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2
, the largest
Despite of maximizing the size of the chosen fixed-shape region
(
),
(
)
(
)
provable domain of attraction (DA) estimate (for fixed degree of
V
x
s
x
and
r
x
in Theorem5.1) depends also on the modeling-region size. Indeed, the smaller
is,
the smaller the provable DA is. However, the larger
is, the more conservative the
fuzzy model becomes, so the provable DA might be small too (large
might result
in infeasible conditions). If the objective is obtaining the largest DA estimate of the
original nonlinear system, an exploration in the modeling region size is required
(Pitarch et al.
2010
).
Example 5.9
Consider the nonlinear system (
5.32
) from Example 5.8. The goal is
to find an estimate of
D
as large as possible in the modeling region
2
x
1
≥
2
x
2
≥
={
x
:
ρ
−
0
,ρ
−
0
}
(5.46)
where 0
4 is a size parameter. By following the fuzzy-polynomial modeling
methodology with Taylor series decomposition up to degree 3, a 2-rule model (
5.19
)
can be obtained. For instance, if
<ρ
≤
ρ
=
4, the model consequents are:
074325
x
1
sin
(
x
1
)
−
x
1
+
0
.
μ
1
(
x
)
=
(5.47)
0923417
x
1
−
0
.
μ
2
(
x
)
=
1
−
μ
1
(
x
)
(5.48)
−
3
x
1
+
0
.
5
x
2
p
1
(
x
)
=
(5.49)
166667
x
1
)
−
2
x
2
+
3
x
2
(
x
1
−
0
.
−
3
x
1
+
0
.
5
x
2
p
2
(
x
)
=
(5.50)
074325
x
1
)
−
2
x
2
+
3
x
2
(
x
1
−
0
.
In order to avoid ill-shaped solutions, a prefixed-shape spherical region
2
x
1
+
x
2
)
≥
2
={
x
:
1
−
λ
(
0
}
(5.51)
=
λ
−
1
/
2
, is chosen in order to maximize the radius. The largest sphere
is obtained iterating with modeling region size
of radius
rad
ρ
and solving the SOS problem of
Theorem5.1 for each region
.
Different Lyapunov-function degrees have been analyzed. Quadratic Lyapunov
functions are analyzed first and the maximum sphere is proved for
ρ
=
2
.
1having
radius
rad
1 ellipsoids start to be ill-conditioned and
their maximum contained spheres become smaller (see Fig.
5.1
).
With higher Lyapunov function degree, larger spheres are found. For instance, with
degrees 2, 4 and 5 the maximum probable radius were 1.96, 2.514, 2.83 respectively.
=
1
.
96. Indeed, for
ρ>
2
.
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