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Comparing with (
3.27
), the new
R
rules (
R
N
) of the fuzzy system will be,
Rule
R
j
:
IF
x
1
is
x
1
j
,...,
and
x
n
(
k
)
is
x
n
j
,
THEN:
y
j
=
h
0
j
+
h
1
j
x
1
+···+
h
nj
x
n
where
x
i
j
is the fuzzy set respective to
x
i
(
k
)
on the rule
j
.
μ
ij
(
x
i
)
, will be the new
membership degree of
x
i
to the set
ij
. For each rule, we must have
σ
j
(
x
)
=
μ
1
j
(
x
1
)
·
μ
2
j
(
x
2
)
·
...
·
μ
nj
(
x
n
)
(3.38)
or
σ
j
(
x
)
=
min
{
μ
1
j
(
x
1
), μ
2
j
(
x
2
),...,μ
nj
(
x
n
)
}
(3.39)
and
)
i
=
1
σ
i
(
σ
i
(
x
ξ
i
(
x
)
=
(3.40)
)
x
ij
should be
prototypical
to be implemented in a controller, using the
standard Fuzzy Control Language (FCL), i.e. with membership functions satisfying
that
The new
1 and convex. The Standard IEC-1131-7 permits
non-prototypical shapes for the membership function (CENELEC
2000
) defined by
a set of points. Anyway, PLCs which use the norm (Schneider
2009
) don't include
this functionality now days, but they have functions to operate with the fuzzyfication
result, permitting calculation of the eigenfunctions (Eq.
3.30
).
ξ
i
(
x
)
should be 0
<ξ
i
(
x
)<
3.5 Illustrative Examples
3.5.1 Pilot Plant
In order to clarify the ideas developed here, we will see two examples. The first one
is a pilot plant site in the Department of System Engineering and Automatic Control
of University of Seville (Fig.
3.3
). The plant is used to emulate exothermic chemical
reactions based on temperature changes. It has previously been used as a benchmark
for control by researchers (Gruber et al.
2010
). The main elements of the pilot plant
is the reactor, the heat exchanger, the cooling jacket and the valve to manipulate the
flow rate through the cooling jacket (Fig.
3.4
).
The emulated chemical reaction represents a refinement process. At the same flow
at the inlet in the reactor outlet and a constant volume, the model of the chemical
reaction can be defined as:
dT
dt
=−
F
j
V
(
T
j
,
out
)
+
(
−
H
)
V
k
0
e
−
E
/(
RT
)
C
A
T
j
,
in
−
(3.41)
M
·
C
p
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