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

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“Principles of Soft Computing, 2 nd 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 “Principles of Soft Computing, 2 nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

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 “Principles of Soft Computing, 2 nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

LINGUISTIC HEDGES (LINGUISTIC QUANTIFIERS)  Hedges modify the shape of a fuzzy set. “Principles of Soft Computing, 2 nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

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. “Principles of Soft Computing, 2 nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

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. “Principles of Soft Computing, 2 nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

 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  ) ALSO (make MEDIUM reduction in speed v).  Three antecedents (or premises) in this example give rise to two outputs (consequences). “Principles of Soft Computing, 2 nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

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] “Principles of Soft Computing, 2 nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

FORMATION OF FUZZY RULES Three general forms are adopted for forming fuzzy rules. They are:  Assignment statements,  Conditional statements,  Unconditional statements. “Principles of Soft Computing, 2 nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

Assignment Statements Conditional Statements Unconditional Statements “Principles of Soft Computing, 2 nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

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). “Principles of Soft Computing, 2 nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

DECOMPOSITION OF FUZZY RULES Multiple Conjunctive Antecedants Conditional Statements ( With Else and Unless) Multiple disjunctive antecedent “Principles of Soft Computing, 2 nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

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. “Principles of Soft Computing, 2 nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

AGGREGATION OF FUZZY RULES Conjunctive system of rules “Principles of Soft Computing, 2 nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

Disjunctive system of rules “Principles of Soft Computing, 2 nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

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. “Principles of Soft Computing, 2 nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

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. “Principles of Soft Computing, 2 nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

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. “Principles of Soft Computing, 2 nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

“Principles of Soft Computing, 2 nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

“Principles of Soft Computing, 2 nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

“Principles of Soft Computing, 2 nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

 The cumulative fuzzy output is obtained by OR-ing the output from each rule.  Cumulative fuzzy output (weight at 6’1”). “Principles of Soft Computing, 2 nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

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. “Principles of Soft Computing, 2 nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

FUZZY REASONING There exist four modes of fuzzy approximate reasoning, which include: 1.Categorical reasoning, 2.Qualitative reasoning, 3.Syllogistic reasoning, 4.Dispositional reasoning. “Principles of Soft Computing, 2 nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

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). “Principles of Soft Computing, 2 nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

SINGLE RULE WITH SINGLE ANTECEDANT “Principles of Soft Computing, 2 nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

“Principles of Soft Computing, 2 nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

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. “Principles of Soft Computing, 2 nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

MULTIPLE RULE WITH MULTIPLE ANTECEDANTS “Principles of Soft Computing, 2 nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

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. “Principles of Soft Computing, 2 nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

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. “Principles of Soft Computing, 2 nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

BLOCK DIAGRAM OF FIS “Principles of Soft Computing, 2 nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

TYPES OF FIS There are two types of Fuzzy Inference Systems:  Mamdani FIS(1975)  Sugeno FIS(1985) “Principles of Soft Computing, 2 nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

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). “Principles of Soft Computing, 2 nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

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. “Principles of Soft Computing, 2 nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

SUGENO FIS “Principles of Soft Computing, 2 nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

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. “Principles of Soft Computing, 2 nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

BLOCK DIAGRAM OF FUZZY EXPERT SYSTEMS Examples of Fuzzy Expert System include Z-II, MILORD. “Principles of Soft Computing, 2 nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

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. “Principles of Soft Computing, 2 nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.