State Space Transformations and Feedback

  • Henk Nijmeijer
  • Arjan van der Schaft


This chapter deals with some preliminaries which are basic to controller and observer design for nonlinear systems. In particular we discuss the possibility of linearizing a system by state space transformations and we introduce various types of nonlinear feedback.


Linear System Nonlinear System Coordinate Transformation State Feedback Output Feedback 
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.


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  1. [Ban88]
    S.P. Banks. Mathematical Theories of Nonlinear Systems. Prentice Hall, Hertfordshire, 1988.Google Scholar
  2. [Bro78]
    R.W. Brockett. Feedback invariants for nonlinear systems. In Preprints 6th IFAC Congress, Helsinki, pages 1115–1120, 1978.Google Scholar
  3. [Bro80]
    R.W. Brockett. Global descriptions of nonlinear control problems; vector bundles and nonlinear control theory. Notes for a CBMS conference, manuscript, 1980.Google Scholar
  4. [BZ83]
    D. Bestle and M. Zeitz. Canonical form observer design for nonlinear time-variable systems. Int. J. Control, 38:419–431, 1983.Google Scholar
  5. [FK83]
    M. Fliess and I. Kupka. A finiteness criterion for nonlinear input-output differential systems. SIAM J. Contr. Optimiz., 21:721–728, 1983.Google Scholar
  6. [Fre75]
    E. Freund. The structure of decoupled nonlinear systems. Int. J. Contr., 21:443–450, 1975.Google Scholar
  7. [Kai80]
    T. Kailath. Linear Systems. Prentice Hall, Englewood Cliffs, N.J., 1980.Google Scholar
  8. [KI83]
    A.J. Krener and A. Isidori. Linearization by output injection and nonlinear observers. Systems Control Lett., 3:47–52, 1983.Google Scholar
  9. [KR85]
    A.J. Krener and W. Respondek. Nonlinear observers with linearizable error dynamics. SIAM J. Contr. Optimiz., 23:197–216, 1985.Google Scholar
  10. [Kre73]
    A.J. Krener. On the equivalence of control systems and linearization of nonlinear systems. SIAM J. Contr. Optimiz., 11:670–676, 1973.Google Scholar
  11. [MC75]
    G. Meyer and L. Cicolani. A formal structure for advanced automatic flight-control systems. NASA Technical Note TND-7940, Ames Research Center, Moffett Field (Ca), 1975.Google Scholar
  12. [Nij81]
    H. Nijmeijer. Observability of a class of nonlinear systems: a geometric approach. Ricerche di Automatica, 12:1–19, 1981.Google Scholar
  13. [Nij84]
    H. Nijmeijer. State-space equivalence of an affine nonlinear system with outputs to a minimal linear system. Int. J. Contr., 39:919–922, 1984.Google Scholar
  14. [Por69]
    W.A. Porter. Decoupling of and inverses for time-varying linear systems. IEEE Trans. Aut. Contr., 14:378–380, 1969.Google Scholar
  15. [Res85a]
    W. Respondek. Geometric methods in linearization of control systems. In C. Olech, B. Jakubczyk, and J. Zabczyk, editors, Mathematical Control Theory, volume 14 of Banach Center Publications, pages 453–467. PWN, Warsaw, 1985.Google Scholar
  16. [Res85b]
    W. Respondek. Linearization, feedback and Lie brackets. In B. Jakubczyk, W. Respondek, and K. Tchoń, editors, Geometric theory of nonlinear control systems, pages 131–166. Technical University of Wroclaw, Poland, 1985.Google Scholar
  17. [SR72]
    S.N. Singh and W.J. Rugh. Decoupling in a class of nonlinear systems by state variable feedback. J. Dynamic Systems, Measurement and Control, pages 323–329, 1972.Google Scholar
  18. [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. Birkhäuser, Boston, 1983.Google Scholar
  19. [Wil79]
    J.C. Willems. System theoretic models for the analysis of physical systems. Ricerche di Automatica, 10:71–106, 1979.Google Scholar
  20. [XG88]
    Xiao-Hua Xia andWei-Bin Gao. Nonlinear observer design by observer canonical forms. Int. J. Contr., 47:1081–1100, 1988.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

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