fuzzylogic basic

8/7/2019 fuzzylogic basic

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Fuzzy Logic

Presented by: Mahesh Todkar


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Content What is Fuzzy?

Sets Theory

What is Fuzzy Logic?

Why use Fuzzy Logic?

Theory of Fuzzy Sets

Vocabulary Fuzzy if-then Rules

Fuzzy Logic Operations

Fuzzy Inference Systems (FIS) Fuzzy Inference Process

References


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What is Fuzzy? Fuzzy means

not clear, distinct or precise;

not crisp (well defined);

blurred (with unclear outline).


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Sets TheoryClassical Set: An element either belongs or does not

belong to a sets that have been defined.

Fuzzy Set: An element belongs partially or gradually tothe sets that have been defined.


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What is Fuzzy Logic? It has two different meanings as,

In narrow sense: Fuzzy logic is a logical system,

which is an extension of multi-valued logic.

In a wider sense: Fuzzy logic (FL) is almostsynonymous with the theory of fuzzy sets, a theory

which relates to classes of objects with unsharpboundaries in which membership is a matter ofdegree.

Fuzzy logic (FL) should be interpreted in its widersense


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What is Fuzzy Logic? A way to represent variation or imprecision in logic

A way to make use of natural language in logic

Approximate reasoning

e n on o uzzy og c:

A form of knowledge representation suitable for

notions that cannot be defined precisely, but whichdepend upon their contexts.

Superset of conventional (Boolean) logic that has beenextended to handle the concept of partial truth - the truthvalues between "completely true & completely false".


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Why use Fuzzy Logic? Conceptually easy to understand

Flexible

Tolerant of imprecise data

can mo e non near unc ons o ar rary comp ex y

FL can be built on top of the experience of experts

FL can be blended with conventional control techniques

FL is based on natural language


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Theory of Fuzzy SetsClassical Set Fuzzy Set


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Theory of Fuzzy Sets Theory which relates to classes of objects with unsharp

boundaries in which membership is matter of degree

Thus every problem can be presented in terms ofFuzzy Sets

A set without crisp

Fuzz set describes va ue conce ts

Fuzzy set admits the possibility of partial membershipin it

Degree of an object belongs to Fuzzy Set is denoted bymembership value between 0 to 1

Membership Function (MF) associated with a givenFuzzy Set maps an input value to its appropriatemembership value


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Vocabulary Linguistic Variable:Variable whose values are words

or sentences rather than numbers

It represent qualities spanning a particular spectrum

Example: Speed, Service, Tip, Temperature, etc.

Linguistic Value or Term:Values or Terms used to

describe Linguistic Variable

Example: For Speed (Slowest, Slow, Fast, Fastest), ForService (Poor, Good, Excellent), For Temperature(Freezing, Cool, Warm, Hot), etc.


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Vocabulary Universe of Discourse or Universe or Input Space (U):

Set of all possible elements that can come intoconsideration, confer the set U in (1).

It depends on context.

Elements of a fuzz set are taken from a Universe of

Discourse.

An application of the universe is to suppress faultymeasurement data.

Example:

Set of x >> 1 could have as a universe of all real numbers,alternatively all positive integer.


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Vocabulary Membership Function (MF) is a curve that defines how

each point in the input space is mapped to a membership

value between 0 and 1. It is denoted by .

Membership value is also called as degree of membership.

Types of Membership Functions:

Piece-wise linear functions

Gaussian distribution function

Sigmoid curve

Quadratic and cubic polynomial curves

Singleton Membership Function


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Membership Functions


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Syntax of Fuzzy Set

A = {x, A(x) | x X}

Where,

A Fuzzy Set

x Elements of XX Universe of Discourse

A(x) Membership Function of x in A


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Fuzzy if-then Rules Statements used to formulate the conditional statements

that comprise fuzzy logic

Example:if x is A then y is B

where,

A & B Linguistic valuesx Element of Fuzzy set X

y Element of Fuzzy set Y

In above example,

Antecedent (or Premise) if part of rule (i.e. x is A)

Consequent (or Conclusion) then part of rule (i.e. y is B)

Antecedent is interpretation & Consequent is assignment


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Fuzzy if-then Rules Antecedent is combination of proposals by AND, OR, NOT

operators

Consequent is combination of proposals linked by ANDoperators. OR and NOT operators are not used inconsequents as these are cases of uncertainty.

Example:

If it is early, then John can study.

Universe: U = {4,8,12,16,20,24}; time of day

Input Fuzzy set: early = {(4,0),(8,1),(12,0.9),(16,0.7),(20,0.5),(24,0.2)}

Output Fuzzy set: can study=singleton Fuzzy set (assume) so study =1

i.e. at 20 (8 pm), early (20) = 0.5


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Fuzzy if-then Rules Interpreting if-then rule is a threepart process

1) Fuzzify Input: Resolve all fuzzy statements in the

antecedent to a degree of membership between 0 and 1.2)Apply fuzzy operator to multiple part antecedents:

If there are multiple parts to the antecedent, apply fuzzy

number between 0 and 1.

3)Apply implication method: The output fuzzy sets

for each rule are aggregated into a single output fuzzy

set. Then the resulting output fuzzy set is defuzzified, orresolved to a single number.


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Fuzzy if-then RulesInterpreting if-then rule is a threepart process:


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Fuzzy Logic Operations Fuzzy Logic Operators are used to write logic

combinations between fuzzy notions (i.e. to performcomputations on degree of membership)

Zadeh operators

1) Intersection: The logic operator corresponding tothe intersection of sets is AND.

(A AND B) = MIN((A), (B))

2) Union: The logic operator corresponding to theunion of sets is OR.

(A OR B) = MAX((A), (B))

3) Negation: The logic operator corresponding to thecomplement of a set is the negation.

(NOT A) = 1 - (A)


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Fuzzy Logic Operations


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Fuzzy Inference Systems (FIS) Fuzzy Inference is the process of formulating the mapping

from a given input to an output using fuzzy logic.

Process of fuzzy inference involves Membership Functions(MF), Logical Operations and If-Then Rules.

FIS having multidisciplinary nature, so cab called as- - , ,

modeling, fuzzy associative memory, fuzzy logiccontrollers, and simply (and ambiguously) fuzzy systems.

Types of FIS:

1) Mamdani-type: Most commonly used. Expects the output MFs tobe fuzzy sets.

2) Sugeno-type: Output MFs are either linear or constant.


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Fuzzy Inference ProcessTo describe the fuzzy inference process, lets consider theexample of two-input, one-output, two-rule valve controlproblem.


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Fuzzy Inference ProcessStep 1: Fuzzify Input (Fuzzification)

Take the inputs and determine the degree to which theybelong to each of the appropriate fuzzy sets viamembership functions.

Input is always a crisp numerical value limited to the.

Output is a fuzzy degree of membership in thequalifying linguistic set.

Each input is fuzzified over all the qualifyingmembership functions required by the rules.


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Fuzzy Inference ProcessStep 1: Fuzzify Input (Fuzzification)


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Fuzzy Inference ProcessStep 2 : Apply Fuzzy Operator

If the antecedent of a given rule has more than one

part, the fuzzy operator is applied to obtain onenumber that represents the result of the antecedent

for that rule.

The input to the fuzzy operator is two or moremembership values from fuzzified input variables.

The output is a single truth value.


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Fuzzy Inference ProcessStep 2 : Apply Fuzzy Operator


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Fuzzy Inference ProcessStep 3: Apply Implication Method

First must determine the rules weight.

Operation in which the result of fuzzy operator is used todetermine the conclusion of the rule is called as

im lication.

The input for the implication process is a single numbergiven by the antecedent.

The output of the implication process is a fuzzy set.

Implication is implemented for each rule.


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Fuzzy Inference ProcessStep 3: Apply Implication Method

Antecedent Consequent


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Fuzzy Inference ProcessStep 4 : Aggregate All Outputs

Aggregation is the process by which the fuzzy sets that

represent the outputs of each rule are combined into asingle fuzzy set.

A re ation onl occurs once for each out ut variable.

The input of the aggregation process is the list oftruncated output functions returned by the implicationprocess for each rule.

The output of the aggregation process is one fuzzy setfor each output variable.


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Fuzzy Inference ProcessStep 4 : Aggregate All Outputs


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Fuzzy Inference ProcessStep 5: Defuzzify

Move from the fuzzy world to the real world is

known as defuzzification. The input for the defuzzification process is a fuzzy set.

The output is a single number.

The most popular defuzzification method is thecentroid calculation, which returns the center of areaunder the curve

Other methods are bisector, middle of maximum (the

average of the maximum value of the output set),largest of maximum, and smallest of maximum.


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Fuzzy Inference ProcessStep 5: Defuzzify


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References Fuzzy Logic Toolbox 2 Users Guide

Tutorial On Fuzzy Logic by Jan Jantzen

Fuzzy Logic by Cahier Technique Schneider


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