Skip to main content
Log in

Stabilization of nonlinear systems: A bilinear approach

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

Abstract

This paper gives a necessary and sufficient condition for the stabilization of planar bilinear systems. Local stabilization results are obtained for other systems.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. D. Aeyels, Local and global stabilizability for nonlinear systems, inTheory and Applications of Nonlinear Control Systems (C. I. Byrnes and A. Lindquist, eds.), Elsevier, Amsterdam, 1986, pp. 93–105.

    Google Scholar 

  2. Z. Artstein, Stabilization with relaxed controls,Nonlinear Anal.,7 (1983), 1163–1173.

    Google Scholar 

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

    Google Scholar 

  4. A Bacciotti and P. Boieri, A characterization of single input planar bilinear systems which admit a smooth stabilizer,Systems Control Lett.,16 (1991), 139–143.

    Google Scholar 

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

    Google Scholar 

  6. R. W. Brockett. Asymptotic stability and feedback stabilization, inDifferential Geometric Control Theory (R. W. Brockett, R. S. Millman, and H. J. Sussmann, eds.), Birhäuser, Boston, 1983, pp. 181–191.

    Google Scholar 

  7. C. Byrnes and A. Isidori, Local stabilization of minimum phase nonlinear systems,Systems Control Lett.,11 (1988), 9–17.

    Google Scholar 

  8. C. Byrnes, A. Isidori, and J. C. Willems, Passivity, feedback equivalence, and the global stabilization of minimum phase nonlinear systems,IEEE Trans. Automat. Control,36 (1991), 1228–1240.

    Google Scholar 

  9. R. Chabour and J. C. Vivalda, Stabilisation des systèmes bilinéaires dans le plan par une commande non régulière,Proceedings of the European Control Conference, Hermes, 1991, pp. 485–487.

  10. W. P. Dayawansa and C. F. Martin, Asymptotic stabilization of two-dimensional real analytic systems,Systems Control Lett,12 (1989), 205–211.

    Google Scholar 

  11. 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 

  12. J. P. Gauthier and G. Bornard, Stabilisation des systèmes non linéaires, inOutils et Modeles Mathematiques pour l'Automatique, l'Analyse de Systemes et le Traitement du Signal, Editions du CNRS, Paris, 1981, pp. 307–324.

    Google Scholar 

  13. H. Hermes, Homogeneous coordinates and continuous stabilizing feedback controls, inDifferential Equations: Stability and Control (S. Elaydi, ed.), Marcel Dekker, New York, 1991.

    Google Scholar 

  14. M. W. Hirsch and S. Smale,Differential Equations, Dynamical Systems and Linear Algebra, Academic Press, New York, 1974.

    Google Scholar 

  15. V. Jurdjevic and J. P. Quinn, Controllability and stability,J. Differential Equations,28 (1978), 381–389.

    Google Scholar 

  16. M. Kawski, Homogeneous stabilizing feedback laws,Control Theory Adv. Tech.,6 (1990), 497–516.

    Google Scholar 

  17. M. Kawski, Stabilization of nonlinear systems in the plane,Systems Control Lett.,12 (1990), 169–175.

    Google Scholar 

  18. J. L. Massera, Contribution to stability theory,Ann of Math.,64 (1956), 182–206.

    Google Scholar 

  19. L. Praly, B. d'Andréa-Novel, and J.-M. Coron, Lyapunov design of stabilizing controllers,Proceedings of the 28th IEEE Conference on Decision and Control, Tampa, FL, 1989, pp. 1047–1052.

  20. N. Rouche and J. Mawhin,Equations Différentielles Ordinaires, vol. 2, Masson, Paris, 1973.

    Google Scholar 

  21. A. Saberi, P. V. Kokotović, and H. J. Sussmann. Global stabilization of partially linear composed systems.SIAM J. Control Optim.,28 (1990), 1491–1503.

    Google Scholar 

  22. E. D. Sontag, A Lyapunov-like characterization of asymptotic controllability,SIAM J. Control Optim.,21 (1983), 462–471.

    Google Scholar 

  23. E. D. Sontag and H. J. Sussmann, Further comments on the stabilizability of the angular velocity of the rigid body,Systems Control Lett.,12 (1988), 213–217.

    Google Scholar 

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

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chabour, R., Sallet, G. & Vivalda, J.C. Stabilization of nonlinear systems: A bilinear approach. Math. Control Signal Systems 6, 224–246 (1993). https://doi.org/10.1007/BF01211621

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01211621

Key words

Navigation