EngineeringEducationMethodology on Intelligent Control Fuzzy Logic and Fuzzy

Engineering　Education　Methodology on Intelligent Control (Fuzzy Logic and Fuzzy Control) M. Yamakita Dept. of Mechanical and Control Systems Eng. Tokyo Inst. Of Tech. 11/22/2020 1

Natural Reasoning IF he/she is Tall, THEN his/her foot is Big. IF his/her foot is Big, THEN his/her shoe’s are Expensive. IF he is Tall, THEN his shoe’s are Expensive. IF he/she is Tall, THEN his/her foot is Big. Mr. Smith is Tall. Mr. Smith’s foot is Big. 11/22/2020 2

Crisp Expert System Inference (Reasoning) Condition part (Antecedent part) Conclusion (Operation part) A　→　B B　→　C A　→　C Formal Logic IF　A　THEN　B IF　B　THEN　C IF　A　THEN　C Implication operator Hypothetical　Syllogism 11/22/2020 A　→　B A B IF　A　THEN B A　is true B is true Modus　Ponens 3

Crisp Logic Tall 181ｃｍ （] Mr. A （] Mrs. B １７０ｃｍ 177ｃｍ Short 11/22/2020 ( 4

IF he is Tall, THEN his foot is Big. Mrs. B is Tall. Mrs. B’s foot is Big. Mr. A is Very-Tall. × Mr. A is Very-Tall. Mr. A’s foot is Very-Big. Natural Reasoning 11/22/2020 5

Fuzzy Logic （Fuzzy Inference) B’ A’ A　→　B A’ B’ A　→　B IF　A　THEN B A’　is true B’ is true A　→　B B　→　C IF　A　THEN　B IF　B　THEN　C A’　→　C’ IF　A’　THEN　C’ A　→　B A’ B’ B 　→　C B’ A’　→　C’ 11/22/2020 6

How To Realize Fuzzy Inference ? Introduction of membership function ! We consider a member of a set as well as the degree of the membership. Degree of property 100% 50% 30% ) 170 11/22/2020 Tall 180 190 Very Tall x Height 7

Representation of Fuzzy Set １．Countable Set 2. Uncountable Set 11/22/2020 8

Example 1. Countable Case Membership Function 1. 0 0. 5 170 11/22/2020 Tall 180 190 Very Tall x Height 9

1. Uncountable Case Membership Function 1. 0 0. 5 170 11/22/2020 Tall 180 190 Very Tall x Height 10

Fuzzy Set Operations 1. Implication 2. Union 3. Intersection 4. Compliment 11/22/2020 11

Fuzzy　Relation Definition [Fuzzy Relation] Let　assume that X and Y are sets. Fuzzy relation R of X and Y is a fuzzy subset of X x Y as In general, 11/22/2020 fuzzy relation R of is 12

Composition of Relations Definition [Composition of Fuzzy Relations] Let R and S are fuzzy relations, i. e. , Composition of fuzzy relations, R and S, is a fuzzy set defined by R X If A is a fuzzy set and R is a fuzzy relation, 11/22/2020 S Z Y is 13

Fuzzy Inference Direct Method (Mamdani) （Max－Min　Composition) A　→　B A’ B’ IF　A　THEN B A’　is true B’ is true Caution! A’ and B’ are Fuzzy Sets. 11/22/2020 14

If he/she is tall then his/her foot is big. He is very tall. If he/she is tall then his/her foot is big. He is 178 cm tall. A’ is not fuzzy set or Defuzzy value 11/22/2020 A=Tall B=Big A’=Very Tall A=Tall B=Big A’=178 B’ is still Fuzzy Set 15

Fuzzy Control Rules A 1 and y is = B 1 then z is C 1 Rule 2 : If x is A 2 and y is = B 2 then z is C 2 Rule 1 : If x is . Rule n : If x is Input x is Output o A 11/22/2020 . A nand y is = B n and y is then z is C n o B 16

Defuzzication Control Input is Number Defuzzication If x and y are defuzzy values, 11/22/2020 17 This operation is sometimes replaced by x (multiplication)

Triangular Membership Function NB NM NS ZO PS PM PB 1 -1 Example R 1: If x is NS, and y is PS, then z is PS R 2: If x is ZO, and y is ZO, then z is ZO NS PS PS R 1 PS ZO ZO ZO R 2 11/22/2020 18

Simplification NS PS PS R 1 PS ZO ZO ZO R 2 Further Simplification (Height Method) NS PS PS R 1 PS ZO ZO ZO R 2 11/22/2020 19

TS（Takegaki－Sugeno）Model 1. 2. 3. Singleton Fuzzifier Product　Inference Weighted Average Deffuzifier PM PS PS R 3 PS PS PS R 4 11/22/2020 20

References 1. 2. 3. S. Murakami: Fuzzy Control , Vol. 22, Computer and Application’s Mook, Corona Pub. (1988) in Japanese K. Hirota: Fuzzy !? , Inter AI (Aug, 88 -June, 90) in Japanese S. S. Farinwata et. Ed. : Fuzzy Control, Wiley (2000) 11/22/2020 21