Output frequency response function-based analysis for nonlinear ...

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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 .