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5 Simulation Examples
In this section, we aim at illustrating the performance of the proposed methods with
three simulation examples. First of all, we take an academic PWARX model
where the proposed methods are compared with the well known k-means one
(Ferrari-Trecate et al.
2001
,
2003
). After that, a nonlinear model is considered to
show the ef
ciency of the proposed methods in approximating any nonlinear model.
Finally, a pH neutralization process is simulated in order to prove their ability to
model complex systems and to determine the number of sub-models.
5.1 Quality Measures
To achieve the purpose of these simulations, we consider the following quality
measures (Juloski et al.
2006
):
The maximum of relative error of parameter vectors is de
ned by
2
h
if
h
i
2
D
h
¼
max
i¼1
;...;
s
ð
18
Þ
h
i
where
h
if
are the true and the estimated parameter vectors for sub-model i,
respectively. The identi
h
if
and
ed model is deemed acceptable if
D
h
is small or close to
zero.
The averaged sum of the squared residuals is de
ned by
s
X
s
1
SSR
i
D
jj
2
e
r
¼
ð
19
Þ
i
¼
1
h
i
Þ
P
ð
y
ð
k
Þ;uð
k
ÞÞ2
D
i
ð
Þ
0
1
2
ð
Þ uð
where SSR
if
¼
y
k
k
and D
jj
is the cardinality of
cluster Di.
i
.
The identi
2
e
ed model is considered acceptable if
r
is small and/or close to the
expected noise variance of the true system.
The percentage of the output variation that is explained by the model is de
ned
by
^
k
y
y
k
2
¼
ð
Þ
FIT
100
1
20
k
y
y
k
2
where
y and y are the estimated and the real outputs
vectors, respectively, and y
'
is the mean value of y.
The identi
ed model is considered acceptable if FIT is close to 100.
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