Skip to main content
SpringerLink
Log in

Linear stabilization of nonlinear cascade systems

  • Published:
Mathematics of Control, Signals and Systems Aims and scope Submit manuscript

Abstract

In this paper we consider the cascade connection of a nonlinear system and a system of integrators. Under suitable conditions we prove that if the nonlinear subsystem is stabilizable by means of a linear feedback, then a linear stabilizer exists for the overall system as well. In particular, we point out the role of the classical notion ofk-asymptotic stability.

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. D. Aeyels, Stabilization of a class of nonlinear systems by a smooth feedback control,Systems Control Lett.,5 (1985), 289–294.

    Google Scholar 

  2. D. Aeyels and M. Szafranski, Comments on the stabilizability of the angular velocity of a rigid body,Systems Control Lett.,10 (1988), 35–39.

    Google Scholar 

  3. V. Andriano, Global feedback stabilization of the angular velocity of a symmetric rigid body,Systems Control Lett.,20 (1993), 361–364.

    Google Scholar 

  4. A. Bacciotti,Local Stabilizability of Nonlinear Control Systems, Advances in Mathematics for Applied Sciences, Vol. 8, World Scientific, Singapore, 1991.

    Google Scholar 

  5. A. Bacciotti, Linear feedback: the local and potentially global stabilization of cascade systems,Proceedings of the 1992 IF AC Symposium on Nonlinear Control (NOLCOS), Bordeaux, Pergamon, Oxford, 1993.

    Google Scholar 

  6. A. Bacciotti and P. Boieri, Linear stabilizability of planar nonlinear systems,Math. Control Signals Systems,3 (1990), 183–193.

    Google Scholar 

  7. A. Bacciotti and P. Boieri, Stabilizability of oscillatory systems: a classical approach supported by symbolic computation,Le Matematiche,XLV (1990), 319–335.

    Google Scholar 

  8. S. Bernfeld and L. Salvadori, Generalized Hopf bifurcation andh-asymptotic stability,Nonlinear Anal.,4 (1980), 1091–1107.

    Google Scholar 

  9. W. M. Boothby and R. Marino, Feedback stabilization of planar nonlinear systems,Systems Control Lett.,12 (1989), 87–92.

    Google Scholar 

  10. S. Bromberg and R. Suarez, Stabilization of two-dimensional C* control systems,Proceedings of the MTNS-91 International Symposium, Kobe, MITA Press, 1991.

    Google Scholar 

  11. S. Caprino and P. Negrini, Attractivity properties of closed orbits in a case of generalized Hopf bifurcation,J. Math. Anal. Appl.,80 (1981), 92–101.

    Google Scholar 

  12. J. M. Coron and L. Praly, Adding an integrator for the stabilization problem,Systems Control Lett.,17 (1991), 89–104.

    Google Scholar 

  13. W. P. Dayawansa, C. F. Martin, and G. Knowles, Asymptotic stabilization of a class of smooth two-dimensional systems,SIAM J. Control Optim.,28 (1990), 1321–1349.

    Google Scholar 

  14. J. H. Fu, Linear feedback stabilization of critical bilinear systems,Proceedings of the 28th IEEE Conference on Decision and Control, Tampa, FL, 1989, pp. 1034–1037.

  15. J. H. Fu and E. H. Abed, Linear feedback stabilization of bifurcations and critical nonlinear systems,Proceedings of the 28th IEEE Conference on Decision and Control, Tampa, FL, 1989, pp. 1363–1365.

  16. J. H. Fu and E. H. Abed, Linear feedback stabilization of nonlinear systems,Proceedings of the 30th IEEE Conference on Decision and Control, Brighton, 1991, pp. 58–63.

  17. J. Guckenheimer and P. Holmes,Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983.

    Google Scholar 

  18. W. Hahn,Stability of Motion, Springer-Verlag, Berlin, 1967.

    Google Scholar 

  19. H. Hermes, Asymptotically stabilizing feedback controls,J. Differential Equations,92 (1991), 76–89.

    Google Scholar 

  20. H. Hermes, Asymptotic stabilization of planar systems,Systems Control Lett.,17 (1991), 437–443.

    Google Scholar 

  21. A. Isidori,Nonlinear Control Systems: An Introduction, 2nd edn., Springer-Verlag, New York, 1989.

    Google Scholar 

  22. M. Kawski, Stabilization of nonlinear systems in the plane,Systems Control Lett.,12 (1989), 169–175.

    Google Scholar 

  23. I. G. Malkin,Theorie der StabilitÄt einer Bewegung, Oldenburg, Munich, 1959.

    Google Scholar 

  24. V. Moauro, A stability problem for holonomic systems,Ann. Mat. Pura Appl.,139 (1985), 227–235.

    Google Scholar 

  25. P. Negrini and L. Salvadori, Attractivity and Hopf bifurcation,Nonlinear Anal.,3(1979), 87–99.

    Google Scholar 

  26. J. Palis, Jr., and W. de Melo,Geometric Theory of Dynamical Systems: An Introduction, Springer-Verlag, New York, 1982.

    Google Scholar 

  27. E. D. Sontag, A “universal” construction of Artstein's theorem on nonlinear stabilization,Systems Control Lett.,13 (1989), 117–123.

    Google Scholar 

  28. H. J. Sussmann and P. V. Kokotovic, The peaking phenomenon and the global stabilization of nonlinear systems,IEEE Trans. Automat. Control,36 (1991), 424–440.

    Google Scholar 

  29. J. Tsinias, Sufficient Lyapunov-like conditions for stabilizability,Math. Control Signals Systems,2 (1989), 343–357.

    Google Scholar 

  30. J. Tsinias, A theorem on global stabilization of nonlinear systems by linear feedback,Systems Control Lett.,17 (1991), 357–362.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Torino, Italy

    A. Bacciotti, P. Boieri & L. Mazzi

Authors
  1. A. Bacciotti
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. Boieri
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. L. Mazzi
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This work has been partially supported by the Ministero dell'Università e delia Ricerca Scientifica e Tecnologica (Italy).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bacciotti, A., Boieri, P. & Mazzi, L. Linear stabilization of nonlinear cascade systems. Math. Control Signal Systems 6, 146–165 (1993). https://doi.org/10.1007/BF01211745

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01211745

Key words

Search

Navigation