OUTLINE. Introduction History and basic concepts. Fuzzy sets and fuzzy logic. Fuzzy clustering. Fuzzy inference. Fuzzy systems. Application examples

Transcription

1 OUTLINE Introduction History and basic concepts Fuzzy sets and fuzzy logic Fuzzy clustering Fuzzy inference Fuzzy systems Application examples

2 "So far as the laws of mathematics refer to reality, they are not certain And so far as they are certain, they do not refer to reality" Albert Einstein Geometry and Experience

3 Fuzzy Logic Mathematical formalism for representing imprecise knowledge in a human like way Knowledge representation Mathematically formal Deals with imprecision

4 some history Multi valued logic Jan Lukasiewicz Three valued logic (1920) Other multi valued logics: Kurt Gödel, John von Neumann, Donald Kleene Proto fuzzy Sets Max Black Vagueness, a logic analysis (1937) Fuzzy Logic Lotfi Zadeh Fuzzy Sets (1965)

5 Jan Lukasiewicz Three valued logic (1920) 0 = False 1/2 = Possible 1 = True Sentence: It is possible that tomorrow will rain Negation: It is possible that tomorrow will not rain Paradoxes: This sentence is false True or false?

6 Max Black Proto fuzzy Sets (1937) Bald Hairy 1E3 Number of hairs 1E5 Falakros paradox: (belongs to the "sorites paradox" class) Would you describe a man with one hair on his head as bald? Yes Would you describe a man with two hairs on his head as bald? Yes You must refrain from describing a man with ten thousand hairs on his head as bald, so where do you draw the line?

7 Max Black s sets vs actual fuzzy sets Bald Hairy 1E3 Number of hairs 1E5 Based on degree of usage Based on (subjective) degree of truth

8 Lotfi Zadeh: Father of modern fuzzy logic (1965) Key concept: Partial membership ( µ ) "Tall" membership Height Crisp Fuzzy

9 Fuzzy numbers and fuzzy quantities about 4 kg of beans between 5 and 8 weeks As they assign a membership value µ to a given real value x, they are called membership functions

10 Fuzzy set operations Emptiness: Sets with no members (ie x, µ (x)=0 ) Complement: How much do items not belong Containment: What groups belong to other groups Intersection: How much are items in BOTH sets Union: How much are items in EITHER sets A

11 Containment: What groups belong to other groups ie each element on the subset belong to the larger set B C A iff x, (x) <= (x) µ B µ A

12 Complement: How much do items not belong? ie how far are them from full membership? µ F F(x) = 1 µ (x)

13 Intersection: How much are items in both sets? ie the degrees of membership both sets share µ A B C µ B µ A (x) = min{ (x), (x)}

14 Union: How much are items in either set? CµA B µ B µ A (x) = max{ (x), (x)}

15 Equivalence between sets and logic Sets Membership John s height belongs to "Tall" Complement Logic Truth John is tall Negation Intersection And Union Or

16 1 D vs 2 D fuzzy logic operations (1) 1 D: Temperature is cool AND warm

17 1 D vs 2 D fuzzy logic operations (2) 2 D: Temperature is Cool AND Pressure is High Temperature is Warm and Pressure is Ok

18 Fuzzy Clustering Cluster analysis: To partition a given set of data into clusters having the following properties: Intra cluster homogeneity Inter cluster heterogeneity Data in the same cluster, Data in different clusters, as similar as possible as different as possible

19

20

21

22 How do humans explain their decisions? * By enumerating some (partially) fullfilled conditions Why did you choose this car? It is big enough for my family, but not too much Its gas consumption is not high, but the car is still fast I prefer clear colors, but not white This mark is well known

23 How do humans explain their decisions? * By proposing some (fuzzy) behavior rules How do you decide to push or to release your car s accelerator? Release if car is fast(er than desired) and speed is constant or rising, push if car is slow(er) and speed is constant or slowing down; else do nothing Fast Ok Slow Slowing Constant Rising N R R N N N P P N

24 Labels: Membership functions: Linguistic or "fuzzy" variables Name: The "thing" being qualified (eg speed) "Linguistic" values or adjectives (eg slow, fast) assign membership values to real, measured values Slow Ok Fast 20% Speed +15%

25 Fuzzy Inference System Knowledge base Database Rule base Fuzzifier Inference Engine Defuzzifier Carlos Andres Pena Reyes

26 Inverted Pendulum Problem Θ w State of the System S = {x,v, θ,w} v F = {-10,10} x ECOLE POLYTECHNIQUE FEDERALE DE LAUSANNE ANDRES PEREZ URIBE - LSL -EPFL

27 Inverted Pendulum Control: Rules IF angle is zero AND angular velocity is zero THEN speed is zero IF angle is zero AND angular velocity is pos_low THEN speed is pos_low IF angle is neg_low AND angular velocity is zero THEN speed is neg_low speed ang velocity NH NL Z PL PH angle NH NL Z PL PH NH NL Z NH NL Z PL PH Z PL PH Carlos Andres Pena Reyes

28 Inverted Pendulum Control: Variables Input1: Angular velocity Input2: Angle Output: Speed Carlos Andres Pena Reyes

29 Inverted Pendulum Control: Input measures Carlos Andres Pena Reyes

30 Inverted Pendulum Control: Inference 1 (Fuzzyfication) IF angle is zero AND angular velocity is zero THEN speed is zero Carlos Andres Pena Reyes

31 Inverted Pendulum Control: Inference 2 (Fuzzyfication) IF angle is zero AND angular velocity is zerothen speed is zero Carlos Andres Pena Reyes

32 Inverted Pendulum Control: Inference 3 (Implication) IF angle is zero AND angular velocity is zero THEN speed is zero Carlos Andres Pena Reyes

33 Inverted Pendulum Control: Inference 4 (Aggregation) Carlos Andres Pena Reyes

34 Inverted Pendulum Control: Inference 5 (Defuzzyfication) Carlos Andres Pena Reyes

35 Inverted Pendulum Control: Response Carlos Andrés Peña-Reyes Logic Systems Laboratory - Swiss Federal Institute of Technology Lausanne

36 Fuzzy Inference System Knowledge base Database Rule base Fuzzifier Inference Engine Defuzzifier Small Big Slow Angle Fast Angle Small Small Big Big Ang vel Speed Slow Zero Fast Low Slow High Fast Zero Zero Low High z Z Ang vel Carlos Andrés Peña-Reyes Logic Systems Laboratory - Swiss Federal Institute of Technology Lausanne

37 3 types of fuzzy rule consequents IF angle is zero AND angular velocity is pos_low THEN speed is pos_low pos_low Mamdani type: membership functions Sugeno type: linear function of inputs pos_low = 01*angle + 02* velocity Singleton type constant values pos_low = 10

38 Fuzzy temperature controller Temperature High Wished Low Pressure Low Correct High NS Z PB NB Z PS NB NB NB Low Temperature Wished High Low Pressure Correct High

39 Mamdani-type fuzzy temperature controller NB NS Z PS PB Carlos Andrés Peña-Reyes Logic Systems Laboratory - Swiss Federal Institute of Technology Lausanne

40 Sugeno-type fuzzy temperature controller PB = -01T +1 PS = -001T + 05 Z = 0 NS = -001T - 01P -05 NB = -01T -05P -1 Carlos Andrés Peña-Reyes Logic Systems Laboratory - Swiss Federal Institute of Technology Lausanne

41 Singleton-type fuzzy temperature controller NB NS Z PS PB Carlos Andrés Peña-Reyes Logic Systems Laboratory - Swiss Federal Institute of Technology Lausanne

42 Comparison among the three systems Carlos Andrés Peña-Reyes Logic Systems Laboratory - Swiss Federal Institute of Technology Lausanne