» Download Article

Authors
Jose Borges; Vincent Verdult; Miguel Ayala Botto

Digital Object Identifier (DOI)
10.3182/20050703-6-CZ-1902.00031

Page Numbers:
30-30

Index Terms
identification,nonlinear systems,state-space models,least squares algorithm,efficient algorithms

Abstract
The efficient implementation of separable least squares identification of non-linear systems using composite local linear state-space models is discussed in this paper. A full parametrization of system matrices combined with projected gradient search is used to identify the model. This combined approach reduces the number of iterations and improves the numerical condition of the optimization algorithm. Further enhancements result from using numerical tools, e.g. the QR-decomposition, and efficient approaches compute the separable least squares matrices and gradients of the cost functions. Monte-Carlo simulations are used to show the effectiveness of the proposed approach.

References

[1] N. Bergboer, V. Verdult, and M. Verhaegen. An efficient implementation
     of maximum likelihood identification of LTI state-space
    models by local gradient search. In Proceedings of the 41st IEEE
    Conference on Decision and Control, pages 616-621, Las Vegas,
    Nevada USA, December 2002.
[2] J. Borges, V. Verdult, M. Verhaegen, and M. Ayala Botto. Separable
    least squares for projected gradient identification of composite
    local linear state-space models. In Proceedings of the 16th
    International Symposium on Mathematical Theory of Networks
    and Systems, Leuven, Belgium, July 2004.
[3] J. Bruls, C. Chou, B. Haverkamp, and M. Verhaegen. Linear and
    non-linear system identification using separable least-squares.
    European Journal of Control, 5(1):116-128, 1999.
[4] G. H. Golub and V. Pereyra. The differentiation of pseudo-inverse
    and nonlinear least squares problems whose variables separate.
    SIAM J. of Numerical Analysis, 10(2):413-432, April 1973.
[5] G. H. Golub and C. F. Van Loan. Matrix computations. Johns
    Hopkins, Baltimore, 3rd edition, 1996.
[6] L. H. Lee and K. Poolla. Identification of linear parameter-varying
    systems using nonlinear programming. Journal of Dynamic
    Systems, Measurement and Control, 121(1):71-78, March 1999.
[7] T. McKelvey and A. Helmersson. System identification using
    an over-parametrized model class: Improving the optimization
    algorithm. In Proceedings of the 36th IEEE Conference on
    Decision and Control, pages 2984-2989, San Diego, California,
    December 1997.
[8] R. Murray-Smith and T. A. Johansen. Multiple Model Approaches
    to Modelling and Control. Taylor and Francis, London, 1997.
[9] K. S. Narendra and K. Parthasarathy. Identification and control of
    dynamical systems using neural networks. IEEE Transactions
    on Neural Networks, 1(1):4-27, 1990.
[10] T. Ribarits, M. Deistler, and B. Hanzon. On new parametrization
     methods for the estimation of linear state-space models. forthcoming
      in the International Journal of Adaptive Control and
     Signal Processing, 2004.
[11] V. Verdult. Nonlinear System Identification: A State-Space Approach
     . PhD thesis, University of Twente, Faculty of Applied
     Physics, Enschede, The Netherlands, 2002.
[12] V. Verdult, L. Ljung, and M. Verhaegen. Identification of composite
     local linear state-space models using a projected gradient search.
     International Journal of Control, 75(16/17):1385-1398, 2002.
[13] M. Verhaegen. Identification of the deterministic part of MIMO
     state space models given in innovations form from input-output
     data. Automatica, 30(1):61-74, 1994.