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˜
˜
˜
∀
j
∈
Ξ
J
, ∃
σ
j
≥
0
, ∃
τ
j
≥
0:
∀
(
x
,
χ
)
∈
X×X
χ
,
φ
4
(
x
)
N
1
,
j
−
)
φ
4
(
x
˜
,
χ
σ
N
0
(
P
,
χ
)
≤
0
,
j
)
N
2
,
j
−
)
φ
4
(
x
(10.29)
φ
4
(
x
˜
,
χ
τ
j
N
0
(
P
,
χ
)
≤
0
.
˜
For each
j
, the multipliers ˜
σ
j
and ˜
τ
j
cannot be zero as
X
is not wholly admissible.
1
˜
1
˜
Setting
τ
j
leads to the following equivalent formulation of (10.29),
which is also a sufficient condition to (10.9):
σ
j
and
τ
j
σ
j
˜
∀
j
∈
Ξ
J
, ∃
σ
j
>
0
, ∃
τ
j
>
0:
∀
(
x
,
χ
)
∈
X×X
χ
,
φ
4
(
x
)
σ
j
N
1
,
j
−N
0
(
)
φ
4
(
x
,
χ
P
,
χ
)
≤
0
,
)
τ
j
N
2
,
j
−N
0
(
)
φ
4
(
x
(10.30)
φ
4
(
x
,
χ
P
,
χ
)
≤
0
.
X
0
into
˜
˜
Inclusion of a given ellipsoidal set
E
. Consider the problem of enclosing
˜
˜
a given ellipsoid
,where
X
0
,X
c
0
respectively term some sets of initial sensor-target situations and initial
controller state vectors. Let
X
0
X
0
×X
c
0
by the multicriteria basin of attraction
E
˜
X
0
write as
˜
x
:
x
=
x
0
+
n
z
X
0
{
E
z
, |
z
|≤
1
,
z
∈
R
}
(10.31)
with
x
0
its center and
(
1 φ
1
(
x
,
χ)
z
)
leads
E
its “shape matrix”. Setting
φ
5
(
x
,
χ
)
˜
˜
to the following trivial formulation of the inclusion
X
0
⊂
E
:
X×X
χ
,
φ
5
(
x
0
⇒
φ
5
(
x
0
,
(10.32)
˜
∀
(
x
,
χ
)
∈
,
χ
)
O
0
φ
5
(
x
,
χ
)
≤
,
χ
)
O
1
φ
5
(
x
,
χ
)
≤
−
1
00
0
and
−
1
00
0
. By the S-procedure (Lemma 10.1), a
with
O
0
O
1
OO
PO
0
OI
0
OO
sufficient condition to (10.32) is
,
χ)
O
1
−
κ
O
0
φ
5
(
x
˜
X×X
χ
,
φ
5
(
x
∃
κ
≥
0:
∀
(
x
,
χ)
∈
,
χ)
≤
0
,
(10.33)
where
κ
is readily seen to be nonzero. We are now ready for the main theorem.
˜
Theorem 10.1 (Multicriteria Analysis via BQLFs).
defined within Definition
10.5 is a multicriteria basin of attraction for the visual servo
(10.4)
subject to the
constraints
(10.5)
and encloses the ellipsoid of initial conditions defined by
(10.31)
if the LMIs on the matrices
E
L
,
W
,
{
Y
k
}
k
∈
Ξ
K
,
{
F
j
}
j
∈
Ξ
J
,
{
G
j
}
j
∈
Ξ
J
,
K
, on the pos-
itive scalars
{
η
k
}
k
∈
Ξ
K
,
{
σ
j
}
j
∈
Ξ
J
,
{
τ
j
}
j
∈
Ξ
J
,
κ
, and on the matrix
P
defining the
(
˜
BQLF
(10.18)
, are in effect at all the vertices
V
X×X
χ
)
:
(
˜
∀
(
x
,
χ
)
∈V
X×X
χ
)
,
Ψ
1
(
x
L
>
P
+
LΨ
1
(
x
,
χ
)+
,
χ
)
0
,
(10.34)
Γ
1
A
1
Θ
(
x
,
χ)
P
+
P Θ
(
x
,
χ)
A
1
Γ
1
+
Ψ
2
(
x
W
<
WΨ
2
(
x
,
χ
)+
,
χ
)
0
,
(10.35)
A
2
Θ
(
x
,
χ
)
P
O
(2η
k
−
1)
+
Ψ
3
(
x
Y
k
≥
∀
k
∈
Ξ
K
,
Y
k
Ψ
3
(
x
,
χ
)+
,
χ
)
0
,
(10.36)
−
η
k
Γ
1
a
k
P
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