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the choice of some particular solutions is obtained thanks to convex optimization
schemes.
15.4.1
Theoretical Issues
Let us now consider a positive definite function
V
(
x
)
= 0, with
V
(0)=
0, such that its time-derivative along the trajectories of the closed-loop saturating
system (15.21) verifies
>
0
,∀
x
V
(
x
)
≤
ω
ω
,
(15.29)
∀
∈D
D
for all
is some bounded domain inside
the basin of attraction of (15.21). If (15.29) is verified, it follows that
ω
satisfying (15.11) and
x
,where
t
0
ω
)
ω
V
(
x
(
t
))
−
V
(
x
(0))
≤
(
τ
(
τ
)
d
τ
,
(15.30)
1
δ
1
∀
∈D
∀
≥
≤
x
and
t
0. From (15.11), we have
V
(
x
(
t
))
V
(
x
(0)) +
and, hence,
1
ζ
1
ζ
1
δ
1
.
V
(
x
(0))
≤
=
⇒
V
(
x
(
t
))
≤
+
(15.31)
1
ζ
1
δ
1
}
1
ζ
}
6
;
V
(
x
)
6
;
V
(
x
)
Define the sets
S
1
=
{
x
∈
ℜ
≤
+
and
S
0
=
{
x
∈
ℜ
≤
.Pro-
vided that
S
1
is included in the basin of attraction of (15.21), from (15.31) it follows
that the closed-loop system trajectories remain bounded in
S
1
, for any initial condi-
tion
x
(0)
∈S
0
,andany
ω
satisfying (15.11) [3]. A general result can then be stated
in the following theorem.
Theorem 15.1.
If there exist a positive definite function V
(
x
)
(V
(
x
)
>
0
,
∀
x
= 0
), a
gain
K
, a positive definite diagonal matrix M, a matrix
G
and two positive scalars
ζ
and
δ
1
satisfying, for any admissible z and i
= 1
,
2
,
3
,
∂
V
∂
x
[(
x
)
M
(
A
(
z
,
x
)+
B
K
)
x
+
B
φ
(
K
x
)+
B
2
(
z
,
x
)
ω
]
−
2
φ
(
K
φ
(
K
x
)+
G
x
)
1
1
−
ω
ω
<
0
,
(15.32)
1
ζ
+
1
δ
1
u
0(
i
)
x
(
K
(
i
)
−
G
(
i
)
)
V
(
x
)
−
(
K
(
i
)
−
G
(
i
)
)
x
≥
0
,
(15.33)
1
ζ
1
δ
1
+
x
R
(
i
)
V
(
x
)
−
R
(
i
)
x
≥
0
,
(15.34)
2
β
1
ζ
+
1
δ
1
u
1(
i
)
C
(
i
)
x
x
C
(
i
)
V
(
x
)
−
≥
0
,
(15.35)
6
;
V
(
x
)
1
ζ
+
1
then the gain
K
and the sets
S
1
(
V
,
ζ
,
δ
1
)=
{
x
∈
ℜ
≤
δ
1
}
and
S
0
(
V
,
ζ
)=
6
;
V
(
x
)
≤
ζ
−
1
{
x
∈
ℜ
}
are solutions to Problem 15.1.
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