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)
A
i
(
)
ρ
T
R
−
ρ
(
)
(
˜
)
−
B
i
(
)
M
j
(
)
+
A
j
(
)
(
˜
)
−
B
j
(
)
M
i
(
2
x
x
X
x
x
x
x
X
x
x
x
k
T
dX
(
˜
x
)
i
:
1
,...,
r
+
ρ
A
i
(
x
)
z
(
x
)ρ
−
s
ijc
(
x
,ρ)
g
c
(
x
)
∈
x
,ρ
(5.59)
r
≥
j
>
i
dx
c
=
1
dz
(
x
)
∈
R
v
×
n
x
where
ε>
0
acts as a tolerance, R
(
x
)
=
and s
i
(
x
)
∈
x
,
dx
(
s
ic
(
x
,ρ),
s
ijc
(
x
,ρ))
∈
x
,ρ
, then controller
(
5.55
)
stabilizes system
(
5.54
)
in
a region of the state space
{
x
:
V
(
x
)
≤
v
0
}⊂
(see Remark 5.2). Controller gains
)
−
1
.
can be obtained by K
j
(
x
)
=
M
j
(
x
)
X
(
˜
x
Proof
Conditions (
5.58
) and (
5.59
) together mean (
5.28
) after carrying out some
operations with the change of variable
)
−
1
and the
ρ
=
P
(
˜
x
)
z
(
x
),
X
(
˜
x
)
=
P
(
˜
x
evident fact of
dP
(
˜
x
)
dX
(
˜
x
)
P
(
˜
x
)
X
(
˜
x
)
=
I
,
X
(
˜
x
)
+
P
(
˜
x
)
=
0
.
dx
dx
So, jointly with (
5.57
), they make
V
(
x
)
to be a Lyapunov function for system (
5.54
),
with controller (
5.55
), locally in
by Lemma 5.3 and 5.2. The use of
X
(
˜
x
)
instead
of
X
(
x
)
allows conditions (
5.58
)-(
5.59
) to be convex due to the fact that term
T
dX
(
˜
x
)
0in
V
v
(
B
i
(
x
)
K
j
(
x
)
z
(
x
))v
=
(
x
)
.
dx
Remark 5.3
Note that conditions (
5.59
) may be relaxed via dimensionality expan-
sion or via artificial decision variables by using Polya's theorem (Sala and Ariño
2007
).
, etc) may also be
adapted to the fuzzy polynomial case. Details and examples omitted for brevity, see
Prajna et al. (
2004b
); Tanaka et al. (
2009a
).
The discrete-time case can be addressed by using the equivalent discrete-time
fuzzy polynomial model and replacing conditions (
5.58
) and (
5.59
)by
Other state-feedback design criteria (such as decay rate,
H
∞
T
)
−
c
=
1
s
ic
(
X
(
˜
x
x
,ρ)
g
c
(
x
)
I
(
∗
)
ρ
ρ
∈
x
,ρ
i
:
1
,...,
r
(
A
(
A
T
(
x
)
x
)(
A
i
(
x
)
X
(
˜
x
)
−
B
i
(
x
)
M
i
(
x
))
X
(
x
)
x
)
(5.60)
T
X
)
−
c
=
1
s
ijc
(
:
,...,
(
˜
x
x
,ρ)
g
c
(
x
)
I
(
∗
)
i
1
r
ρ
ρ
∈
x
,ρ
(
A
(
A
1
r
≥
j
>
i
(5.61)
(
)
)(
ij
(
)
+
ji
(
))
(
)
)
2
T
x
x
x
x
X
x
x
respectively, where
ij
(
x
)
=
A
i
(
x
)
X
(
˜
x
)
−
B
i
(
x
)
M
j
(
x
)
,
(
s
i
(
x
),
s
ic
(
x
,ρ),
s
ijc
(
x
,ρ))
are as in Theorem5.2,
x
are the states whose time-derivative does not depend on
u
and does not contain non-polynomial nonlinear terms,
˜
A
(
x
)
isamatrixformedby
˜
the rows of
x
and
T
(
˜
x
)
is a matrix defined by
z
(
x
)
=
T
(
˜
x
)
x
. Proof omitted for
brevity (Tanaka et al.
2008
).
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