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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 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 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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