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∂ϕ
q
(
d
ϕ
q
(
x
)
x
)
,
∂ϕ
q
(
x
)
,...,
∂ϕ
q
(
x
)
=
(4.42)
d
x
∂
x
1
∂
x
2
∂
x
n
(
1
×
n
)
and
⎛
⎞
x
1
˙
˙
⎝
⎠
x
2
.
˙
f
(
x
)
=
,
(4.43)
x
n
(
n
×
1
)
V
so that
(
x
)
can be written as
f
J
f
s
T
d
ϕ
q
(
x
)
V
T
T
P
q
+
(
x
)
=
(
x
)
ϕ
q
(
x
)
(
x
)
P
q
J
(
x
)
(
x
)
+
f
(
x
)
f
(
x
)
P
q
f
(
x
)
d
x
q
=
1
(4.44)
T
to the right, and
extracting these functions of the summation (they do not depend on
q
), the final
expression of
Finally, taking
f
(
x
)
as a common factor on the left, and
f
(
x
)
V
(
x
)
is obtained:
P
q
f
s
J
d
ϕ
q
(
x
)
V
T
T
P
q
+
(
x
)
=
f
(
x
)
ϕ
q
(
x
)
(
x
)
P
q
J
(
x
)
+
f
(
x
)
(
x
)
(4.45)
d
x
q
=
1
Given
F
(
x
)
as defined in (
4.28
), the above equation can be written as:
V
T
F
(
x
)
=
f
(
x
)
(
x
)
f
(
x
)
(4.46)
V
(
)
(
)
Therefore, it is demonstrated that if
F
x
is negative definite,
x
is as well, and
V
(
x
)
is a Lyapunov function for the system (
4.27
).
Obviously, if the dynamics of the plant is only known through its fuzzy model,
and if this was obtained by input/output data, it is impossible to guarantee the global
asymptotic stability of the system, since it is unknown outside its behavior the uni-
verse of discourse of the variables. Thus, asymptotic stability can only be guaranteed
for those trajectories that meet the Theorem 4.1 and remain entirely within the uni-
verse of discourse. That is:
Let
X
the universe of discourse of the state variables and
∂
X
its edge, and are
C
2
∈ R
+
, so that
C
1
=
)
|
V
C
1
,
min
{
V
(
x
(
x
)
=
0
}
and
C
2
=
min
{
V
(
x
)
|
x
∈
∂
X
}
.
The most conservative estimate of the region of attraction
which contains the
equilibrium state at the origin, is that region contained in the universe of discourse
of the state variables,
⊆
X
, in which
V
(
x
)
behaves as a Lyapunov function and
satisfying (
4.47
).
⊆ {
∈
|
(
)<
{
C
1
,
C
2
}}
x
X
V
x
min
(4.47)
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