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R
1
:
−
IF
z
1
is
around
1
⊧
⊨
−
x
20
x
˙
=
THEN
1
−
3
⊩
y
=
x
1
R
2
:
IF
z
1
is
around
1
⊧
⊨
00
1
x
x
˙
=
THEN
−
3
⊩
y
=
x
1
It can be easily seen that this is an exact representation of the nonlinear system
in the compact set
x
1
,
, when expressions
i
=
1
h
i
(
x
2
∈[−
1
,
1
]
z
(
t
))
A
i
x
(
t
)
and
i
=
1
h
i
(
z
(
t
))
C
i
x
(
t
)
are developed,
−
x
1
x
2
00
1
x
1
x
2
−
1
−
x
2
1
+
x
2
20
1
x
1
+
x
1
x
2
+
=
−
3
−
3
x
1
−
3
x
2
2
2
1
−
x
2
1
+
x
2
x
1
+
x
1
=
x
1
=
y
.
2
2
The main idea of the controller design based on TS fuzzy systems is to derive
each control rule so as to compensate each subsystem or rule of the fuzzy system.
So, for every subsystem
x
˙
=
A
i
x
+
B
i
u
a control signal of the form
u
=−
F
i
x
is selected, resulting in a closed-loop system formed by a set of subsystems such as,
x
˙
=
(
A
i
−
B
i
F
i
)
x
.
The design procedure is conceptually simple and natural. The stability analysis and
control design problems can be reduced to linear matrix inequality (LMI) problems.
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