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CHAPTER I

INTRODUCTION

1.1 FUZZY LOGIC

Fuzzy logic refers to a logical system that generalizes the

classical two value logic for reasoning under uncertainty. It is a

system of computing and approximate reasoning based on collection

of theories and technologies that employ fuzzy sets, which are classes

of objects without sharp boundaries where membership is a matter of

degree [Yen and Langari, 1999]. The mathematical background of

fuzzy logic is associated to the fuzzy set theory, which is the extension

of the classical set theory.

Fuzzy logic was first proposed by Lotfi A. Zadeh of the University

of California at Berkeley United States in 1965. Zadeh had observed

that conventional computer logic could not manipulate data that

represented subjective or vague ideas, so he created fuzzy logic to

allow computers to determine the distinctions among data with

shades of gray, similar to the process of human thinking. In 1973 he

introduced the concept of linguistic variables. Hence it should be

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possible to improve the performance of electromechanical controllers

by modeling the way of handling this type of information. The theory

developed slowly at first, but by the early 1970's it had attracted a

small international group of scientists. This included a number of

westerners, mostly mathematicians, and a small number of Japanese

engineers.

Fuzzy logic has been implemented in many applications

including household appliance, consumer-electronic goods, transit

systems, automobiles, and industrial processes. Many consumer

products using fuzzy technology are currently available in Japan, and

some are now being marketed in the US and Europe.

Although US was the first to introduce fuzzy Logic in different

areas of practical applications but it is Japan where a leadership was

taken in widely implementing the associated technology in the fields

mentioned above. They use this approach in problems that involve

knowledge-based decision making. So we can say that Japan

benefited the global community, industry, academia, and various

professions.

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The first applications of fuzzy theory were primly industrial, such as

process control for cement kilns built in Denmark, coming on line in

1975. However, as the technology was further embraced, fuzzy logic

was used in more useful applications.

In 1987, the first fuzzy logic-controlled subway was opened in

Sendai in northern Japan. Here, fuzzy-logic controllers make subway

journeys more comfortable with smooth braking and acceleration.

Best of all, all the driver has to do is push the start button. Fuzzy

logic was also put to work in elevators to reduce waiting time. Since

then, the applications of Fuzzy Logic technology have virtually

exploded, affecting things we use every day.

Another event in 1987 helped to promote interest in fuzzy

systems. During a international meeting of fuzzy researchers in Tokyo

that year, Takeshi Yamakawa demonstrated the use of fuzzy control,

through a set of simple dedicated fuzzy logic chips, in an "inverted

pendulum" experiment. This is a classic control problem, in which a

vehicle tries to keep a pole mounted on its top by a hinge upright by

moving back and forth. Observers were impressed with this

demonstration, as well as later experiments by Yamakawa in which he

mounted a wine glass containing water or even a live mouse to the top

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of the pendulum. The system maintained stability in both cases.

Yamakawa eventually went on to organize his own fuzzy-systems

research lab to help exploit his patents in the field. Following such

demonstrations, the Japanese became infatuated with fuzzy systems,

developing them for both industrial and consumer applications. In

1988 they established the Laboratory for International Fuzzy

Engineering (LIFE), a cooperative arrangement between 48 companies

to pursue fuzzy research. Japanese companies developed a wide range

of products using fuzzy logic, ranging from washing machines to

autofocus cameras and industrial air conditioners.

Fuzzy logic can be used in two different views one is in narrow

view, fuzzy logic, as it is based on multi-valued logic, concentrates on

approximate reasoning. Although fuzzy logic is rooted on multivalued

logic, it has some contradiction with traditional multivalued logical

system e.g. Lukasiewicz’s logic. Similarly, the concepts which made

fuzzy logic effective in approximate reasoning are also not from the

traditional multivalued system. These are linguistic variable,

canonical form, fuzzy rule, fuzzy graph, and fuzzy quantifiers.

The second approach of looking at fuzzy logic, i.e. in broad view,

is almost equivalent to fuzzy set theory. In this sense fuzzy logic

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serves mainly as an apparatus for fuzzy control, analysis of vagueness

in natural language and several other application domains.

It is one of the techniques of soft-computing that is

computational methods tolerant to sub optimality and impreciseness

and giving simple and sufficiently good solutions. Linguistic variable

plays important role in applications of fuzzy logic. Fuzzy logic as it

makes it easy to understand and interpret the physical variables, i.e.

linguistic variables, to be processed in simple way. Linguistic

variables are nothing but variables with values of words.

Although words are often less precise than numbers and their

use is closet to human perception. For example, “height” is linguistic

variable with values “short”, “tall”, “very tall”. “Age” is linguistic

variable with values “young”, “old”, “not very old”. In the universe of

discourse of fuzzy set, these values are used as label to the fuzzy set.

Another basic concept in fuzzy logic is “if-then” rule called a fuzzy rule

or fuzzy inference system which plays an important role in most of its

applications and it is based on natural language used by people on a

daily basis.

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Fuzzy logic provides a simple way to arrive at definite conclusion

based upon vague, ambiguous, imprecise and noisy, or missing

information. Fuzzy logic technology has achieved the impressive

success in diverse engineering applications ranging from mass market

consumer products to sophisticated decision and control problems.

1.1.1 FUZZY SET

A fuzzy set is a set containing elements that have varying

degrees of membership in the set. Fuzzy set provides a suitable point

of departure for the construction of conceptual framework which

parallels in many respects of the framework used in the case of

ordinary sets, but is more general than the latter and potentially, may

prove to have a much wider scope of applicability, particularly in the

field of pattern classification and information processing.

When a set A is stated and if we consider any object in the

universe, then there are only two possibilities. The object chosen

satisfies the condition hence it is an element of A or the object chosen

does not satisfy the condition and hence it is not an element of A. So

the entire set theoretical concept is based on the assumption that a

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given condition divides the entire universe exactly into two parts, one

part contains elements satisfying the condition and another part

contains elements not satisfying the condition. So, if we consider any

object and if we ask the question whether the condition is satisfied by

the object or not then the answer must be either YES or NO. This is

called the Binary logic.

In general there are some attributes, the existence of which an

individual cannot be determined by YES or NO. Let X be the

considered universe. A be an attribute for any x in X we have a

measure say (x). Hence any attribute gives a map : X!R where "(x)

measures the existence of the attribute in x. This is defined as a fuzzy

set.

A is fuzzy set in X means given any x in X we cannot say x

belongs to A, or x does not belong to A, but we say that x belongs to A

with measure of "(x), " is called the membership function. We assume

"(x) lies in [0, 1] for all x. In general a fuzzy set on X is nothing but a

map (function) X! [0, 1].

For example, let ‘A’ be the collection of all talented students in a

class having 100 students. Suppose a particular student always

scores 40% in any exam then is he talented or not? From this

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statement it is not possible to conclude that he is talented, and at the

same time it is also not possible to close the matter by saying that he

is not talented. Suppose a student consistently scores 40% marks, he

is talented or not? In this case also there is no conclusion. So the two

way logic (YES or NO logic) (True or False logic) doesn’t work. The

drawback of conventional sets is that many concepts encountered in

the real world cannot always be described exclusively by their

membership and non-membership in sets.

What best can be said about this student is that he is 40%

talented. So he belongs to the set of all talented students with a

measure of say 40%. Like that all the 100 students can be given the

percentage. So consider a universe which contains all the objects of

consideration, given an attribute, each member in the considered

universe is given a number (percentage) which measures the existence

of attribute in it.

The first publication on fuzzy set theory by [Zadeh, 1965]

shows the intention to generalize the classical notion of a set and a

statement to accommodate fuzziness. The word “imprecision” here is

meant in the sense of vagueness rather than the lack of knowledge

about the value of a parameter. Fuzzy set theory provides a strict

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mathematical framework in which vague conceptual phenomena can

be precisely and rigorously studied [Zimmermann, 1991]. The notion

of a fuzzy set is completely non-statistical in nature Zadeh, 1965].

Fuzzy set theory takes the same logical approach as researchers use

the concept of classical set theory. In the classical set theory, as soon

as the two-valued characteristic function has been defined and

adopted, rigorous mathematics follows. However, in the fuzzy set case,

as soon as a multi-valued membership function has been chosen, a

rigorous mathematical theory can be developed.

1.1.2 UNIVRSE OF DISCOURSE

All elements in a set are taken from a universe of discourse or

universe set that contains all the elements that can be taken into

consideration when the set is formed. In reality there is no such thing

as a set or a fuzzy set because all sets are subsets of some universe

set, even though the term ‘set’ is predominantly used. In the fuzzy

case, each element in the universe set is a member of the fuzzy set to

some degree, even zero. The set of elements that have a non-zero

membership is referred to as the support. We will use the notation U

for the universe set.

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1.1.3 LINGUISTIC HEDGES

Humans communicate with their own natural language by

referring to previous mental images with rather vague but simple

terms. Therefore any attempt to model the human thought process as

expressed in our communications with one another must be preceded

by models that attempt to emulate our natural language. Natural

language consists of fundamental terms called “atomic terms”.

Examples of some atomic terms are “medium”, “young” and

“beautiful”, etc. Collection of atomic terms are called composite terms.

Examples of composite terms are “Very slow car”, “Slightly Young

student”, “fairly beautiful lady”, etc. Suppose we define the atomic

terms and sets of atomic terms to exist as elements and sets on a

universe of natural language terms, say universe X. Furthermore let

us define another universe, called Y, as a universe of cognitive

interpretations, or meanings. Though it may seem easy to envision the

universe of terms, it may be difficult to ponder a universe of

“interpretations”. The atomic terms are called linguistic variable in

Fuzzy set theory.

In linguistics, fundamental atomic terms are often modified with

noun or verbs like very, low, slightly, more-or-less, fairly, almost,

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barely, mostly, roughly, approximately, etc. These modifiers are called

“linguistic hedges”, that is, the singular meaning of an atomic term is

modified or hedged from its original interpretation. When we use fuzzy

set for interpretation, the linguistic hedges have the effect of

modifying the membership function for a basic atom term.

The following are examples of hedges used in different types of

variables:

Very, quite, extremely-general modifiers

Quite true, mostly true-truth value modifiers

Likely, not very likely-probability modifiers

Low, medium, high-quantity modifier

Consider as example the Linguistic Variable humidity defined

on a universe [0%, 100%] with Linguistic Terms labeled as low, below

average, average, above average, high.

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Figure 1.1: Representation of the Linguistic Variable “humidity”

The representation of the Linguistic Variable will certainly be

different for an inhabitant of Serbia than for an inhabitant of north

Finland. Assume that the one used here is adequate. Notice that if

the prevailing physical humidity were 30%, the corresponding

linguistic representation would be given as a linear combination: 0.35

below average + 0.65 averages. By defining proper operations upon

linguistic variables, it is possible to have the basics for a fuzzy logic,

which in its turn provides an adequate formalism for inferences, fuzzy

modeling, fuzzy decision making and finally for fuzzy control, possibly

the best known application of fuzzy systems.

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1.1.4 FUZZY SET OPERATIONS

In crisp set theory we have the concepts of union and

intersection of two sets. This concept should be extended to fuzzy

sets. Also the concept of complement of a set in crisp set theory

should be extended to fuzzy sets [Sivanandam et al., 2007].

Definition: Let A and B be two fuzzy sets on a nonempty set X. The

union of A and B denoted as A# $ is defined as "A (x) = max ("A(x),

"B(x)), % x&X, where

" AUB is the membership function of A#B is map from X to [0, 1].

Hence A# $ is a fuzzy set on X. This is called the standard union of

two fuzzy sets.

Figure 1.2: Union of Fuzzy Sets ' ()* $

Definition: Let A and B be two fuzzy sets on a nonempty set X. The

intersection of A and B denoted as A+ $ is defined by

"A+ B (x) = min ("A(x), "B(x)), % x&X

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Where "A+B is the membership function of A+B is a fuzzy set on X.

This is called the standard intersection of two fuzzy sets.

Figure 1.3: Intersection of Fuzzy sets ' and $

Definition: Let A be a fuzzy set on nonempty set X. The complement

of A denoted as Ac is defined as 'c = 1 – A(x), % x& X. Clearly Ac is a

map from X to [0, 1]. Hence Ac is a fuzzy set. This is called the

standard complement.

Figure 1.4: Complement of Fuzzy set'.

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Definition: Let A and B be two fuzzy sets on a nonempty set X. The

difference denoted by A – B is defined as A – B = A+Bc

1.1.5 MEMBERSHIP FUNCTIONS

A membership function is a curve that defines how each point in

the input space is mapped to a membership value between 0 and 1.

Sometimes the input space refers to universe of discourse.

Definition: For any set X, a membership function A on X is any

function from X to the real unit interval [0, 1]. Membership functions

on X represent fuzzy subsets of X. The membership function which

represents a fuzzy set ' is usually denoted by A. For an element x of

X, the value A(x) is called the membership degree of x in the fuzzy

set '. The membership degree A(x) quantifies the grade of

membership of the element x to the fuzzy set ' . The value 0 means

that x is not a member of the fuzzy set; the value 1 means that x is

fully a member of the fuzzy set. The values between 0 and 1

characterize fuzzy members, which belong to the fuzzy set partially.

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Figure 1.5: Membership function of a fuzzy set

The general membership function of a fuzzy set can be defined as

the functions which take values in an arbitrary fixed algebra or

structure L; usually it is required that L be at least a poset or lattice.

The usual membership functions with values in [0, 1] are then called

[0, 1]-valued membership functions.

Mathematically, the above membership function can be defined as,

!

""#

""$

%

&

''((

)

*

bx1,

bxa,ab

ax

ax0,

x A

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In the case where a fuzzy set A is a conventional (crisp) set, the

corresponding membership function can be reduced to

!#$%

+

,*

Ax0,

Ax1,x A

The above function has only two outputs, 0 or 1. Whenever A(x)

=1, x is a member of A, if A(x) =0, x is declared a non-member of A.

Triangular Fuzzy Number: A triangular fuzzy number A is a fuzzy

number with a piecewise linear membership function A(x) and

is defined by !

"""

#

"""

$

%

''(

''(

(

*

otherwise0,

axa,aa

-xa

axa,aa

ax

x 32

23

13

21

12

1

A

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Figure 1.6: Triangular Membership function of a fuzzy set

Triangular fuzzy numbers are more often used when fuzziness exist

on both sides of a single value/parameter/factor.

Trapezoidal Fuzzy Number: A trapezoidal fuzzy number A is a fuzzy

number with a piecewise linear membership function A and is defined

by !

"""

#

"""

$

%

''(

''

''(

(

&)

*

43

34

4

32

21

12

1

41

A

axa,aa

x-a

axa1,

axa,aa

ax

axandaxwhen0,

x

-

1.0

a2 a3 x a1

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Figure 1.7: Trapezoidal Membership function of a fuzzy set

Trapezoidal fuzzy numbers are generally used when fuzziness exists

on both sides of an interval.

Gaussian Fuzzy Number

A Gaussian membership function is defined by

! !. / 012

345 ((*

2

!2/ xexp! ":xG

where the parameters and , control the center and width of the

membership function.

1.0

1a a2 a4

x a3

CORE

-

20

Figure 1.8: Gaussian Membership function of a fuzzy set

1.1.6 DEFUZZIFICATION METHOD

Defuzzification is the conversion of a fuzzy quantity to precise

quantity, just as fuzzification is the conversion of a precise quantity to

a fuzzy quantity. In practice, the output of the defuzzifier process is a

single value from the set. There are several built-in defuzzifier

methods. The centre of gravity method is the most commonly used for

extracting a crisp value from a fuzzy set. This method calculates the

weighted average of the elements in the support set. The bisector

method focuses on the axis of the vertical line which divides the area

under the diagram into two equal parts. The mean of maxima method

chooses the point by taking the mean of the maximal memberships.

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The smallest maximum and largest maximum methods choose either

the lower or upper boundary of the maximal membership. For

computational complexity, the gravity bisector methods are

categorized.

1.2 FUZZY RULES

Fuzzy rules are used to express piece of knowledge in fuzzy logic.

A fuzzy rule is a linguistic expression of causal dependencies between

linguistic variables in form of if-then statements.

If X is A then Y is B

where X and Y are linguistic variables and A and B are their linguistic

values determined by a fuzzy sets on the universe of discourse X and

Y, respectively.

In fuzzy logic, fuzzy rules are used for decision making.

Linguistic variables are used for the construction of language referred

as FDCL (Fuzzy Dependency and Command Language). FDCL gives

structure to estimated dependencies and commands using fuzzy if-

then rules or simply fuzzy rules. FDCL, like other language defined by

its syntax and semantics, and they define form of rules and their

meaning. The following example best describes the FDCL language.

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The following set of “if – then” rules constituting a fuzzy control-model

for an extreme simple irrigation system for a particular kind of cereal:

R1 : if the amount of rainfall last night was scarce then the

watering should be gallonwise

R2 : if the amount of rainfall last night was regular then the

watering should be literwise

R3 : if the amount of rainfall last night was large then the

watering should be dropwise

To use the rules, the meaning of “scarce”, “regular” and “large” in

a universe with a liter/m2 scale as well as that of “watering

gallonwise”, “watering literwise” and “watering dropwise” in a universe

with a scale in volume of water is needed. Assume that the

representation shown in the first two columns of Figure 1.9 is

adequate and corresponds to the use of these concepts by the

farmers. Furthermore assume that the universes [0, K1] and [0, K2]

have been defined by agricultural experts.

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Figure 1.9: Representation of the rule base of a fuzzy model-rainfall

If the measured (or estimated) rainfall in a given night is “r”

liter/m2, then as may be seen on Figure 1.9, “r” is not considered to

be “scarce”; accordingly, R1 does not apply. On the other hand, “r” is

close to “regular”. Its degree of membership is given by reg(r).

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Similarly, large(r) gives the degree of membership of “r” to “large”. It

is clear that reg(r)> large(r). This means that R2 is more satisfied

than R3. It is reasonable to expect that the behaviour of the system

should be closer to “literwise” than to “dropwise”. This way of

approximate reasoning may be formalized (among other alternatives),

as follows: use the degree of membership of the actual input to the

corresponding fuzzy sets representing the premises, to scale the fuzzy

sets of the corresponding conclusions (see the second column in

Figure 1.9) and combine the scaled conclusions by means of a point

wise maximum. This will lead to the shaded fuzzy set shown on the

third column of Figure 1.9. Its equation is:

irrigation (x) = max ! "#$(r)_ g liter(x) , large(r)_ g drops(x) ] _ x _

[0, K2 ]

This resulting fuzzy set represents the flexible predicate

assignable to the required watering. As mentioned earlier, this might

quite well be “closer to literwise than to dropwise”. However, in the

scenario under consideration, farmers will not be interested in the

kind of needed watering, but on the amount of water to be spread on

the field. This implies the conversion of the obtained fuzzy set into a

numerical value of the same universe [0, K2 ]. One way (but not the

only one) of doing this, quite often used in fuzzy control, is to

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calculate the abscise “g” of the gravity center of the fuzzy set (as

shown in Figure 1.9) and use it as the numerical answer of the model.

1.3 FUZZY DECISION MAKING PROCEDURE

Mechanism of fuzzy decision making involves manipulation of

fuzzy variable through linguistic equations or fuzzy rules. The below

Figure 1.10 explains the whole mechanism.

Figure 1.10: Fuzzy decision making procedure

Fuzzification: In fuzzification, membership degree is computed for

each input variable with respect to its linguistic term.

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Rule matching: In rule matching, the firing strength (degree of

satisfaction) of individual rule is calculated.

Fuzzy Inference: The recommendation of rules according to firing

strengths and rule conclusions are determined in fuzzy inference.

Fuzzy Aggregation: Fuzzy aggregation combines recommendations

from individual rules into an overall implied fuzzy set.

Defuzzification: Defuzzification involves determination of a crisp

value based on implied fuzzy sets derived from the rules, as final

result or solution.

1.4 APPLICATIONS

Fuzzy logic has wide area of applications, from control theory

to medical diagnosis. One of the well known application of fuzzy logic

can be referred as Hitachi’s first automated train operation for Sendai

subway system in Japan that has been in daily operation since 1987.

The train controlled by fuzzy predictive controller, which consume less

electric energy and ride more comfortably than the case where the

control was done by non-fuzzy controller. Another Hitachi’s product

was group fuzzy control operation for elevator. In this application, the

elevator which is used in rush hours, the fuzzy control could reduce

the waiting time as well as idle time and also make the riding and

stopping smoother as compared to elevators controlled by traditional

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controllers. The use of fuzzy control in consumer product was started

from 1989, the first home appliance products which include fuzzy

control were washing machines, these were fully automatic and

washing clothes was much simpler for users. User had to press start

button and machine was taking care of the rest of the task. It could

automatically select the wash, rinse and spin cycles, and the results

were cleaner clothes and efficient wash cycle.

Fuzzy processing was available on the chip which has made it

feasible to use these chips within individual products. The product

range was increased because of this, with fuzzy controlled rice

cookers, vacuum cleaners and home climate control system. The

major application area is automotive industry. Most of the automotive

manufacturers are pursuing fuzzy control concepts. Fuzzy control has

been applied to control automatic transmission system, suspension

system, engine system, climate system and antilock brake system.

These systems are used to make the vehicle better, more efficient and

safer to ride.

Many commercial products use fuzzy system technology. It has

been used to enhance processing of digital image and signals.

The Canon camera’s autofocus system is the best example of

the efficient implementation of fuzzy technology. The autofocus, auto

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zoom and auto exposure systems for Minolta cameras use fuzzy

controllers. Using fuzzy logic techniques, Sanyo and Canon

camcorders are better in auto-white balancing system, auto exposure

and autofocus system. Image stabilizer for camcorder of Matsushita

has been implemented with fuzzy technology; fuzzy inference was also

used to improve image quality. Because of Fuzzy system technology,

electro photography process of photocopying machine has been

improved. The image quality of Sanyo copies has been improved by

better toner supply control based on fuzzy control and Matsushita

copies are also improved by better auto exposure fuzzy based control.

Some other successful applications are hand written language

recognition and voice recognition.

Like all other areas, fuzzy system technology has larger impact

on healthcare industry. The biomedical applications are less due to

inherent complexity and uncertainty of the systems as well as the risk

involved. In biomedicine science, human knowledge, skills and

experience are more important in diagnosis and treatment of diseases.

This biomedical system is difficult to manage as it has time delay and

nonlinearity.

World’s first fuzzy control in medicine is drug delivery system,

which was developed and implemented to regulate blood pressure in

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post-surgical open heart patients at cardiac intensive care unit (ICU).

Some other applications include determining the disease risk using

fuzzy expert systems, medical control system, and determination of

drug dose. To control muscle immobility and hyper tension during

general anesthesia, assessment of cardiovascular dynamics during

ventricular assistance, diagnosis of artery lesions and coronary

stenosis, support for seriodignosis, intelligent medical alarms, fuzzy

control system has been applied. Classification of tissue and structure

in electrocardiograms, classification of normal and cancerous tissues

in brain magnetic resonance image are the other successful

application of fuzzy systems in medicine.

1.5 FUZZY LOGIC CONTROL

Control systems theory, or what is called modern control

systems theory today, can be traced back to the age of World War II,

or even earlier, when the design, analysis, and synthesis of

servomechanisms were essential in the manufacturing of

electromechanical systems. The development of control systems

theory has since gone through an evolutionary process, starting from

some basic, simplistic, frequency-domain analysis for single-input

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single-output (SISO) linear control systems, and generalized to a

mathematically sophisticated modern theory of multi-input multi-

output (MIMO) linear or nonlinear systems described by differential

and/or difference equations.

It is believed that the advances of space technology in the

1950s completely changed the spirit and orientation of the classical

control systems theory: the challenges posed by the high accuracy

and extreme complexity of the space systems, such as space vehicles

and structures, stimulated and promoted the existing control theory

very strongly, developing it to such a high mathematical level that can

use many new concepts like state-space and optimal controls.

The theory is still rapidly growing today; it employs many

advanced mathematics such as differential geometry, operation theory

and functional analysis, and connects to many theoretical and applied

sciences like artificial intelligence, computer science, and various

types of engineering. This modern control systems theory, referred to

as conventional or classical control systems theory, has been

extensively developed. The theory is now relatively complete for linear

control systems, and has taken the lead in modern technology and

industrial applications where control and automation are

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fundamental. The theory has its solid foundation built on

contemporary mathematical sciences and electrical engineering. As a

result, it can provide rigorous analysis and often perfect solutions

when a system is defined in precise mathematical terms. In addition

to these advances, adaptive and robust as well as nonlinear systems

control theories have also seen very rapid development in the last two

decades, which have significantly extended the potential power and

applicable range of the linear control systems theory in practice.

Conventional mathematics and control theory exclude

vagueness and contradictory conditions. As a consequence,

conventional control systems theory does not attempt to study any

formulation, analysis, and control of what has been called fuzzy

systems, which may be vague, incomplete, linguistically described, or

even inconsistent. Fuzzy set theory and fuzzy logic play a central role

in the investigation of controlling such systems.

The main contribution of fuzzy control theory, a new

alternative and branch of control systems theory that uses fuzzy logic,

has its ability to handle many practical problems that cannot be

adequately managed by conventional control techniques. At the same

time the results of fuzzy control theory are consistent with the existing

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classical ones when the system under control reduces from fuzzy to

nonfuzzy. In other words, many well-known classical results can be

extended in some natural way to the fuzzy setting.

Basically, the aim of fuzzy control systems theory is to extend

the existing successful conventional control systems techniques and

methods as much as possible, and to develop many new and special-

purposed ones, for a much larger class of complex, complicated, and

ill-modeled systems- fuzzy systems.

1.6 FUZZY MATRICES

Matrices with entries in [0, 1] and matrix operation defined by

fuzzy logical operations are called fuzzy matrices. All fuzzy matrices

are matrices but every matrix is not a fuzzy matrix. Fuzzy matrices

play a fundamental role in fuzzy set theory. They provide us with a

rich framework within which many problems of practical applications

of the theory can be formulated. Fuzzy matrices can be successfully

used when fuzzy uncertainty occurs in a problem. These results are

extensively used for cluster analysis and classification problem of

static patterns under subjective measure of similarity. On the other

hand, fuzzy matrices are generalized Boolean matrices which have

been studied for fruitful results. And the theory of Boolean matrices

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can be back to the theory of matrices with non negative contents, for

which most famous classical results were obtained from 1907 to 1912

by Parren and Frobenius. So the theory of fuzzy matrices is

interesting in its own right. An important connection between fuzzy

sets and fuzzy matrices has been recognized and this has led us to

define fuzzy matrices in a quite different way. This will inevitably play

an important role in any problem area that involves complementation

of fuzzy matrices.

Applications of the theory of fuzzy matrixes are of fundamental

importance in the formulation and analysis of many classes of

discrete structural models which arise in physical, biological, medical,

social and engineering sciences. Atanassov introduced the concept of

intutionistic fuzzy set (IFS). Later, Turksen introduced the concept of

interval-valued intutionistic fuzzy set (IVIFS), which is a generalization

of the IFS. The fundamental characteristic of the IVFS and IVIFS is

that the values of its membership function and non-membership

function are intervals rather than exact numbers.

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1.7 ORGANIZATION OF THE THESIS

This thesis consists of seven chapters. In Chapter I, a brief

introduction about the basic concepts of Fuzzy logic, Interval-valued

fuzzy logic, fuzzy matrices and the related topics are also discussed.

Chapter II explores the literature review of fuzzy logic and its

applications in the field of Bio-informatics, Data Mining, Image

Processing, Adaptive fuzzy rules for Image Segmentation, Fuzzy logic

in industrial applications and other related areas.

Chapter III discusses the extension of Sanchez’s approach for

medical diagnosis adopting the representation of an interval valued

fuzzy matrices. In this approach, geometric mean of interval valued

fuzzy matrices is introduced and the proposed method is applied to

the medical diagnosis system. The results are strengthened with

numerical computations.

Chapter IV deals with the technique of fuzzy matrices to

analyze the knowledge gathering attitude of research scholars.

A linguistic based information is gathered from selected universities of

various scholars in Tamilnadu. These informations are transformed

into fuzzy membership values. The computed results are compared

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numerically and graphically. In addition, using decision making

approach in a fuzzy environment, membership values are defined

separately for different category of scholars and performance of the

models are evaluated.

Chapter V analyses the algorithmic approaches of some fuzzy

techniques. Recent researchers have attempted different algorithmic

approaches for fuzzy pattern recognition. Furthermore, fuzzy pattern

recognition with an application to diagnose diseases in an early stage

through fuzzy C-means clustering.

Chapter VI focuses on fuzzy logic in the field of medicine. It

reveals the concepts and techniques underlying the application of

fuzzy concepts related to medical diagnosis. Different algorithms

based on the arithmetic operators: addition and multiplication are

analyzed and adapted the same to diagnose diseases. A Numerical

computation revels that both algorithms leads to the same

conclusion.

Chapter VII concludes with the summary and findings of the

research work.