» Download Article

Authors
Linda Brus; Torbjorn Wigren

Identifier
10.3182/20060329-3-AU-2901.00159

Index Terms
nonlinear systems,identification,Kalman filter

Abstract
A recursive identification algorithm for systems described by nonlinear ordinary differential equation (ODE) models is proposed. The ODE model is parameterized with coefficients of a polynomial in the state variables and inputs, that describes one component of the right hand side function of the ODE. This avoids over-parameterization problems. The model is then discretized with an Euler integration method. The algorithm exploits a Kalman filter, where the state variables needed in the right hand side function are derived by numerical differentiation. This approach makes a standard Kalman filter applicable to the identification problem. Contrary to a previously described RPEM algorithm, the proposed Kalman filter scheme cannot converge to false local minima of the criterion function. The proposed algorithm is therefore suitable for generation of initial values for the RPEM. The performance of the Kalman filter based algorithm is illustrated using a numerical example.

References

[1] Bohlin, T. (1994). A case study of grey box
    identification. Automatica 30, 307-318.
[2] Brus, L. (2005a). Nonlinear identification of a
    solar heating system. In: Proc. CCA. Toronto,
    Canada.
[3] Brus, L. (2005b). Nonlinear identification of an
    anaerobic digestion process. In: Proc. CCA.
    Toronto, Canada.
[4] Chen, S. and S. A. Billings (1989). Representation
    of nonlinear systems: the NARMAX model.
    Int. J. Control 49, 1013-1032.
[5] Khalil, H. K. (1996). Nonlinear System. 2nd ed.,
    Prentice Hall. Upper Saddle River, NJ.
[6] Nijmeijer, H. and A. J. van der Schaft
    (1990). Nonlinear Dynamic Control Systems.
    Springer Verlag. New York, NY.
[7] Söderström, T. (2002). Descrete-time stochastic
     systems: estimation and control, 2nd ed.,
    Springer-Verlag. London, U.K.
[8] Söderström, T. and P. Stoica (1989). System Identification
    . Prentice-Hall. Hemel Hempstead,
    UK.
[9] Wahlberg, B. and L. Ljung (1986). Design variables
     for bias distribution in transfer function
    estimation. IEEE Trans. Automat. Contr.
    31(2), 134-144.
[10] Wigren, T. (2005a). Matlab software for recursive
      identification and scaling using a structured
      nonlinear black-box model - revision 2.
     Technical Report 2005-022. Uppsala University.
      Uppsala, Sweden.
     http://www.it.uu.se/research/reports/2005-
     022/NRISoftwareRev2.zip.
[11] Wigren, T. (2005b). Scaling of the sampling period
     in nonlinear system identification. In: Proc. of
     ACC 2005. IEEE. Portland, Oregon, U.S.A.,
     pp. 5058-5065.
[12] Wigren, T. (2006). Recursive prediction error
     identification and scaling of nonlinear state
     space models using a restricted black box parameterization.
      Automatica 42(1), 159-168.