Fakultät ETIT Institut für Automatisierungstechnik, Professur für Prozessleittechnik. Dr. Engin YEŞİL. Introduction to Fuzzy Modeling & Control

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1 Fakultät ETIT Institut für Automatisierungstechnik, Professur für Prozessleittechnik Dr. Engin YEŞİL Introduction to Fuzzy Modeling & Control

2 1998- Istanbul Technical University Faculty of Electrical and Electronics Engineering, Control Engineering Department Programs of our Department: Control Engineering (Turkish) Control Engineering (English) NEW- Control Engineering + Michigan State Uni. NEW- PostDoc ( ) Fuzzy Modeling & Control (SS 2010) Wednesday; 4.DS; BAR/213/H Folie 2

3 Dr. Engin YESIL Room: BAR 277 Tel: Webpage: Office hours: Anytime (just before!!!) Exam ( 20%) Folie 3

4 FUZZY SETS & FUZZY LOGIC FUZZY SYSTEMS FUZZY PD CONTROLLER Folie 4

5 Evolution of development 1965 Seminal Papers Fuzzy Sets by Prof. Lotfi Zadeh, U.C. Berkeley. Folie 5

6 Why Fuzzy? fuzzy adj. 1 unclear or confused and lacking details. 2 not clearly thought out or expressed. 3 indistinct, unclear, distorted or imprecise. Folie 6

7 Cantor: Set Theory at the end of 19th century. (CRISP SETS) Sanders Peirce ( ): Uncertainty theory Bertrand Russel ( ): "All language is vague Jan Lukasiewicz ( ): Many-valued logic Max Black ( ) proto-fuzzy sets Lotfi Zadeh Founder of fuzzy sets & logic Folie 7

8 1975 Fuzzy control was first introduced by E. Mamdani in Advances in the linguistic synthesis of fuzzy controllers Introduction of Fuzzy Logic in Japan 1980 Empirical Verification of Fuzzy Logic in Europe 1985 Broad Application of Fuzzy Logic in Japan 1990 Broad Application of Fuzzy Logic in Europe 1995 Broad Application of Fuzzy Logic in the U.S Fuzzy logic becomes a standard technique for multivariable control and is also applied in data and sensor signal analysis. Application of Fuzzy Logic in business and finance. Folie 8

9 Rice Cooker Washing Machine Folie 9

10 Automatic gear shift Folie 10

11 Recovery Boiler Fuzzy Logic Control Folie 11

12 Folie 12

13 Propositions, Sets and Characteristic Functions In classical logic a proposition is a statement that is either true or false. A proposition is represented by a set, a collection of elements that share a common property. These elements are referred to as members of the set. A set can be specified by specifying its members, i.e., a set A is the set of all elements x in U that have the property P. We write Folie 13

14 Examples: Set of natural numbers smaller than 5: A = {0,1, 2, 3, 4} Unit disk in the complex plane: A = {z z C, z 1} A line in R 2 : A = {(x, y) ax + by + c = 0, (x, y, a, b, c) R} Folie 14

15 Propositions, Sets and Characteristic Functions Another way to characterize a set is to use a characteristic function, defined by where we have introduced the truth value truth value 1 for a true statement, and 0 for a false statement. Folie 15

16 Propositions, Sets and Characteristic Functions There are three special sets; 1. the universal set (or universe of discourse), U, which contains all elements 2. the empty set,, that contains no elements 3. the singleton set,, that contains only one element. Folie 16

17 Operations on sets To derive the set representation of a compound proposition, the set operations corresponding to the logic connectives and, or and not must be defined. These operations are called intersection, union and complement respectively. Let A and B be two sets defined on the same universe of discourse, U. Folie 17

18 Operations on sets We can then define our three basic set operations as follows: Folie 18

19 EXAMPLE? Folie 19

20 Operations on sets The following formulas give the characteristic functions resulting from the set operations above. Folie 20

21 Why Fuzzy Sets? Classical sets are good for well-defined concepts (maths, programs, etc.) Less suitable for representing commonsense knowledge in terms of vague concepts such as: a tall person, slippery road, nice weather,... want to buy a big car with moderate consumption If the temperature is too low, increase heating a lot Folie 21

22 Classical Set Approach set of tall people A = {h h 180} Logical Propositions Engin is tall... true or false? Engin s height: h Engin = μ A (180.0) = 1 (true) h Engin = μ A (179.5) = 0 (false) Folie 22

23 Fuzzy Set Approach Folie 23

24 Fuzzy Logic Propositions Engin is tall... degree of truth Engin s height: h Engin = μ A (180.0) = 0.6 h Engin = μ A (179.5) = 0.56 h Engin = μ A (201.0) = 1 Folie 24

25 Subjective and Context Dependent Tall in China Tall in Europe Tall in NBA Folie 25

26 Can you give any more example? Folie 26

27 Expensive Cars: U= {Bugatti, Ferrari, BMW, Porsche, Mercedes, Ford, FIAT, Rolls-Royce, Lamborghini} Expensive Cars 1 Bugatti 0 FIAT 1 Ferrari 0.4 BMW 0.7 Rolls - Royce 0.9 Porsche 0.65 Lamborghini 0.7 Mercedes 0.1 Ford Folie 27

28 Shapes of Membership Functions A fuzzy set, A, is a set whose characteristic function μ A (a) takes values in the interval [0, 1]. In fuzzy logic literature, the characteristic function of a fuzzy set is always called the membership function of the fuzzy set. We interpret the membership value of an element to a specific set as the degree to which the corresponding proposition applies. Folie 28

29 The trapezoidal membership function: Trapezoidal membership functions with varying widths and centers (left), and varying slopes (right). Folie 29

30 The triangular membership function: Triangular membership function with varying centers (left) and slopes (right). Folie 30

31 The Gaussian membership function: Gaussian membership functions for different :s (left) and different x centers (right). Folie 31

32 The sigmoidal membership function: determines the crossover point of the membership function (the element whose membership value equals 0.5), and controls the slope of the function at this point. Sigmoidal membership functions for different :s (left) and different :s (right). Folie 32

33 The singleton membership function: Folie 33

34 Linguistic Variable 27 Is 27 C high? Folie 34

35 Support of a Fuzzy Set supp(a) = {x μ A (x) > 0} support is an ordinary set Folie 35

36 Core (Kernel) of a Fuzzy Set core(a) = {x μ A (x) = 1} core is an ordinary set Folie 36

37 -cut of a Fuzzy Set A α = {x μ A (x) α} A α is an ordinary set Folie 37

38 Convex and Non-Convex Fuzzy Sets A fuzzy set is convex all its α-cuts are convex sets. A = {x μ A (x) } Folie 38

39 Non-Convex Fuzzy Set: an Example High-risk age for car insurance policy. Folie 39

40 Fuzzy Numbers and Singletons Fuzzy linear regression: Folie 40

41 Fuzzy Set Operations In fuzzy logic the classical set operations are extended to produce an intuitively correct result on fuzzy sets. This is done through the introduction of T-norms and S-norms. It is important to notice that a compound set is formed by applying the set operations point-wise over all elements on the universe of discourse. Folie 41

42 INTERSECTION OPERATOR Folie 42

43 UNION OPERATOR Folie 43

44 COMPLEMENT OPERATOR Folie 44

45 Linguistic Modifiers (Hedges) Modify the meaning of a fuzzy set. For instance, very can change the meaning of the fuzzy set tall to very tall. Other common hedges: slightly, more or less, rather, etc. Usual approach: powered hedges: Folie 45

46 Folie 46

47 Fuzzy Logic Folie 47

48 Example? NOT VERY YOUNG AND NOT OLD! Folie 48

49 Fuzzy Set in Multidimensional Domains Folie 49

50 Cylindrical Extension Projection onto X 1 Projection onto X 2 Folie 50

51 Intersection on Cartesian Product Space An operation between fuzzy sets are defined in different domains results in a multi-dimensional fuzzy set. Example: A 1 A 2 on X 1 X 2 : Folie 51

52 Fuzzy Relations Classical relation represents the presence or absence of interaction between the elements of two or more sets. With fuzzy relations, the degree of association (correlation) is represented by membership grades. An n-dimensional fuzzy relation is a mapping R : X 1 X 2 X 3... X n [0, 1] which assigns membership grades to all n-tuples (x 1, x 2,..., x n ) from the Cartesian product universe. Folie 52

53 Fuzzy Relations: Example Example: R : x y ( x is approximately equal to y ) Folie 53

54 Relational Composition Given fuzzy relation R defined in X Y and fuzzy set A defined in X, derive the corresponding fuzzy set B defined in Y : max-min composition: Analogous to evaluating a function. Folie 54

55 crisp function interval function fuzzy function Folie 55

56 Guessing Game: LEON (a fictitious name) is nearsighted and colorblind. When he goes to a local grocery where fruits are laced on high shelves, he cannot see them very well. He can only recognize the size and blurred shape of the fruits. He has lived in such a world for some 20 years and now he is a houseman, and he has some knowledge about the features of the fruits. For example, tangerines are round and relatively small. Folie 56

57 Fruit={tangerine, apple, pineapple, watermelon, strawberry} Shape={long, round, large} tangerine apple pineapple watermelon strawberry long round large Let s guess a fruit that LEON sees. If he recognizes a fruit that is round and big and if interpret this as long round large Folie 57

58 o tangerine apple pineapple watermelon strawberry = From this result, the possibility of watermelon is the highest and tangerine, apple, and pineapple come next at an equal possibility. Folie 58

59 If LEON recognizes another fruit as relatively long, somewhat round, and not very large and if we can interpret his observation as long round large Answer: tangerine apple pineapple watermelon strawberry Folie 59

60 Folie 60