, L., An SIS Epidemic Model with Stage Structure and a Delay, Acta Math. Appl. Sinica. Engl. Ser., 2002, vol. 18, no. 4, pp. 607–618.

    Article  MATH  MathSciNet  Google Scholar 

  21. Zubov, V.I., Lektsii po teorii upravleniya (Lectures on the Control Theory), Moscow: Nauka, 1975.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Ul’yanovsk State University, Ul’yanovsk, Russia

    N. O. Sedova

Authors
  1. N. O. Sedova
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Original Russian Text © N.O. Sedova, 2008, published in Avtomatika i Telemekhanika, 2008, No. 6, pp. 70–81.

This work was supported by the Russian Foundation for Basic Research, project no. 08-01-00741(a), and the Program “Leading Scientific Schools,” project no. NSh-6667.2006.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sedova, N.O. Local and semiglobal stabilization in a cascade with delay. Autom Remote Control 69, 968–979 (2008). https://doi.org/10.1134/S0005117908060076

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0005117908060076

PACS number

Search

Navigation