Advertisement

Automation and Remote Control

, Volume 78, Issue 1, pp 1–15 | Cite as

Stabilization of one class of nonlinear systems

  • V. I. KorobovEmail author
  • M. O. Bebiya
Nonlinear Systems
  • 61 Downloads

Abstract

Consideration was given to the problem of stabilization for one class of systems that are nonlinear and uncontrollable in the first approximation. The stabilization problem was solved by considering the nonlinear approximation. The stabilizing control was obtained by the method of Lyapunov function which was constructed as a quadratic form. To determine matrix of this quadratic form, a singular matrix Lyapunov equation was solved. For the system of nonlinear approximation, the stabilizing control was determined explicitly. It was proved that the resulting control solves the problem of stabilizing the original nonlinear system. An ellipsoidal estimate of the attraction domain of the zero stationary point was given.

Key words

stabilization with respect to nonlinear approximation Lyapunov function method singular Lyapunov matrix equation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Krasovskii, N.N., Problems of Stabilization of the Controlled Motions, in Malkin, I.G., Teoriya ustoichivosti dvizheniya (Theory of Motion Stability), Moscow: Nauka, 1966, pp. 475–514.Google Scholar
  2. 2.
    Polyak, B.T. and Shcherbakov, P.S., Robastnaya ustoichivost’ i upravlenie (Robust Stability and Control), Moscow: Nauka, 2002.Google Scholar
  3. 3.
    Korobov, V.I., Metod funktsii upravlyaemosti (Method of the Controllability Function), Moscow–Izhevsk: NITs “Regulyarnaya i Khaoticheskaya Dinamika,” Izhev. Inst. Komp’yut. Issled., 2007.Google Scholar
  4. 4.
    Khalil, N.K., Nonlinear Systems, Upper Saddle River: Prentice Hall, 2002.zbMATHGoogle Scholar
  5. 5.
    Kawski, M., Stabilization of Nonlinear Systems in the Plane, SCL, 1989, no. 12, pp. 169–175.MathSciNetzbMATHGoogle Scholar
  6. 6.
    Korobov, V.I. and Bebiya, M.O., Stabilization of One Class of Nonlinear Systems Uncontrollable in the First Approximation, Dokl. Nats. Akad. Nauk Ukrainy, 2014, no. 2, pp. 20–25.CrossRefzbMATHGoogle Scholar
  7. 7.
    Korobov, V.I. and Lutsenko, A.V., Robust Stabilization of One Class of Nonlinear Systems, Autom. Remote Control, 2014, vol. 75, no. 8, pp. 1433–1444.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Saberi, A., Kokotovic, P.V., and Sussmann, H.J., Global Stabilization of Partially Linear Composite Systems, SIAM J. Control Optim., 1990, vol. 28, no. 6, pp. 1491–1503.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Krstic, M., Kanellakopoulos, I., and Kokotovic, P.V., Nonlinear and Adaptive Control Design, NewYork: Wiley, 1995.zbMATHGoogle Scholar
  10. 10.
    Coron, J.-M. and Praly, L., Adding an Integrator for the Stabilization Problem, Syst. Control Lett., 1991, vol. 17, no. 2, pp. 89–104.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Lin, W. and Quan, C., Adding One Power Integrator: A Tool for Global Stabilization of High Order Lower-triangular Systems, Syst. Control Lett., 2000, vol. 39, no. 5, pp. 339–351.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lin, W. and Quan, C., Adaptive Regulation of High-order Lower-triangular Systems: An Adding a Power Integrator Technique, Syst. Control Lett., 2000, vol. 39, no. 5, pp. 353–364.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Zhao, G. and Duan, N., A Continuous State Feedback Controller Design for High-order Nonlinear Systems with Polynomial Growth Nonlinearities, IJAC, 2013, vol. 10, no. 4, pp. 267–274.Google Scholar
  14. 14.
    Sun, Z. and Liu, Y., State-feedback Adaptive Stabilizing Control Design for a Class of High-order Nonlinear Systems with Unknown Control Coefficients, J. Syst. Sci. Complexity, 2007, vol. 20, no. 3, pp. 350–361.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Sklyar, G.M., Sklyar, K.V., and Ignatovich, S.Yu., On the Extension of the Korobov’s Class of Linearizable Triangular Systems by Nonlinear Control Systems of the Class C 1, Syst. Control Lett., 2005, vol. 54, no. 11, pp. 1097–1108.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.V.N. Karazin National UniversityKharkovUkraine

Personalised recommendations