Unit - III

Unit - III

FUZZY LOGIC & CONTROL

V.BALAJI, AP/EEE , DCE

Classical sets (crisp sets)

• A classical set is defined by crisp boundaries; i.e. there is no uncertainty in the prescription or location of the boundaries of the set.the set.


Operations on classical set

• Four operations are possible on classical sets. They are:

• Union• Intersection• Intersection• Complement and• Difference


Union:

• Let A and B be two sets on universe X. The union between the two sets, denoted,

• represents all those elements in the universe • represents all those elements in the universe that reside in (or belong to) either the set A, the setB or both the sets.

• This operation is also called the logical OR. The union operation can be expressed as



Intersection:

• The intersection of two sets, denoted by

• represents all those elements in the universe X that simultaneously reside in universe X that simultaneously reside in (or belong to) both the sets A and B.

• This operation is also called the logical AND


The intersection operation can be expressed as


Complement:

• The complement of a set A denoted , is defined as the collection of all elements in the universe that do not reside in the set A. A.

• The complement operation is expressed as



Difference:

• The difference of a set A with respect to B, denoted, is defined as the collection of all elements in the universe that reside in A but not reside in B. this operation can be but not reside in B. this operation can be expressed





Fuzzy sets:

• In classical (crisp) sets, the transition for an element in the universe between membership and non-membership in a given set is abrupt and well-defined.given set is abrupt and well-defined.

For a fuzzy set, this transition is gradual due to the fact that the boundaries of the fuzzy sets are vague and ambiguous.




Operations on fuzzy sets:

• Four operations are possible on classical sets. They are:

• Union• Intersection• Intersection• Complement and• Difference





Difference:





Properties of Fuzzy sets

• The properties of fuzzy sets are same as crisp sets except for the Excluded middle laws.

• The excluded middle laws do not hold • The excluded middle laws do not hold good for fuzzy sets because the fuzzy sets and its complement can overlap



Fuzzy Relations

• A fuzzy relation, f is a mapping from the Cartesian space to the interval [0,1], where the strength of the mapping is expressed by the membership function of expressed by the membership function of the relation


Cardinality of fuzzy relation

• Since the cardinality of a fuzzy set on any universe is infinity, the cardinality of fuzzy relation between two or more universes is also infinityalso infinity

• The cardinality of a crisp relation













Fuzzification

• Fuzzification is the process of making a crisp quantity fuzzy.


The various methods used for fuzzification are

• Intuition• Inference• Rank ordering• Angular fuzzy sets• Neural networks• Neural networks• Genetic algorithms• Inductive reasoning• Soft-partitioning• Meta rules and• Fuzzy statics


Intuition

• The innate intelligence and understandingof human beings is used to develop fuzzysets from crisp data in this method.Intuition involves contextual and semanticIntuition involves contextual and semanticknowledge about an issue; it can alsoinvolve linguistic truth values about thisknowledge.



Inference:

• In the inference method, knowledge isused to perform deductive reasoning i.e.we wish to deduce or infer a conclusion,given a body of facts and knowledge.given a body of facts and knowledge.


Rank Ordering:

• Assessing preferences by a singleindividual, a committee, a poll and otheropinion methods can be used to assignmembership values to a fuzzy variable.membership values to a fuzzy variable.Preference is determined by pair wisecomparisons, and these determine theordering of the membership


Angular Fuzzy Sets:

• Angular fuzzy sets differ from ordinaryfuzzy sets in the coordinate description.Angular fuzzy sets are defined on auniverse of angles, hence are repeatinguniverse of angles, hence are repeatingshapes every 2 cycles.

• These are usually used in the quantitativedescription of linguistic variables known astruth values


Neural Networks:

• In neural network method of determiningmembership function, a number of input datavalues are selected and are divided into training-data set and checking-data set.

• The training dataset is used to train the neural• The training dataset is used to train the neuralnetwork. Let us consider an input training-dataset as shown in fig.11. The tables 1 and 2 showcoordinate values of the different data pointsconsidered.

• Thedata points are first divided into differentclasses by conventional clustering techniques.







Defuzzification:• Defuzzification is the conversion of a fuzzy

quantity to a precise (crisp) quantity. The methods available for converting a fuzzy quantity to crisp quantity are:

• Max-membership principle• Centroid method• Centroid method• Weighted average method• Mean-max membership method• Center of sums• Center of largest area• First (or last) of maxima


DEFUZZIFICATION


DEFUZZIFICATION

• Defuzzification is the conversion of a fuzzy quantity to a precise (crisp) quantity. quantity.


DefuzzifierFuzzy

Knowledge baseFuzzy

Knowledge base

Input FuzzifierInferenceEngine

Defuzzifier OutputInput FuzzifierInferenceEngine

Defuzzifier Output

Converts the fuzzy output of the inference Converts the fuzzy output of the inference

engine to crisp using membership functions

analogous to the ones used by the fuzzifier.


DefuzzifierFuzzy

Knowledge baseFuzzy

Knowledge base



Defuzzifier Output


DefuzzifierFuzzy

Knowledge baseFuzzy

Knowledge base



Defuzzifier Output


The methods available for converting a

fuzzy quantity to crisp quantity are

• Max-membership principle• Centroid method• Centroid method• Weighted average method• Mean-max membership method• Center of sums• Center of largest area


Max-membership principle

• This method also known as height method is limited to peaked

output functions. This method is given by the algebraic expression:


Centroid method

• This method also called center of area or center of gravity method is the most prevalent and physically appealing of all the defuzzification methods. the defuzzification methods.

• It is given by the algebraic expression:


Weighted average method

• This method is valid only for symmetrical output membership functions.

• It is given by the algebraic expression



Mean-max membership method

• This method also called as middle of maxima is closely related to the mean-maxmembership principle. maxmembership principle.

• The difference is that, here the location of maximum membership can benon-unique i.e. the maximum membership can be a plateau rather than a single point.

• Thismethod is given by the expressionV.BALAJI, AP/EEE , DCE





Criteria for selecting defuzzification method:

• The selection of a defuzzification method is context or problem dependent. The criteria formeasurement of the methods are:are:

• Continuity: small change in the input of fuzzy process should not produce a large changein the output


• Disambiguity: a unique value for should be produced.

• (Not satisfied by center of largest area method)method)



FUZZY RULE BASED SYSTEMSYSTEM














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Unit - III