Advertisement

Controlled Invariant Distributions and the Disturbance Decoupling Problem

  • Henk Nijmeijer
  • Arjan van der Schaft
Chapter

Abstract

In this chapter, Section 7.1, we will introduce and discuss the concept of controlled invariance for nonlinear systems. Controlled invariant distributions play a crucial role in various synthesis problems like for instance the disturbance decoupling problem and the input-output decoupling problem. A detailed account of the disturbance decoupling problem together with some worked examples will be given in Section 7.2. Later, in Chapter 9 we will exploit controlled invariant distributions in the input-output decoupling problem.

Keywords

Nonlinear System Control Invariance Invariant Distribution Constant Dimension Output Invariance 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [BK83]
    C.I. Byrnes and A.J. Krener. On the existence of globally (f, g)-invariant distributions. In R.W. Brockett, R.S. Millman, and H.J. Sussmann, editors, Differential Geometric Control Theory, pages 209–225. Birkhäuser, Boston, 1983.Google Scholar
  2. [BM69]
    G. Basile and G. Marro. Controlled and conditioned invariant subspaces in linear systems theory. J. Optimiz. Th. Applic, 3:306–315, 1969.Google Scholar
  3. [Cla82]
    D. Claude. Decoupling of nonlinear systems. Syst. Contr. Lett., 1:242–248, 1982.Google Scholar
  4. [CT89]
    D. Cheng and T.J. Tarn. New results on (f, g)-invariance. Syst. Contr. Lett., 12:319–326, 1989.Google Scholar
  5. [dBI86]
    M.D. di Benedetto and A. Isidori. The matching of nonlinear models via dynamic state feedback. SIAM J. Contr. Optimiz., 24:1063–1075, 1986.Google Scholar
  6. [GBBI83]
    J.P. Gauthier, G. Bornard, S. Bacha, and M. Idir. Rejet des perturbations pour un modèle non-linéaire de colonne à distiller. In I.D. Landau, editor, Outils et Modèles Mathématiques pour l’Automatique, l’Analyse de Systèmes et le Traitement du Signal, volume III, pages 459–573. Editions du CNRS, Paris, 1983.Google Scholar
  7. [Hir81]
    R.M. Hirschorn. (A, B)-invariant distributions and disturbance decoupling of nonlinear systems. SIAM J. Contr. Optimiz, 19:1–19, 1981.Google Scholar
  8. [IKGGM81a]
    A. Isidori, A.J. Krener, C. Gori-Giorgi, and S. Monaco. Locally (f, g)-invariant distributions. Syst. Contr. Lett., 1:12–15, 1981.Google Scholar
  9. [IKGGM81b]
    A. Isidori, A.J. Krener, C. Gori-Giorgi, and S. Monaco. Nonlinear decoupling via feedback: a differential geometric approach. IEEE Trans. Aut. Contr., AC-26:331–345, 1981.Google Scholar
  10. [Isi81]
    A. Isidori. Sur la théorie structurelle et la problème de la rejection des perturbations dans les systèmes non linéaires. In I.D. Landau, editor, Outils et Modèles Mathématiques pour l’Automatique, l’Analyse de Systèmes et le Traitement du Signal, volume I, pages 245–294. Editions du CNRS, Paris, 1981.Google Scholar
  11. [Isi85]
    A. Isidori. Nonlinear Control Systems: An Introduction, volume 72 of Lect. Notes Contr. Inf. Sci. Springer, Berlin, 1985.Google Scholar
  12. [Kre81]
    A.J. Krener. (f, g)-invariant distributions, connections and Pontryagin classes. In Proceedings 20th IEEE Conf. Decision Control, San Diego, pages 1322–1325, 1981.Google Scholar
  13. [Kre85]
    A.J. Krener. (Ad f, g), (ad f, g) and locally (ad f, g) invariant and controllability distributions. SIAM J. Contr. Optimiz., 23:523–549, 1985.Google Scholar
  14. [MG83]
    C.H. Moog and A. Glumineau. Le problème du rejet de perturbations mesurables dans les systèmes non linéaires. Application à l’amarrage en un seul point des grands petroliers. In I.D. Landau, editor, Outils et Modèles Mathématiques pour l’Automatique, l’Analyse de Systèmes et le Traitement du Signal, volume III, pages 689–698. Editions du CNRS, Paris, 1983.Google Scholar
  15. [MW78]
    S.H. Mikhail and W.M. Wonham. Local decomposability and the disturbance decoupling problem in nonlinear autonomous systems. Allerton Conf. Comm. Contr. Comp., 16:664–669, 1978.Google Scholar
  16. [Nij81]
    H. Nijmeijer. Controlled invariance for affine control systems. Int. J. Contr., 34:824– 833, 1981.Google Scholar
  17. [NvdS82a]
    H. Nijmeijer and A.J. van der Schaft. Controlled invariance by static output feedback. Syst. Contr. Lett., 2:39–47, 1982.Google Scholar
  18. [NvdS82b]
    H. Nijmeijer and A.J. van der Schaft. Controlled invariance for nonlinear systems. IEEE Trans. Aut. Contr., AC-27:904–914, 1982.Google Scholar
  19. [NvdS84]
    H. Nijmeijer and A.J. van der Schaft. Controlled invariance for nonlinear systems: two worked examples. IEEE Trans. Aut. Contr., AC-29:361–364, 1984.Google Scholar
  20. [WM70]
    W.M. Wonham and A.S. Morse. Decoupling and pole assignment in linear multivariable systems: a geometric approach. SIAM J. Contr. Optimiz., 8:1–18, 1970.Google Scholar
  21. [Won79]
    W.M. Wonham. Linear multivariable control: a geometric approach. Springer, Berlin, 1979.Google Scholar

Copyright information

© Springer Science+Business Media New York 1990, Corrected printing 2016 1990

Authors and Affiliations

  1. 1.Dynamics and Control GroupEindhoven University of TechnologyEindhovenThe Netherlands
  2. 2.Johann Bernoulli Institute for Mathematics and Computer ScienceUniversity of GroningenGroningenThe Netherlands

Personalised recommendations