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Journal of Mathematical Sciences

, Volume 212, Issue 4, pp 397–411 | Cite as

Robust Stabilization of Nonlinear Mechanical Systems

  • L. V. Kupriyanchyk
Article

We study the problems of robust stabilization and optimization of the equilibrium states of nonlinear mechanical systems. Sufficient conditions for the stabilization of a linear system with measured output feedback are formulated by means of the full-order state observers. The solution of the general problem of robust stabilization and the estimates of the quadratic performance criterion are presented for a family of nonlinear systems on examples of a one-link pendulum on a moving platform in the upper equilibrium position and a pendulum with flywheel control. The application of the obtained results is reduced to the solution of systems of linear matrix inequalities.

Keywords

Linear Matrix Inequality Robust Stabilization Ukrainian National Academy Nonlinear Control System Dynamic Feedback 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    B. T. Polyak and P. S. Shcherbakov, Robust Stability and Control [in Russian], Nauka, Moscow (2002).Google Scholar
  2. 2.
    K. Zhou, J. C. Doyle, and K. Glover, Robust and Optimal Control, Prentice Hall, Englewood (1996).zbMATHGoogle Scholar
  3. 3.
    D. V. Balandin and M. M. Kogan, Synthesis of the Regularities of Control on the Basis of Linear Matrix Inequalities [in Russian], Fizmatlit, Moscow (2007).Google Scholar
  4. 4.
    B. T. Polyak and P. S. Shcherbakov, “Complex problems of the linear control theory. Some approaches to the solution,” Avtomat. Telemekh., No. 5, 7–46 (2005).Google Scholar
  5. 5.
    V. L. Syrmos, C. T. Abdallah, P. Dorato, and K. Grigoriadis, “Static output feedback: a survey,” Automatica, 33, No. 2, 125–137 (1997).Google Scholar
  6. 6.
    F. A. Aliev and V. B. Larin, “Problems of stabilization of the system with feedback in the output variable (a survey),” Prikl. Mekh., 47, No. 3, 3–49 (2011).MathSciNetGoogle Scholar
  7. 7.
    A. G. Mazko and V. V. Shram, “Stability and stabilization of the family of pseudolinear differential systems,” Nelin. Kolyv., 14, No. 2, 227–237 (2011); English translation: Nonlin. Oscillat., 14, No. 2, 237–248 (2011).Google Scholar
  8. 8.
    O. H. Mazko and L. V. Bohdanovych, “Robust stability and optimization of nonlinear control systems,” in: “Analytic Mechanics and Its Application,” Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences [in Ukrainian], 9, No. 1 (2012), pp. 200–218.Google Scholar
  9. 9.
    O. H. Mazko and L. V. Bohdanovych, “Stabilization of mechanical systems with uncertain parameters,” in: Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences [in Ukrainian], 10, No. 3 (2013), pp. 123–144.Google Scholar
  10. 10.
    S. A. Krasnova and V. A. Utkin, Cascade Synthesis of State Observers for Dynamical Systems [in Russian], Nauka, Moscow (2006).Google Scholar
  11. 11.
    I. Petersen, “A stabilization algorithm for a class of uncertain linear systems,” Syst. Control Lett., 8, No. 4, 351–357 (1987).zbMATHCrossRefGoogle Scholar
  12. 12.
    N. M. Singh, J. Dubey, and G. Laddha, “Control of pendulum on a cart with state dependent Riccati equations,” World Acad. Sci. Eng. Technol., 41, 671–675 (2008).Google Scholar
  13. 13.
    A. A. Grishin, A. V. Lenskii, D. E. Okhotsimskii, D. A. Panin, and A. M. Formal’skii, “On the synthesis of control by an unstable object. Inverted pendulum,” Izv. Ros. Akad. Nauk., Teor. Sist. Upravlen., No. 5, 14–24 (2002).Google Scholar
  14. 14.
    B. R. Andrievskii, “Global stabilization of an unstable pendulum with flywheel control,” Upravl. Bol. Sist., Issue 24, 258–280 (2009).Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Institute of MathematicsUkrainian National Academy of SciencesKyivUkraine

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