Digital Signal Processing Reference
In-Depth Information
low
low
1
1
β
Rule 1
1
1
β
y
+
β
y
y
=
1
1
2
2
y
1
=f(
β
1
)
y
x
β
+
β
1
2
high
high
1
1
y
Rule 2
y
β
2
output
y
2
=f(
β
2
)
x
y
x
antecedents
consequents
Fig. 11.14
Tsukamoto reasoning scheme with two rules and one input
rules are to be aggregated so that the final fuzzy output is generated and then the
output of the scheme ought to be defuzzified to produce equivalent crisp output
value.
The process of rules aggregation can be done in various ways. The most popular
ones are:
• weighting factor method (applicable for rules with crisp or singleton outputs),
• compositional approach (max-min or max-product schemes, where min/product
refers to the way how the implication is performed).
The examples of FIS belonging to the first family are Tsukamoto or Sugeno
schemes, while the Mamdani FIS is a representative of the second family of
reasoning schemes.
The Tsukamoto reasoning scheme [
22
] is a monotonic selection scheme, where
the value of the rule output (truth membership grade of the rule consequent) can be
estimated directly from a corresponding truth membership grade in the antecedent.
The illustration of the process is given in Fig.
11.14
for a scheme with two rules.
For the input value x the rule antecedents' membership grades are found (b
1
, b
2
),
which is followed by looking for the consequents' output values (y
1
, y
2
read out
from particular consequents' MFs). The final output of the Tsukamoto FIS is
defined as a weighted sum of the rules' outputs, as shown in Fig.
11.14
.
Another, quite frequently used example of fuzzy system with weighting factor
method aggregation of the rules' outputs, is the Takagi-Sugeno controller [
15
,
22
].
Here, the input part of the controller only is based on fuzzy processing (signal
fuzzification), whereas further signal processing is non-fuzzy, determined clearly
with some functional relationships.
In Fig.
11.15
the Sugeno reasoning structure is shown for a case of two rules
and two inputs. The consequentce of the rules are here defined as singleton values
being linear combinations of the input values x
i
, where the coefficients of (
11.30
)
are different for particular rule k:
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