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n
r
j
r
r
ω
=
1
μ
kj
(
x
k
,
kj
)
(4.18)
k
=
Proceeding in a similar way as performed to obtain (
4.7
) and (
4.8
), can be obtained
(
4.19
) and (
4.20
). In this case, the variable coefficients
c
kj
(
are obtained via (
4.21
).
As in the previous case, the dependence of the variable coefficients of the controller
with respect to the state vector is omitted in order to simplify the notation.
x
)
N
j
r
N
j
r
N
j
r
r
c
0
j
r
c
1
j
r
c
r
nj
1
ω
j
(
x
)
1
ω
j
(
x
)
1
ω
j
(
x
)
=
=
=
u
j
=
+
+···+
(4.19)
N
j
r
N
j
r
N
j
r
r
r
r
1
ω
j
(
x
)
1
ω
j
(
x
)
1
ω
j
(
x
)
=
=
=
u
j
=
c
0
j
(
x
)
+
c
1
j
(
x
)
x
1
+···+
c
nj
(
x
)
x
n
(4.20)
N
j
r
r
c
kj
1
ω
j
(
x
)
=
c
kj
(
x
)
=
(4.21)
N
j
r
r
1
ω
j
(
x
)
=
Equation (
4.20
) can be grouped, resulting in (
4.22
), which represents the equiva-
lent mathematical model of the fuzzy controller.
n
u
j
=
c
0
j
+
c
kj
x
k
(4.22)
k
=
1
Using the state vector extension, the fuzzy controller can be represented in a
compact way by (
4.23
).
n
u
j
=
c
kj
˜
x
k
(4.23)
k
=
0
The expression (
4.23
) represents the equivalent mathematical model of a com-
pletely general multivariable controller, without limitations on the size of the state
vector nor the control vector. Each output of the controller is characterized by a
independent number of rules, and each of their membership functions can be defined
completely independently, and can even be of different types in the same rule.
4.2.3 Fuzzy Model of the Closed Loop Control System
Given a closed loop control system of as shown in Fig.
4.1
, if both the plant and the
controller are modeled by TS type fuzzy systems, it is possible to obtain an equivalent
mathematical model of the system.
Substituting
u
j
from (
4.23
)in(
4.12
):
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