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Wan, Y.; Dodd, T.J.; Harrison, R.F. (2003)
Publisher: Automatic Control and Systems Engineering, University of Sheffield
Languages: English
Types: Book
Volterra series expansions are widely used in analyzing\ud and solving the problems of non-linear dynamical\ud systems. However, the problem that the number of\ud terms to be determined increases exponentially with the\ud order of the expansion restricts its practical application.\ud In practice, Volterra series expansions are truncated\ud severely so that they may not give accurate representations\ud of the original system. To address this problem,\ud kernel methods are shown to be deserving of exploration.\ud In this report, we make use of an existing result\ud from the theory of approximation in reproducing kernel\ud Hilbert space (RKHS) that has not yet been exploited in\ud the systems identification field. An exponential kernel\ud method, based on an RKHS called a generalized Fock\ud space, is introduced, to model non-linear dynamical systems\ud and to specify the corresponding Volterra series\ud expansion. In this way a non-linear dynamical system\ud can be modelled using a finite memory length, infinite\ud degree Volterra series expansion, thus reducing the\ud source of approximation error solely to truncation in\ud time. We can also, in principle, recover any coefficient\ud in the Volterra series.
  • The results below are discovered through our pilot algorithms. Let us know how we are doing!

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    • [2] Boyd, Stephen and Chua, Leon O. (1985). Fading memory and the problem of approximating nonlinear operators with Volterra series. IEEE Transactions on Circuits and Systems, Vol. Cas-32, No. 11.
    • [3] Dodd, T. J. and Harrison, R. F. (2002a). A new solution to Volterra series estimation. CD-Rom Proceedings of the 2002 IFAC World Congress.
    • [4] Dodd, T. J. and Harrison, R. F. (2002b). Estimating Volterra filters in Hilbert space. Proceedings of IFAC International Conference on Intelligent Control Systems and Signal Processing, Faro.
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    • [6] De Figueiredo, Rui J. P. (1983). A generalized Fock space framework for non-linear system and signal analysis. IEEE Transactions on Circuits and Systems. Vol. Cas-30, No. 9.
    • [7] De Figueiredo, Rui J. P. and Dwyer, III, Thomas A. W. (1980). A best approximation framework and implementation for simulatoin of large-scale non-linear systems. IEEE Transactions on Circuits and Systems. Vol. Cas-27, No. 11.
    • [8] Harrison, R. F. (1999). Computable Volterra filters of arbitrary degree. MAE Technical Report No. 3060, Princeton University.
    • [9] Schetzen, Martin (1980). The Volterra and Wiener Theories of Non-linear Systems. New York; Chichester (etc.): Wiley.
    • [10] Wahba, G. (1990). Spline Models for Observational Data. SIAM. Series in Applied Mathematics. Vol. 50. Philadelphia.
    • [11] Zyla, L. V. and De Figueiredo, Rui J. P. (1983). Non-linear system identification based on a Fock space framework. SIAM J. Control and Optimization. Vol. 21. No. 6.
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  • Discovered through pilot similarity algorithms. Send us your feedback.

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