Dept. of Mechanical and Control Systems Eng. Engineering　Education　Methodology on Intelligent Control (Fuzzy Logic and Fuzzy Control) M.Yamakita Dept. of Mechanical and Control Systems Eng. Tokyo Inst. Of Tech. 2019/2/24

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. 2019/2/24

Crisp Expert System Inference (Reasoning) Formal Logic A → B A → B Condition part (Antecedent part) Implication operator Conclusion (Operation part) A　→　B B　→　C IF　A　THEN　B A　→　B IF　A　THEN B A　is true IF　B　THEN　C A A　→　C IF　A　THEN　C B B is true Hypothetical　Syllogism Modus　Ponens 2019/2/24

Crisp Logic （ （ Mr.A Mrs.B Tall ] 181ｃｍ １７０ｃｍ ] 177ｃｍ Short ( 2019/2/24

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 2019/2/24

Fuzzy Logic （Fuzzy Inference) A’ B’ A　→　B A　→　B A’ B’ IF　A　THEN B A’　is true B’ is true A　→　B A’ A　→　B B　→　C IF　A　THEN　B B’ IF　B　THEN　C B 　→　C B’ A’　→　C’ IF　A’　THEN　C’ A’　→　C’ 2019/2/24

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% ) x 170 180 190 Height Tall Very Tall 2019/2/24

Representation of Fuzzy Set １．Countable Set 2. Uncountable Set 2019/2/24

Example 1.0 0.5 x 170 180 190 Height Tall Very Tall 1. Countable Case Membership Function 1.0 0.5 x 170 180 190 Height Tall Very Tall 2019/2/24

1.0 0.5 x 170 180 190 Height Tall Very Tall 1. Uncountable Case Membership Function 1.0 0.5 x 170 180 190 Height Tall Very Tall 2019/2/24

Fuzzy Set Operations 1. Implication 2. Union 3. Intersection 4. Compliment 2019/2/24

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, fuzzy relation R of is 2019/2/24

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 S X Y Z If A is a fuzzy set and R is a fuzzy relation, is 2019/2/24

Fuzzy Inference A → B A’ B’ Caution! A’ and B’ are Fuzzy Sets. Direct Method (Mamdani) A　→　B A’ B’ IF　A　THEN B A’　is true B’ is true （Max－Min　Composition) Caution! A’ and B’ are Fuzzy Sets. 2019/2/24

If he/she is tall then his/her foot is big. He is very tall. A=Tall B=Big A’=Very Tall 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 178cm tall. A=Tall B=Big A’=178 B’ is still Fuzzy Set A’ is not fuzzy set or Defuzzy value 2019/2/24

Fuzzy Control B A A B C A B C A B C Rules Input Output x is is and y 1 : If x is A and y is = B then z is C 1 1 1 Rule 2 : If x is A and y is = B then z is C 2 2 2 . . is z then y and x If : n Rule = A B C n n n Input x is B A is and o y Output 2019/2/24

Defuzzication Control Input is Number Defuzzication If x and y are defuzzy values, This operation is sometimes replaced by x (multiplication) 2019/2/24

Triangular Membership Function -1 1 NM NS NB ZO PS PM PB Example R1: If x is NS, and y is PS, then z is PS If x is ZO, and y is ZO, then z is ZO R2: NS PS PS R1 PS ZO ZO ZO R2 2019/2/24

Further Simplification (Height Method) NS PS PS R1 PS ZO ZO ZO R2 Further Simplification (Height Method) NS PS PS R1 PS ZO ZO ZO R2 2019/2/24

TS（Takegaki－Sugeno）Model Singleton Fuzzifier Product　Inference Weighted Average Deffuzifier PM PS PS R3 PS PS PS R4 2019/2/24

References 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) 2019/2/24