Fuzzy Inference System1

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FUZZY INFERENCE SYSTEM

It is a popular computing framework based on

fuzzy set theory, fuzzy if then rules , and fuzzy

reasoning.

3 components:

1. Rule Base: a selection of fuzzy rules

2. Database (or Dictionary): defines the MFs

3. Inference Engine: a reasoning mechanism

which performs the inference procedure

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Cont..

It can take either fuzzy inputs or crisp inputs

but the output it produces are always fuzzy

sets.

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MODELS OF FUZZY INFERENCE SYSTEM

Three types

Mamdani fuzzy model

Sugeno fuzzy model

Larsan fuzzy model

They differ in the consequences of their fuzzy

rules and defuzzification methods

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MAMDANI FUZZY MODEL

Proposedd to control a steam engine and

boiler combination

Mamdani rule base can model the system

using rules that have a high correctness.

Correctness-measure of how close our model

is to the real one

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Cont..

Mamdani model is a crisp model of a system.

It can model a real system where the relationbetween the inputs and outputs are known.

2 common operators that we use are T-normand T-conorm operators.

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MAMDANI RULE BASE

Can be broken down into 4 part

Fuzzification

Determining the output of each rule given its fuzzy

antecedent.

Aggregation

defuzzification

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Fuzzification

Mamdani rule base models a crisp system,it

has crisp inputs and outputs.

Fuzzifier converts the crisp input into fuzzy

variables.

The membership of each fuzzy input variable

is evaluated for a given crisp input.

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EVALUATING THE RULES

Rules are evaluated based on the membership

values.

The rule base let the user to determine which

type of operation to use like minimum ,

maximum, product

If we use min for T-norm and T-conorm

(implication )operators respectively and use

maxmin for composition then the resulting

fuzzy reasoning is as,

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max-min T-conorm/norm

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AGGREGATING THE RULES

The output of the rule base should be the

maximum of the output of each rule.

Can use any one of the operators defined on

fuzzy sets like maximum , algebraic sum or

min.

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DEFUZZIFICATION

a method to extract a representative crispvalue from a fuzzy set.

5 methods

1. Centroid of areazCOA :

zCOA =ZA(z) z dz /ZA(z) dz

Most widely used strategy.

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Cont.

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Cont..

2.Bisector of areazBOA :It generates the value z0 which partitions

the area into two area with same area.

zBOA A(z) dz =zBOA A(z) dz

3.Mean of maximumzMOM :

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BOA

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CONT

Is the average of the maximizing z at which the MF reaches a

maximum .

IfA(z) has a single maximum at z=z then ZMOM=z

IfA(z) reaches its maximum whenever z[zleft,zright] then

ZMOM=(ZLEFT+ZRIGHT)/2.

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CONT.

4.SMALLEST OF MAXIMUM ZSOM

It is the minimum of Z of the maximizing .

5.LARGEST OF MAXIMUM ZSOM

ZSOM is the maximum z of the maximizing z

ZSOM and LOM are not used as often as the other 3 defuzzificationmethods.

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Example: Mamdanis Fuzzy Model

Single-input single-output Mamdani fuzzy

model

If X is small then Y is small.

If X is medium then Y is medium.If X is large then Y is large.

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ADVANTAGES AND

DISADVANTAGES

Advantages

Its well suited to human input

Rules are having high correctness

Disadvantage

Defuzzification requires a large amount of

mathematical calculations.

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II.LARSEN MODEL

Product operator for a fuzzy implication

Max-product operator for the composition

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Contd..

The output of Larsen model are also fuzzy

sets.

Need defuzzification methods to get final

output.

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II.SUGENO FUZZY MODELS

Also known as the TSK fuzzy model

Introduced in 1984 by

T.Takagi

M.Sugeno and

K.T.Kang

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Motivation of TSK

To reduce the number of rules required by the

Mamdani model.

For complex and high dimensional problems

Develop a systematic approach to generate fuzzy

rules from a given input-output data set.

TSK model replaces the fuzzy set (then part) of

mamdani rule with function(equation) of theinput variables.

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TSK fuzzy rule

If x is A and y is B then z=f(x,y)

Where A and B are fuzzy sets in the antecedent

,and

Z=f(x,y) is a crisp function in the consequences .

Usually f(x,y) is a polynomial in the input variables x

and y.

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First order TSK fuzzy model

F(x,y) is a first order polynomial

Example: a two-input one-output TSK If x is Aj and y is Bk then z= px+qy+r

The degree the input matches ith rule is typically

computed using min operator:

wi=min(Aj(x), Bk(y))

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Cont..

Each rule has a crisp output

Overall output is obtained via weighted

average (reduce computation time of

defuzzification required in a Mamdani model)

Z=i wi zi/ i wi

to further reduce computation ,weighted

sum may be used. i.e

Z= i wi zi

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TSK fuzzy model

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example

Example: a single input TSK fuzzy model

can be expressed as,

If X is small then Y = 0.1X + 6.4 If X is medium then Y = - 0.5X + 4

If X is large then Y = X2.

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Example-2

Two-input single-output Sugeno fuzzy model

If X is small and Y is small then z=-x+y+1.

If X is small and Y is large then z=-y+3.

If X is large and Y is small then z=-x+3.

If X is large and Y is large then z=x+y+2

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Zero order TSK

When f is a constant , we have a zero order

TSK fuzzy model.

It has minimum computation time.

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summary

Overall output via either weighted average or

weighted sum is always crisp.

Without the time consuming defuzzification

operation the TSK fuzzy model is by the far

most popular candidate for sample-data-

based fuzzy modeling.

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Advantages

Computationally efficient

Works well with linear techniques

Has continuity of the output surface

Well suited to mathematical analysis.

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Reference

Neuro-fuzzy and soft computingJ.Jang,C.Sun

and E.Mizutani, Prentice Hall 1997

System modelling using a Mamdani Rule Base-

Bryan Davis,University of Florida

Fuzzy systems toolbox,M.Beale and

H.Demuth.

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Fuzzy Inference System1