Advertisement

Adaptive Estimation of Nonlinear Distributed Parameter Systems

  • Joseph Kazimir
  • I. G. Rosen
Conference paper
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 118)

Abstract

The adaptive (on-line) estimation of parameters for a class of nonlinear distributed parameter systems is considered. A combined state and parameter estimator is constructed as an initial value problem for an infinite dimensional evolution equation. State convergence is established via a Lyapunov-like estimate. The finite dimensional notion of persistence of excitation is extended to the infinite dimensional case and used to establish parameter convergence. A finite dimensional approximation theory is presented and a convergence result is proven. An example involving the identification of a nonlinear heat equation is discussed and results of a numerical study are presented.

1991 Mathematics Subject Classification

93B30 93C25 93C20 65J10 

Key words and phrases

On-line estimation adaptive identification parameter convergence persistence of excitation distributed parameter systems infinite dimensional systems finite dimensional approximation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    H. W. Alt and K.-H. Hoffmann and J. Sprekels, A numerical procedure to solve certain identification problems ,Intern. Series Numer. Math., 68 (1984), pp. 11–43.MathSciNetGoogle Scholar
  2. 2.
    J. Baumeister and W. Scondo, Adaptive methods for parameter identification ,Methoden und Verfahren der Mathematischen Physik, (1987), pp. 87–116.Google Scholar
  3. 3.
    J. Baumeister and W. Scondo,Asymptotic embedding methods for parameter estimation ,Proceedings of the 26th IEEE Conference on Decision and Control, (1987), pp. 170–174.CrossRefGoogle Scholar
  4. 4.
    A. Belleni-Morante, Applied Semigroups and Evolution Equations ,Clarendon Press, Oxford, 1979.zbMATHGoogle Scholar
  5. 5.
    M. A. Demetriou and I. G. Rosen, On-line parameter estimation for infinite dimensional dynamical systems ,CAMS Report 93-2, Center for Applied Mathematical Sciences, Department of Mathematics, University of Southern California, Los Angeles, CA 90089-1113, February 1993.Google Scholar
  6. 6.
    M. A. Demetriou and I. G. Rosen,On the persistence of excitation in the adaptive identification of distributed parameter systems ,CAMS Report 93-4, Center for Applied Mathematical Sciences, Department of Mathematics, University of Southern California, Los Angeles, CA 90089-1113, March 1993 and IEEE Transactions on Automatic Control ,to appear.Google Scholar
  7. 7.
    K. -H. Hoffmann and J. Sprekels, The method of asymptotic regularization and restricted parameter identification problems in variational inequalities , in Free Boundary Problems: Application and Theory, IV, Maubuisson, (1984), pp. 508–513.Google Scholar
  8. 8.
    K. -H. Hoffmann and J. Sprekels, On the identification of coefficients of elliptic problems by asymptotic regularization,Numer. Funct. Anal. and Optimiz. 7 (1984-85), pp. 157–177.MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    K. -H. Hoffmann and J. Sprekels, On the identification of parameters in general variational inequalities by asymptotic regularization ,SIAM J. Math. Anal. 17 (1986), pp. 1198–1217.MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    T. Kato, Perturbation Theory for Linear Operators ,Second Edition, Springer-Verlag, New York, 1984.zbMATHGoogle Scholar
  11. 11.
    A. P. Morgan and K. S. Narendra, On the stability of nonautonomous differential equations x = [A + B(t)]x, with skew symmetric matrix B(t) ,SIAM J. Control and Optimization 15 (1977), pp. 163–176.MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    K. S. Narendra and P. Kudva, Stable adaptive schemes for system identification and control, Parts I and II ,IEEE Trans. Systems, Man and Cybernetics, SMC-4 (1974), pp. 542–560.MathSciNetCrossRefGoogle Scholar
  13. 13.
    A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations ,Springer-Verlag, New York, 1983.zbMATHCrossRefGoogle Scholar
  14. 14.
    V. M. Popov, Hyper stability of Control Systems ,Springer-Verlag, Berlin, Heidelberg, and New York, 1973.Google Scholar
  15. 15.
    M. H. Schultz, Spline Analysis ,Prentice Hall, Englewood Clffis, N.J., 1973.zbMATHGoogle Scholar
  16. 16.
    W. Scondo, Ein Modellabgleichsverfahren zur adaptiven Parameteridentifikation in Evolutionsgleichungen ,PhD thesis, Johann Wolfgang Goethe-Universitat zu Frankfurt am Main, Frankfurt am Main, Germany, 1987.zbMATHGoogle Scholar
  17. 17.
    R. E. Showalter, Hilbert Space Methods for Partial Differential Equations ,Pitman, London, 1977.zbMATHGoogle Scholar
  18. 18.
    H. Tanabe, Equations of Evolution ,Pitman, London, 1979.zbMATHGoogle Scholar

Copyright information

© Springer Basel AG 1994

Authors and Affiliations

  • Joseph Kazimir
    • 1
  • I. G. Rosen
    • 1
  1. 1.Center for Applied Mathematical Sciences Department of MathematicsUniversity of Southern CaliforniaLos AngelesUSA

Personalised recommendations