Fuzzy Control Lecture 2 Fuzzy Set Basil Hamed

Fuzzy Control Lecture 2 Fuzzy Set Basil Hamed Electrical Engineering Islamic University of Gaza

Content § Crisp Sets § Fuzzy Sets § Set-Theoretic Operations § Extension Principle § Fuzzy Relations Dr Basil Hamed 2

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. Dr Basil Hamed 3

Fuzzy Logic is suitable to Very complex models Judgmental Reasoning Perception Decision making Dr Basil Hamed 4

Crisp Set and Fuzzy Set Dr Basil Hamed 5

Information World Crisp set has a unique membership function A(x) = 1 0 x A A(x) {0, 1} Fuzzy Set can have an infinite number of membership functions A [0, 1] Dr Basil Hamed 6

Fuzziness Examples: A number is close to 5 Dr Basil Hamed 7

Fuzziness Examples: He/she is tall Dr Basil Hamed 8

Classical Sets Dr Basil Hamed 9

CLASSICAL SETS Define 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. Dr Basil Hamed 10

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 Set Difference: A | B = {x | x A and x B} Set difference is also denoted by A - B Dr Basil Hamed 11

Operations on Classical Sets Union of sets A and B (logical or). Intersection of sets A and B. Dr Basil Hamed 12

Operations on Classical Sets Complement of set A. Difference operation A|B. Dr Basil Hamed 13

Properties of Classical Sets A B=B A A (B C) = (A B) C A (B C) = (A B) (A C) A A=A A X=X A X=A A = Dr Basil Hamed 14

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. Dr Basil Hamed 15

Fuzzy Sets Dr Basil Hamed 16

Fuzzy Sets q. A fuzzy set, is a set containing elements that have varying degrees of membership in the set. q. Elements in a fuzzy set, because their membership need not be complete, can also be members of other fuzzy sets on the same universe. q Elements of a fuzzy set are mapped to a universe of membership values using a function-theoretic form. Dr Basil Hamed 17

Fuzzy Set Theory l l An object has a numeric “degree of membership” Normally, between 0 and 1 (inclusive) l 0 membership means the object is not in the set l 1 membership means the object is fully inside the set l In between means the object is partially in the set Dr Basil Hamed 18

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: membership function Dr Basil Hamed U : universe of discourse. 19

Fuzzy Sets Characteristic function X, belongingness of x to the set A X(x) = 1 x A 0 x A or called membership indicating the 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 Dr Basil Hamed 20 addition!

Fuzzy Sets A B XA B(x) = XA(x) XB(x) = min(XA(x), XB(x)) A’ XA’(x) = 1 – XA(x) A’’ = A Dr Basil Hamed 21

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’ But, in general A A’ Dr Basil Hamed 22

Fuzzy Set Operations Union of fuzzy sets A and B∼. Intersection of fuzzy sets A and B∼. Dr Basil Hamed 23

Fuzzy Set Operations Complement of fuzzy set A ∼ . Dr Basil Hamed 24

Operations A B A B A Dr 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). Dr Basil Hamed 26

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). Dr Basil Hamed 27

Set-Theoretic Operations Dr Basil Hamed 28

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. n A = {1. 0, 0. 20, 0. 75} n B = {0. 2, 0. 45, 0. 50} n A B = {MAX(1. 0, 0. 2), MAX(0. 20, 0. 45), MAX(0. 75, 0. 50)} = {1. 0, 0. 45, 0. 75} Dr Basil Hamed 29

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. n A B = {MIN(1. 0, 0. 2), MIN(0. 20, 0. 45), MIN(0. 75, 0. 50)} = {0. 2, 0. 20, 0. 50} Dr Basil Hamed 30

Examples of Fuzzy Set Operations Dr Basil Hamed 31

Properties of Fuzzy Sets A B=B A A (B C) = (A B) C A (B C) = (A B) (A C) A A=A A X=X A =A A A=A A X=A A = If A B C, then A C A’’ = A Dr Basil Hamed 32

Fuzzy Sets Note (x) [0, 1] not {0, 1} like Crisp set A = { A(x 1) / x 1 + A(x 2) / x 2 + …} = { A(xi) / xi} Note: ‘+’ add ‘/ ’ divide Only for representing element and its membership. Also some books use (x) for Crisp Sets too. Dr Basil Hamed 33

Example (Discrete Universe) # courses a student may take in a semester. appropriate # courses taken 1 0. 5 0 2 4 6 Dr Basil Hamed x : # courses 8 34

Example (Discrete Universe) # courses a student may take in a semester. appropriate # courses taken Alternative Representation: Dr Basil Hamed 35

Example (Continuous Universe) U : the set of positive real numbers possible ages about 50 years old Alternative Representation: Dr Basil Hamed x : age 36

Alternative Notation U : discrete universe U : continuous universe Note that and integral signs stand for the union of membership grades; “ / ” stands for a marker and does not imply division. Dr Basil Hamed 37

Fuzzy Disjunction A B max(A, B) A B = C "Quality C is the disjunction of Quality A and B" • (A B = C) (C = 0. 75) Dr Basil Hamed 38

Fuzzy Conjunction A B min(A, B) A B = C "Quality C is the conjunction of Quality A and B" • (A B = C) (C = 0. 375) Dr Basil Hamed 39

Example: Fuzzy Conjunction Calculate A B given that A is. 4 and B is 20 Dr Basil Hamed 40

Example: Fuzzy Conjunction Calculate A B given that A is. 4 and B is 20 • Determine degrees of membership: Dr Basil Hamed 41

Example: Fuzzy Conjunction Calculate A B given that A is. 4 and B is 20 0. 7 • Determine degrees of membership: • A = 0. 7 Dr Basil Hamed 42

Example: Fuzzy Conjunction Calculate A B given that A is. 4 and B is 20 0. 9 0. 7 • Determine degrees of membership: • A = 0. 7 B = 0. 9 Dr Basil Hamed 43

Example: Fuzzy Conjunction Calculate A B given that A is. 4 and B is 20 0. 9 0. 7 • Determine degrees of membership: • A = 0. 7 B = 0. 9 • Apply Fuzzy AND • A B = min(A, B) = 0. 7 Dr Basil Hamed 44

Generalized Union/Intersection Generalized Union t-norm Or called triangular norm. Generalized Intersection t-conorm Or called s-norm. Dr Basil Hamed 45

T-norms and S-norms l l And/OR definitions are called T-norms (S-norms) l Duals of one another l A definition of one defines the other implicitly Many different ones have been proposed l Min/Max, Product/Bounded-Sum, etc. l Tons of theoretical literature l We will not go into this. Dr Basil Hamed 46

Examples: T-Norm & T-Conorm Minimum/Maximum: Lukasiewicz: Dr Basil Hamed 47

Classical Logic &Fuzzy Logic Hypothesis : 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 propositions P: a person is an engineer Q: a person is a mathematician R: a person is a logical thinker S: a person believes in magic The statements can now be expressed as algebraic propositions as ((P Q) (R S) (Q R)) (P S) Dr Basil Hamed 48

Fuzzy Relations … Dr Basil Hamed 49

Crisp Relation (R) b 1 b 2 b 3 b 4 b 5 a 1 A a 2 a 3 a 4 Dr Basil Hamed B 50

Crisp Relation (R) b 1 b 2 b 3 b 4 b 5 a 1 A a 2 a 3 a 4 Dr Basil Hamed B 51

Crisp Relations Example: If X = {1, 2, 3} Y = {a, b, c} R = { (1 a), (1 c), (2 a), (2 b), (3 c) } R= 1 2 3 a 1 1 0 b 0 1 1 c 1 0 1 Using a diagram to represent the relation Dr Basil Hamed 52

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 Dr Basil Hamed 53

Fuzzy Relations Triples 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 … Dr Basil Hamed 54

Fuzzy Relations Matrices Example: Color-Ripeness relation for tomatoes R 1(x, y) unripe semi ripe green 1 0. 5 0 yellow 0. 3 1 0. 4 Red 0 0. 2 1 Dr Basil Hamed 55

Composition Let 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. Dr Basil Hamed 56

Composition If R is a fuzzy relation on the space X x Y S is a fuzzy relation on the space Y x Z Then, fuzzy composition is T = R S There are two common forms of the composition operation: 1. Fuzzy max-min composition T(xz) = ( R(xy) s(yz)) y Y 2. Fuzzy max-production composition T(xz) = ( R(xy) s(yz)) y Y Note: R S S R multiplication Dr Basil Hamed 57

Max-Min Composition X Y Z R: fuzzy relation defined on X and Y. S: fuzzy relation defined on Y and Z. R 。S: the composition of R and S. A fuzzy relation defined on X an Z. Dr Basil Hamed 58

Example min max Dr Basil Hamed 59

. Max-Product Composition X Y Z R: fuzzy relation defined on X and Y. S: fuzzy relation defined on Y and Z. R。S: the composition of R and S. A fuzzy relation defined on X an Z. Dr Basil Hamed 60

Example Product max. 09 . 04 0. 0 0. 4 R S 1 0. 4 0. 2 0. 3 2 0. 27 0. 3 0. 24 3 0. 8 0. 9 0. 7 Dr Basil Hamed 61

Properties of Fuzzy Relations Example: R = y 1 y 2 x 1 0. 7 0. 5 x 2 0. 8 0. 4 z 1 z 2 z 3 S = y 1 0. 9 0. 6 0. 2 y 2 0. 1 0. 7 0. 5 z 1 z 2 z 3 Using max-min, T = x 1 0. 7 0. 6 0. 5 x 2 0. 8 0. 6 0. 4 Using max-product, T = z 1 z 2 z 3 x 1 0. 63 0. 42 0. 25 x 2 0. 72 0. 48 0. 20 Dr Basil Hamed 62

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. Dr Basil Hamed 63

Example 3. 8 Initially, 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 = k. Nφ, 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. Dr Basil Hamed 64

Example 3. 8 Let 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. 8 Let 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. Dr Basil Hamed 66

Example 3. 8 The operations needed to develop these relations are as follows – two fuzzy Cartesian products and one composition: Dr Basil Hamed 67

Example 3. 8 Suppose 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, Dr Basil Hamed 68