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Fuzzy Inference Systems

Fuzzy Logic and Approximate Reasoning

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Outline

Introduction

Mamdani Fuzzy models

Sugeno Fuzzy Models

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Introduction

Fuzzy inference is a computer paradigmbased on fuzzy set theory, fuzzy if-then-rulesand fuzzy reasoning

Applications:data classification, decisionanalysis, expert systems, times seriespredictions, robotics & pattern recognition

Different names; fuzzy rule-based system, fuzzy

model, fuzzy associative memory, fuzzy logic

controller & fuzzy system

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Introduction

Structure

Rule base selects the set of fuzzy rules

Database (or dictionary) defines the membershipfunctions used in the fuzzy rules

A reasoning mechanism performs the inferenceprocedure (derive a conclusion from facts & rules!)

Defuzzification: extraction of a crisp value that bestrepresents a fuzzy set

Need: it is necessary to have a crisp output in somesituations where an inference system is used as acontroller

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Fuzzy Inference System

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Rules

Fuzzy rule base: collection of IF-THEN rules.

The lth rule has form:

Where l= 1,..,M; Fil, Gl: fuzzy sets.

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Example

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Generate Rules

Expert Knowledge.

Data

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Extraction Rules from Data

Divide the input/output space into fuzzy

regions (2N+1).

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Extraction Rules from Data

Generate fuzzy rules: x1(i), x2(i), and y(i)

maximum degree of membership

R1: ifx1 is B1 and x2 is CE, then yis B1

Assign a degree to each rule

Create a combined FAM bank

If there is more than one rule in any cell, then

the rule with the maximum degree is used

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Fuzzification & Inference Engine

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Defuzzification

It refers to the way a crisp value is extracted from a fuzzy set as arepresentative value

There are five methods of defuzzifying a fuzzy set A of auniverse of discourse Z

Centroid of area zCOA

Bisector of area zBOA

Mean of maximum zMOM

Smallest of maximum zSOM

Largest of maximum zLOM

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Defuzzification Centroid of area zCOA

where A(z) is the aggregated output MF.

Bisector of area zBOA this operator satisfies the

following;

where = min {z; z Z} & = max {z; z Z}.

The vertical line z = zBOA partitions the regionbetween z = , z = , y = 0 & y = A(z) into two

regions

,dz)z(

zdz)z(

z

Z

A

Z

A

COA

BOAz

BOAz

AA ,dz)z(dz)z(

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Defuzzification Mean of maximum zMOM

This operator computes the average of the maximizing z at

which the MF reaches a maximum . It is expressed by :

})z(;z{Z'where

,dz

zdz

z

A

'Z

'ZMOM

2

zzzthenz,z)z(maxif:However

zzthen

zzatmaximumsingleahas)z(if:definitionBy

21

MOM21Az

MOM

A

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Defuzzification

Various defuzzification schemes for obtaining a crisp output

Smallest of maximum zSOM

Amongst all z that belong to [z1, z2], the smallest is called zSOM Largest of maximum zLOM

Amongst all z that belong to [z1, z2], the largest value is called

zLOM

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Example

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Example

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Example

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Example

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Sugeno Fuzzy Models

Goal: Generation of fuzzy rules from a given input-

output data set

A TSK fuzzy rule is of the form:

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

Where A & B are fuzzy sets in the antecedent, while z =

f(x, y) is a crisp function in the consequent

f(.,.) is very often a polynomial function x & y

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Sugeno Fuzzy Models

If f(.,.) is a first order polynomial, then the resulting fuzzyinference is called a first order Sugeno fuzzy model

If f(.,.) is a constant then it is a zero-order Sugeno fuzzy

model (special case of Mamdani model)

Case of two rules with a first-order Sugeno fuzzy model Each rule has a crisp output

Overall output is obtained via weighted average No defuzzyfication required

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Sugeno Fuzzy Models

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Example 1

Single output-input Sugeno fuzzy model with threerules

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

If X is large then Y = X 2

If small, medium & large are nonfuzzy setsthen the overall input-output curve is a piecewise linear

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Example 1

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Example 1

However, if we have smooth membership functions(fuzzy rules) the overall input-output curve becomesa smoother one

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Example 2

Two-input single output fuzzy model with 4 rules

R1: if X is small & Y is small then z = -x +y +1R2: if X is small & Y is large then z = -y +3R3: if X is large & Y is small then z = -x +3

R4: if X is large & Y is large then z = x + y + 2

R1 (x s) & (y s) w1R2 (x s) & (y l) w2R3 (x l) & (y s) w3

R4 (x l) & (y l) w4Aggregated consequent F[(w1, z1); (w2, z2); (w3, z3);(w4, z4)]

= weighted average

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Example 2

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Tsukamoto fuzzy model

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single-input Tsukamoto fuzzy modelwith 3 rules

if X is small then Y is C1

if X is medium then Y is C2

if X is large then Y is C3

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