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M
i
l
M
i
l
M
i
l
1
w
i
(
a
l
0
i
1
w
i
(
a
l
1
i
1
w
i
(
a
l
ni
x
,
u
)
x
,
u
)
x
,
u
)
=
=
=
x
i
˙
=
+
+···+
M
i
l
M
i
l
M
i
l
1
w
i
(
1
w
i
(
1
w
i
(
x
,
u
)
x
,
u
)
x
,
u
)
=
=
=
M
i
l
M
i
l
1
w
i
(
b
l
1
i
1
w
i
(
b
l
mi
x
,
u
)
x
,
u
)
=
=
+
+···+
(4.7)
M
i
l
M
i
l
1
w
i
(
1
w
i
(
x
,
u
)
x
,
u
)
=
=
or, more simplified:
x
i
˙
=
a
0
i
(
x
,
u
)
+
a
1
i
(
x
,
u
)
x
1
+···+
a
ni
(
x
,
u
)
x
n
+
b
1
i
(
x
,
u
)
u
1
+···+
b
mi
(
x
,
u
)
u
m
(4.8)
where
a
ki
(
x
,
u
)
and
b
ji
(
x
,
u
)
are variable coefficients (Wong et al.
1997
) given by:
M
i
l
M
i
l
k
=
0
,
1
,...,
n
1
w
i
(
b
l
ji
1
w
i
(
a
l
ki
x
,
u
)
x
,
u
)
=
=
a
ki
(
x
,
u
)
=
,
b
ji
(
x
,
u
)
=
,
i
=
1
,...,
n
M
i
l
M
i
l
1
w
i
(
1
w
i
(
x
,
u
)
x
,
u
)
j
=
1
,...,
m
=
=
(4.9)
Since there is no risk of confusing the previous variable coefficients with the
parameters of the consequent of the rules, since these coefficients do not depend on
the index
l
, in the subsequent expressions the dependence of these coefficients with
respect the state and control vectors is omitted in order to simplify the notation.
The Eq. (
4.8
) can then be grouped in the form:
n
m
x
i
˙
=
a
0
i
+
a
ki
x
k
+
b
ji
u
j
(4.10)
k
=
1
j
=
1
If the state vector is extended in a coordinate (Andújar and Barragán
2005a
)by
x
0
=
˜
˜
1, the extended state vector,
x
, takes the form:
T
T
x
˜
=
(
˜
x
0
,
˜
x
1
,...,
˜
x
n
)
=
(
1
,
x
1
,...,
x
n
)
(4.11)
Finally, using (
4.11
) over (
4.10
), the final expression is obtained. This equation,
more compact, represents the state equation of a continuous-time completely general
plant modeled by fuzzy logic:
n
m
x
i
˙
=
a
ki
˜
x
k
+
b
ji
u
j
(4.12)
k
=
0
j
=
1
In (
4.12
), each of the state equations is characterized by a of independent number
of rules, and each of their membership functions can be defined completely inde-
pendently, even allowing the mixture of different types of membership functions in
a single rule.
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