Information Technology Reference
In-Depth Information
˜
˜
definitions of
Γ
1
and
a
k
,
k
∈
Ξ
K
, let each convex cell
X
s
of
X
,
s
∈
Ξ
S
, be charac-
∈
Ξ
∈
Ξ
terized by
n
v
(
s
)
vectors
v
(
s
,
k
v
)
,
k
v
n
v
(
s
)
,and
n
z
(
s
)
vectors
z
(
s
,
k
z
)
,
k
z
n
z
(
s
)
:
˜
:
v
(
s
,
k
v
)
x
z
(
s
,
k
z
)
x
n
X
s
=
{
x
∈
R
≤
1
,
≤
0
,
k
v
∈
Ξ
n
v
(
s
)
,
k
z
∈
Ξ
n
z
(
s
)
}.
(10.42)
˜
Let the vectors
e
s
a
s
b
“support” the boundaries
∂
X
s
a
s
b
of adjacent polytopes,
i.e.
˜
˜
˜
x
:
e
s
a
s
b
x
= 0
∀
(
s
a
,
s
b
)
∈
Π
,
∂
X
s
a
s
b
X
s
a
∩
X
s
b
⊂{
},
(10.43)
(
n
+
n
)
×
(
n
+
n
)
with
Π
as in (10.19). Define a constant matrix
E
s
a
s
b
∈
R
such that
Θ
Θ
˜
∀
(
x
,
χ
)
∈
∂
X
s
a
s
b
×X
χ
,
E
s
a
s
b
φ
1
(
x
,
χ
)=
0
.
(10.44)
1
e
s
a
s
b
e
s
a
s
b
Γ
A simple choice of
E
is
E
s
a
s
b
=
Γ
1
. However, less conservative re-
s
a
s
b
sults can be obtained for other definitions of
E
s
a
s
b
depending on the choice of the
partition and the matrix
,
χ), see end of Section 10.5.
The following steps enable the definition of LMI sufficient conditions for
Θ
(
x
˜
de-
fined within Definition 10.6 to be a multicriteria basin of attraction for (10.4) subject
to (10.5):
E
∈
}
×X
χ
,
V
s
(
x
˜
•∀
s
∈
Ξ
S
,
∀
(
x
,
χ
)
X
s
\{
0
,
χ
)
>
0;
,
χ)
∈
}
×X
χ
,
V
s
(
x
˜
•∀
s
∈
Ξ
S
,
∀
(
x
X
s
\{
0
,
χ)
<
0;
⇒
a
k
x
1
;
˜
•∀
s
∈
Ξ
S
,
∀
k
∈
Ξ
K
,
∀
(
x
,
χ
)
∈
X×X
χ
, (
V
s
(
x
,
χ
)
≤
1)
≤
˜
•∀
s
∈
Ξ
S
,
∀
j
∈
Ξ
J
,
∀
(
x
,
χ
)
∈
X
s
×X
χ
,
⎧
⎨
Z
j
(
x
)
x
V
s
(
x
,
χ
)
≤
1
≤
ζ
j
;
v
(
s
,
k
v
)
x
≤
1
, ∀
k
v
∈
Ξ
n
v
(
s
)
k
z
∈
Ξ
n
z
(
s
)
⇒
Z
j
(
x
)
x
≥
ζ
j
;
⎩
z
(
s
,
k
z
)
x
≤
0
, ∀
˜
•∀
(
s
a
,
s
b
)
∈
Π
, ∀
(
x
,
χ
)
∈
∂
X
s
a
s
b
×X
χ
,
V
s
a
(
x
,
χ
)=
V
s
b
(
x
,
χ
);
˜
V
s
a
(
x
)=
V
s
b
(
x
•∀
(
s
a
,
s
b
)
∈
Π
, ∀
(
x
,
χ
)
∈
∂
X
s
a
s
b
×X
χ
,
,
χ
,
χ
).
Theorem 10.2 (Multicriteria Analysis via PW-BQLFs).
˜
defined within Defini-
tion 10.6 is a multicriteria basin of attraction for the visual servo
(10.4)
subject
to the constraints
(10.5)
if the following LMIs on the matrices
E
{
P
s
}
s
∈
Ξ
S
,
{
L
s
}
s
∈
Ξ
S
,
{
W
s
}
s
∈
Ξ
S
,
{
Y
s
,
k
}
s
∈
Ξ
S
,
k
∈
Ξ
K
,
{
F
s
,
j
}
s
∈
Ξ
S
,
j
∈
Ξ
J
,
{
G
s
,
j
}
s
∈
Ξ
S
,
j
∈
Ξ
J
,
{
M
s
a
s
b
}
(
s
a
,
s
b
)
∈
Π
, and
{
N
s
a
s
b
}
(
s
a
,
s
b
)
∈
Π
, on the positive scalars
{
η
s
,
k
}
s
∈
Ξ
S
,
k
∈
Ξ
K
,
{
σ
0(
s
,
j
)
}
s
∈
Ξ
S
,
j
∈
Ξ
J
, and
{
τ
0(
s
,
j
)
}
s
∈
Ξ
S
,
j
∈
Ξ
J
, and on the nonnegative scalars
{
σ
v
(
s
,
j
,
k
v
)
}
s
∈
Ξ
S
,
j
∈
Ξ
J
,
k
v
∈
Ξ
n
v
(
s
)
,
{
σ
z
(
s
,
j
,
k
z
)
}
s
∈
Ξ
S
,
j
∈
Ξ
J
,
k
z
∈
Ξ
n
z
(
s
)
,
{
τ
v
(
s
,
j
,
k
v
)
}
s
∈
Ξ
S
,
j
∈
Ξ
J
,
k
v
∈
Ξ
n
v
(
s
)
,
{
τ
z
(
s
,
j
,
k
z
)
}
s
∈
Ξ
S
,
j
∈
Ξ
J
,
k
z
∈
Ξ
n
z
(
s
)
,
are in effect:
˜
Ψ
1
(
x
L
s
>
∀
s
∈
Ξ
S
, ∀
(
x
,
χ
)
∈V
(
X
s
×X
χ
)
,
P
s
+
L
s
Ψ
1
(
x
,
χ
)+
,
χ
)
0
,
(10.45)
(
˜
∀
s
∈
Ξ
S
, ∀
(
x
,
χ
)
∈V
X
s
×X
χ
)
,
Ψ
2
(
x
W
s
<
M
(
x
,
χ)
(
P
s
)+
W
s
Ψ
2
(
x
,
χ
)+
,
χ
)
0
,
(10.46)
Search WWH ::
Custom Search