Chapter 7 Fuzzy Systems Concepts and Paradigms

1. Chapter 7Fuzzy Systems Concepts and Paradigms

2. “Fuzzification is a kind of scientific permisiveness; it tends to result in socially appealing slogans unaccompanied by the discipline of hard work.” R. E. Kalman, 1972

3. Fuzzy Logic History: Buddha vs. Aristotle • Should fuzzy logic history start with Gautama Buddha (b. 563 BC)? • Shades of gray • Contradictions and paradoxes • x is not-x • Aristotle (b. ~350 BC) • Crisp logic • 1 or 0 • True or false • Aristotle has dominated Western thought for >2,000 years • Probability overlays logic • Axioms derived from assumptions • Math and science built on logic

4. Jan Lukasiewicz *First paper on 3-valued logic in 1920 * Described many-valued logic in 1923 book > Wrote of a “bundle of many-valued logics” and said, “Symbols other than 0 and 1...would thus correspond to the various degrees of truth...” > Established that each theorem of 3-valued logic is also a theorem of 2-valued logic * Others (including Godel and von Neumann) also developed multi-valued logics

5. Max Black • Worked at Cornell • * Recognized that a continuum implies vagueness, and that vagueness • has degrees • * Assigned numbers to objects based on the degree to which it was • perceived to belong to the class • * “Degree” was defined as the percent of people who would assign • the object to the class or category • * Membership was thus linked with probability...a different concept • than is used now

6. Early Computer Deficiencies It almost immediately turned out, however, that computers did not live up to expectations. … These “brilliant” machines weren’t very good at solving real problems, problems having to do with real people and real business, and things with moving parts. It seemed that no matter how many variables were added to the decision process, there was always something else. Systems didn’t work the same when they were hot, or cold, or stressed, or dirty, or cranky, or in the light or in the dark, or when two things went wrong at the same time. There was always something else. The problem was that the computer was unable to make accurate inferences. It couldn’t very well tell what would happen, given some preconditions, no matter how precisely specified they were.

7. Early Computer Deficiencies, Continued • So, early computers: • Had trouble solving “real-world” problems • Computer always seemed to need more information • Couldn’t handle unexpected conditions (brittle) or multiple faults • Major problem: inability to make accurate inferences

8. Lotfi Zadeh • The single most significant developer and champion of fuzzy logic • theory and applications • Born in Baku in Soviet Azerbaijan in 1921 • MSEE MIT 1946, Ph.D. Columbia 1949, moved to Cal. Berkeley in 1959 • Significant contributions to field of systems theory • Text on linear systems theory in 1963

9. Lotfi Zadeh, Cont’d. • “Fuzzy Sets” paper published in 1965 • Comprehensive - contains everything needed to implement FL • Key concept is that of membership values: extent to which an • object meets vague or imprecise properties • Membership function: membership values over domain of interest • Fuzzy set operations • Endured criticism with grace • Awarded the IEEE Medal of Honor in 1995

10. Ebrahim Mamdani • First fuzzy control system, work done in 1973 with Assilian (1975) • Developed for boiler-engine steam plant • 24 fuzzy rules • Developed in a few days • Laboratory-based • Served as proof-of-concept

11. Early European Researchers • Hans Zimmerman, Univ. of Aachen • Founded first European FL working group in 1975 • First Editor of Fuzzy Sets and Systems • First President of Int’l. Fuzzy Systems Association • Didier Dubois and Henri Prade in France • Charter members of European working group • Developed families of operators • Co-authored a textbook (1980)

12. Early U. S. Researchers • K. S. Fu (Purdue) and Azriel Rosenfeld (U. Md.) (1965-75) • Encouraged students • Worked during FL’s “unpopular” period • Enrique Ruspini at SRI • Theoretical FL foundations • Developed fuzzy clustering • Chaired 2nd Fuzz/IEEE Conference in 1993 • James Bezdek, Univ. of West Florida • Developed fuzzy pattern recognition algorithms • Proved fuzzy c-means clustering algorithm • Combined fuzzy logic and neural networks • Chaired 1st Fuzz/IEEE Conf. in 1992 and others • President of IEEE NNC 1997-1999

13. Holmblad and Ostergaard • First industrial application late 1970s in Denmark • Control system for cement kiln • Similar systems in Sweden and elsewhere

14. The Dark Age • Lasted most of 1980s • Funding dried up, in US especially • As recently as 1991, • “...Fuzzy logic is based on fuzzy thinking. It fails to distinguish • between the issues specifically addressed by the traditional methods of logic, definition and statistical decision-making...” • - J. Konieki (1991) in AI Expert • Symbolics ruled: “fuzzy” label amounted to the ‘kiss of death’ • In Japan, however, faaji was welcomed and implemented

15. Michio Sugeno • Secretary of Terano’s FL working group, est. in 1972 • 1974 Ph.D. dissertation: fuzzy measures theory • Worked with Mamdani in UK • First commercial application of FL in Japan: control system for water purification plant (1983)

16. Other Japanese Developements • 1st consumer product: shower head using FL circuitry to control temperature (1987) • Fuzzy control system for Sendai subway (1987) • 2d annual IFSA conference in Tokyo was turning point for FL (1987) • Laboratory for Int’l. Fuzzy Engineering Research (LIFE) founded in Yokohama with Terano as Director, Sugeno as Leading Advisor in 1989.

17. Fuzzy Systems Theory and Paradigms * Variation on 2-valued logic that makes analysis and control of real (non-linear) systems possible * Crisp “first order” logic is insufficient for many applications because almost all human reasoning is imprecise * We will discuss fuzzy sets, approximate reasoning, and fuzzy logic issues and applications

18. Fuzzy versus Crisp • Fuzzy logic comprises fuzzy sets and approximate reasoning • * A fuzzy “fact” is any assertion or piece of information, and can have • a “degree of truth”, usually a value between 0 and 1 • * Fuzziness: “A type of imprecision which is associated with ... classes • in which there is no sharp transition from membership to non- • membership” - Zadeh (1970)

19. Fuzziness is not probability • Probability is used, for example, in weather forecasting • Probability is a number between 0 and 1 that is the certainty that an • event will occur • The event occurrence is usually 0 or 1 in crisp logic, but fuzziness • says that it happens to some degree • Fuzziness is more than probability; probability is a subset of fuzziness • Probability is only valid for future/unknown events • Fuzzy set membership continues after the event

20. Probability • * Probability is based on a closed world model in which it is • assumed that everything is known • Probability is based on frequency; Bayesian on subjectivity • Probability requires independence of variables • In probability, absence of a fact implies knowledge • Probability goes away once the event is observed • If you have everything you need to develop a probabilistic system, • do so...it may be the best approach.

21. Fuzzy Logic • FL is not based on a closed world model (we don’t assume • everything is known) • FL (& crisp logic) state objective descriptions/measures • FL does not require statistical independence of variables • In FL, the absence of a fact doesn’t imply anything • A FL membership value persists after observation • The more complex a system is, the more it involves intelligent • behavior, the more likely it is that fuzzy logic will provide • a good approach.

22. A Pair of Bottles for the Weary Traveler(from Bezdek) Potable (drinkable)

23. The Pair of Bottles Unmasked(from Bezdek)

24. Crisp Logic Venn Diagrams A point is either in the set or not; it’s either in the intersection or not.

25. Set Membership • In fuzzy logic, set membership occurs by degree • Set membership values are between 0 and 1 • Consider the set tall American male professional basketball players (TAMPBP) Shaquille O’Neal (7’ 1”), and Travis Best (5’ 11”) • We might assign TAMPBP = 1.0/O’Neal, 0.1 Best • What about Reggie Miller, who is 6” 7”? Perhaps he is a 0.6 member. • We can now reason by degree, and apply logical operations to fuzzy sets • We usually write . or, the membership value of x in the fuzzy set A is m, where

26. Fuzzy Set Membership Functions * Fuzzy sets have “shapes”: the membership values plotted versus the variable * Fuzzy membership function: the shape of the fuzzy set over the range of the numeric variable > Can be any shape, including arbitrary or irregular > Is normalized to values between 0 and 1 > Often uses triangular approximations to save computation time

27. Fuzzy Sets Are Membership Functions from Bezdek

28. Representations of Membership Functions

29. Two Types of Fuzzy Membership Function

30. Linguistic Variables * Linguistic variable: “a variable whose values are words or sentences in a natural or artificial language.” – Zadeh * Linguistic variables translate ordinary language into logical or numerical statements * Imprecision of linguistic variables makes them useful for reasoning

31. Linguistic Variable Categories • Quantification terms: all, most, many, about one-fourth, some • Usuality terms: always, sometimes, seldom, never • Likelihood terms: certain, likely, possible, certainly not

32. Hedges * Linguistic variables (LVs) can modify or qualify one another * Hedges: LVs that change the shape or position of a membership function * “Very” and “sort of” are examples that can shift membership functions in opposite directions (i.e., tall), or can change the membership function width (i.e., medium).

33. Kinds of Hedges * Intensify a fuzzy set (very, extremely) * Dilute a fuzzy set (somewhat, sort of) * Express probabilities (probably, not likely) * Approximate a scalar or single number (exactly) * Express vague quantities (most, seldom)

34. Implementing Hedges * “Very” can be the mathematical square (.5 tall --> .25 very tall) * “Somewhat” can be the square root (.81 tall --> .90 somewhat tall) * Other conventions are possible

35. Approximate Reasoning * Fuzzy reasoning involves different processes than binary logic * Relations and operators have similar names (AND, OR, etc.) but have different meanings * In Aristotlian logic: Law of Noncontradiction: (intersection same as ‘and’) This law means “A” can’t simultaneously be true and false. Law of Excluded Middle: (union same as ‘or’) This law means “A” must be either true or false. * Neither law holds in (is relevant to) fuzzy logic

36. Equality of Fuzzy Sets * In traditional logic, sets containing the same members are equal: {A,B,C} = {A,B,C} * In fuzzy logic, however, two sets are equal if and only if all elements have identical membership values: {.1/A,.6/B,.8C} = {.1/A,.6/B,.8/C}

37. Fuzzy Containment * In traditional logic, if and only if all elements in A are also in B. * In fuzzy logic, containment means that the membership values for each element in a subset is less than or equal to the membership value of the corresponding element in the superset. * Adding a hedge can create a subset or superset.

38. Fuzzy Complement * In traditional logic, the complement of a set is all of the elements not in the set. * In fuzzy logic, the value of the complement of a membership is (1 - membership_value) Remember that the law of the excluded middle doesn’t hold! “Reality flourishes on ambiguity.” – L. Zadeh

39. Fuzzy Intersection * In standard logic, the intersection of two sets contains those elements in both sets. * In fuzzy logic, the weakest element determines the degree of membership in the intersection (Law of Noncontradiction does not hold)

40. Fuzzy Union * In traditional logic, all elements in either (or both) set(s) are included * In fuzzy logic, union is the maximum set membership value

41. Summary of Fuzzy Relations and Operators

42. Alternative Fuzzy Operators • Operators other than those defined above can be used. • * For example, intersection can be defined as the product of • the membership values, and union can be the sums of the • membership values.

43. Compensatory Operators These are alternatives to Zadeh’s operators. Consider the membership values .9, .8, .6, .1; the Zadeh AND (intersection) is .1, which may be a little too extreme for these values Mean Operator: Intersection is defined as the average (mean) of the membership values. For the above values, the mean operator gives 0.6 as the intersection (AND) value. We implement the mean operator in our software, in addition to the traditional Zadeh operators.

44. Compensatory Operators, Cont’d. Gamma Operator: where ,and m is the number of fuzzy membership values. , then If .

45. Fuzzy Rules Have antecedent part and consequent part Mamdani-type fuzzy rule: If X1 is A1 and ... and Xn is An then Y is Bj TSK-type fuzzy rule: If X1 is A1 and ... and Xn is An then Y = p0 + p1 X1 + ... + pn Xn The default type of rule we use is the Mamdani type.

46. Fuzzification • Fuzzification is the combining of the antecedent sets (the if- side of the rule) • We use a gas flow regulator for a furnace as an example. • Input parameters: indoor temp., outdoor temp., 5-min. delta temp. • Output parameter: change in gas flow

47. Fuzzy Set Definition Now we must define the fuzzy sets over each parameter. We decide to use triangular membership functions. First decide how many sets per parameter, then decide range for each: For the InTemp parameter, we define three fuzzy sets: cool, comfortable, and too_warm. For OutTemp, we have five fuzzy sets defined: very_cold, chilly, warm, very_warm, and hot. For DeltaInTemp, we define five fuzzy sets: large_negative, small_negative, near_zero, small_positive, and large_postive. For our output parameter FlowChange, we define five fuzzy sets: decrease_greatly, decrease_small, no_change, increase_small, and increase_greatly. Number of membership functions for a parameter depends on the situation.

48. Define the Rule Set Possible rules: Rule 1: If InTemp is comfortable and DeltaInTemp is near_zero, then FlowChange is no_change. Rule 2: If OutTemp is chilly and DeltaInTemp is small_negative, then FlowChange is increase_small. Rule 3: If InTemp is too_warm and DeltaInTemp is large­_positive, then FlowChange is decrease_greatly. Rule 4: If InTemp is cool and DeltaInTemp is near_zero, then FlowChange is increase_small. …and so on.

49. Define the Membership Functions (We define only those we need for our example rules.) For inside temperature (intemp):

50. Define the Membership Functions, Cont’d. For DeltaInTemp: For OutTemp: (These are all we need for our rules.)