Modeling, Identification and Control of Irrigation Station with Sprinkling: Takagi-Sugeno Approach - Complex System Modelling and Control Through Intelligent Soft Computations - page 469

almost Boolean i.e., l ik 2 fg 0 ; 1 . Whereas when m becomes very large, the parti-

tion becomes fuzzier and = c

Generally m is selected between 1.5 and 2.5 but in several applications, it is
selected between 2 and 4.
In the following section, we present the fuzzy c-means algorithm.

l ik ¼ 1

3.2 Fuzzy c-Means (FCM) Algorithm

This method is based on minimization of the criterion obtained by the addition of
the standardization constraint (Troudi et al. 2011 ).

" #

X

N

X

c
i¼1 ð l ik Þ

X

k¼1 k k X

N c

i¼1 l ik

m T M

J ð X ; U ; V Þ ¼ ð x k v i Þ ð x k v i Þþ 1 ð 8 Þ

k¼1

In this case the minimization of the criterion 8 can be solved by cancelling the
derivative of J where the variables are U, V and

λ . The solution of this criterion is

given by:

P k¼1 ð l ik Þ

m

x k

v i ¼

P k¼1 ð l ik Þ

m

ð 9 Þ

1
P j¼1 d ik = d jk

l ik ¼

2
m 1

where d i k: represent the distance enters X k and v i

T M

d ik ¼ ð x k v i Þ ð x k v i Þ ð 10 Þ

M: generally selected equal to the identity. The prototype vector of the clusters is
given by:

d ik ¼ ð

T

x k v i Þ ð x k v i Þ i ¼ 1 ; ...; c ; k ¼ 1 ; ...:; N ð 11 Þ

The iteration count of c-means algorithm is selected according to the precise
details required by the expert and according to the type of application considered.
The criterion of the stop is selected by satisfying the following condition:

\ d

U ð l Þ U ð l 1 Þ ð 12 Þ

where l is the iteration count.
Fuzzy c-means algorithm (FCM): Being given a whole of data X, FCM algo-
rithm is described by the following stages (Fig. 5 ):