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Definition 10.5 (Biquadratic Lyapunov Functions (BQLFs)).
BQLFs
candidates
for the analysis of
(10.4)
subject to
(10.5)
are defined by
˜
(
Θ
(
x
,
χ)
I
n
)
x
φ
1
(
x
∀
(
x
,
χ
∈
X×X
χ
,
V
(
x
,
χ
,
χ
P
φ
1
(
x
,
χ
,
φ
1
(
x
,
χ
,
(10.18)
)
)=
)
)
)
n
Θ
×
n
where
Θ
(
.,.
)
∈
R
terms a linear matrix function selected beforehand, and
P
=
P
∈
R
+
n
)
is a constant matrix to be tuned. Importantly, BQLFs can lead
to asymmetric nonconvex basins
˜
(
n
Θ
+
n
)
×
(
n
Θ
E
{
x
:
V
(
x
,
χ)
≤
1
, ∀
χ
∈X
χ
}
.
Definition 10.6 (Piecewise-biquadratic Lyapunov Functions (PW-BQLFs) [9]).
Partition
˜
˜
˜
n
X ⊂
R
into S convex polytopic regions
X
1
,...,
X
S
enclosing
0
, so that
˜
˜
˜
˜
X
=
∪
s
∈
Ξ
S
X
s
. Assume that the boundary
∂
X
s
a
s
b
of two adjacent cells
X
s
a
and
˜
X
s
b
,s
a
=
s
b
∈
Ξ
S
,isa
(
n
−
1)
-dimensional polytope. Denote by
Π
the set of pairs
of indexes of adjacent cells arranged in increasing order, i.e.
Π
(
s
a
,
}
.
˜
˜
s
b
)
∈
Ξ
S
×
Ξ
S
:
s
a
<
s
b
and
X
s
a
∩
X
s
b
=
{
0
(10.19)
˜
To each cell
X
s
, associate a BQLF
˜
φ
1
(
x
(
Θ
(
x
,
χ)
I
n
)
x
∀
(
x
,
χ
)
∈
X
s
×X
χ
,
V
s
(
x
,
χ
)=
,
χ
)
P
s
φ
1
(
x
,
χ
)
,
φ
1
(
x
,
χ
)
,
(10.20)
P
s
∈
R
(
n
+
n
)
×
(
n
Θ
+
n
)
a matrix to be determined, and
n
Θ
×
n
s
∈
Ξ
S
, with
P
s
=
Θ
(
.,.
)
∈
R
Θ
˜
a predefined linear matrix function. Let V
(
.,.
) :
X×X
χ
−→
R
be the piecewise
˜
function whose restriction to each
X
s
,s
∈
Ξ
S
, is equal to V
s
(
.,.
)
. If, in addition,
˜
)
and V
s
a
(
x
)=
V
s
b
(
x
∀
(
s
a
,
s
b
)
∈
Π
,∀
(
x
,
χ
)
∈
∂
X
s
a
s
b
×X
χ
,
V
s
a
(
x
,
χ
)=
V
s
b
(
x
,
χ
,
χ
,
(10.21)
,
χ
)
˜
˜
then V
(
.,.
)
is said a
PW-BQLF
on
X×X
χ
. Further, a basin of attraction
E
is
defined as the union
˜
˜
˜
˜
E
∪
s
∈
Ξ
S
E
,
∀
∈
Ξ
S
,
E
{
x
∈
X
s
:
V
s
(
x
,
χ
≤
,∀
χ
∈X
χ
}.
with
s
)
1
(10.22)
s
s
10.4.2
LMI Conditions for Multicriteria Analysis Based on
BQLFs
is assumed constant
2
. The vector function
To simplify, the uncertainty vector
χ
,
χ)=(
Θ
(
x
,
χ)
I
n
)
x
being introduced in (10.18), define
φ
1
(
x
Γ
1
=(
O
n
×
n
Θ
I
n
) so that
)=
x
,and
Θ
˜
˙
)=
Θ
)
x
.Let
(
n
Θ
+
n
)
×
n
Γ
1
φ
1
(
x
,
χ
(
.,.
) :
X×X
χ
→
R
so that
φ
1
(
x
,
χ
(
x
,
χ
˜
˜
x
:
a
k
x
the vectors
a
k
,
k
∈
Ξ
K
, define the edges of
X
by
X
=
{
≤
1
,
k
∈
Ξ
K
}
.
˜
Some preliminaries are needed to establish LMI sufficient conditions for
de-
fined within Definition 10.5 to be a multicriteria basin of attraction for the visual
servo (10.4) subject to the constraints (10.5).
E
2
The method can be easily extended in order to handle time-varying smooth parametric
uncertainties in χ, once polytopes enclosing χ and χ are given.
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