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performance of this approach depends on the ef
cation
algorithm (Lassoued and Abderrahim 2013a , b , c , d , 2014a , b ). The early methods
have favored the simplicity of implementation. In fact, they present several draw-
backs, which can be summarized as follows:
ciency of the used classi
Most of them are based on the optimization of nonlinear criteria. Consequently,
they may converge to local minima in the case of poor initializations.
￿
Their performances degrade in the case of the presence of outliers in the data to
be classi
￿
ed.
Most of them assume that the number of sub-models is a priori known.
￿
To overcome these problems, we have proposed the use of other clustering
algorithms such as Chiu
s method (Chiu 1997 ) and Density Based Spatial Clus-
tering of Applications with Noise (DBSCAN) method (Chaitali 2012 ; Sander et al.
1998 ). This choice is justi
'
ed by the fact
that
these algorithms automatically
generate the number of models. In addition,
they are characterized by their
robustness to the classi
cation of noisy measurements that containing also outliers.
This chapter is organized as follows. Section 2 presents the assumptions for
PWARX model identi
-
cation of PWARX systems based on clustering algorithm and its main drawbacks.
Section 4 proposes two solutions to overcome the main problems of the existing
methods. In Sect. 5 , we present three simulation examples in order to illustrate the
performance of the proposed solutions and to compare their ef
cation. In Sect. 3 , we recall the main steps of the identifi-
ciency with the
modi
ed k-means method. Section 6 proposes an application of the developed
approach to an olive oil esteri
cation reactor.
2 Piecewise Affine System Identification
Consider a discrete-time PieceWise Auto-Regressive eXogenous model (PWARX)
with input u
ð
k
Þ2 R
, output y
ð
k
Þ2 R
de
ned in the bounded polyhedron regressor
d
space H
R
(d
¼
n a þ
n b þ
1). The system is decomposed in s different modes
s
i¼1 , in each one an ARX model is associated:
H fg
y
ð
k
Þ ¼
f
ðuð
k
ÞÞ þ
e
ð
k
Þ:
ð
1
Þ
f is a piecewise af
ne function de
ned by:
8
<
T
1
u
h
if
u 2
H 1
.
h
ðuÞ ¼
ð
Þ
f
2
:
T
s u
if
u 2
H s
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