صفحه 1:
Fuzzy logic
Introduction 3
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Aleksandar Rakié
تور تقد لقن [-و
2
صفحه 2:
Contents
= Mamdani Fuzzy Inference
Fuzzification of the input variables
Rule evaluation
Aggregation of the rule outputs
Defuzzification
= Sugeno Fuzzy Inference
= Mamdani or Sugeno?
صفحه 3:
Mamdani Fuzzy
Inference
= The most commonly used fuzzy inference technique is the so-called
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= Jn 1975, Professor Ebrahim Mamdani of London University built one of
the first fuzzy systems to control a steam engine and boiler
combination. He applied a set of fuzzy rules supplied by experienced
human operators.
= The Mamdani-style fuzzy inference process is performed in four steps:
Fuzzification of the input variables
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Defuzzification.
صفحه 4:
Mamdani Fuzzy
Inference
We examine a simple two-input one-output problem that includes
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Real-life example for these kinds of rules:
Rule: 1 IF project funding is adequate OR project staffing is small THEN _risk
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Rule: 2 IF project funding is marginal AND project staffingis large THEN risk
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Rule: 3 IF project funding is inadequate 0
صفحه 5:
Step 1: Fuzzification
= The first step is to take the crisp inputs, x1 and y1 (project
funding and jroject staffing), and determine the degree to
which these inputs belong to each of the appropriate fuzzy
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صفحه 6:
Step 2: Rule Evaluation
= The second step is to take the fuzzified inputs,
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and apply them to the antecedents of the fuzzy rules.
Ifa given fuzzy rule has multiple antecedents, the fuzzy
operator (AND or OR) is used to obtain a single number that
represents the result of the antecedent evaluation.
RECALL: To evaluate the disjunction of the rule antecedents, we
use the OR fuzzy operation. Typically, fuzzy expert systems
make use of the classical fuzzy operation union:
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Similarly, in order to evaluate the conjunction of the rule
antecedents, we apply the AND fuzzy operation intersection:
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صفحه 7:
Step 2: Rule Evaluation
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Rule1: IF xis A3(0.0) OR yis Bl (0.1) THEN zis C1 (0.1)
تا( 7
zis C2(0.2)
Rule3: IF xis Al (0.5) THEN zis C3 (0.5)
صفحه 8:
Step 2: Rule Evaluation
™ Now the result of the antecedent evaluationegree of
can be applied to the membership functiorMembership
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consequent membership function at the level of
the antecedent truth. This method is called
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= However, Clipping is still often preferred because 0
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Generates an aggregated output surface that is
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consequent is adjusted by multiplying all its
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صفحه 9:
Step 3: Aggregation of
the Rule Outputs
= Aggregation is the process of unification of the outputs
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™ We take the membership functions of all rule
consequents previously clipped or scaled and combine
them into a single fuzzy set.
= The input of the aggregation process is the list of clipped
or scaled consequent membership functions, and the
output is one fuzzy set for each output variable.
8
صفحه 10:
Step 4: Defuzzification
The last step in the fuzzy inference process is defuzzification
Fuzziness helps us to evaluate the rules, but the final output of a
fuzzy system has to be a crisp number.
The input for the defuzzification process is the aggregate output
fuzzy set and the output is a single number.
There are several defuzzification methods, but probably the most
popular one is the centroid technique. It finds the point where a
vertical line would slice the aggregate set into two equal masses.
Mathematically this centre of gravity (COG) can be expressed as:
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صفحه 11:
Step 4: Defuzzification
= Centroid defuzzification method finds a point representing the
centre of gravity of the aggregated fuzzy set A, on the interval [a,
62
= A reasonable estimate can be obtained by calculating it over a
sample of ooints.
COG =0+10+ 20) x0.1+ (30+ 40+ 50+ 60) x0.2+ (70+ 804 90+ 100) x0.5
0.1+ 0.1+ 0.14 0.2+ 0.2+ 0.2+0.2+ 0.5+ 0.5+0.5+ 0.5
=67.4
0.
صفحه 12:
Sugeno Fuzzy Inference
= Mamdani-style inference, as we have just seen, requires us
to find the centroid of a two-dimensional shape by
integrating across a continuously varying function. In
general, this process is not computationally efficient.
™ Michio Sugeno suggested to use a single spike, a singleton,
as the membership function of the rule consequent.
= A singleton, or more precisely a fuzzy singleton, is a fuzzy set
with a membership function that is unity at a single particular
point on the universe of discourse and zero everywhere else.
صفحه 13:
Sugeno Fuzzy Inference
Sugeno-style fuzzy inference is very similar to the Mamdani method.
Sugeno changed only a rule consequent: instead of a fuzzy set, he used
a mathematical function of the input variable.
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The most commonly used zero-order Sugeno fuzzy model applies
fuzzy rules in the following form:
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membership functions are represented by singleton spikes.
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صفحه 14:
Sugeno Rule Evaluation
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1: IF xis A3 (0.0) yis Bi (0.1)
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: IF xis A2 (0.2) AND yis B2 (0.7) zis k2 (0.2)
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ule 3: IF xis Al (0.5)
صفحه 15:
Sugeno Aggregation and
Defuzzification
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صفحه 16:
Mamdani or Sugeno?
= Mamdani method is widely accepted for capturing expert.
knowledge. It allows us to describe the expertise in more
intuitive, more human-like manner, However, Mamdani-
type fuzzy inference entails a substantial computational
burden.
™ On the other hand, Sugeno method is computationally
effective and works well with optimization and adaptive
techniques, which makes it very attractive in control
problems, particularly for dynamic nonlinear systems.