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2.1.1 Problem Formulation
Let
n
be the number of input variables and
m
the number of output variables
of a completely general system to model; a discrete Multiple Input Multiple
Output—MIMO—TS fuzzy model can be represented by the following set of rules
(Babuška
1995
; Babuška et al.
1996
; Nguyen et al.
1995
; Takagi and Sugeno
1985
):
R
(
l
,
i
)
:
is
A
l
1
i
and
is
A
l
ni
If
x
1
(
k
)
...
and
x
n
(
k
)
j
=
1
n
(2.1)
Then
y
i
(
a
l
0
i
+
a
l
ji
x
j
(
k
)
=
k
),
where
l
M
i
is the index of the rule and
M
i
the number of rules that model
the evolution of the
i
th system output,
y
i
(
=
1
...
.
a
ij
,
j
)
=
...
n
, represents the set of
adaptive parameters of the consequents of the rules, thus they must be determined
by the modeling process.
k
indicates the current sampling time.
Note that in using the above representation, there are multiple inputs and outputs,
and each output can be modeled by a different number of rules. This representation
facilitates the reduction of the total number of rules needed to model correctly a
complex system, and, therefore, facilitates the modeling process by reducing the
number of model parameters.
If input vector is extended in a coordinate (Andújar and Barragán
2005
; Andújar
et al.
2009
)by
k
0
x
0
=
˜
1, extended vector
x
takes the form:
˜
T
T
x
˜
=
(
˜
x
0
,
˜
x
1
,...,
˜
x
n
)
=
(
1
,
x
1
,...,
x
n
)
(2.2)
Using the weighted average as a method of aggregation and the extension of the
state vector given in (
2.2
) , the output
y
i
generated by the set of rules
R
(
l
,
i
)
, can be
calculated by (Wang
1994
,
1997
):
n
y
i
(
k
)
=
h
i
(
x
(
k
))
=
a
ji
(
x
)
˜
x
j
(
k
),
(2.3)
j
=
0
being
a
ji
(
x
)
variables coefficients (Wong et al.
1997
) defined by
M
i
l
1
w
i
(
a
l
ji
x
)
=
a
ji
(
x
)
=
,
(2.4)
M
i
l
1
w
i
(
x
)
=
where
w
i
(
is calculated by (
2.5
) and represents the degree of activation of the rules
of the fuzzy model:
x
)
n
j
=
1
μ
w
i
(
l
l
x
)
=
ji
(
x
j
(
k
),
σ
ji
).
(2.5)
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