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As memberships add one, without loss of generality, the polynomials
p
i
may be
assumed to be
homogeneous
in
μ
,
i.e.
, composed only of monomials whose degree
in the variables
μ
is the same. Indeed, any monomial in
p
i
can be multiplied by
(
i
μ
i
)
q
, for any
q
, in order to incorporate as many powers of
μ
as required to
make
p
i
homogeneous (Sala and Ariño
2007
).
The sector-nonlinearity methodology in Tanaka andWang (
2001
) can be extended
to a polynomial case. A first idea is bounding by constants the non-polynomial non-
linearities (Tanaka et al.
2009b
). But there exists the possibility to bound a nonlin-
earity within two polynomials (instead of two
constants
): the result would still be a
polynomial fuzzy system.
Lemma 5.1
(Chesi
2009
; Sala and Ariño
2009
)
Consider a sufficiently smooth func-
tion of one real variable, f
, so that its Taylor expansion of degree “n” exists, i.e.,
there exists an intermediate point
(
x
)
ψ(
x
)
∈[
0
,
x
]
, so that:
n
−
1
f
[
i
]
(
f
[
n
]
(ψ(
0
)
x
))
x
i
x
n
f
(
x
)
=
+
(5.4)
i
!
n
!
i
=
0
where f
[
i
]
(
denotes the i th derivative of f and f
[
0
]
(
x
)
x
)
is defined, plainly, as f
(
x
)
.
Assume also that f
[
n
]
(
x
)
is continuous in a compact region of interest
. Then, an
equivalent fuzzy representation exists in the form:
f
(
x
)
=
μ
1
(
x
)
·
p
1
(
x
)
+
μ
2
(
x
)
·
p
2
(
x
)
∀
x
∈
(5.5)
μ
1
(
)
+
μ
2
(
)
=
1
and p
1
(
)
,p
2
(
)
where
x
x
x
x
are polynomials of degree n.
Proof
Denote the Taylor approximations as:
n
−
1
f
[
i
]
(
0
)
x
i
f
n
(
x
)
:=
(5.6)
(
i
)
!
i
=
0
For later developments, denoting
f
0
(
x
)
=
0, denote
f
[
n
]
(ψ(
f
(
x
)
−
f
n
(
x
)
x
))
T
n
(
x
)
:=
=
(5.7)
x
n
n
!
x
n
. In the region of interest
so the Taylor remainder is
f
(
x
)
−
f
n
(
x
)
=
T
n
(
x
)
,
is bounded because
f
[
n
]
(
T
n
(
)
)
x
x
is continuous in
, as assumed in the lemma.
Denoting:
ψ
1
:=
sup
x
∈
T
n
(
x
),
ψ
2
:=
inf
x
T
n
(
x
),
(5.8)
∈
we may write
T
n
(
x
)
=
(μ(
x
)ψ
1
+
(
1
−
μ(
x
))ψ
2
)
with:
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