Controllability and Observability, Local Decompositions



In the first two sections of this chapter we will give some basic concepts and results in the study of controllability and observability for nonlinear systems. Roughly speaking we will restrict ourselves to what can be seen as the nonlinear generalizations of the Kalman rank conditions for controllability and observability of linear systems. The reason for this is that in the following chapters we will not need so much the notions of nonlinear controllability and observability per se, but only the “structural properties” as expressed by these nonlinear “controllability” and “observability” rank conditions that will be obtained. In the last section of this chapter we will show how the geometric interpretation of reachable and unobservable subspaces for linear systems as invariant subspaces enjoying some maximality or minimality properties can be generalized to the nonlinear case, using the notion of invariant distributions. In this way we make contact with the nonlinear generalization of linear geometric control theory as dealt with in later chapters, where this last notion plays a fundamental role.


Nonlinear System Neighborhood Versus Rank Condition Invariant Distribution Integral Manifold 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Bai81]
    J. Baillieul. Controllability and observability of polynomial dynamical systems. Nonlinear Anal., Theory, Meth. and Appl., 5:543–552, 1981.Google Scholar
  2. [Bro72]
    R.W. Brockett. System theory on group manifolds and coset spaces. SIAM J. Contr., 10:265–284, 1972.Google Scholar
  3. [Cho39]
    W.L. Chow. Uber Systemen von linearen partiellen Differentialgleichungen erster Ordnung. Math. Ann., 117:98–105, 1939.Google Scholar
  4. [Cro81a]
    P.E. Crouch. Dynamical realizations of finite Volterra series. SIAM J. Contr., 19:177–202, 1981.Google Scholar
  5. [Cro81b]
    P.E. Crouch. Lecture notes on geometric non linear systems theory. University of Warwick. Control Theory Centre, 1981.Google Scholar
  6. [Cro84]
    P.E. Crouch. Spacecraft attitude control and stabilization. IEEE Trans. Aut. Contr., AC-29:321–331, 1984.Google Scholar
  7. [Ell70]
    D.L. Elliott. A consequence of controllability. J. Diff. Eqns., 10:364–370, 1970.Google Scholar
  8. [GB81]
    J.P. Gauthier and G. Bornard. Observability for any u(t) of a class of nonlinear systems. IEEE Trans. Autom. Contr., AC-26:922–926, 1981.Google Scholar
  9. [Hel62]
    S. Helgason. Differential geometry and symmetric spaces. Academic, New York, 1962Google Scholar
  10. [Her63]
    R. Hermann. On the accessibility problem in control theory. In J.P. La Salle and S. Lefschetz, editors, Int. Symp. on Nonlinear Differential Equations and Nonlinear Mechanics, pages 325–332, New York, 1963. Academic.Google Scholar
  11. [Her82]
    H. Hermes. Control systems which generate decomposable Lie Algebras. J. Diff. Eqns., 44:166–187, 1982.Google Scholar
  12. [HH70]
    G.W. Haynes and H. Hermes. Nonlinear controllability via Lie theory. SIAM J. Contr., 8:450–460, 1970.Google Scholar
  13. [Hir81]
    R.M. Hirschorn. (A, B)-invariant distributions and disturbance decoupling. SIAM J. Contr., 19:1–19, 1981.Google Scholar
  14. [HK77]
    R. Hermann and A.J. Krener. Nonlinear controllability and observability. IEEE Trans. Aut. Contr., AC-22:728–740, 1977.Google Scholar
  15. [IKGM81]
    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
  16. [Isi85]
    A. Isidori. Nonlinear Control Systems: An Introduction, volume 72 of Lect. Notes Contr. Inf. Sci. Springer, Berlin, 1985.Google Scholar
  17. [Kre74]
    A.J. Krener. A generalization of Chow’s theorem and the Bang-bang theorem to nonlinear control systems. SIAM J. Contr., 12:43–52, 1974.Google Scholar
  18. [Kre77]
    A.J. Krener. A decomposition theory for differentiable systems. SIAM J. Contr., 15:289–297, 1977.Google Scholar
  19. [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
  20. [LM67]
    E.B. Lee and L. Markus. Foundations of Optimal Control Theory. Wiley, New York, 1967.Google Scholar
  21. [Lob70]
    C. Lobry. Contrôlabilité des syst`emes non linéaires. SIAM J. Contr., 8:573–605, 1970.Google Scholar
  22. [Nel67]
    E. Nelson. Tensor analysis. Princeton University Press, Princeton, 1967.Google Scholar
  23. [Res82]
    W. Respondek. On decomposition of nonlinear control systems. Syst. & Contr. Lett., 1:301–308, 1982.Google Scholar
  24. [SJ72]
    H.J. Sussmann and V. Jurdjevic. Controllability of nonlinear systems. J. Diff. Eqns., 12:95–116, 1972.Google Scholar
  25. [Son85]
    E. Sonntag. A concept of local observability. Systems Control Lett., 5:41–47, 1985.Google Scholar
  26. [Ste86]
    G. Stefani. On the local controllability of a scalar input-control system. In C.I. Byrnes and A. Lindquist, editors, Theory and Applications of Nonlinear Control Systems, pages 167–182. North-Holland, Amsterdam, 1986.Google Scholar
  27. [Sus73]
    H.J. Sussmann. Orbits of families of vectorfields and integrability of distributions. Trans. American Math. Soc., 180:171–188, 1973.Google Scholar
  28. [Sus77]
    H.J. Sussmann. Existence and uniqueness of minimal realizations of nonlinear systems. Math. Syst. Theory, 10:263–284, 1977.Google Scholar
  29. [Sus83]
    H.J. Sussmann. Lie brackets, real analyticity and geometric control. In R.W. Brockett, R.S. Millman, and H.J. Sussmann, editors, Differential Geometric Control Theory, pages 1–116. Birkh¨auser, Boston, 1983.Google Scholar
  30. [Sus87]
    H.J. Sussmann. A general theorem on local controllability. SIAM J. Contr. Optimiz., 25:158–194, 1987.Google Scholar
  31. [vdS82]
    A.J. van der Schaft. Observability and controllability for smooth nonlinear systems. SIAM J. Contr. Optimiz., 20:338–354, 1982.Google Scholar
  32. [vdS84]
    A.J. van der Schaft. System theoretic descriptions of physical systems. CWI Tract 3, CWI, Amsterdam, 1984.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