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`
P
x
proj
P
,
n
1
gj Gjs
j
gs
where “proj” is the point-wise projection operator (Kruse et al. , 1994). The point-
wise fuzzy sets G gj are typically non-convex. However, the core and the
corresponding left and right parts of the set can be recognized.
Mu(x)
1.0
0.5
x
Figure 4.5. Parametric function fitting (solid line) to obtain one-dimensional antecedent
fuzzy sets from point-wise projection (dots) of rows of fuzzy partition matrix
To obtain convex (unimodal) fuzzy sets, for the computation of
gj Gj
P for any
value of x j , the fuzzy sets are approximated by fitting suitable parametric
membership functions (say, Gaussian type) to the point-wise projection (Babuška,
1996) as illustrated in Figure 4.5. After determination of the antecedent fuzzy sets,
the LSE estimate is applied, as usual, to determine the rule consequent parameters.
4.7.5 Modelling of a Nonlinear Plant
In order to demonstrate the efficiency of the clustering-based fuzzy model, the
second-order nonlinear plant (4.25) that was studied by Wang and Yen (1999) and
Roubos and Setnes (2001) is considered here.
yk
g yk
1,
yk
2
uk
,
with,
yk
1
yk
2
yk
1
0.5
gyk
1,
yk
2
(4.25)
1
yk
2
1
yk
2
2
The goal is to approximate the nonlinear component g ( y ( k -1), y ( k -2)) of the
plant with the fuzzy model. For this experiment, 400 data points were available, of
which 200 samples of identification data were obtained with a random input signal
u ( k ) uniformly distributed in [-1.5, 1.5], followed by 200 samples of evaluation
data obtained by using a sinusoidal input signal.
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