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

Content Crisp Sets Fuzzy Sets Set-Theoretic Operations Fuzzy Relations Dr Basil Hamed

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

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

Crisp Set and Fuzzy Set Dr Basil Hamed

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] Dr Basil Hamed

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

Fuzziness Examples: He/she is tall Dr Basil Hamed

Classical Sets Dr Basil Hamed

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

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

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

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

Properties of Classical Sets A  B = B  A A  B = B  A A  (B  C) = (A  B)  C A  (B  C) = (A  B)  C A  (B  C) = (A  B)  (A  C) A  (B  C) = (A  B)  (A  C) A  A = A A  A = A A  X = X A  X = A A   = A A   =  Dr Basil Hamed

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

Fuzzy Sets Dr Basil Hamed

Fuzzy Sets 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. Dr Basil Hamed

Fuzzy Set Theory 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 Dr Basil Hamed

U : universe of discourse. 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 U : universe of discourse. Dr Basil Hamed

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! Dr Basil Hamed

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

Fuzzy Set Operations A  B(x) = A(x)  B(x) = max(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 general A  A’ A  A’ Dr Basil Hamed

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

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

Operations A B A  B A  B A Dr Basil Hamed

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

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). Dr Basil Hamed

Set-Theoretic Operations Dr Basil Hamed

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} Dr Basil Hamed

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} Dr Basil Hamed

Examples of Fuzzy Set Operations A = {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: 𝐴 , = {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

Properties of Fuzzy Sets A  B = B  A A  B = B  A A  (B  C) = (A  B)  C A  (B  C) = (A  B)  C A  (B  C) = (A  B)  (A  C) A  (B  C) = (A  B)  (A  C) A  A = A A  A = A A  X = X A  X = A A   = A A   =  If A  B  C, then A  C A’’ = A Dr Basil Hamed

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. Dr Basil Hamed

Example (Discrete Universe) # courses a student may take in a semester. appropriate # courses taken 0.5 1 2 4 6 8 x : # courses Dr Basil Hamed

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

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

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

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

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

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

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

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

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

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

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

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. Dr Basil Hamed

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

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

Fuzzy Relations … Dr Basil Hamed

Crisp Relation (R) A a1 a2 a3 a4 B b1 b2 b3 b4 b5 Dr Basil Hamed

Crisp Relation (R) A a1 a2 a3 a4 B b1 b2 b3 b4 b5 Dr Basil Hamed

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 c 1 1 0 1 R = 2 1 1 0 3 0 1 1 Using a diagram to represent the relation Dr Basil Hamed

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

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

Fuzzy Relations Matrices Example: Color-Ripeness relation for tomatoes R1(x, y) unripe semi ripe ripe green 1 0.5 yellow 0.3 0.4 Red 0.2 Dr Basil Hamed

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

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: Fuzzy max-min composition T(xz) =  (R(xy)  s(yz)) 2. Fuzzy max-production composition T(xz) =  (R(xy)  s(yz)) Note: R  S  S  R multiplication y  Y y  Y Dr Basil Hamed

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 and Z. Dr Basil Hamed

Example min max Dr Basil Hamed

Max-Product Composition . 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

Example Product max .09 .04 0.0 0.4 R  S    1 0.4 0.2 0.3 2 0.27 .09 .04 0.0 0.4 R  S    1 0.4 0.2 0.3 2 0.27 0.24 3 0.8 0.9 0.7 Dr Basil Hamed

Properties of Fuzzy Relations Example: y1 y2 z1 z2 z3 R = x1 0.7 0.5 S = y1 0.9 0.6 0.2 x2 0.8 0.4 y2 0.1 0.7 0.5 z1 z2 z3 Using max-min, T = x1 0.7 0.6 0.5 x2 0.8 0.6 0.4 z1 z2 z3 Using max-product, T = x1 0.63 0.42 0.25 x2 0.72 0.48 0.20 Dr Basil Hamed

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

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 = 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. Dr Basil Hamed

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

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

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

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

Example 3.8 The following relation then result from use of the Cartesian product to determine R: Dr Basil Hamed

Example 3.8 Cartesian product to determine S: Dr Basil Hamed

Example 3.8 The following relation results from a max–min composition for T: Dr Basil Hamed

HW 1 2.4, 2.5,2.7, 2.11, 3.2, 3.4, 3.8 Due 2/ 10/ 2017 Good Luck Dr Basil Hamed