Fuzzy Control

Lecture 2 Fuzzy Set

Basil HamedElectrical Engineering Islamic University of Gaza

Content Crisp Sets Fuzzy Sets Set-Theoretic Operations Extension Principle Fuzzy Relations

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Introduction

Fuzzy set theory provides a means for representing uncertainties.

Natural Language is vague and imprecise.

Fuzzy set theory uses Linguistic variables, rather than quantitative variables to represent imprecise concepts.

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Fuzzy Logic

Fuzzy Logic is suitable toVery complex modelsJudgmentalReasoningPerceptionDecision making

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Crisp Set and Fuzzy Set

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Information World

Crisp set has a unique membership function

A(x) = 1 x A 0 x A

A(x) {0, 1}

Fuzzy Set can have an infinite number of membership functions

A [0,1]

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Fuzziness

Examples:

A number is close to 5

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Fuzziness

Examples:

He/she is tall

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Classical Sets

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CLASSICAL SETSDefine a universe of discourse, X, as a collection of objects all having the same characteristics. The individual elements in the universe X will be denoted as x. The features of the elements in X can be discrete, or continuous valued quantities on the real line. Examples of elements of various universes might be as follows:

the clock speeds of computer CPUs;the operating currents of an electronic motor;the operating temperature of a heat pump;the integers 1 to 10.

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Operations on Classical Sets

Union:A B = {x | x A or x B}

Intersection:A B = {x | x A and x B}

Complement:A’ = {x | x A, x X}

X – Universal SetSet Difference:

A | B = {x | x A and x B} Set difference is also denoted by A - B

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Union of sets A and B (logical or).

Intersection of sets A and B.

Operations on Classical Sets

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Operations on Classical Sets

Complement of set A.

Difference operation A|B.

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Properties of Classical Sets

A B = B AA B = B AA (B C) = (A B) CA (B C) = (A B) C

A (B C) = (A B) (A C)A (B C) = (A B) (A C)

A A = AA A = A

A X = XA X = AA = AA =

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Mapping of Classical Sets to Functions

Mapping is an important concept in relating set-theoretic forms to function-theoretic representations of information. In its most general form it can be used to map elements or subsets in one universe of discourse to elements or sets in another universe.

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Fuzzy Sets

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A fuzzy set, is a set containing elements that have varying degrees of membership in the set.

Elements in a fuzzy set, because their membership need not be complete, can also be members of other fuzzy sets on the same universe.

Elements of a fuzzy set are mapped to a universe of membership values using a function-theoretic form.

Fuzzy Sets

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An object has a numeric “degree of membership” Normally, between 0 and 1 (inclusive)

0 membership means the object is not in the set 1 membership means the object is fully inside the set In between means the object is partially in the set

Fuzzy Set Theory

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If U is a collection of objects denoted generically by x, then a fuzzy set A in U is defined as a set of ordered pairs:

membershipfunction

U : universe of discourse.

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Fuzzy Sets

Characteristic function X, indicating the belongingness of x to the set A

X(x) = 1 x A 0 x A

or called membership

Hence,A B XA B(x)

= XA(x) XB(x)= max(XA(x),XB(x))

Note: Some books use + for , but still it is not ordinary addition!

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Fuzzy Sets

A B XA B(x)= XA(x) XB(x)= min(XA(x),XB(x))

A’ XA’(x) = 1 – XA(x)

A’’ = A

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Fuzzy Set Operations

A B(x) = A(x) B(x) = max(A(x), B(x))

A B(x) = A(x) B(x) = min(A(x), B(x))

A’(x) = 1 - A(x)

De Morgan’s Law also holds: (A B)’ = A’ B’ (A B)’ = A’ B’

But, in generalA A’ A A’

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Union of fuzzy sets A and B∼

.

Intersection of fuzzy sets A and B∼

.

Fuzzy Set Operations

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Complement of fuzzy set A∼

.

Fuzzy Set Operations

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Operations

A B

A B A B ADr Basil Hamed 25

A A’ = X A A’ = Ø

Excluded middle axioms for crisp sets. (a) Crisp set A and its complement; (b) crisp A ∪ A = X (axiom of excluded middle); and (c) crisp A ∩ A = Ø (axiom of contradiction).

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A A’ A A’

Excluded middle axioms for fuzzy sets are not valid. (a) Fuzzy set A∼

and its complement; (b) fuzzy A ∪ A∼ = X (axiom of excluded middle); and (c) fuzzy A ∩ A = Ø (axiom of contradiction).

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Set-Theoretic Operations

A B

A B

A

A B

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Examples of Fuzzy Set Operations

Fuzzy union (): the union of two fuzzy sets is the maximum (MAX) of each element from two sets.E.g. A = {1.0, 0.20, 0.75} B = {0.2, 0.45, 0.50} A B = {MAX(1.0, 0.2), MAX(0.20, 0.45),

MAX(0.75, 0.50)} = {1.0, 0.45, 0.75}

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Examples of Fuzzy Set Operations

Fuzzy intersection (): the intersection of two fuzzy sets is just the MIN of each element from the two sets.E.g. A B = {MIN(1.0, 0.2), MIN(0.20,

0.45), MIN(0.75, 0.50)} = {0.2, 0.20, 0.50}

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Examples of Fuzzy Set OperationsA = {1/a, 0.3/b, 0.2/c 0.8/d, 0/e}B = {0.6/a, 0.9/b, 0.1/c, 0.3/d, 0.2/e}Complement: = {0/a, 0.7/b, 0.8/c 0.2/d, 1/e}Union:A B = {1/a, 0.9/b, 0.2/c, 0.8/d, 0.2/e}Intersection:A B = {0.6/a, 0.3/b, 0.1/c, 0.3/d, 0/e} Dr Basil Hamed 31

Properties of Fuzzy Sets

A B = B AA B = B AA (B C) = (A B) CA (B C) = (A B) C

A (B C) = (A B) (A C)A (B C) = (A B) (A C)

A A = A A A = AA X = X A X = AA = A A =

If A B C, then A C

A’’ = ADr Basil Hamed 32

Fuzzy Sets

Note (x) [0,1] not {0,1} like Crisp set

A = {A(x1) / x1 + A(x2) / x2 + …} = { A(xi) / xi}Note: ‘+’ add

‘/ ’ divide

Only for representing element and its membership.

Also some books use (x) for Crisp Sets too.

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Example (Discrete Universe)

{1, 2,3,4,5,6,7,8}U # courses a student may take in a semester.

(1,0.1) (2,0.3) (3,0.8) (4,1)(5,0.9) (6,0.5) (7,0.2) (8,0.1)

A

appropriate # courses taken

0.5

1

02 4 6 8

x : # courses

( )A x

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Example (Discrete Universe)

{1, 2,3,4,5,6,7,8}U # courses a student may take in a semester.

(1,0.1) (2,0.3) (3,0.8) (4,1)(5,0.9) (6,0.5) (7,0.2) (8,0.1)

A

appropriate # courses taken

Alternative Representation:

1 2 3 40.1/ 0.3 / 0.8 / 1.0 / 0.9 / 0.5 / 0.2 / 0.1/5 6 7 8A

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Example (Continuous Universe)

possible ages

U : the set of positive real numbers

( , ( ))BB x x x U

4

1( )501

5

B xx

about 50 years old

00.20.40.60.8

11.2

0 20 40 60 80 100

x : age

( )B x

4505

11 xR

B x

Alternative Representation:

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Alternative Notation

( , ( ))AA x x x U

U : discrete universe

U : continuous universe

( ) /i

A i ix U

A x x

( ) /AUA x x

Note that and integral signs stand for the union of membership grades; “ / ” stands for a marker and does not imply division.

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Fuzzy DisjunctionAB max(A, B)AB = C "Quality C is the disjunction of Quality A and B"

0

1

0.375

A

0

1

0.75

B

• (AB = C) (C = 0.75) Dr Basil Hamed 38

Fuzzy ConjunctionAB min(A, B)AB = C "Quality C is the conjunction of Quality A and B"

0

1

0.375

A

0

1

0.75

B

• (AB = C) (C = 0.375) Dr Basil Hamed 39

Example: Fuzzy Conjunction

Calculate AB given that A is .4 and B is 20

0

1A

0

1B

.1 .2 .3 .4 .5 .6 .7 .8 .9 1 5 10 15 20 25 30 35 40

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Example: Fuzzy Conjunction

Calculate AB given that A is .4 and B is 20

0

1A

0

1B

.1 .2 .3 .4 .5 .6 .7 .8 .9 1 5 10 15 20 25 30 35 40

• Determine degrees of membership:

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Example: Fuzzy Conjunction

Calculate AB given that A is .4 and B is 20

0

1A

0

1B

.1 .2 .3 .4 .5 .6 .7 .8 .9 1 5 10 15 20 25 30 35 40

• Determine degrees of membership:• A = 0.7

0.7

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Example: Fuzzy Conjunction

Calculate AB given that A is .4 and B is 20

0

1A

0

1B

.1 .2 .3 .4 .5 .6 .7 .8 .9 1 5 10 15 20 25 30 35 40

• Determine degrees of membership:• A = 0.7 B = 0.9

0.70.9

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Example: Fuzzy Conjunction

Calculate AB given that A is .4 and B is 20

0

1A

0

1B

.1 .2 .3 .4 .5 .6 .7 .8 .9 1 5 10 15 20 25 30 35 40

• Determine degrees of membership:• A = 0.7 B = 0.9

• Apply Fuzzy AND• AB = min(A, B) = 0.7

0.70.9

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Generalized Union/Intersection

Generalized Union Or called triangular norm.

Generalized Intersection

t-norm

t-conorm Or called s-norm.

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T-norms and S-norms

And/OR definitions are called T-norms (S-norms) Duals of one another A definition of one defines the other implicitly

Many different ones have been proposed Min/Max, Product/Bounded-Sum, etc. Tons of theoretical literature We will not go into this.

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Examples: T-Norm & T-Conorm

Minimum/Maximum:

Lukasiewicz:

( , ) min( , )T a b a b a b

( , ) max( , )S a b a b a b

( , ) max( 1,0) ( , )T a b a b LAND a b

( , ) min( ,1) ( , )S a b a b LOR a b

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Classical Logic &Fuzzy LogicHypothesis : Engineers are mathematicians. Logical thinkers do not believe in magic. Mathematicians are logical thinkers.Conclusion : Engineers do not believe in magic.Let us decompose this information into individual propositionsP: a person is an engineerQ: a person is a mathematicianR: a person is a logical thinkerS: a person believes in magicThe statements can now be expressed as algebraic propositions as((PQ)(RS)(QR))(PS)

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Fuzzy Relations

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Aa1

a2

a3

a4

Bb1

b2

b3

b4

b5

Crisp Relation (R)

R A B Dr Basil Hamed 50

Aa1

a2

a3

a4

Bb1

b2

b3

b4

b5

Crisp Relation (R)R A B

1 1 1 3 2 5

3 1 3 4 4 2

( , ), ( , ), ( , )( , ), ( , ), ( , )a b a b a b

Ra b a b a b

1 0 1 0 00 0 0 0 11 0 0 1 00 1 0 0 0

RM

1 1a Rb 1 3a Rb 2 5a Rb

3 1a Rb 3 4a Rb 4 2a RbDr Basil Hamed 51

Crisp Relations

Example:

If X = {1,2,3} Y = {a,b,c}R = { (1 a),(1 c),(2 a),(2 b),(3 b),(3 c) }

a b c1 1 0 1

R = 2 1 1 03 0 1 1

Using a diagram to represent the relation

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The Real-Life Relation x is close to y

x and y are numbers x depends on y

x and y are events x and y look alike

x and y are persons or objects If x is large, then y is small

x is an observed reading and y is a corresponding action

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Fuzzy RelationsTriples showing connection between two sets:

(a,b,#): a is related to b with degree #

Fuzzy relations are set themselves

Fuzzy relations can be expressed as matrices

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Fuzzy Relations MatricesExample: Color-Ripeness relation for tomatoesR1(x, y) unripe semi ripe ripe

green 1 0.5 0

yellow 0.3 1 0.4

Red 0 0.2 1

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CompositionLet R be a relation that relates, or maps, elements from universe X to universe Y, and let S be a relation that relates, or maps, elements from universe Y to universe Z.

A useful question we seek to answer is whether we can find a relation, T, that relates the same elements in universe X that R contains to the same elements in universe Z that S contains. It turns out that we can find such a relation using an operation known as composition.

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CompositionIf R is a fuzzy relation on the space X x Y S is a fuzzy relation on the space Y x ZThen, fuzzy composition is T = R SThere are two common forms of the composition operation: 1. Fuzzy max-min composition

T(xz) = (R(xy) s(yz))

2. Fuzzy max-production compositionT(xz) = (R(xy) s(yz))

Note: R S S R multiplication

y Y

y Y

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A fuzzy relation defined on X an Z.

Max-Min Composition

X Y ZR: fuzzy relation defined on X and Y.

S: fuzzy relation defined on Y and Z.R 。 S: the composition of R and S.

( , ) max min ( , ), ( , )R S y R Sx z x y y z

( , ) ( , )y R Sx y y z Dr Basil Hamed 58

Example

1 0.1 0.2 0.0 1.02 0.3 0.3 0.0 0.23 0.8 0.9 1.0 0.4

R a b c d0.9 0.0 0.30.2 1.0 0.80.8 0.0 0.70.4 0.2 0.3

Sabcd

1 0.4 0.2 0.32 0.3 0.3 0.33 0.8 0.9 0.8

R S

0.1 0.2 0.0 1.00.9 0.2 0.8 0.4min0.1 0.2 0.0 0.4max

( , ) max min ( , ), ( , )S R v R Sx y x v v y

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Max-Product Composition

( , ) max ( , ) ( , )R S v R Sx y x v v y

A fuzzy relation defined on X an Z.

X Y ZR: fuzzy relation defined on X and Y.

S: fuzzy relation defined on Y and Z.R。 S: the composition of R and S.

.

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Example

1 0.1 0.2 0.0 1.02 0.3 0.3 0.0 0.23 0.8 0.9 1.0 0.4

R a b c d0.9 0.0 0.30.2 1.0 0.80.8 0.0 0.70.4 0.2 0.3

Sabcd

0.1 0.2 0.0 1.00.9 0.2 0.8 0.4Product

max .09 .04 0.0 0.4

R S

1 0.4 0.2 0.32 0.27 0.3 0.243 0.8 0.9 0.7

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Properties of Fuzzy Relations

Example: y1 y2 z1 z2 z3

R = x1 0.7 0.5 S = y1 0.9 0.6 0.2x2 0.8 0.4 y2 0.1 0.7 0.5

z1 z2 z3Using max-min, T = x1 0.7 0.6 0.5

x2 0.8 0.6 0.4

z1 z2 z3Using max-product, T = x1 0.63 0.42 0.25

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Example 3.8 (Page 59)

Suppose we are interested in understanding the speed control of the DC shunt motor under no-load condition, as shown.

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Example 3.8Initially, the series resistance Rse in should be kept in the cut-in position for the following reasons:1. The back electromagnetic force, given by Eb = kNφ, where k is a constant of proportionality, N is the motor speed, and φ is the flux (which is proportional to input voltage, V ), is equal to zero because the motor speed is equal to zero initially.2. We have V = Eb + Ia(Ra + Rse), therefore Ia = (V − Eb)/(Ra + Rse), where Ia is the armature current and Ra is the armature resistance. Since Eb is equal to zero initially, the armature current will be Ia = V/(Ra + Rse), which is going to be quite large initially and may destroy the armature.

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Example 3.8Let Rse be a fuzzy set representing a number of possible values for series resistance, say sn values, given as

and let Ia be a fuzzy set having a number of possible values of the armature current, say m values, given as

The fuzzy sets Rse and Ia can be related through a fuzzy relation, say R, which would allow for the establishment of various degrees of relationship between pairs of resistance and current.Dr Basil Hamed 65

Example 3.8Let N be another fuzzy set having numerous values for the motor speed, say v values, given as

Now, we can determine another fuzzy relation, say S, to relate current to motor speed, that is, Ia to N.Using the operation of composition, we could then compute a relation, say T, to be used to relate series resistance to motor speed, that is, Rse to N.

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Example 3.8The operations needed to develop these relations are as follows – two fuzzy Cartesian products and one composition:

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Example 3.8Suppose the membership functions for both series resistance Rse and armature current Ia are given in terms of percentages of their respective rated values, that is,

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Example 3.8

The following relation then result from use of the Cartesian product to determine R:

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Example 3.8Cartesian product to determine S:

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Example 3.8

The following relation results from a max–min composition for T:

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HW 1 2.4, 2.5,2.7, 2.11, 3.2, 3.4, 3.8 Due 30/ 9/ 2012Good Luck

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