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Nonlinear system theory: the Volterra-Wiener approach - download pdf or read online

By Rugh W.J.

ISBN-10: 0801825490

ISBN-13: 9780801825491

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Extra resources for Nonlinear system theory: the Volterra-Wiener approach

Sample text

In other words, G (p) can be viewed as a polynomial truncation of H −1 , assuming of course that H −1 exists. The expression (87) can be used to determine G (p) in a homogeneousterm-by-homogeneous-term fashion by using the cascade formulas developed earlier. That is, (87) can be written, through degree 3, in the form 32 (p) (p) . . )* (H 1 + H 2 + H 3 + + . . ) G (p) *H = (G (p) 1 + G2 + G3 + (p) (p) = (G (p) 1 *H 1 ) + (G 2 *H 1 + G 1 *H 2 ) (p) (p) (p) (p) ... + [G (p) 3 *H 1 G 1 *H 3 − G 2 *H 1 − G 2 *H 2 + G 2 * (H 1 + H 2 )] + where the terms have been grouped according to degree.

Dtn (3) 0 Of course, this definition also is subject to convergence considerations.

Also it is obvious from earlier results that degree (Gn *Hm ) = degree (Hm *Gn ) = mn (62) To consider cascade connections of polynomial or Volterra systems in terms of the notation in (50) requires further development. 11, an operator expression of the form ∞ y= Σ Fn [u ] (64) n =1 where each homogeneous operator Fn is specified in terms of the Hn ’s and Gn ’s. It is convenient in this regard to consider the input signal αu (t), where α is an arbitrary real number. Then Hn [αu ] = αn wn , and ∞ w= Σ αn w n (65) n =1 so that 27 ∞ ∞ Σ y= Gm [ Σ αn wn ] m =1 (66) n =1 The general term of interest is ∞ Gm [ Σ αn wn ] (67) n =1 and to analyze this further it is necessary to bring in the kernel representation for Gm .

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Nonlinear system theory: the Volterra-Wiener approach by Rugh W.J.


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