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⎡
⎤
W
u
0(
i
)
⎣
Y
(
i
)
−
Z
(
i
)
ζ
⎦
≥
0
,
(15.37)
δ
1
u
0(
i
)
Y
(
i
)
−
Z
(
i
)
0
⎡
⎤
W
⎣
⎦
≥
2
R
(
i
)
W
ζβ
0
,
(15.38)
2
R
(
i
)
W
0
δ
1
β
⎡
⎤
W
u
1(
i
)
⎣
C
(
i
)
W
ζ
⎦
≥
0
,
(15.39)
δ
1
u
1(
i
)
C
(
i
)
W
0
3
×
6
given by
=
YW
−
1
∀
i
= 1
,
2
,
3
, then, the control gain
K
∈
ℜ
K
is such that:
(i)
when
ω
= 0
, the trajectories of the closed-loop system (15.18) remain bounded
in the set
,
ζ
,
δ
1
)=
x
1
ζ
1
δ
1
6
;
x
W
−
1
x
E
1
(
W
∈
ℜ
≤
+
,
(15.40)
for any x
(0)
∈E
0
(
W
,
ζ
)
,
)=
x
1
ζ
6
;
x
W
−
1
x
E
0
(
W
,
ζ
∈
ℜ
≤
,
(15.41)
and for any
ω
satisfying (15.11);
(ii)
,
ζ
,
δ
1
)
is included in the basin of at-
traction of the closed-loop system (15.18) and is contractive.
when
ω
= 0
,theset
E
0
(
W
,
ζ
)=
E
1
(
W
Proof.
The proof mimics the one of Theorem 15.1. The satisfaction of relation
(15.37) means that the set
,
ζ
,
δ
1
) defined in (15.40) is included in the set
S
(
u
0
)
defined in (15.22). Thus, one can conclude that for any
x
E
1
(
W
∈E
1
(
W
,
ζ
,
δ
1
) the nonlin-
=
ZW
−
1
. Furthermore, the
satisfaction of relations (15.38) and (15.39) implies that the set
earity
φ
(
K
x
) satisfies the sector condition (15.23) with
G
E
1
(
W
,
ζ
,
δ
1
) is in-
cluded in the set
(
x
) defined in (15.19). Hence, for any admissible uncertain vector
z
(see (15.27)) and any admissible vector belonging to
Ω
,
ζ
,
δ
1
), the closed-loop
system (15.21) or (15.24) can be written through the polytopic model (15.28).
The time-derivative of
V
(
x
)=
x
W
−
1
x
along the trajectories of system (15.28)
writes:
E
1
(
W
4
j
=1
λ
j
{
[
A
1
j
+
R
T
(2)
B
2
j
R
+
R
T
(3)
(
B
3
+
D
(
e
))
R
+
B
1
YW
−
1
]
x
V
(
x
)
−
ω
ω
≤
2
x
W
−
1
R
[
B
4
j
+
D
(
e
)
B
5
j
]
+ 2
x
W
−
1
+
ω
}
B
1
φ
(
K
x
)
x
)
M
(
x
)+
ZW
−
1
x
)
−
ω
ω
.
−
2
φ
(
K
φ
(
K
By convexity one can prove that the right term of the above inequality is negative
definite if
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