Skip to main content
Springer Nature Link
Log in

Control of systems with aftereffect in scales of linear controllers with respect to the type of feedback

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

Abstract

For control systems with deviating argument, we consider basic problems of qualitative control theory such as problems of stabilization and modal control in scales of linear controllers with respect to the type of feedback, from the simplest difference controllers to integral controllers of general form. We analyze the results obtained in this direction. Special attention is paid to the stabilization of two-dimensional systems by feedback in the form of difference controllers. For the case in which the construction of a difference controller is impossible or too difficult, an integral feedback is used. Unlike the well-known Krasovskii-Osipov method for the construction of integral feedback in delay systems, the suggested method is based on the Paley-Wiener theorem for entire functions of exponential type.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. Krasovskii, N.N. and Osipov, Yu.S., On the Stabilization of Motions of a Plant with Delays in the Control System, Izv. Akad. Nauk SSSR Tehn. Kibernet., 1963, no. 6, pp. 3–15.

  2. Osipov, Yu.S., On the Stabilization of Control Systems with Delay, Differ. Uravn., 1965, vol. 1, no. 5, pp. 606–618.

    Google Scholar 

  3. Gabelaya, A.G., Ivanenko, V.I., and Odarich, O.N., Stabilization of Linear Autonomous Control Systems, Kibernetika, 1975, no. 3, pp. 69–72.

  4. Pandolfi, L., Stabilization of Neutral Functional-Differential Equations, J. Optim. Theory Appl., 1976, vol. 20, pp. 191–204.

    Article  MathSciNet  MATH  Google Scholar 

  5. Olbrot, A.W., Stabilizability, Detectability and Spectrum Assignment for Linear Autonomous Systems with General Time Delays, IEEE Trans. Automat. Control, 1978, vol. AC-23, pp. 887–890.

    Article  MathSciNet  Google Scholar 

  6. Watanabe, K., Stabilization of Linear Systems with Non-Commensurate Delays, Internat. J. Control, 1988, vol. 48, no. 81, pp. 333–342.

    Article  MathSciNet  MATH  Google Scholar 

  7. Borkovskaya, I.M. and Marchenko, V.M., On an Approach to the Problem of the Stabilization of Systems with Delay, Prikl. Mat. Mekh., 1996, vol. 60, no. 4, pp. 531–541.

    MathSciNet  Google Scholar 

  8. Rabah, R., Sklyar, G.M., and Rezounenko, A.V., On Strong Stability and Stabilizability of Linear Systems of Neutral Type, Adv. Time-Delay Systems, Berlin, 2004, pp. 257–268.

  9. Marchenko, V.M. and Yakimenko, A.A., On the Stabilizing Controller Design for Delay Systems of Neutral Type, Differ. Uravn., 2007, vol. 43, no. 11, pp. 1480–1486.

    MathSciNet  Google Scholar 

  10. Letov, A.M., Analytic Construction of Controllers, Avtomat. i Telemekh., 1960, no. 4, pp. 436–441; no. 5, pp. 561–568; no. 6, pp. 661–665; 1961, no. 4, pp. 425–435.

  11. Rosenbrock, H.H., Distinctive Problems of Process Control, Chem. Engng. Progr., 1962, vol. 58, no. 9, pp. 43–50.

    Google Scholar 

  12. Wonham, W.M., On Pole Assignment in Multi-Input Controllable Systems, IEEE Trans. Automat. Control, 1967, vol. AC-12, no. 6, pp. 660–665.

    Article  Google Scholar 

  13. Porter, B. and Grossley, T.R., Modal Control, London, 1972.

  14. Kuzovkov, N.T., Modal’noe upravlenie i nablyudayushchie ustroistva (Modal Control and Observing Devices), Moscow, 1976.

  15. Andreev, Yu.N., Algebraic State Space Methods in Linear Plant Control Theory (Survey of Foreign Literature), Avtomat. i Telemekh., 1977, no. 3, pp. 5–50.

  16. Bulatov, V.I., Kalyuzhnaya, T.S., and Naumovich, R.F., Control of the Spectrum of Differential Equations, Differ. Uravn., 1974, vol. 10, no. 11, pp. 1946–1952.

    MATH  Google Scholar 

  17. Morse, A.S., Ring Models for Delay-Differential Systems, Automatica, 1976, vol. 12, no. 5, pp. 529–531.

    Article  MathSciNet  MATH  Google Scholar 

  18. Asmykovich, I.K. and Marchenko, V.M., Spectrum Control in Systems with Delay, Avtomat. i Telemekh., 1976, no. 7, pp. 5–14.

  19. Marchenko, V.M., On the Problem of Modal Control in Linear Systems with Delays, Dokl. Akad. Nauk BSSR, 1978, vol. 22, no. 5, pp. 401–404.

    MathSciNet  MATH  Google Scholar 

  20. Watanabe, K., Finite Spectrum Assignment of Linear Systems with a Class of Noncommensurate Delays, Internat. J. Control, 1987, vol. 47, no. 5, pp. 1277–1289.

    Article  Google Scholar 

  21. Marchenko, V.M., Modal Control in Systems with Aftereffect, Avtomat. i Telemekh., 1988, no. 11, pp. 73–83.

  22. Gorecki, H., Fuksa, S., Grabowski, P., and Korytowski, A., Analysis and Synthesis of Time Delay Systems, Warszawa, 1989.

  23. Borkovskaya, I.M. and Marchenko, V.M., Modal Control of Systems with Distributed Delay under Incomplete Information Conditions, Differ. Uravn., 1993, vol. 29, no. 11, pp. 1928–1936.

    MathSciNet  MATH  Google Scholar 

  24. Marchenko, V.M. and Yakimenko, A.A., On the Modal Control of Multi-Input Systems with Retarded Argument of Neutral Type, Differ. Uravn., 2008, vol. 44, no. 11, pp. 1534–1543.

    MathSciNet  Google Scholar 

  25. Kalman, R., Falb, P., and Arbib, M., Topics in Mathematical System Theory, New York: McGraw-Hill, 1969. Translated under the title Ocherki po matematicheskoi teorii sistem, Moscow: Mir, 1971.

    MATH  Google Scholar 

  26. Kamen, E.W., Linear Systems with Commensurate Time Delays: Stability and Stabilization Independent of Delay, IEEE Trans. Automat. Control, 1982, vol. AC-27, no. 2, pp. 367–375.

    Article  MathSciNet  Google Scholar 

  27. Datko, R., Remarks Concerning the Asymptotic Stability and Stabilization of Linear Delay Differential Equations, Math. Anal. Appl., 1985, vol. 111, no. 2, pp. 571–584.

    Article  MathSciNet  MATH  Google Scholar 

  28. Filimonov, A.B., Controlling the Spectrum of Linear Stationary Objects with Lags, Avtomat. i Telemekh., 1983, no. 4, pp. 34–42.

  29. Shilov, G.E., Matematicheskii analiz. Spetsial’nyi Kurs (Mathematical Analysis. Special Course), Moscow, 1960.

  30. Pearson, J.B. and Ding, C.Y., Compensator Design for Multivariable Linear Systems, IEEE Trans. Automat. Control, 1985, vol. AC-14, no. 2, pp. 130–134.

    MathSciNet  Google Scholar 

  31. El’sgol’ts, L.E. and Norkin, S.B., Vvedenie v teoriyu differentsial’nykh uravnenii s otklonyayushchimsya argumentom (Introduction to the Theory of Differential Equations with Deviating Argument), Moscow: Nauka, 1971.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Belarus State Technological University, Minsk, Belarus

    V. M. Marchenko

  2. Białystok University of Technology, Białystok, Poland

    V. M. Marchenko

Authors
  1. V. M. Marchenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Original Russian Text © V.M. Marchenko, 2011, published in Differentsial’nye Uravneniya, 2011, Vol. 47, No. 7, pp. 1003–1017.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Marchenko, V.M. Control of systems with aftereffect in scales of linear controllers with respect to the type of feedback. Diff Equat 47, 1014–1028 (2011). https://doi.org/10.1134/S0012266111070111

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0012266111070111

Keywords

Search

Navigation