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(2) The OFRF is varying with different values of c1, c2 and c3. The effect on the output spectrum from any two nonlinear terms is not necessarily the simple combination of the contributions from each term respectively. Thus the parameters can be analyzed in order to get the best output frequency response performance. The OFRF provides a useful basis for this kind of analysis and optimization. From the discussions above, it can be concluded that the OFRF-based analysis provides a novel, effective and useful approach to the analysis and design of nonlinear Volterra systems in the frequency domain. 6. Conclusions and discussions ARTICLE IN PRESS X.J. Jing et al. / Mechanical Systems and Signal Processing 22 (2008) 102–120 117 Fig. 7. Output spectrums with respect to any two combinations of c1, c2 and c3. Based on some recently developed results, the OFRF-based analysis for nonlinear Volterra systems is proposed in this paper. The OFRF-based analysis provides a novel and effective approach to the analysis and design of nonlinear Volterra systems in the frequency domain by using the explicit relationship between the system output frequency response and model parameters. The OFRF is characterized by its parametric characteristic timing a complex valued frequency dependent function vector. Thus in stead of the direct analytical computation of the OFRF, the proposed method simplifies the computation of the OFRF by splitting the computation procedure into two parts—one is the computation of the parametric characteristics for the OFRF, which is analytical in the determination of the relationship between the output spectrum and model parameters, and simpler to be carried out, and the other is the determination of the complex valued frequency-dependent function vectors, which are obtained by using the least-square method. Some
118 fundamental results, techniques, and a general procedure for the determination of the OFRF for a given NDE or NARX model subject to any specific input signal are provided. Although the proposed method needs r(N) simulation data for the numerical method of Process C, and the OFRF obtained by the proposed method is not analytical with respect to the input signal and frequency variants at present, the case study for a simple mechanical system shows that the OFRF-based analysis is a useful approach to the analysis and design of nonlinear Volterra systems in the frequency domain. Acknowledgements The authors take their gratitude to the anonymous reviewers for their useful and constructive comments and suggestions on the manuscript, and also gratefully acknowledge the support of the EPSRC-Hutchison Whampoa Dorothy Hodgkin Postgraduate Award, for this work. Appendix A. Proof of Proposition 1 CEðHnðjo1; ...; jonÞÞ ¼ C0;n n p¼2 n 1 n q q¼1 p¼1 0 B @ Cp;0 0 B @ Cp;q 0 B @ n pþ1 0 B @ p r1 rp¼1 P i¼1 ri ¼n n q pþ1 p r1 rp¼1 P i¼1 ri ¼n q CE Hri ðjorXþ1 ; ...; jorXþr i Þ CE H ri ðjorXþ1 ; ...; jorXþr i Þ 11 CC AA 11 CC AA. ðC1Þ C0,n is the first term in Eq. (C1). For clarity, consider a simpler case that there is only output nonlinearities in Eq. (C1), then Eq. (C1) is reduced to only the last term of Eq. (C1), i.e., n CEðHn;pð ÞÞ ¼ p¼2 n Cp;0 n pþ1 p r1 rp¼1 P i¼1 ri ¼n [24] in order to finish the proof. & Appendix B. Proof of Proposition 2 Cp;0 p¼2 CEðHri ð ÞÞ. At this stage, the reader is recommended to refer to Jing et al. Regard all other nonlinear parameters as constants or 1. From Proposition 1, if p+q4n then the parameter has no contribution to CE(Hn(.)), in this case CE(Hn(.)) ¼ 1 with respect to this parameter. Similarly, if p+q ¼ n then the parameter is an independent contribution in CE(Hn(.)), thus CE(Hn(.)) ¼ [1c] with respect to this parameter in this case. If p+qon and p40, then the independent contribution in CE(Hn(.)) for this ðn1Þ= ðpþq1Þ parameter should be cb c , and the monomials c x for 0pxo n 1=p þ q 1 are all not independent contributions in CE(Hn(.)). Hence CEðHnð ÞÞ ¼ 1; c; c2 h i ðn1Þ= ðpþq1Þ ; ...; cb c for this case. The similar result is held for the case p+qon and p ¼ 0. However, since there should be at least one p40 ina complete monomial, thus in this latter case c x for any x are not complete, which follows CEðHnð ÞÞ ¼ 1; c; c2 h i ðn1Þ= ðpþq1Þ ; ...; cb c 1 . The parametric characteristic vector for the nonlinear parameter c for all the cases above can be summarized into CEðHnð ÞÞ ¼ 1; c; c 2 ; ...; c This completes the proof. & ARTICLE IN PRESS X.J. Jing et al. / Mechanical Systems and Signal Processing 22 (2008) 102–120 n 1 n 1 pþq 1 dðpÞ posðn qÞ d pþq 1 n 1 pþq 1 .
Page 1 and 2: Abstract Mechanical Systems and Sig
Page 3 and 4: 104 experimental data, which is giv
Page 5 and 6: 106 Alternatively, CE(Hn( )) can al
Page 7 and 8: 108 proposed method above as demons
Page 9 and 10: 110 except that Cp,q ¼ c is non-ze
Page 11 and 12: 112 For clarity in discussion, let
Page 13 and 14: 114 Magnitude Magnitude of the sens
Page 15: 116 ARTICLE IN PRESS X.J. Jing et a
Page 19: 120 ARTICLE IN PRESS X.J. Jing et a

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