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must be proved (Boyd et al.
1994
; Tanaka and Wang
2001
) in the region defined by
(
5.33
). Hence, using Lemma 5.2, a SOS program is set up in order to find decision
variables proving
V
(
x
)
−
ε(
x
)
∈
x
(5.37)
2
x
1
)
−
2
x
2
)
∈
x
,σ
Q
(
x
,σ)
−
s
1
(
x
, σ )(π
−
s
2
(
x
, σ )(π
−
(5.38)
where
(
s
1
,
s
2
)
∈
x
,σ
are to-be-found polynomial Positivstellensatz multipliers. As
Q
(
x
,σ)
is degree 3 in
x
and degree 2 in
σ
, multipliers of degree 4 in
x
and
σ
are proved powerful enough to obtain a feasible result. The tolerance
ε(
x
)
is set to
x
1
+
x
2
)
0
.
01
.
As a result, tools like SOSTOOLS+SeDuMi are able to find a Lyapunov function
(
8982
x
1
−
2087
x
2
proving a decay rate of
V
(
x
)
=
3
.
0
.
0096
x
1
x
2
+
0
.
α
=
0
.
272.
The code appears in the Appendix B. A larger
α
resulted into numerical problems
or infeasibility.
If a third-order approximation of the sinusoid in
16667
x
1
≤
[−
π, π
]
is used,
x
1
−
0
.
1012
x
1
, we get a decay
sin
(
x
1
)
≤
x
1
−
0
.
α
=
0
.
309 with also a quadratic Lyapunov
function.
This example has illustrated that polynomial fuzzy modeling may provide sat-
isfactory results in situations where classical TS methods fail and that increasing
the precision (degree of Taylor expansion) of the polynomial fuzzy modeling may
improve the results.
Furthermore, if the shape of the obtained stable regions is also taken into account
(to avoid unsatisfactory ill-conditioned solutions), the following definitions are
needed in order to address the problem of fitting a region of predefined shape
(user-defined a priori, usually circles or squares) with the largest possible size
in the Lyapunov level set. For instance, a circle of radius
ρ
may be defined by
2
x
T
x
1
={
,etc.
Definition 5.2
Given a region with predefined shape defined by polynomial con-
straints:
x
:
ρ
−
≥
0
}
1
={
x
:
q
1
(
x
)
≥
0
,...,
q
m
q
(
x
)
≥
0
}
(5.39)
q
1
(λ
−
1
x
q
m
q
(λ
−
1
x
the set
2
={
x
:
)
≥
0
,...,
)
≥
0
}
will be named as
scaled
region
and
λ
will be denoted as
scale factor
. For simplicity, the notation
2
=
λ
1
,
q
i
(λ
−
1
x
q
i
(
¯
x
will be sometimes used in later developments.
Then, the overall problem will consider the full region of interest
)
=
)
where the
model is valid plus a predefined-shape one in its interior to be enlarged as much as
possible.
Consider a region of the state space
defined by
m
polynomial inequalities as
follows:
={
x
:
g
1
(
x
)
≥
0
,...,
g
m
(
x
)
≥
0
}
(5.40)
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