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3 Identi
cation and Modeling of the Irrigation Station
The implementation of a mathematical model of a complex real process operating in
a stochastic environment draw the attention of many researchers in various disci-
plines of science and technology. In this context the use of traditional methods of
modeling and identi
cation in order to estimate the parameters of such a type of
process cannot satisfy the desired performance indices (speed, accuracy and sta-
bility). To overcome this problem, other techniques such as fuzzy logic (Azar
2010b
,
2012
) and more particularly the T-S fuzzy model showed a good result in
the identi
cation of these processes types.
3.1 Fuzzy Coalescence Algorithms for System Identi
cation
Let us consider a system described by the following differential equation:
y
ð
k
Þ
¼
f
NL
ð
x
k
Þ
ð
5
Þ
R
n
. The most used algorithms of
with x
k
represent the observation vector, x
k
2
fuzzy coalescence for the identi
cation parameters of
5
are as follows:
The algorithm of the fuzzy C-averages, or fuzzy c-Means (FCM) (Bezdek
1981
;
Chen et al.
1998
),
The algorithm of Gustafson-Kessel (GK) (Gustafson and Kessel
1979
),
The NRFCM algorithm (Soltani et al.
2012
).
All these algorithms are based on their minimization of a function objecti
es
form (Troudi et al.
2012
):
X
X
N
c
m
T
M
J
ð
X
;
U
;
V
Þ
¼
1
ð
l
lik
Þ
ð
x
k
v
i
Þ
ð
x
k
v
i
Þ
ð
6
Þ
k
¼
1
i
¼
where: X ={x
k
/k =1,2,
…
, N}, such that N donate the number of observations;
[0, 1]
(c
×
N)
], the fuzzy partition matrix of data vector X: with
U =[
μ
lik
∊
X
c
i¼1
l
lik
¼
11
i
c
ð
7
Þ
V: The prototype clusters vector,
V ={v
1
, v
2
,
…
, v
c
}, where c represents the rule number (or of clusters) and
R
n
,
m: represent the weighting degree
This parameter in
v
i
2
uences directly on the form of cluster in data space. Indeed,
when m is close to 1, the function of the membership of each cluster becomes
fl
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