Skip to main content
SpringerLink
Log in

Optimization of nonlinear cascade systems in Lurie form with bounded external disturbances

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

Abstract

Optimization of nonlinear cascade systems in the form of Lurie with bounded disturbances is considered. For the solution of the problem the method of invariant ellipsoids is used. The obtained result is compared with the previously obtained classic result based on Lyapunov function.

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. Andrievskii, B.R. and Fradkov, A.L., Izbrannye glavy teorii avtomaticheskogo upravleniya s primerami na yazyke MATLAB (Selected Chapters of the Theory of Automatic Control with Examples in the MATLAB Language), St. Petersburg: Nauka, 1999.

    MATH  Google Scholar 

  2. Andrievskii, B.R. and Frakov, A.L., Elementy matematicheskogo modelirovaniya v programmnykh sredakh MATLAB 5 i Scilab (uchebnoe posobie) (Elements of Mathematical Modeling in Program Media of MATLAB 5 and Scilab (Manual)), St. Petersburg: Nauka, 2001.

    Google Scholar 

  3. Andrievskii, B.R. and Fradkov, A.L., Method of Passification in Problems of Adaptive Control, Estimation, and Synchronization, Autom. Remote Control, 2006, vol. 67, no. 11, pp. 1699–1731.

    Article  MathSciNet  MATH  Google Scholar 

  4. Bobtsov, A.A. and Nikolaev, N.A., Fradkov Theorem-Based Design of the Control of Nonlinear Systems with Functional and Parametric Uncertainties, Autom. Remote Control, 2005, vol. 66, no. 1, pp. 108–118.

    Article  MathSciNet  MATH  Google Scholar 

  5. Miroshnik, I.V., Nikiforov, V.O., and Fradkov, A.L., Nelineinoe i adaptivnoe upravlenie slozhnymi sistemami (Nonlinear and Adaptive Control of Complex Systems), St. Petersburg: Nauka, 2000.

    MATH  Google Scholar 

  6. Polushin, I.G., Fradkov, A.L., and Hill, D.J., Passivity and Passification of Nonlinear Systems, Autom. Remote Control, 2000, vol. 61, no. 3, part 1, pp. 355–388.

    Google Scholar 

  7. Polyak, B.T., Khlebnikov, M.V., and Shcherbakov, P.S., Nonlinear Systems with Bounded or Multiplicative Disturbances. http://premolab.ru/publication/14/

  8. Usik, E.V., Synchronization of Nonlinear Lurie Systems on the Basis of Passification and Backstepping, Autom. Remote Control, 2012, vol. 73, no. 8, pp. 1305–1315.

    Article  MathSciNet  MATH  Google Scholar 

  9. Fradkov, A.L., Synthesis of Adaptive System of Stabilization of Linear Dynamic Plants, Autom. Remote Control, 1974, vol. 35, no. 12, part 2, pp. 1960–1966.

    Google Scholar 

  10. Fradkov, A.L., Lyapunov Quadratic Functions in the Problem of Adaptive Stabilization of a Linear Dynamic Object, Sib. Mat. Zh., 1976, no. 2, pp. 436–446.

    MATH  Google Scholar 

  11. Khlebnikov, M.V., Polyak, B.T., and Kuntsevich, V.M., Optimization of Linear Systems Subject to Bounded Exogenous Disturbances: The Invariant Ellipsoid Technique, Autom. Remote Control, 2011, vol. 72, no. 11, pp. 2227–2275.

    Article  MathSciNet  MATH  Google Scholar 

  12. Khlebnikov, M.V., Settling Time in a Linear Dynamic System with Bounded External Disturbances, Autom. Remote Control, 2012, vol. 73, no. 6, pp. 937–948.

    Article  MathSciNet  MATH  Google Scholar 

  13. Yakubovich, V.A., Method of Matrix Inequalities in the Theory of Stability of the Nonlinear Controllable Systems. I. Absolute Stability of Forced Oscillations, Avtom. Telemekh., 1964, vol. 25, no. 7, pp. 1017–1029.

    Google Scholar 

  14. Byrnes, C.I., Isidori, A., and Willems, J.C., Passivity, Feedback Equivalence, and the Global Stabilization of Minimum Phase Nonlinear Systems, IEEE Trans. Automat. Control, 1991, vol. AC-36, no. 11, pp. 1228–1240.

    Google Scholar 

  15. Fradkov, A.L., Passification of Non-square Linear Systems and Feedback Yakubovich–Kalman–Popov Lemma, Eur. J. Control, 2003, vol. 9, no. 11, pp. 573–582.

    MATH  Google Scholar 

  16. Gusev, S.V., Paromtchik, I.E., Makarov, I.A., and Yakubovich, V.A., Adaptive Motion Control of Nonholonomic Vehicle, Proc. IEEE Int. Conf. Robot. Automat., Belgium, 1998, vol. 4, pp. 3285–3290.

    Google Scholar 

  17. Kokotovich, P. and Arcak, M., Constructive Nonlinear Control: A Historical Perspective, Automatica, 2001, vol. 37, no. 5, pp. 637–662.

    Article  MathSciNet  MATH  Google Scholar 

  18. Krstic, M., Kanellakopoulas, I., and Kokotovich, P., Nonlinear and Adaptive Control Design, New York: Wiley, 1995.

    Google Scholar 

  19. Latombe, J.C., Robot Motion Planning, Boston: Kluwer, 1991.

    Book  MATH  Google Scholar 

  20. Nikiforov, V.O. and Voronov, K.V., Nonlinear Adaptive Controller with Integral Action, IEEE Trans. Automat. Control, 2001, vol. 46, no. 12, pp. 2035–2037.

    Article  MathSciNet  MATH  Google Scholar 

  21. Polyak, B.T., Convexity of Quadratic Transformations and Its Use in Control and Optimization, J. Optim. Theory Appl., 1998, vol. 99, pp. 553–583.

    Article  MathSciNet  MATH  Google Scholar 

  22. Pyrkin, A., Bobtsov, A., Kolyubin, S., et al., Output Control Approach “Consecutive Compensator” Providing Exponential and L-Infinity-Stability for Nonlinear Systems with Delay and Disturbance, IEEE Multi-Conf. Syst. Control, Denver, 2011, pp. 1499–1504.

    Google Scholar 

  23. Seron, M.M., Hill, D.J., and Fradkov, A.L., Adaptive Passification of Nonlinear Systems, Proc. 33rd IEEE Conf. Decision Control, Orlando, 1994, pp. 190–195.

    Google Scholar 

  24. Seron, M.M., Hill, D.J., and Fradkov, A.L., Nonlinear Adaptive Control of Feedback Passive Systems, Automatica, 1995, vol. 31, no. 7, pp. 1053–1060.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    E. V. Usik

Authors
  1. E. V. Usik
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. V. Usik.

Additional information

Original Russian Text © E.V. Usik, 2015, published in Upravlenie Bol’shimi Sistemami, 2015, No. 57, pp. 37–52.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Usik, E.V. Optimization of nonlinear cascade systems in Lurie form with bounded external disturbances. Autom Remote Control 78, 1350–1359 (2017). https://doi.org/10.1134/S0005117917070165

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0005117917070165

Search

Navigation