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⎧
⎨
u
0(
i
)
−
K
(
i
)
x
if
K
(
i
)
x
>
u
0(
i
)
0
if
|
K
(
i
)
x
|≤
u
0(
i
)
φ
(
K
(
i
)
x
)=
(15.20)
⎩
−
u
0(
i
)
−
K
(
i
)
x
if
K
(
i
)
x
<−
u
0(
i
)
∀
,
,
i
= 1
2
3, the closed-loop system (15.18) can be described by
x
=(
A
(
z
,
x
)+
B
1
K
)
x
+
B
1
φ(
K
x
)+
B
2
(
z
,
x
)ω
.
(15.21)
3
×
6
and define the set
Consider a matrix
G
∈
ℜ
6
;
S
(
u
0
)=
{
x
∈
ℜ
−
u
0
(
K
−
G
)
x
u
0
}.
(15.22)
The following lemma, issued from [12], can then be stated.
Lemma 15.1.
Consider the nonlinearity
φ
(
K
x
)
defined in (15.20). If x
∈
S
(
u
0
)
,then
the relation
x
)
M
(
φ
(
K
φ
(
K
x
)+
G
x
)
≤
0
(15.23)
×
3
3
.
is satisfied for any diagonal positive definite matrix M
∈
ℜ
Let us first write in a more tractable way matrices
x
), and therefore
the closed-loop system (15.21). In this sense, we define the matrices
A
(
z
,
x
) and
B
2
(
z
,
⎡
⎤
z
1
Y
1
/
z
1
1 +(
Y
1
)
2
−
1
/
⎣
⎦
;
z
2
Y
2
/
z
2
1 +(
Y
2
)
2
B
1
(
z
)=
−
1
/
z
3
Y
3
/
z
3
1 +(
Y
3
)
2
−
1
/
⎡
⎤
⎡
⎤
2
Y
1
00
02
Y
2
0
00
Y
3
1
/
z
1
00
01
⎣
⎦
;
B
3
=
⎣
⎦
;
B
2
(
z
)=
z
2
0
001
/
/
z
3
⎡
⎤
Y
1
/
Y
1
/
1
/
z
1
−
z
1
1
/
z
1
−
z
1
⎣
⎦
;
Y
2
/
B
4
(
z
)=
1
/
z
2
−
z
2
0
0
Y
3
/
Y
3
/
1
/
z
3
−
z
3
−
1
/
z
3
z
3
⎡
⎤
⎡
⎤
0
−
1
/
z
1
0
−
1
/
z
1
e
1
00
0
e
2
0
00
e
3
⎣
⎦
;
D
(
e
)=
⎣
⎦
;
B
5
(
z
)=
0
−
1
/
z
2
00
−
/
/
0
1
z
3
01
z
3
=
I
3
0
;
=
0
I
3
.
R
C
Then, the closed-loop system reads
R
B
1
(
z
)
R
T
(2)
B
2
(
z
)
R
T
(3)
(
B
3
+
D
(
e
))
x
=(
C
+
R
+
B
1
K
+
R
)
x
+
B
1
φ
(
K
x
)
R
(
B
4
(
z
)+
D
(
e
)
B
5
(
z
))
+
ω
.
(15.24)
According to the notation previously specified,
T
(2)
and
T
(3)
respectively denote the
2th and 3th components of vector
T
. Furthermore, in (15.6) and (15.7), the con-
straints on the vector
z
=
z
1
z
2
z
3
∈
ℜ
3
follow the description given in Figure
15.1. From relations (15.1), (15.2), (15.4) and (15.5), the depth of target points
E
1
and
E
3
can be expressed in terms of the depth
z
2
of the central point
E
2
:
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