Differential Algebra for Nonlinear Control Theory

  • Rafael Martínez-GuerraEmail author
  • Oscar Martínez-Fuentes
  • Juan Javier Montesinos-García
Part of the Mathematical and Analytical Techniques with Applications to Engineering book series (MATE)


This chapter focus is to show the application of the commutative algebra, algebraic geometry and differential algebra concepts to nonlinear control theory, it begins with necessary information of differential algebra, it continues with definitions of single-input single-output systems, invertible systems, realization and canonical forms, finally we present methods for stabilization of nonlinear systems throughout linearization by dynamical feedback and some examples of such processes.


  1. 1.
    Byrnes, C.I., Falb, P.L.: Applications of algebraic geometry in system theory. Am. J. Math. 101(2), 337–363 (1979)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Dieudonn, J. (1974). Cours de gomtrie algbrique (Vol. 2). Presses universitaires de FranceGoogle Scholar
  3. 3.
    Samuel, P., Serge, B.: Mthodes d’algbre abstraite en gomtrie algbrique. Springer (1967)Google Scholar
  4. 4.
    Kolchin, E. R.: Differential algebra & algebraic groups. Academic Press (1973)Google Scholar
  5. 5.
    Mumford, D.: Introduction to algebraic geometry. Department of Mathematics, Harvard University (1976)Google Scholar
  6. 6.
    Hartshorne, R.: Algebraic Geometry. Springer Science & Business Media (1977)Google Scholar
  7. 7.
    Borel, A.: Linear algebraic groups, vol. 126. Springer Science & Business Media (2012)Google Scholar
  8. 8.
    Fliess, M.: Nonlinear control theory and differential algebra. In: Modelling and Adaptive Control, pp. 134–145. Springer, Berlin (1988)Google Scholar
  9. 9.
    Simmons, G.F.: Differential Equations with Applications and Historical Notes. CRC Press (2016)Google Scholar
  10. 10.
    Boyce, W.E., DiPrima, R.C.: Differential Equations Elementary and Boundary Value Problems. Willey & Sons (1977)Google Scholar
  11. 11.
    Falb, P.: Methods of Algebraic Geometry in Control Theory: Part I. Scalar Linear Systems and Affine Algebraic Geometry, Birkhuser (1990)Google Scholar
  12. 12.
    Diop, S., Fliess, M.: Nonlinear observability, identifiability, and persistent trajectories. In: Proceedings of the 30th IEEE Conference on Decision and Control, pp. 714–719. IEEE (1991)Google Scholar
  13. 13.
    Kawski, M.: Stabilization of nonlinear systems in the plane. Syst. Control Lett. 12(2), 169–175 (1989)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Aeyels, D.: Stabilization of a class of nonlinear systems by a smooth feedback control. Syst. Control Lett. 5(5), 289–294 (1985)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Fliess, M., Messager, F.: Vers une stabilisation non linaire discontinue. In: Analysis and Optimization of System, pp. 778–787. Springer, Berlin (1990)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Rafael Martínez-Guerra
    • 1
    Email author
  • Oscar Martínez-Fuentes
    • 1
  • Juan Javier Montesinos-García
    • 1
  1. 1.Departamento de Control AutomáticoCINVESTAV-IPNMexico CityMexico

Personalised recommendations