NNFL 12- Guru Nanak Dev Engineering College

“Principles of Soft Computing, 2nd
Edition”
by S.N. Sivanandam & SN Deepa
Copyright © 2011 Wiley India Pvt. Ltd. All rights reserved.
CHAPTER 12
FUZZY RULE BASE AND
APPROXIMATE
REASONING

The degree of an element in a fuzzy set corresponds to the truth value
of a proposition in fuzzy logic systems.
FUZZY RULES AND REASONING
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LINGUISTIC VARIABLES
 A linguistic variable is a fuzzy variable.
• The linguistic variable speed ranges between 0 and 300 km/h and
includes the fuzzy sets slow, very slow, fast, …
• Fuzzy sets define the linguistic values.
 Hedges are qualifiers of a linguistic variable.
• All purpose: very, quite, extremely
• Probability: likely, unlikely
• Quantifiers: most, several, few
• Possibilities: almost impossible, quite possible
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LINGUISTIC HEDGES (LINGUISTIC
QUANTIFIERS)
 Hedges modify the shape of a fuzzy set.
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TRUTH TABLES
Truth tables define logic functions of two propositions. Let X and Y be
two propositions, either of which can be true or false.
The operations over the propositions are:
1.Conjunction (∧): X AND Y.
2.Disjunction (∨): X OR Y.
3.Implication or conditional (⇒): IF X THEN Y.
4.Bidirectional or equivalence (⇔): X IF AND ONLY IF Y.
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FUZZY RULES
A fuzzy rule is defined as the conditional statement of the form
If x is A
THEN y is B
where x and y are linguistic variables and A and B are linguistic values
determined by fuzzy sets on the universes of discourse X and Y.
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 The decision-making process is based on rules with sentence
conjunctives AND, OR and ALSO.
 Each rule corresponds to a fuzzy relation.
 Rules belong to a rule base.
 Example: If (Distance x to second car is SMALL) OR (Distance y
to obstacle is CLOSE) AND (speed v is HIGH) THEN (perform
LARGE correction to steering angle θ) ALSO (make MEDIUM
reduction in speed v).
 Three antecedents (or premises) in this example give rise to two
outputs (consequences).
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FUZZY RULE FORMATION
IF height is tall
THEN weight is heavy.
Here the fuzzy classes height and weight have a given range (i.e., the
universe of discourse).
range (height) = [140, 220]
range (weight) = [50, 250]
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FORMATION OF FUZZY RULES
Three general forms are adopted for forming fuzzy rules. They are:
Assignment statements,
Conditional statements,
Unconditional statements.
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Assignment Statements
Conditional Statements
Unconditional Statements
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DECOMPOSITION OF FUZZY RULES
A compound rule is a collection of several simple rules combined
together.
Multiple conjunctive antecedent,
Multiple disjunctive antecedent,
Conditional statements (with ELSE and UNLESS).
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DECOMPOSITION OF FUZZY RULES
Multiple
Conjunctive
Antecedants
Conditional Statements ( With Else and
Unless)
Multiple disjunctive
antecedent
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AGGREGATION OF FUZZY RULES
Aggregation of rules is the process of obtaining the overall
consequents from the individual consequents provided by each rule.
 Conjunctive system of rules.
 Disjunctive system of rules.
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AGGREGATION OF FUZZY RULES
Conjunctive system of rules
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Disjunctive system of rules
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FUZZY RULE - EXAMPLE
Rule 1: If height is short then weight is light.
Rule 2: If height is medium then weight is medium.
Rule 3: If height is tall then weight is heavy.
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Problem: Given
(a) membership functions for short, medium-height, tall, light,
medium-weight and heavy;
(b) The three fuzzy rules;
(c)the fact that John’s height is 6’1”
estimate John’s weight.
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Solution:
(1) From John’s height we know that
John is short (degree 0.3)
John is of medium height (degree 0.6).
John is tall (degree 0.2).
(2) Each rule produces a fuzzy set as output by truncating the
consequent membership function at the value of the antecedent
membership.
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 The cumulative fuzzy output is obtained by OR-ing the output from
each rule.
 Cumulative fuzzy output (weight at 6’1”).
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1. De-fuzzify to obtain a numerical estimate of the output.
2. Choose the middle of the range where the truth value is maximum.
3. John’s weight = 80 Kg.
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FUZZY REASONING
There exist four modes of fuzzy approximate reasoning, which include:
1.Categorical reasoning,
2.Qualitative reasoning,
3.Syllogistic reasoning,
4.Dispositional reasoning.
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REASONING WITH FUZZY RULES
 In classical systems, rules with true antecedents fire.
 In fuzzy systems, truth (i.e., membership in some class) is relative,
so all rules fire (to some extent).
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SINGLE RULE WITH SINGLE
ANTECEDANT
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MULTIPLE ANTECEDANTS
IF x is A AND y is B THEN z is C
IF x is A OR y is B THEN z is C
Use unification (OR) or intersection (AND) operations to calculate a
membership value for the whole antecedent.
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MULTIPLE RULE WITH MULTIPLE
ANTECEDANTS
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MULTIPLE CONSEQUENTS
IF x is A THEN y is B AND z is C
Each consequent is affected equally by the membership in the
antecedent class(es).
E.g., IF x is tall THEN x is heavy AND x has large feet.
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FUZZY INFERENCE SYSTEMS (FIS)
 Fuzzy rule based systems, fuzzy models, and fuzzy expert systems
are also known as fuzzy inference systems.
 The key unit of a fuzzy logic system is FIS.
 The primary work of this system is decision-making.
 FIS uses “IF...THEN” rules along with connectors “OR” or “AND”
for making necessary decision rules.
 The input to FIS may be fuzzy or crisp, but the output from FIS is
always a fuzzy set.
 When FIS is used as a controller, it is necessary to have crisp
output.
 Hence, there should be a defuzzification unit for converting fuzzy
variables into crisp variables along FIS.
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BLOCK DIAGRAM OF FIS
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TYPES OF FIS
There are two types of Fuzzy Inference Systems:
 Mamdani FIS(1975)
 Sugeno FIS(1985)
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MAMDANI FUZZY INFERENCE SYSTEMS
(FIS)
 Fuzzify input variables:
• Determine membership values.
 Evaluate rules:
• Based on membership values of (composite) antecedents.
 Aggregate rule outputs:
• Unify all membership values for the output from all rules.
 Defuzzify the output:
• COG: Center of gravity (approx. by summation).
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SUGENO FUZZY INFERENCE SYSTEMS
(FIS)
The main steps of the fuzzy inference process namely,
1.fuzzifying the inputs and
2.applying the fuzzy operator are exactly the same as in MAMDANI
FIS.
The main difference between Mamdani’s and Sugeno’s methods is that
Sugeno output membership functions are either linear or constant.
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SUGENO FIS
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FUZZY EXPERT SYSTEMS
An expert system contains three major blocks:
Knowledge base that contains the knowledge specific to the domain
of application.
Inference engine that uses the knowledge in the knowledge base for
performing suitable reasoning for user’s queries.
User interface that provides a smooth communication between the
user and the system.
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BLOCK DIAGRAM OF FUZZY EXPERT
SYSTEMS
Examples of Fuzzy Expert System include Z-II, MILORD.
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SUMMARY
 Advantages of fuzzy logic
• Allows the use of vague linguistic terms in the rules.
 Disadvantages of fuzzy logic
• Difficult to estimate membership function
• There are many ways of interpreting fuzzy rules, combining the
outputs of several fuzzy rules and de-fuzzifying the output.
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NNFL 12- Guru Nanak Dev Engineering College

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