UNIT III

1UNIT-3

FUZZY SYSTEMS

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Overview

Fuzzy Logic A Definition

A Little History

Fuzzy Sets

Fuzzy Sets Operations

Fuzzy Applications

An Example

Summary

References

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Fuzzy Systems

Based on Human Thought Processes

Systems that use objective and subjective knowledge of a problem

Objective knowledge mathematical models

Subjective knowledge linguistic information

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Fuzzy Logic A Definition

Fuzzy logic provides a method to formalize reasoning when dealing with vague terms.

Traditional computing requires finite precision which is not always possible in real world scenarios. Not every decision is either true or false, or as with Boolean logic either 0 or 1.

Fuzzy logic allows for membership functions, or degrees of truthfulness and falsehoods. Or as with Boolean logic, not only 0 and 1 but all the numbers that fall in between.

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A little History

The idea behind fuzzy logic dates back to Plato, who recognized not only the logic system of true and false, but also an undetermined area the uncertain.

In the 1960s Lotfi A. Zadeh Ph.D,. University of California, Berkeley, published an obscure paper on fuzzy sets . His unconventional theory allowed for approximate information and uncertainty when generating complex solutions; a process that previously did not exist.

Fuzzy Logic has been around since the mid 60s but was not readily accepted until the 80s and 90s. Although now prevalent throughout much of the world, China, Japan and Korea were the early adopters

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Fuzzy Set Basics

Classical (crisp) sets:

Membership in a set is all or nothing

Membership function cS: Universe {0, 1}

cS(x) = 1 iff x S

Fuzzy sets:

Membership in a set is a degree

membership function cS: Universe [0, 1]

range of real numbers

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Linguistic Characterizations of Degree of

Membership

Consider the set of hot days in Coimbatore in

2006.

Was July 17 hot? It might have been called one of:

very hot

sort of hot

not hot

The answer depends on the observer, time, etc.

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Fuzzy Sets

Sets with fuzzy boundaries

A = Set of tall people

Heights510

1.0

Crisp set A

Membershipfunction

Heights510 62

.5

.9

Fuzzy set A1.0

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Example Fuzzy Variable

Each function tells us how

much we consider a

character in the set if it has a

particular aggressiveness

value

Or, how much truth to

attribute to the statement:

The character is nasty (or

meek, or neither)?

Aggressiveness

Membership (Degree of Truth)

1.0

0.0 10.5

MediumNastyMeek

-1 -0.5

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Fuzzy-Set Operationsexpressed using membership functions

cA U B(x) = max(cA (x), cB(x))1

0

cA B(x) = min(cA (x), cB(x))1

0

A U B

1

0

A B

A B

Fuzzy OR (union) Fuzzy AND (intersection)

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Fuzzy Complement

(not the only possible model)

1

0A

cA(x) = 1 - cA(x).

A

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Fuzzy Anomaly?

1

0

AcA(x) = 1 - cA(x).

A

A A

The intersection of a set with its complement is not necessarily empty.

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Fuzzification Fuzzification can be considered as a

mapping from an observed input space to fuzzy sets.

1. Measure the values of input variables at each sampling instant.

2. Normalizes the measured variables to within the universe of discourse (UOD).

3. Performs the function of fuzzification that converts the input data into suitable linguistic variables.


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Fuzzification

The input variables are

Error e(t)

(-2 pH to +2 pH)

'Change in Error e(t), e(t) = e(t) - e(t-1) (-0.5 pH to +0.5 pH)

The output variable is

change in output u(t)(-0.03 L/s to +0.03 L/s)


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Fuzzification Cont.

The FLC output is given by

u(t) = u(t-1) + u(t)The fuzzy input and output variables,

namely Error, Change in Error and Change in

Output, are divided into seven linguistic

(fuzzy) variables namely NL, NM, NS, ZE, PS,

PM and PL


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Defuzzification

The defuzzifier converts the output fuzzy

set into a crisp solution variable.

The most commonly used method is the

'Center of Area' method.

m mU = C

i

i/

i i=1 i=1


Fuzzy Rules

If our distance to the car in front is small, and the distance is decreasing slowly, then decelerate quite hard

Fuzzy variables in blue

Fuzzy sets in red

Conditions are on membership in fuzzy sets

Actions place an output variable (decelerate) in a fuzzy set (the quite hard deceleration set)

We have a certain belief in the truth of the condition, and hence a certain strength of desire for the outcome

Multiple rules may match to some degree, so we require a means to arbitrate and choose a particular goal - defuzzification

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Fuzzy Rules Example

Rules for controlling a car:

Variables are distance to car in front and how fast it is changing, delta, and acceleration to apply

Sets are:

Very small, small, perfect, big, very big - for distance

Shrinking fast, shrinking, stable, growing, growing fast for delta

Brake hard, slow down, none, speed up, floor it for acceleration

Rules for every combination of distance and delta sets, defining an acceleration set

Assume we have a particular numerical value for distance and delta, and we need to set a numerical value for acceleration

Extension: Allow fuzzy values for input variables (degree to which we believe the value is correct)

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Set Definitions for Example

distance

v. small small perfect big v. big

delta

< = > >>

acceleration

slow present fast fastest

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UNIT III