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(
)
Theorem 5.1
(Pitarch et al.
2012
)
Consider system
(
5.19
)
and let V
be a
predefined-degree candidate polynomial Lyapunov function. The region with pre-
fixed shape
x
q
i
(λ
−
1
x
2
={
x
:
)
≥
0
}
and user-defined scale factor
λ
, according
to Definition 5.2, belongs to the local domain of attraction of x
=
0
inaregionof
interest
, defined on
(
5.40
)
, if the following SOS problem is feasible:
m
q
q
i
(λ
−
1
x
1
−
V
(
x
)
−
s
1
i
(
x
)
)
∈
x
(5.41)
i
=
1
m
(
)
−
−
s
2
ij
(
)
g
j
(
)
+
r
2
i
(
)
g
i
(
)
∈
x
i
:
,...,
V
x
1
x
x
x
x
1
m
(5.42)
j
=
1
m
(
)
−
ε(
)
−
s
3
i
(
)
g
i
(
)
∈
x
V
x
x
x
x
(5.43)
i
=
1
m
dV
(
x
)
−
p
i
(
x
)
−
ε(
x
)
−
s
4
ij
(
x
)
g
j
(
x
)
∈
x
i
:
1
,...,
r
(5.44)
dx
j
=
1
where
(
s
1
i
,
s
2
ij
,
s
3
i
,
s
4
ij
)
∈
x
and r
2
i
∈
R
x
. Moreover, the region
={
x
∈
:
V
(
x
)
≤
1
}
belongs also to
D
.
Proof
Conditions (
5.43
)-(
5.44
)imply(
5.27
) and (
5.28
) locally in
by Lemma 5.2
0,
i
=
1
μ
i
and the fact that
μ
i
≥
=
1. Therefore,
V
(
x
)
is a Lyapunov function by
Lemma 5.3 and, jointlywith conditions (
5.42
), region
={
x
∈
:
V
(
x
)
≤
1
}⊂
D
because conditions (
5.42
) force
V
(
x
)
≥
1 over the boundary of
by Lemma 5.2.
That is achieved by forcing
V
(
x
)
≥
1 in each
g
i
(
x
)
=
0
,
g
1
(
x
)
≥
0
,...,
g
m
(
x
)
≥
0,
i
:
1
m
.
By condition (
5.41
), the region
,...,
2
with size
λ
is contained in the region
=
{
x
∈
:
V
(
x
)
≤
1
}
again by Lemma 5.2. Again by conditions (
5.42
), the region
2
⊂
⊂
and being
V
(
x
)
a Lyapunov function,
2
⊂
D
.
The discrete-time case can be also addressed with Theorem5.1 by using the discrete
representation in (
5.19
) instead of the continuous one and replacing (
5.44
) by:
m
V
(
x
)
−
V
(
p
i
(
x
))
−
ε(
x
)
−
s
4
ij
(
x
)
g
j
(
x
)
∈
x
i
:
1
,...,
r
(5.45)
j
=
1
However, the degree in the involved polynomials increases considerably by the com-
position of functions in
V
dV
(
x
)
(
p
j
(
x
))
instead of the product of them in
−
p
j
(
x
)
.
dx
The above result can be used in order to obtain a “well-shaped” estimate of
D
,
maximizing the scale factor
λ
by bisection or similar methods. Higher degrees for
V
(
x
),
s
(
x
),
r
(
x
)
will give larger estimates, as more degrees of freedom are available.
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