Identification of the dynamic characteristics of nonlinear structures

 
. . . .
(3-61)
r = l
the Wiener G-functionals are a set of nonhomogeneous Volterra 
defined as:
n
 
 +
 
. . .
r = l
s = l
(3-62)
where 
is known as the 
Wiener kernel and 
. . . . 
are known
as the derived Wiener kernels of the Wiener G-functional. 
[ 
satisfy
for m n
(3-63)
where the over bar means taking the average of the process. Theoretically, all the derived
Wiener kernels 
. . . .
of 
can be determined uniquely by the leading
Wiener kernel 
when (3-63) is satisfied for all integer values of 
and,
therefore, in the notation 
only the leading term 
is specified as in the case
of Volterra functional 
f(t)]. The first few 
f(t)] of a general nonlinear
system, are given as 
is a constant)
f(t) =
(3-65)
(3-66)
=
J J J

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3
Identification of Nonlinearity Using Higher-order 
9 4
- 3 A
J J
(3-67)
The relationship between the Volterra kernel 
and the Wiener kernel
is that the system’s n*-order Volterra kernel is equal to the system’s 
order Wiener kernel plus the sum of all the (even or odd order) derived Wiener kernels
that are of the 
that is
(3-68)
From 
it can be seen that since the derived Wiener kernels are determined uniquely
by their leading Wiener kernel, a given system’s Volterra kernels can be obtained
uniquely from the system’s Wiener kernels (leading Wiener kernels). Also, it should be
noted from equations (3-66) and (3-67) that as the input level 
the derived kernels
approach zero and the leading Wiener kernels approach the Volterra kernels. On the other
hand, it should be pointed out that unlike the Volterra kernels, which are mathematically
unique, the Wiener kernels are input-output dependent and since the Volterra kernels
which uniquely determine the system are uniquely determined by the
Wiener kernels
the measured Wiener kernels 
also
uniquely determine the system.
3.4.2 DETERMINATION OF WIENER KERNELS BY
CROSS-CORRELATION
As in the case of Volterra series representation, under Wiener series representation, the
problem of identifying a nonlinear system becomes one of determining all the Wiener
kernels which describe the system. The orthogonality property of 
and the statistical
properties of Gaussian noise enable the Wiener kernels to be determined using a 
correlation technique. The first four kernels are given 
as
= x(t)
(3-70)
(3-7 1)

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3
Identification of Nonlinearity Using Higher-order 
9 5
(3-72)
Equations 
serve as a basis for the measurement of Wiener kernels. To
illustrate the derivation of these equations, consider the calculation of the second-order
kernel 
From equation 
becomes