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form. The overall fuzzy model of the system is achieved by fuzzy “blending” of the
linear systems. A TS fuzzy model representing a continuous dynamical system has
the following form,
Rule i:
IF
z
1
(
t
)
is
M
i
1
and
···
and
z
p
(
t
)
is
M
ip
THEN
˙
x
(
t
)
=
A
i
x
(
t
)
+
B
i
u
(
t
)
y
(
t
)
=
C
i
x
(
t
)
for
i
r
.
Here,
M
ij
is the fuzzy set and
r
is the number of model rules;
x
=
1
,
2
,...,
R
n
is the state
(
t
)
∈
R
m
R
q
R
n
×
n
,
vector,
u
(
t
)
∈
is the input vector,
y
(
t
)
∈
is the output vector,
A
i
∈
R
n
×
m
R
q
×
n
;
z
are known premise variables
that may be functions of the state variables and/or external disturbances.
Each linear consequent equation represented by
A
i
x
B
i
∈
,
and
C
i
∈
(
t
)
={
z
1
(
t
), . . . ,
z
p
(
t
)
}
(
t
)
+
B
i
u
(
t
)
is called a “sub-
system”.
˙
)
=
i
=
1
h
i
(
x
(
t
z
(
t
))
{
A
i
x
(
t
)
+
B
i
u
(
t
)
}
)
=
i
=
1
h
i
(
y
(
t
z
(
t
))
C
i
x
(
t
)
where,
))
i
=
1
w
i
(
w
i
(
z
(
t
h
i
(
z
(
t
))
=
,
(
))
z
t
p
w
i
(
z
(
t
))
=
M
ij
(
z
(
t
))
j
=
1
for all
t
.Theterm
M
ij
(
z
(
t
))
is the grade of membership of
z
j
(
t
)
in
M
ij
.
We have
i
=
1
h
i
(
r
and all
t
.
Thanks to the normalized membership functions, the linear dynamic TS model
is in fact a convex combination of local linear models. This property facilitates the
stability analysis of the fuzzy system. Fuzzy systems are universal function approx-
imators and hence can be used to model a wide class of processes.
z
(
t
))
=
1, and
h
i
(
z
(
t
))
0, for
i
=
1
,
2
,...,
Example 8.3.2
Consider the following nonlinear dynamic system
⊧
⊨
x
1
=−
˙
x
1
+
x
1
x
2
x
2
=
˙
x
1
−
3
x
2
⊩
y
=
x
1
with
x
1
,
. This system can be exactly represented, using the sector
nonlinearity approach. In this model, the scheduling variable
z
1
is chosen as
x
2
,the
fuzzy sets are
M
11
x
2
∈[−
1
,
1
]
=
'
around
−
1',
M
12
=
'
around
1', and the corresponding
membership functions are
h
11
=
(
2, respectively. The
following TS fuzzy system with linear consequents is the exact representation of the
original model,
1
−
z
1
)/
2 and
h
12
=
(
1
+
z
1
)/
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