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Remark 5.1
As each nonlinearity results in a two-rule polynomial fuzzy model, the
number of rules will still be a power of 2, keeping a tensor-product structure (Ariño
and Sala
2007
) analogous to those in classical sector-nonlinearity models.
5.2.1 Homogeneous Expression of Polynomial Fuzzy Models
Consider a continuous-time polynomial fuzzy system expressed as
x
˙
(
t
)
=
p
(
x
(
t
), μ(
t
))
(5.15)
or its discrete-time equivalent
x
k
+
1
=
p
(
x
k
,μ
k
)
(5.16)
where
k
∈ N
is the sample number, fulfilling
t
=
kT
s
and
T
s
is the considered sample
time.
For later operational purposes (Sala
2009
), the polynomials should bemade homo-
geneous in the membership functions
μ
. As in the TS case, it is trivial.
Example 5.2
Consider a nonlinear system
⎧
⎨
x
1
−
(
95 sin
2
x
1
x
3
−
0
.
05
+
0
.
(
x
1
))
sin
2
x
1
)
x
˙
=
−
(
1
+
(
x
1
)
+
x
2
(5.17)
⎩
3
x
1
−
(
4
)
x
3
which, using the methodology presented on Sect.
5.2
, can be modelled as:
⎧
⎨
x
1
−
μ
1
(
x
1
x
3
−
x
1
x
3
−
x
1
)
0
.
05
μ
2
(
x
1
)
x
1
x
2
x
˙
=
−
μ
1
(
x
1
)
x
2
−
2
μ
2
(
x
1
)
x
2
−
(5.18)
⎩
3
x
1
x
3
−
4
x
3
where the sector-nonlinearity methodology is used to model sin
2
expressed as an
interpolation between 0 and 1. Multiplying the terms
x
1
,
x
1
x
2
,
x
1
x
3
and
x
3
by
(
x
1
)
μ
1
+
μ
2
the system is expressed as a degree-1 homogenous polynomial in the memberships.
From the above example, it is easy to see that an arbitrary polynomial fuzzy system
can be expressed without loss of generality as (continuous: left, discrete: right):
r
r
˙
=
1
μ
i
(
)
p
i
(
),
x
k
+
1
=
1
μ
i
(
)
p
i
(
x
k
)
x
z
x
z
(5.19)
i
=
i
=
So, the above expressions will be used instead of equivalent ones from
Definition 5.1.
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