The Important State Coordinates of a Nonlinear System

  • Arthur J. Krener
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 353)


We offer an alternative way of evaluating the relative importance of the state coordinates of a nonlinear control system. Our approach is based on making changes of state coordinates to bring the controllability and observability functions into input normal form. These changes of coordinates are done degree by degree and the resulting normal form is unique through terms of degree seven.


Nonlinear Control Systems Model Reduction 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Al’brecht E G (1961) On the optimal stabilization of nonlinear systems, PMM-Journal of Applied Mathematics and Mechanics, 25:1254–1266CrossRefGoogle Scholar
  2. 2.
    Fujimoto K, Scherpen J M A (2005) Nonlinear Input-Normal Realizations Based on the Differential Eigenstructure of Hankel Operators, IEEE Transaction on Automatic Control, 50:2–18CrossRefMathSciNetGoogle Scholar
  3. 3.
    Grey W S, Scherpen J M A (2001) On the Nonuniqueness of Singular Value Functions and Balanced Nonlinear Realizations, Systems and Control Letters, 44:219–232CrossRefMathSciNetGoogle Scholar
  4. 4.
    Jonckheere E A, Silverman L M (1983) A New Set of Invariants for Linear Systems-Application to Reduced Order Compensator Design, IEEE Transaction on Automatic Control, 28:953–964zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Krener A J (2006) Normal Forms for Reduced Order Modeling of Nonlinear Control Systems, In preparationGoogle Scholar
  6. 6.
    Moore B C (1981) Principle Component Analysis in Linear Systems: Controllability, Observability and Model Reduction, IEEE Transaction on Automatic Control, 26:17–32zbMATHCrossRefGoogle Scholar
  7. 7.
    Mustafa D, Glover K (1991) Controller Reduction by H Balanced Truncation, IEEE Transaction on Automatic Control, 36:668–682CrossRefMathSciNetGoogle Scholar
  8. 8.
    Scherpen J M A (1993) Balancing for Nonlinear Systems, Systems and Control Letters, 21:143–153zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Scherpen J M A (1996) H Balancing for Nonlinear Systems, International Journal of Robust and Nonlinear Control, 6:645–668zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Scherpen J M A, van der Schaft A J (1994) Normalized Coprime Factorizations and Balancing for Unstable Nonlinear Systems, International Journal of Control, 60:1193–1222zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Arthur J. Krener
    • 1
    • 2
  1. 1.University of CaliforniaDavis
  2. 2.Naval Postgraduate SchoolMonterey

Personalised recommendations