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x
2
{
∈ R :
<
}
is positive semi-definite for all
x
such that
x
1
, can be done by proving
T
L
(
,v)
=
v
(
)v
that the polynomial
H
x
x
below is SOS in the augmented set of
variables (
x
,
v
)
H
(
x
,v)
=
(
1
−
x
2
1
+
2
(
0
.
1
x
+
b
)v
1
v
2
+
(
3
−
0
.
1
x
3
+
ax
2
2
+
s
(
x
,v)(
1
−
x
2
)v
)v
)
(5.25)
2
2
x
2
by setting a multiplier with arbitrary user-defined structure, say
s
(
x
,v)
=
m
1
v
+
m
2
v
1
v
2
x
2
2
2
2
+
m
3
v
1
v
2
x
+
m
4
v
2
+
m
5
v
1
x
+
m
6
v
1
, and searching for feasible
a
,
b
,
m
1
,
…,
m
6
such that both
H
(
x
,v)
and
s
(
x
,v)
are SOS. The higher the degree and the
more coefficients in
s
(
x
,v)
the less conservative the test is, at the expense of higher
computational cost.
5.4 Stability Analysis and Domain of Attraction Estimation
Once local fuzzy polynomial models are available, fuzzy stability analysis and control
design techniques can be explored on them.
In polynomial fuzzy systems, as memberships are always positive, they will be
described by the change of variable
2
μ
i
=
σ
, resulting in a polynomial model
i
x
.
Example 5.6
For instance, (
5.18
) can be written as:
˙
=˜
p
(
x
,σ)
⎧
⎨
2
1
2
x
1
−
σ
1
x
1
x
3
−
2
2
x
1
x
3
2
−
(σ
+
σ
2
)
0
.
05
σ
1
x
2
−
2
x
2
−
(σ
1
2
)
x
1
x
2
x
˙
=
−
σ
2
σ
+
σ
(5.26)
⎩
2
1
2
3
x
1
x
3
−
(σ
+
σ
2
)(
4
x
3
)
5.4.1 Global stability analysis
Once the systems are in the above form, the following well-known results are derived
from Lyapunov stability theory.
Lemma 5.3
(continuous-time (Tanaka et al.
2007a
))
Global stability of a system
(
5.15
)
will be proved if a polynomial Lyapunov function V
(
x
)
can be found verifying
V
(
x
)
−
ε
1
(
x
)
∈
x
(5.27)
dV
dx
˜
−
p
(
x
,σ)
−
ε
2
(
x
,σ)
∈
x
,σ
(5.28)
where
ε
1
(
x
)
and
ε
2
(
x
,σ)
are radially unbounded (in the state x ) positive polyno-
)
=
i
=
1
x
2
p
)
i
=
1
σ
2
mials, usually
ε
1
(
x
and
ε
2
(
x
)
=
ε
1
(
x
(but not necessarily
i
i
(
)
=
so), in order to avoid trivial solutions V
x
0
and to ensure asymptotic stability.
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