Fuzzy+logic

Fuzzy Logic

Presented by: Mahesh Todkar

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

What is Fuzzy?

� Fuzzy means

not clear, distinct or precise;

not crisp (well defined);

blurred (with unclear outline).blurred (with unclear outline).

Sets Theory

Classical Set: An element either belongs or does not

belong to a sets that have been defined.

Fuzzy Set: An element belongs partially or gradually to

the sets that have been defined.

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 almost

synonymous with the theory of fuzzy sets, a theory

which relates to classes of objects with unsharpwhich relates to classes of objects with unsharp

boundaries in which membership is a matter of

degree.

� Fuzzy logic (FL) should be interpreted in its wider

sense

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

� Definition of Fuzzy Logic:� Definition of Fuzzy Logic:

A form of knowledge representation suitable for

notions that cannot be defined precisely, but which

depend upon their contexts.

� Superset of conventional (Boolean) logic that has been

extended to handle the concept of partial truth - the truth

values between "completely true & completely false".

Why use Fuzzy Logic?

� Conceptually easy to understand

� Flexible

� Tolerant of imprecise data

� FL can model nonlinear functions of arbitrary complexity� FL can model nonlinear functions of arbitrary complexity

� 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

Theory of Fuzzy Sets

Classical Set Fuzzy Set

Theory of Fuzzy Sets

� Theory which relates to classes of objects with unsharpboundaries in which membership is matter of degree

� Thus every problem can be presented in terms of Fuzzy Sets

� A set without crisp

� Fuzzy set describes vague concepts� Fuzzy set describes vague concepts

� Fuzzy set admits the possibility of partial membership in it

� Degree of an object belongs to Fuzzy Set is denoted by membership value between 0 to 1

� Membership Function (MF) associated with a given Fuzzy Set maps an input value to its appropriate membership value

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), For

Service (Poor, Good, Excellent), For Temperature

(Freezing, Cool, Warm, Hot), etc.

Vocabulary

� Universe of Discourse or Universe or Input Space (U): Set of all possible elements that can come into consideration, confer the set U in (1).

� It depends on context.

� Elements of a fuzzy set are taken from a Universe of � Elements of a fuzzy set are taken from a Universe of Discourse.

� An application of the universe is to suppress faulty measurement data.

� Example:

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

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 µ.

� Membership value is also called as degree of membership

or membership grade or degree of truth of proposal.or membership grade or degree of truth of proposal.

� Types of Membership Functions:

Piece-wise linear functions

Gaussian distribution function

Sigmoid curve

Quadratic and cubic polynomial curves

� Singleton Membership Function

Membership Functions

Syntax of Fuzzy Set

� A = {x, µA(x) | x X}

Where,

A – Fuzzy Set

x – Elements of Xx – Elements of X

X – Universe of Discourse

µA(x) – Membership Function of x in A

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 values

x – 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

Fuzzy if-then Rules

� Antecedent is combination of proposals by AND, OR, NOT

operators

� Consequent is combination of proposals linked by AND

operators. OR and NOT operators are not used in

consequents 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 µstudy =1

i.e. at 20 (8 pm), µearly (20) = 0.5

Fuzzy if-then Rules

� Interpreting if-then rule is a three–part 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

logic operators and resolve the antecedent to a single logic operators and resolve the antecedent to a single

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, or

resolved to a single number.

Fuzzy if-then Rules

Interpreting if-then rule is a three–part process:

Fuzzy Logic Operations

� Fuzzy Logic Operators are used to write logic combinations between fuzzy notions (i.e. to perform computations on degree of membership)

� Zadeh operators

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

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

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

µ(A OR B) = MAX(µ(A), µ(B))

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

µ(NOT A) = 1 - µ(A)

Fuzzy Logic Operations

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

fuzzy-rule-based systems, fuzzy expert systems, fuzzy fuzzy-rule-based systems, fuzzy expert systems, fuzzy

modeling, fuzzy associative memory, fuzzy logic

controllers, and simply (and ambiguously) fuzzy systems.

� Types of FIS:

1) Mamdani-type: Most commonly used. Expects the output MF’s to

be fuzzy sets.

2) Sugeno-type: Output MF’s are either linear or constant.

Fuzzy Inference ProcessTo describe the fuzzy inference process, lets consider the

example of two-input, one-output, two-rule valve control

problem.

Fuzzy Inference ProcessStep 1: Fuzzify Input (Fuzzification)

� Take the inputs and determine the degree to which they belong to each of the appropriate fuzzy sets via membership functions.

� Input is always a crisp numerical value limited to the universe of discourse of the input variable.universe of discourse of the input variable.

� Output is a fuzzy degree of membership in the qualifying linguistic set.

� Each input is fuzzified over all the qualifying membership functions required by the rules.

Fuzzy Inference Process

Step 1: Fuzzify Input (Fuzzification)


Step 2 : Apply Fuzzy Operator

� If the antecedent of a given rule has more than one

part, the fuzzy operator is applied to obtain one

number that represents the result of the antecedent

for that rule.

� The input to the fuzzy operator is two or more

membership values from fuzzified input variables.

� The output is a single truth value.


Step 2 : Apply Fuzzy Operator


Step 3: Apply Implication Method

� First must determine the rule’s weight.

� Operation in which the result of fuzzy operator is used to

determine the conclusion of the rule is called as

implication.implication.

� The input for the implication process is a single number

given by the antecedent.

� The output of the implication process is a fuzzy set.

� Implication is implemented for each rule.


Step 3: Apply Implication Method

Antecedent Consequent


Step 4 : Aggregate All Outputs

� Aggregation is the process by which the fuzzy sets that

represent the outputs of each rule are combined into a

single fuzzy set.

� Aggregation only occurs once for each output variable.� Aggregation only occurs once for each output variable.

� The input of the aggregation process is the list of

truncated output functions returned by the implication

process for each rule.

� The output of the aggregation process is one fuzzy set

for each output variable.

Fuzzy Inference ProcessStep 4 : Aggregate All Outputs


Step 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 the � The most popular defuzzification method is the

centroid calculation, which returns the center of area

under 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.


Step 5: Defuzzify

References

� Fuzzy Logic Toolbox™ 2 User’s Guide

� Tutorial On Fuzzy Logic by Jan Jantzen

� Fuzzy Logic by Cahier Technique Schneider

Thank You…Thank You…

Any Questions? Any Questions?

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