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where
a
l
,
b
l
, and
c
l
parameters with
l
= 1, 2, 3 corresponding to Rule-1, Rule-2, and
Rule-3 are constants.
As another example, we again take a single input-single output system and
present it using Takagi-Sugeno rules:
x IF heaterCurrent is HIGH,
THEN temperatureRise =
a
H
(heaterCurrent) +
b
H
x IF heaterCurrent is MEDIUM,
THEN temperatureRise =
a
M
(heaterCurrent) +
b
M
x IF heaterCurrent is LOW,
THEN temperatureRise =
a
L
(heaterCurrent) +
b
L
Using similar kinds of rules, many real systems can be described and modeled very
accurately, where each rule represents a local linear model of the system. Also,
these types of rule enable the system output variables (real valued/crisp valued) to
be very easily inferred, which is an advantage of the presentation.
Note that in the above rules if the first constant parameters are all set to zero
(
i.e
.
a
H
= 0,
a
M
= 0,
a
L
= 0), then the rule's consequents are singleton fuzzy sets.
Similarly, with Mamdani-type fuzzy rules if the consequent fuzzy sets are
singleton type (a real value) then they are identical to the Takagi-Sugeno type
fuzzy rules with singleton consequents (
i.e.
when
a
H
= 0,
a
M
= 0,
a
L
= 0).
4.3.3 Relational Fuzzy Logic System of Pedrycz
In relational fuzzy logic systems, similar to Mamdani-type fuzzy logic system,
both the IF (antecedent) parts as well as the THEN (consequent) parts are fuzzy.
However, there is a slight difference in the rule's representation: in this case, one
particular antecedent proposition is allowed to be associated with several different
consequent propositions via a fuzzy relation (Pedrycz, 1984). This can be
explained, again, on the above single input-single output system, which is
described now by the following rules:
x IF heater current is HIGH,
THEN temperature rise is SLOW (0.0), MODERATE (0.1), FAST (0.9)
x
IF heater current is MEDIUM,
THEN temperature rise is SLOW (0.1), MODERATE (0.95), FAST (0.0)
x IF the heater current is LOW,
THEN temperature rise is SLOW (1.0), MODERATE (0.1), FAST (0.0).
In the first relational fuzzy rule the consequent fuzzy set FAST (0.9) represents the
output variable (temperature rise) belonging partially to the fuzzy set FAST with
degree of affiliation (also called degree of membership) equal to 0.9. Similarly
SLOW (0.0) and MODERATE (0.1) represent respectively that the same output
variable does not belong to fuzzy set SLOW at all (as the degree of membership in
SLOW is 0.0), whereas the same output belongs a little to fuzzy set MODERATE
(partially with degree of membership 0.1). Following the same argument, one can
see that in the third rule SLOW (1.0) indicates that the output variable (temperature
rise) belongs fully to fuzzy set SLOW (as the degree of membership in SLOW is
1.0), whereas it (output) simultaneously belongs to the fuzzy set MODERATE
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