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x
1
(
˙
t
)
=
x
2
(
t
)
g
L
b
ML
2
1
ML
2
τ(
(4.55)
x
2
(
˙
t
)
=
sin
(
x
1
(
t
))
−
x
2
(
t
)
+
t
).
A fuzzy model for this system in the universe of discourse
u
−
10
3
10
3
X
⊆ {
x
|
x
1
∈
[
−
π, π
]
,
x
2
∈
[
−
10
,
10]
}
,
U
⊆
|
u
∈
,
(4.56)
is given by
4
:
IF
x
1
is Trimf(
−∞
;0;
∞
)and
x
2
is Trimf(
−∞
;0;
∞
)and
τ
is Trimf(
−∞
;0;
∞
)
THEN
x
1
=
x
2
IF
x
1
is Gaussmf(1.013; 1.509) and
x
2
is Gaussmf(1.123; 10.06) and
τ
is Gaussmf(106.6; 924.4)
THEN
x
2
=
0
.
9452
−
2
.
6
∗
x
1
−
1
.
5
∗
x
2
+
4
.
015
∗
τ
IF
x
1
is Gaussmf(1.463; 0.9176) and
x
2
is Gaussmf(
−
1.33; 9.816) and
τ
is Gaussmf(
−
782.7;
924.4)
THEN
x
2
=
.
−
.
∗
x
1
−
.
∗
x
2
+
.
∗
τ
65
57
2
805
1
403
4
038
−
τ
IF
x
1
is Gaussmf(
0.876; 0.8156) and
x
2
is Gaussmf(8.382; 12.32) and
is Gaussmf(944.3;
924.4)
THEN
x
2
=−
31
.
73
−
0
.
4918
∗
x
1
−
1
.
379
∗
x
2
+
4
.
013
∗
τ
where
Gaussmf
(
c
,β)
is a gaussian membership function given by (
4.54
), and
Trimf
(
a
,
b
,
c
)
is a triangular membership function given by:
⎧
⎨
x
−
a
if
a
<
x
≤
b
b
−
a
c
−
x
μ
Tri
[
a
,
b
,
c
]
(
x
)
=
(4.57)
if
b
<
x
<
c
⎩
c
−
b
0
in other case
Once initialization has been performed by the Algorithm 4.4, four regions are
defined,
s
4, corresponding to the regions defined by the four rules of the fuzzy
controller. After the initialization, the phases of design and final adjustments are
made, obtaining the following fuzzy controller and
P
q
matrices:
IF
x
1
is Trimf(
=
−∞
;0;
∞
)and
x
2
is Trimf(
−∞
;0;
∞
)
THEN
x
2
IF
x
1
is Gaussmf(1.013; 1.509) and
x
2
is Gaussmf(1.123; 10.06)
THEN
τ
=
5
.
353
−
7
.
417
∗
x
1
−
1
.
85
∗
x
2
IF
x
1
is Gaussmf(1.463; 0.9176) and
x
2
is Gaussmf(
τ
=−
8
.
169
−
0
.
4213
∗
x
1
−
0
.
5807
∗
−
1.33; 9.816)
THEN
τ
=−
20
.
31
−
3
.
055
∗
x
1
−
2
.
173
∗
x
2
IF
x
1
is Gaussmf(
−
0.876; 0.8156) and
x
2
is Gaussmf(8.382; 12.32)
THEN
τ
=
4
.
055
−
0
.
9655
∗
x
1
+
0
.
06538
∗
x
2
0
0
.
467 0
.
009
.
011 0
00
P
1
=
P
2
=
0
.
009 0
.
011
.
0109
0
0
.
024 0
00
.
01
0
P
3
=
P
4
=
.
012
0
.
001 0
.
0352
4
The fuzzy model has been obtained from a set of system data, using the clustering algorithm (Chiu
1994
) and adjusting the parameters using the ANFIS algorithm (Jang
1993
).
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