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where
T
u
¼
u
T
:
ð
Þ
1
3
e(k) is the additive noise and
uð
k
Þ
is the regressor vector, containing past input and
output observations, de
ned as:
T
uð
k
Þ
¼ y
ð
k
½
1
Þ...
y
ð
k
n
a
Þ
u
ð
k
1
Þ...
u
ð
k
n
b
Þ
:
ð
4
Þ
R
d
þ
1
h
i
2
is the parameter vector, valid in Hi,
i
,de
ned as follows:
T
i
h
¼
½
a
1
a
2
...
a
n
a
b
1
b
2
...
b
n
b
g
ð
5
Þ
where a
i
and b
i
are the coef
cients of the model related respectively to the output
and the input data, while n
a
and n
b
are the model orders. g is the independent af
ne
coef
cient.
Problem statement
Given input-output data generated by a PWARX system, we are interested simul-
taneously in identifying the number of submodels s, the parameter vectors
s
i
hfg
¼
1
s
i¼1
and the partitions Hif
fg
taking into account the following assumptions:
The orders n
a
and n
b
of the system are known.
The noise e(k) is assumed to be a Gaussian process independent and identically
distributed with zero mean and
2
.
finite variance
r
s
i¼1
are the polyhedral partitions of a bounded domain H
d
The regions H
fg
R
such that:
S
i¼1
H
i
¼
H
H
i
T
H
j
¼
; 8
ð
6
Þ
i
6
¼
j
3 Clustering Based PWARX Identification
The main steps of the clustering-based approach for the identi
cation of PWARX
models can be summarized as follows: constructing small data set from the initial
data set, estimating a parameter vector for each small data set, classifying the
parameter vectors in s clusters, classifying the initial data set and estimating the
s sub-models with their partitions.
N
k
1. Form
f
uð
k
Þ;
y
ð
k
Þ
g
1
from the given dataset S
¼
ð
u
ð
k
Þ;
y
ð
k
ÞÞ;
k
¼
1
; ...;
N
¼
2. Create local datasets C
k
and identify the local parameter vectors
h
k
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