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Introduction

NN & FL

Artificial Intelligence

Artificial Intelligence (AI) is a branch of science that is concerned with the automation of intelligent behavior.

it is possible to build machines that can demonstrate intelligence similar to human beings.

A I can be obtained in two ways. soft computing methods (NN, FL, GA etc) Hard computing methods (conventional PID cont.

etc)

Artificial Intelligence

Hard computing methods are predominantlybased on mathematical approaches .

Soft computing techniques have drawn their inherent characteristics from biological systems

Soft computing methods are

Neural networks

Fuzzy logic

Genetic algorithms

Combination of above

Neural Networks

Neural Networks (NN) are simplified models of

the biological nervous systems

An NN can be massively parallel and therefore is

said to exhibit parallel distributed processing

NN architectures have been broadly classified as

Single layer feed forward networks.

Multi layer feed forward networks and

Fuzzy Logic

Fuzzy logic is a set of mathematical principles forknowledge representation based on themembership function

Fuzzy logic provides simple way to draw definiteconclusions from vague, ambiguous or impreciseinformation.

Fuzzy logic is similar to that of Boolean logic

Fuzzy Logic

If the Level is low then open V1

If the Level is High then Close V1

If the level is Medium then open V1 by 50%

Need of fuzzy logic controller

Rigorous mathematical model of some linear process.

in the case of complex process, which are difficult to model.

Non-Linear Systems

Comparison of conventional & fuzzy logiccontrollers:

Sets & Fuzzy Sets

Fuzzy Logic

Set (Crisp set)

Well defined collection of objects

If X is Universe of discourse

(Universal set)

A is any set from X.

x is any element in X

Set & membership function

Def: Let X be the universe of discourse and its elements be denoted as x.

In the classical set theory, crisp set A of X is defined as: Called membership function of A

where

)(xfA

1,0:)( XxfA

Axif

AxifxfA

0

,1)(

membership function (crisp set)

Crisp set of “ tall persons”

Degree of

Membership

Figure 1: A crisp way of modeling tallness


Figure 2: The crisp version of short

Crisp set of “ Short persons” or “ NOT tall ”


Different heights have same ‘tallness’

Fuzzy Set

Def: Let X be the universe of discourse and its elements be denoted as x.

In the fuzzy set theory, fuzzy set A of X is defined as: Called membership function of fuzzy set A

Where

1...0:)( toXxA

)( xA

Ainpartiallyisxifx

Ainnotisxif

Aintotallyisxif

x

A

A

,1)(0

,0

,1

)(

Fuzzy Sets & membership

The shape you see is known as the membership function

Degree of A = “Set of TALL persons”

membership

Fuzzy Sets & memberships

Shows two membership functions: ‘tall’

and ‘short’

Degree of

membership

Fuzzy Sets

Fuzzy Operations

Fuzzy Sets ( Notation)

Formal definition:

A fuzzy set A in X is expressed as a set of ordered pairs:

Universe or

universe of discourseFuzzy set

Membership

function

(MF)

}{ XxxxA A /))(,(

ExampleA = { ( x1, 0.2 ) ( x2, 0.8 ) ( x3, 0.4 ) }

Alternative Notation

A fuzzy set A can be alternatively denoted as follows:

i

iA

Xx x

xA

i

)(

i

iA

Xx

xA

)(

X is discrete

X is continuous

Note that S and integral signs stand for the union of

membership grades; “__” stands for a marker and does

not imply division.

Fuzzy Sets with Discrete Universes

Fuzzy set C = “desirable city to live in”

X = {SF, Boston, LA} (discrete and non ordered)

C = {(SF, 0.9), (Boston, 0.8), (LA, 0.6)}

Fuzzy Sets with Cont. Universes

Fuzzy set B = “temperature about 50 degrees”

X = Set of positive real numbers (continuous)

B = {(x, B(x)) / x in X}

0.5

40 50 c60

1

600

6050)5060(

)60(

5040)4050(

)40(

400

)(

xif

xifx

xifx

xif

xA

Fuzzy Sets with Cont. Universes

Fuzzy set B = “Age about 50 Years”

X = Set of positive real numbers (continuous)

B = {(x, B(x)) / x in X}

B xx

( )

1

150

10

2

Fuzzy Partition

Fuzzy partitions formed by the linguistic values “young”, “middle aged”, and “old”:

Set-Theoretic Operations

Union:

Intersection:

Complement:

BAC

BAC

A

Membership

value

Height

1.0

0.0

A

ShortB

Tall

Union (OR)

If you have x degree of faith in statement A, and y degree of faith in statement B, how much faith do you have in the statement A or B?

• Eg: How much faith in “that person is short ortall”

Union (OR)

Membership

value

Height

1.0

0.0

Short Tall

Take the max of your beliefs in each individual statement

The union of two fuzzy sets A and B is a new fuzzy set A B is defined as

))(,)((max)( xxx BABA

Intersection (AND)

If you have x degree of faith in statement A, and y degree of faith in statement B, how much faith do you have in the statement A and B?

• Eg: How much faith in “that person is short and tall”

Membership

value

Height

1.0

0.0

A

Short

B

Tall

Intersection (AND)

Membership

Height

1.0

0.0

A

Short

B

Tall The Intersection of Fuzzy Sets A and B

is a new fuzzy set

A B defined asShort and tall

))(,)((min)( xxx BABA

Complement ( NOT )

The degree to which you believe something is notin the set is 1.0 minus the degree to which you believe it is in the set

The complement of a fuzzy set A is a new fuzzy set A’

Membership

value

Units

1.0

0.0

FS

FS’

)(1)( xx AA C

0.7

0.3

X 1 X 2

Set-Theoretic Operations

Subset:

Union:

Intersection:

Complement:

))(,)((max)( xxx BABA

BA

BAC

BAC

A)(1)( xx AA C

))(,)((min)( xxx BABA

))()( xx BA

Equality of fuzzy sets:

The two fuzzy sets A and B are said to be equal (A =B)if

Example:

A = { (x1, 0.2) (x2, 0.8) }

B = { (x1, 0.6) (x2, 0.8) }

C = { (x1, 0.2) (x2, 0.8) }

A B and A = C

)()( xx BA

Product of two fuzzy sets:

The product of two fuzzy sets A and B is a new fuzzyset A B with a membership function defined as

Example:

A = { ( x1, 0.2 ) ( x2, 0.8 ) ( x3, 0.4 ) }

B = { ( x1, 0.4 ) ( x2, 0 ) ( x3, 0.1) }

A B = { ( x1, 0.08 ) ( x2, 0 ) ( x3, 0.04 ) }

)(.)()(. xxx BABA

Product of a fuzzy set with a crisp number:

Multiplying a fuzzy set A by a crisp number `a’ resultsin a new fuzzy product a. Ã with the membershipfunction

Example:

A = { ( x1, 0.2 ) ( x2, 0.8 ) ( x3, 1 ) } and

a = 0.5

Then a.A = { (x1, 0.1) (x2, 0.4)( x3, 0.5) }

)(.)(. xax AAa

Power of a fuzzy set

The’ k’ power of a fuzzy set A is a new fuzzy set whosemembership function is given by

Example:

A = { (x1, 0.4) (x2, 0.2) ( x3, 0.7) } and k = 2

Then

A2 = { (x1, 0.16) (x2, 0.04) ( x3, 0.49) }

k

A

k

A xx ])([)(

Difference:The difference of two fuzzy sets A and B is a newfuzzy set A - B is defined as

Example:

A = { ( x1, 0.2 ) ( x2, 0.5 ) ( x3, 0.6) }

B = { ( x1, 0.9 ) ( x2, 0.4 ) ( x3, 0.5) }

B’ = { ( x1, 0.1 ) ( x2, 0.6 ) ( x3, 0.5) }

A – B = { ( x1, 0.1) ( x2, 0.5) ( x3, 0.5 ) }

)( CBABA

Cardinality :

The cardinality of a set is the total membership of elements in the set

Example:

A = { ( x1,0.2 ) ( x2,0.8 ) ( x3,1) }

Card(A) = 0.2+0.8+1 = 2

Height:

The height of a fuzzy set is the highest membership value of its membership function.

Example: A = { (x1,0.2)( x2,0.8)( x3,1)}

Height(A) = 1

i

iA xACard )()(

)(max)( iA xAHeight

Support :

The support of a fuzzy set A is the set of elements whose degree of membership in A is greater than 0.

mathematically,

Example:

Support of fuzzy set A is the

open interval (10,20)

0)(/)( xXxASpt A1

0.5

10 15 20

α- Level Cut :

The α- cut of a fuzzy set A at α denoted as Aα, is the set of elements whose degree of membership in A is greater than or equal to α

mathematically,

)(/ xXxA A

10 20 30 40 50 60 70

Temperature

}60,50,40,30,20,10{2.0 A

}50,40,30,20{4.0 A

}40,30{6.0 A

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Properties of fuzzy sets

Commutative:

Associative:

Distributive:

Identity:

ABBA ABBA

)()( CBACBA

)()( CBACBA

)()()( CABACBA

)()()( CABACBA

AA AXA A XXA

Properties of fuzzy sets

Idempotence:

Involution:

Demorgan’s Law:

AAA AAA

AA CC )(

CCC BABA )(CCC BABA )(

Problem:

Two fuzzy sets (discrete) are given as

Calculate the following fuzzy operations

1. :

2. Show that

3. Verify DeMorgan’s Laws

4. Cardinalities of , , and

5

2.0

4

3.0

3

5.0

2

1

1

0A

5

4.0

4

2.0

3

7.0

2

5.0

1

0B

CA

,

BA

,

BA, AB

CCC BABA )(

CB BA

A BAB BA

Crisp & Fuzzy Relations

Crisp/Fuzzy Relations

• A crisp relation represents the presence

or absence of association, interaction, or

interconnections between the elements of

two or more sets.

• Fuzzy relations allow various degrees or

strengths of relations between elements.

• A classical relation can be considered as

a set of pair of elements.

• A binary pair is denoted by (u, v).

Crisp or classical Relation

Notation: Crisp or classical Relation

},|),{( ByAxyxR

Definition

Let A is set defined on X and B is another set defined on Y.

Then a relation R of A and B is

“ Set of element pairs each pair consists elements from

both the sets”

Cartesian Product

Cartesian product X X Y is Set of All possible element pairs

},(|),{( YyXxyxYX

Membership function

• Membership function of a classical relation

is given by

Ryxif

Ryxif

yxR

),(0

),(1

),(

1. Crisp Relations• Example 1: Let X and Y be two sets given as follows.

Present the relation R: “x is smaller than y” in the form of a

relational matrix

X = { 1,2,3 }, Y = { 2,3,4 }, R: x<y

R = { (1,2),(1,3),(1,4),(2,3),(2,4),(3,4) }.

R =

x/y 2 3 4

1

2

3

1 1 1

0 1 1

0 0 1

Fuzzy RelationDefinition

Let assume that X and Y are sets. Fuzzy relation R of

X and Y is a fuzzy subset of X x Y represented as pair

of elements and its membership function

R x y x y x y X YR {(( , ), ( , ))|( , ) }

))(),(min(),( yxyx BAR

Membership functionIf A and B are two fuzzy sets on X and Y

Then membership function is

2. Fuzzy Relations• Example 2: Let X and Y be two sets given as follows.

Present the relation R: “x is approximately equal to y” in

the form of a relational matrix

X = { 1,2,3 }, Y = { 2,3,4 }, R:

R = { ((1,2),0.66),((1,3),0.33),((1,4),0.0),((2,2),1.0),

((2,3),0.66),((2,4), 0.33),((3,2),0.66),((3,3),1.0),((3,4),0.66) }.

R =

x/y 2 3 4

1

2

3

0.66 0.33 0

1 0.66 0.33

0.66 1 0.66

yx

2.2 Fuzzy Relations

• Example 3. Relation R is given as an association or

interconnection between (Tomato) fruit color and state.

Present R as a crisp relational matrix.

X = { green, yellow, red }, Y= { unripe, semi ripe, ripe}

R unripe semiripe ripe

green

yellow

red

1 0 0

0 1 0

0 0 1

R unripe semiripe ripe

green

yellow

red

1 0.5 0

0.3 1 0.4

0 0.2 1

2.3 Fuzzy Relations

Example:

Let A = { (x1, 0.2), (x2, 0.7), (x3, 0.4) } and

B = { (y1, 0.5), (y2, 0.6) } be two fuzzy sets

Find the fuzzy relation R resulting out of the fuzzy Cartesian product A x B

R = A x B =

)(),(min),(),( yxyxyx BABAR

4.04.0

6.05.0

2.02.0

3

2

1

x

x

x

21 yy

Operations on fuzzy Relations

1. Union

2. Intersection

3. Compliment

SR )),(),,(max(),( yxyxyxSRSR

SR

R ),(),( 1 yxyxRR

)),(),,(min(),( yxyxyxSRSR

Composition of crisp Relations

Let X,Y,Z are three sets

= ?

YXR :

ZYS :

ZXT :

SRT

}),(),,(),,(),,({ 33222111 yxyxyxyxR

}),(),,({ 2411 zyzyS }),({ 11 zxSRT

x1

x2

x3

y1

y2

Y3

y4

z1

z2

R S

X Y Z

Composition

}),({ 11 zxSRT

R y1 y2 y3 y4

x1

x2

x3

1 1 0 0

0 1 0 0

0 0 1 0

S z1 z2

y1

y2

y3

y4

1 0

0 0

0 0

0 1

}),(),,(),,(),,({ 33222111 yxyxyxyxR }),(),,({ 2411 zyzyS

R/S z1 z2

x1

x2

x3

1 0

0 0

0 0

Composition

})},(),,(min{

)},,(),,(min{

)},,(),,(min{

}),,(),,(min{{max),(

1441

1331

1221

111111

zyyx

zyyx

zyyx

zyyxzx

SR

SR

SR

SRSR

)},0,0min(),0,0min()0,1min(),1,1min({max),( 11 zxSR

},0,0,0,1{max),( 11 zxSR

Composition of Relations

Definition [Composition of Fuzzy Relations]

Let R and S are fuzzy relations, i.e.,

),/(),( ),,/(),( zyzySyxyxRZY

SYX

R

Composition of fuzzy relations,

R and S, is a fuzzy set defined by

),/(),( zxzxSRZX

SR

X Y Z

R S

SR

Composition

• Max-Min composition:

• The max-min composition of two fuzzy relations R

(defined on X and Y) and S (defined on Y and Z) is

)],(),([),( zyyxzx SRy

SR

Max-product composition:

)],().,([),( zyyxzx SRy

SR

Composition of Fuzzy Relations

• MAX-MIN composition

• MAX-PROD composition

},|)),(),,({(),( ZzXxzxzxzxSR SR

})},(),,(min{{max),( zyyxzx SRy

SR

}),().,({max),( zyyxzx SRy

SR


Example:

R =

S =

R S (x1, z1) = max { (min (0.5, 0.6), min (0.1, 0.5)) }

= max (0.5, 0.1) = 0.5

X = { x1, x2, x3 } Y = { y1, y2 } Z= { z1, z2, z3 }

21 yy

6.08.0

9.02.0

1.05.0

3

2

1

x

x

x

321 zzz

9.08.05.0

7.04.06.0

2

1

y

y

MAX-MIN composition

7.06.06.0

9.08.05.0

5.04.05.0

)6.0,7.0()6.0,4.0()5.0,6.0(

)9.0,2.0()8.0,2.0()5.0,2.0(

)01,5.0()1.0,4.0()1.0,5.0(

)9.0^60(),7.0^8.0()8.0^60(),4.0^8.0()5.0^60(),6.0^8.0(

)9.0^90(),7.0^2.0()8.0^90(),4.0^2.0()5.0^90(),6.0^2.0(

)9.0^10(),7.0^5.0()8.0^10(),4.0^5.0()5.0^10(),6.0^5.0(

MAX

MAX

SRT

9.08.05.0

7.04.06.0

6.08.0

9.02.0

1.05.0

oRoST


Example:

R =

S =

R S (x1, z1) = max { ( 0.5x 0.6), (0.1x 0.5)) }

= max (0.3, 0.01) = 0.3

X = { x1, x2, x3 } Y = { y1, y2 } Z= { z1, z2, z3 }

21 yy

6.08.0

9.02.0

1.05.0

3

2

1

x

x

x

321 zzz

9.08.05.0

7.04.06.0

2

1

y

y

MAX-PROD composition

56.048.048.0

81.072.045.0

35.02.03.0

)54.0,56.0()48.0,32.0()3.0,48.0(

)81.0,14.0()72.0,08.0()45.0,12.0(

)09.0,35.0()08.0,2.0()05.0,3.0(

)9.060(),7.08.0()8.060(),4.08.0()5.060(),6.08.0(

)9.090(),7.02.0()8.090(),4.02.0()5.090(),6.02.0(

)9.010(),7.05.0()8.010(),4.05.0()5.010(),6.05.0(

MAX

MAX

SRT

9.08.05.0

7.04.06.0

6.08.0

9.02.0

1.05.0

oRoST


• Example 4: R is a relation that describes an

interconnection between color x and ripeness y

of a tomato, and S represents an

interconnection between ripeness y and taste z

of a tomato.

Present relational matrices for the MAX-MIN and

MAX-PROD composition

Composition of Fuzzy RelationsThe rational matrix R (x-y connection) is given as

The rational matrix S (y-z connection) is given as

R (x,y) unripe semiripe ripe

green

yellow

red

1 0.5 0

0.3 1 0.4

0 0.2 1

S (y,z) sour sweet-sour sweet

unripe

semiripe

ripe

1 0.2 0

0.7 1 0.3

0 0.7 1


The MAX-MIN composition R= R 。S results in the

relational matrix

R (x,z) sour sweet-sour sweet

green

yellow

red

1 0.5 0.3

0.7 1 0.4

0.2 0.7 1

1)0,5.0,1())0,0min(),7.0,5.0min(),1,1(min( MAXMAX

4.0)4.0,3.0,0())1,4.0min(),3.0,1min(),0,3.0(min( MAXMAX

Composition of Fuzzy RelationsR (x,z) sour sweet-sour sweet

green

yellow

red

1 0.5 0.15

0.7 1 0.4

0.14 0.7 1

A linguistic interpretation in the form of rules for the

relational matrix above is as follows:

Rule1: IF the tomato is green, THEN it is sour, less likely

to be sweet-sour, and unlikely to be sweet.

Rule2: IF the tomato is yellow, THEN it is sweet-sour,

possibly sour, and unlikely to be sweet.

Rule3: IF the tomato is red, THEN it is sweet, possibly

sweet-sour, and unlikely to be sour.

Properties of Fuzzy Relations

x1

x2

x3

Y1

y2

Y3

y4

RX Y

Projection on X ( First projection ) is given by

Similarly calculate for all pairs, the X

projection is

Projection on Y ( Second projection ) is given by

Similarly calculate for all pairs, the Y projection is

Total projection is

Tolerance and Equivalence Relation

Crisp Equivalence Relation

R X x X

Relation has the following properties:

Reflexivity

Symmetry

Transitivity


Reflexivity

X = { x1,x2,x3 } R X x X

(xi , xi ) R

or

µR (xi , xi ) = 1x1

x2

x3


Symmetry (classical)

(xi xJ) R (xJ xi) R

or µR(xi xj) = µR(xj xi)

x1

x3

x2


Transitivity: (Classical)

(xi xJ) R and (xJ xk) R (xi xk) R

or µR(xi xJ) = 1 and µR(xJ xk) = 1 µR(xi xk) = 1

x1 x2

x3

FUZZY LOGIC & FUZZY RULES

Linguistic variables

• Example:

Age = 65

Age is OLD

Temperature = { cold, cool, warm, hot }

Speed = { stop, slow, medium, fast }

Height = { short , medium , Tall }

Linguistic Value is a fuzzy set

Linguistic variable Linguistic Value

Linguistic Hedges

• Modifying the meaning of a fuzzy set using hedges such as very, more or less, slightly, etc.

• Very TALL

• Very very TALL

• More or less TALL

• etc.

tall

More or less tallVery tall

FUZZY LOGIC

What is Fuzzy Logic?• In Propositional Logic, truth values are either True or False

• Fuzzy logic is a type of Many-Valued Logic

• There are more than two truth values

• The interval [0,1] represents the possible truth values

• 0 is absolute falsity

• 1 is absolute truth

CRISP LOGIC

“Logic is a human capacity to reason” or “ Science of reasoning”

Proposition:

A statement which is either ‘True' or 'False' but not both

Example: P: Water boils at 90°C.

Q: Vapor is produced by water.

Connectives:

Disjunction ( ) OR

Conjunction () AND

Negation ( - or ~) NOT

Implication ()

Equivalence (=)

CRISP LOGIC

Let Sets A and B are defined from Universe X

P and Q are two propositions

P: truth that x Є A

Q: truth that x Є B

Disjunction ( ): P Q : x Є A or x Є B

Conjunction (): P Q : x Є A and x Є B

Implication (): P Q : x Є A or x Є B

CRISP LOGIC skip

Propositional logic connectives

Symbol Connective Usage Description

And P Q P and Q are true.

Or P Q Either P or Q is true,

¬ or ~ Not ~P or ¬ P P is not true.

implication P Q P implies Q is true.

= Equality P = Q P and Q are equal (in truth

values) is true.

Implication skip

P Q P Q

T T T

T F F

F T T

F F T

P Qq

p

qp

When in Rome, do like the Romans

Truth table

CRISP LOGIC skip

• The following table illustrates the truth table for the five connectives.

P Q P Q P Q ~ P P Q P = Q

T T T T F T T

T F F T F F F

F F F F T T T

F T F T T T F

Tautology / contradiction skip

• A formula which has all its interpretations recording true is known as a tautology and the one which records false for all its interpretations is known as contradiction.

Example:

• Obtain a truth table for the formula (P v Q) (~P). Is it a tautology?

P Q P Q ~P P Q ~P

T

F

T

F

F

T

T

F

T

T

T

F

F

T

F

T

F

T

F

T

it is not a tautology

Fuzzy logical operations

• AND, OR, NOT, etc.

• NOT A = A’ = 1 - A(x)

• A AND B = A B = min(A (x), B (x))

• A OR B = A B = max(A (x), B (x))

A not A

0.4 0.6

0.8 0.2

1-A

A B A or B

0.4 0.7 0.7

0.8 0.6 0.8

1 0.3 1

0.1 1 1

max(A,B)

A B A and B

0.4 0.7 0.4

0.8 0.6 0.6

1 0.3 0.3

0.1 1 0.1

min(A,B)

From the following

truth tables it is

seen that fuzzy

logic is a superset

of Boolean logic.

FUZZY RULES

Fuzzy Rules

• Rules often of the form:

IF x is A THEN y is B

where A and B are fuzzy sets defined on the universes of discourse X and Y respectively.

• if pressure is high then volume is small;

• if a tomato is red then a tomato is ripe.

where high, small, red and ripe are fuzzy sets.

Fuzzy Rules

• Crisp rule:

• Example:

“If Self is TALL and Enemy is SHORT, Then Attack.”

The Condition of a Rule:

“If Self is Tall and Enemy is Short”

• Fuzzy rule:

“ Self is 0.3 TALL and Enemy is 0.6 SHORT ” ,

Then 0.3 Attack

then this condition is 0.3 True.

So, should we attack?

Decision Making

• For example:

• Rule 1: If Self is TALL and Enemy is SHORT, then Attack

• Rule 2: If Self is BIG and Enemy is LEAN, then Attack.

• Rule 3: If Self’s Power is greater than 10,

and Self’s Health is greater than 5, then Attack.

Average of the 3 rules NOT TO ATTACK

0.6 0.8

0.6 0.9

0

0.6

0.6

0

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