Fuzzy Inference Systems J S Roger Jang CS

Fuzzy Inference Systems J. -S. Roger Jang (張智星) CS Dept. , Tsing Hua Univ. , Taiwan http: //www. cs. nthu. edu. tw/~jang@cs. nthu. edu. tw

Fuzzy Inference Systems Outline Introduction Mamdani fuzzy inference systems Sugeno fuzzy inference systems Tsukamoto fuzzy inference systems Fuzzy modeling 2

Fuzzy Inference Systems What is a fuzzy inference system (FIS)? A nonlinear mapping that derives its output based on fuzzy reasoning and a set of fuzzy if-then rules. The domain and range of the mapping could be fuzzy sets or points in a multidimensional spaces. Also known as • • 3 Fuzzy models Fuzzy associate memory Fuzzy-rule-based systems Fuzzy expert systems

Fuzzy Inference Systems Schematic diagram Rulebase (Fuzzy rules) Database (MFs) input output Fuzzy reasoning 4

Fuzzy Inference Systems Operating block diagram 5

Fuzzy Inference Systems Max-Star Composition Max-product composition: In general, we have max-* composition: where * is a T-norm operator. 6

Fuzzy Inference Systems Linguistic Variables A numerical variables takes numerical values: Age = 65 A linguistic variables takes linguistic values: Age is old A linguistic values is a fuzzy set. All linguistic values form a term set: T(age) = {young, not young, very young, . . . middle aged, not middle aged, . . . old, not old, very old, more or less old, . . . not very yound and not very old, . . . } 7

Fuzzy Inference Systems Linguistic Values (Terms) 8 complv. m

Fuzzy Inference Systems Operations on Linguistic Values Concentration: Dilation: Contrast intensification: 9 intensif. m

Fuzzy Inference Systems Fuzzy If-Then Rules General format: If x is A then y is B Examples: • • 10 If pressure is high, then volume is small. If the road is slippery, then driving is dangerous. If a tomato is red, then it is ripe. If the speed is high, then apply the brake a little.

Fuzzy Inference Systems Fuzzy If-Then Rules Two ways to interpret “If x is A then y is B”: y A coupled with B B y A entails B B x A 11 x A

Fuzzy Inference Systems Fuzzy If-Then Rules Two ways to interpret “If x is A then y is B”: • A coupled with B: (A and B) • A entails B: (not A or B) - Material implication - Propositional calculus - Extended propositional calculus - Generalization of modus ponens 12

Fuzzy Inference Systems Fuzzy If-Then Rules Fuzzy implication function: A coupled with B 13 fuzimp. m

Fuzzy Inference Systems Fuzzy If-Then Rules A entails B 14 fuzimp. m

Fuzzy Inference Systems Compositional Rule of Inference Derivation of y = b from x = a and y = f(x): y y b b y = f(x) a a and b: points y = f(x) : a curve 15 x a and b: intervals y = f(x) : an interval-valued function

Fuzzy Inference Systems Compositional Rule of Inference a is a fuzzy set and y = f(x) is a fuzzy relation: 16 cri. m

Fuzzy Inference Systems Fuzzy Reasoning Single rule with single antecedent Rule: if x is A then y is B Fact: x is A’ Conclusion: y is B’ Graphic Representation: A’ A B w X A’ x is A’ 17 Y B’ X y is B’ Y

Fuzzy Inference Systems Fuzzy Reasoning Single rule with multiple antecedent Rule: if x is A and y is B then z is C Fact: x is A’ and y is B’ Conclusion: z is C’ Graphic Representation: A’ A B’ B T-norm C 2 w X A’ x is A’ 18 Y B’ X Z y is B’ C’ Y z is C’ Z

Fuzzy Inference Systems Fuzzy Reasoning Multiple rules with multiple antecedent Rule 1: if x is 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 Fact: x is A’ and y is B’ Conclusion: z is C’ Graphic Representation: (next slide) 19

Fuzzy Inference Systems Fuzzy Reasoning Graphics representation: A’ A 1 B’ B 1 C 1 w 1 X A’ A 2 Z Y B’ B 2 C 2 w 2 X A’ Y Z T-norm B’ C’ x is A’ 20 X y is B’ Y z is C’ Z

Fuzzy Inference Systems Fuzzy Reasoning: MATLAB Demo >> ruleview mam 21 21

Fuzzy Inference Systems Other Variants Some terminology: • • 22 Degrees of compatibility (match) Firing strength Qualified (induced) MFs Overall output MF