CHAPTER 1

Introduction to Fuzzy Logic Theory(Part A)BY NORLIDA HASSAN

Introduction to Fuzzy Logic

Chapter 1

Part A: The Introduction to FL

History and motivation

Introduction to fuzzy theory and its practical applications

Uncertainty

Part B: Fuzzy Set

Representation of uncertainty by fuzzy sets

Definition and representation of fuzzy sets

Fuzzy sets properties

Operations on fuzzy sets

Probability theory

History and motivation

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Fuzzy, or multi-valued logic,

was introduced in the 1930s

by Jan Lukasiewicz, a Polish

philosopher.

classical logic operates with only two values 1 (true) and 0 (false),

Multi-valued logic introduce the extended range of truth values to all real numbers in the interval between 0 and 1.

For example, the possibility

that a man 181 cm tall is

really tall might be set to a

value of 0.86. It is likely that

the man is tall.

This work led to an inexact reasoning technique often called possibility theory.

In 1965 Lotfi Zadeh, published

his famous paper “Fuzzy sets”.

Zadeh extended the work on possibility theory into a formal system of mathematical logic, and introduced a new concept for applying natural language terms. This new logic for representing and manipulating fuzzy terms was called fuzzy

logic.

Who is Lotfi A. Zadeh ?

Lotfi Aliasker Zadeh

@ Lotfi A. Zadeh

The founder of fuzzy logic

Also the founder of fuzzy mathematics, fuzzy set theory, and fuzzy logic, Z numbers, Z-transform

Born: February 4, 1921, Baku, Azerbaijan

Died: September 6, 2017, (at the age of 96) in Berkeley, California, United States

Today, Fuzzy Logic Has Already Become

the Standard

Technique for Multi-Variable Control !

History, State of the Art, and Future Development

1965 Seminar Paper “Fuzzy Logic” by Prof. LotfiZadeh, Faculty in Electrical Engineering, U.C. Berkeley, Sets the Foundation of the “Fuzzy Set Theory”

1970 First Application of Fuzzy Logic in Control Engineering (Europe)

1975 Introduction of Fuzzy Logic in Japan

1980 Empirical Verification of Fuzzy Logic in Europe

1985 Broad Application of Fuzzy Logic in Japan

1990 Broad Application of Fuzzy Logic in Europe

1995 Broad Application of Fuzzy Logic in the U.S.

2000 Fuzzy Logic Becomes a Standard Technology and Is Also Applied in Data and Sensor Signal Analysis. Application of Fuzzy Logic in Business and Finance.

Why?

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Why fuzzy?As Zadeh said, the term is concrete, immediate and descriptive; we all know what it means.

However, many people in the West were repelled by the word fuzzy, because it is usually used in a negative sense.

Why logic? Fuzziness rests on fuzzy set theory, and

fuzzy logic is just a small part of that theory.

The Term “Fuzzy Logic”

The term of fuzzy logic is used in two senses:

Narrow sense:

Fuzzy logic is a branch of fuzzy set theory, which deals (as logical systems do) with the representation and inference from knowledge.

Fuzzy logic, unlike other logical systems, deals with imprecise or uncertain knowledge. In this narrow, and perhaps correct sense, fuzzy logic is just one of the branches of fuzzy set theory.

Broad Sense: fuzzy logic synonymously with fuzzy set theory

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What is Fuzzy Logic?

Fuzzy logic is a form of many-valued logic in

which the truth values of variables may be any

real number between 0 and 1.

It is employed to handle the concept of partial

truth, where the truth value may range between

completely true and completely false.

By contrast, in Boolean logic, the truth values of

variables may only be the integer values 0 or 1.

What is Fuzzy Logic?

Definition of fuzzy

Fuzzy – “not clear, distinct, or precise; blurred”

Definition of fuzzy logic

A form of knowledge representation suitable for

notions that cannot be defined precisely, but

which depend upon their contexts.

Why Fuzzy Logic?

Fuzzy logic can handle problems with imprecise

and incomplete data, and it can model

nonlinear functions of arbitrary complexity

Fuzzy logic is an approach to computing based

on "degrees of truth" rather than the usual "true

or false" (1 or 0) Boolean logic on which the

modern computer is based.

Fuzzy Approach

• An effective tool to tackle the problem of

uncertainty,

• Support a flexible sense of membership and is

defined to be the pair (X, µA(X))

• Operations used as min, max and

complement and …………

Uncertainty

Most Words and Evaluations We Use in Our Daily Reasoning Are Not Clearly Defined in a Mathematical Manner. This Allows

Humans to Reason on an Abstract Level!

2 types of uncertainty

Stochastic

Lexical

Types of Uncertainty (1)

Stochastic Uncertainty:

Deals with the uncertainty toward the occurrence of a certain event.

Eg:

The Probability of hitting the target is 0.8

The uncertainty in above statement is whether the target is hit or not

The statement can be processes and combined with other statements using stochastic methods such as the Bayesian calculus of conditional probability

Types of Uncertainty (2)

Lexical Uncertainty:

Deals with the imprecision that is inherent in mosr words humans use to evaluate cpncept and derive conclusions.

Eg:

We will probably have a successful financial year.

The event itself is not clearly defined, i.e. successful financial year is subjective category

The statement does not quantify a probability, i.e. a perceived probability (not mathematically defined)

Other examples; "Tall Men", "Hot Days", or "Stable Currencies"

Stochastic and Fuzzy

Logic Complement

Each Other !

Probability and Uncertainty

“... a person suffering from hepatitis shows in 60% of all cases a strong fever, in 45% of all cases yellowish colored skin, and in 30% of all cases suffers from nausea ...”

Fuzzy Applications

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Theory of fuzzy sets and fuzzy logic has been

applied to problems in a variety of fields:

• taxonomy; topology; linguistics; logic; automata theory; game theory; pattern recognition; medicine; law; decision support; Information retrieval; etc.

And more recently fuzzy machines have been developed including:

• automatic train control; tunnel digging machinery; washing machines; rice cookers; vacuum cleaners; air conditioners, etc.

Conclusions

Fuzzy logic provides an alternative way to represent linguistic and subjective attributes of the real world in computing.

FL deals with events and situations with subjectively defined attributes.

a) A Proposition in FL does not have to be either "True' or 'False‘.

b) An event (or situation) can be, for example, 'a bit true', fairly true', 'almost true', 'very true' or'not true' depending on the event (or situation) attributes.

Conclusions

It is able to be applied to control systems and other applications in order to improve the efficiency and simplicity of the design process.

Fuzzy Logic provides way to calculate with imprecision and vagueness.

Fuzzy Logic can be used to represent some kinds of human expertise.

The control stability, reliability, efficiency, and durability of fuzzy logic makes it popular.

The speed and complexity of application production would not be possible without systems like fuzzy logic.

HOW IT REPRESENTS?

TRADITIONAL REPRESENTATION

OF LOGIC

Slow Fast

Speed = 0 Speed = 1

bool speed;

get the speed

if ( speed == 0) {

// speed is slow

}

else {

// speed is fast

}

FUZZY LOGIC REPRESENTATION

For every problem

must represent in

terms of fuzzy sets.

What are fuzzy

sets?

Slowest

Fastest

Slow

Fast

[ 0.0 – 0.25 ]

[ 0.25 – 0.50 ]

[ 0.50 – 0.75 ]

[ 0.75 – 1.00 ]

FUZZY LOGIC REPRESENTATION Cont.

Slowest Fastest

float speed;

get the speed

if ((speed >= 0.0)&&(speed < 0.25)) {

// speed is slowest

}

else if ((speed >= 0.25)&&(speed < 0.5))

{

// speed is slow

}

else if ((speed >= 0.5)&&(speed < 0.75))

{

// speed is fast

}

else // speed >= 0.75 && speed < 1.0

{

// speed is fastest

}

Slow Fast

FUZZY LOGIC IN CONTROL

SYSTEMS

Fuzzy Logic provides a more efficient and

resourceful way to solve Control Systems.

Some Examples

� Temperature Controller

� Anti – Lock Break System ( ABS )

TEMPERATURE CONTROLLER

The problem

� Change the speed of a heater fan, based off the room

temperature and humidity.

A temperature control system has four settings

� Cold, Cool, Warm, and Hot

Humidity can be defined by:

� Low, Medium, and High

Using this we can define

the fuzzy set.

BENEFITS OF USING FUZZY LOGIC

ANTI LOCK BREAK SYSTEM ( ABS )

Nonlinear and dynamic in nature

Inputs for Intel Fuzzy ABS are derived from

� Brake

� 4 WD

� Feedback

� Wheel speed

� Ignition

Outputs

� Pulsewidth

� Error lamp

To be continued…

What is a Fuzzy Set?

A fuzzy set is a set with elements whose attributes are not sharply defined,

i.e. Membership grades of the set can have any real value between 0 and 1.