A kernel method for non-linear systems\ud identification – infinite degree volterra series\ud estimation

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Wan, Y. ; Dodd, T.J. ; Harrison, R.F. (2003)
  • Publisher: Automatic Control and Systems Engineering, University of Sheffield

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.
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