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Fuzzy Logic

Fuzzy Logic. E. Fuzzy Inference Engine. Crisp input. Fuzzification. Rules. Defuzzification. Crisp Output Result. Fuzzy Inference. “antecedent”. “consequent”. Fuzzy Inference Example. Assume that we need to evaluate student applicants based on their GPA and GRE scores.

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Fuzzy Logic

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  1. Fuzzy Logic E. Fuzzy Inference Engine

  2. Crisp input Fuzzification Rules Defuzzification Crisp Output Result Fuzzy Inference “antecedent” “consequent”

  3. Fuzzy Inference Example • Assume that we need to evaluate student applicants based on their GPA and GRE scores. • For simplicity, let us have three categories for each score [High (H), Medium (M), and Low(L)] • Let us assume that the decision should be Excellent (E), Very Good (VG), Good (G), Fair (F) or Poor (P) • An expert will associate the decisions to the GPA and GRE score. They are then Tabulated.

  4. Fuzzy Inference Example • Fuzzy if-then Rules If the GRE is HIGHand the GPA is HIGHthen the student will be EXCELLENT. If the GRE is LOWand the GPA is HIGHthen the student will be FAIR. etc Fuzzy Linguistic Variables Fuzzy Logic Antecedent Consequent

  5. GRE Antecedents L H M E VG F H G G B GPA M B B F L Fuzzy Rule Table Consequents

  6. Fuzzification • Fuzzifier converts a crisp input into a vector of fuzzy membership values. • The membership functions • reflects the designer's knowledge • provides smooth transition between fuzzy sets • are simple to calculate • Typical shapes of the membership function are Gaussian, trapezoidal and triangular.

  7. mGRE Medium Low High 1 800 1200 1800 GRE mGRE = {mL , mM , mH } Membership Functions for GRE

  8. Medium Low High 1 2.2 3.0 3.8 GPA Membership Functions for the GPA mGPA mGPA = {mL , mM , mH }

  9. c VG B G F 1 60 90 80 70 100 Decision Membership Function for the Consequent

  10. mGRE = {mL = 0.8 , mM = 0.2, mH = 0} mGPA = {mL = 0 , mM = 0.6, mH = 0.4} Fuzzification • Transform the crisp antecedents into a vector of fuzzy membership values. • Assume a student with GRE=900 and GPA=3.6. Examining the membership function gives

  11. Table: GRE 0.8 0.2 0.0 L H M E VG F H 0.0 0.6 0.4 G GPA G B M B B L F

  12. Table: GRE 0.8 0.2 0.0 L H M E VG F 0.0 0.6 0.4 H 0.0 0.0 0.0 GPA G G B M 0.6 0.2 0.0 F B B L 0.4 0.2 0.0

  13. GRE 0.8 0.2 0.0 L H M GPA E VG F 0.0 0.6 0.4 H 0.0 0.0 0.0 G G B M 0.6 0.2 0.0 F B B L 0.4 0.2 0.0 Interpretation: The student is GOOD if (the GRE is HIGH and the GPA is MEDIUM) OR (the GRE is MEDIUM and the GPA is MEDIUM) The consequent GOOD has a membership of max(0.6,0.2)=0.6

  14. GRE 0.8 0.2 0.0 L H M GPA E VG F 0.0 0.6 0.4 H 0.0 0.0 0.0 G G B M 0.6 0.2 0.0 F B B L 0.4 0.2 0.0 Interpretation: E = 0.0 VG = 0.0 F = max( 0.0, 0.4) = 0.4 G = max( 0.6, 0.2) = 0.6 B = max( 0,0,0.2) = 0.2

  15. c 1 G 0.6 0.4 0.2 F B E VG 60 70 80 90 100 Decision Weight Consequent Memberships

  16. c 1 G 0.6 0.4 0.2 F B E VG 60 70 80 90 100 Defuzzification • Converts the output fuzzy numbers into a unique (crisp) number • Center of Mass Method: Add all weighted curves and find the center of mass

  17. Mode Method • AnAlternate Approach: Fuzzy set with the largest membership value is selected. • Fuzzy decision: {B, F, G,VG, E} = {0.2, 0.4, 0.6, 0.0, 0.0} • Final Decision (FD) = Fair Student • If two decisions have same membership max, use the average of the two.

  18. CE LN MN SN ZE SP MP LP LN LN LN LN LN MN SN SN MN LN LN LN MN SN ZE ZE SN LN LN MN SN ZE ZE SP E ZE LN MN SN ZE SP MP LP SP SN ZE ZE SP MP LP LP MP ZE ZE SP MP LP LP LP LP SP SP MP LP LP LP LP Example: Fuzzy Table for Control

  19. m LN MN SN ZE SP MP LP 1 E CU 0 -6 -3 -1 0 1 3 6 m LN MN SN ZE SP MP LP 1 CE 0 -3 -2 -1 0 1 2 3 Membership Functions

  20. CE LN MN SN ZE SP MP LP LN LN LN LN LN MN e. SN 0.2 f. SN 0.0 MN LN LN LN MN d. SN 0.5 ZE ZE SN LN LN MN c.SN 0.3 ZE ZE SP E ZE LN MN b.SN 0.4 ZE SP MP LP SP a. SN 0.1 ZE ZE SP MP LP LP MP ZE SP SP MP LP LP LP LP SP SP MP LP LP LP LP Rule Aggregation Consequent is or SN if a or b or c or d or f.

  21. Rule Aggregation Consequent is or SN if a or b or c or d or f. Consequent Membership = max(a,b,c,d,e,f) = 0.5 Use General Mean Aggregation:

  22. 1800 trajectory 900 response rpm 0 -900 -1800 0 3 6 9 12 15 18 21 24 27 Time [sec] Lab Test: Speed Tracking of IM

  23. 5 4 trajectory Turn 3 response 2 1 0 0 3 6 9 12 15 18 21 24 27 Time [sec] Lab Test: Prercision Position Tracking of IM

  24. Commonly Used Variations Clipped vs. Weighted Defuzzification 1 B F G VG 80 90 100 60 1 B F G VG 80 90 100 60

  25. Commonly Used Variations Sum-Product Inferencing Instead of min(x,y) for fuzzy AND... Use  x • y Instead of max(x,y) for fuzzy OR... Use  min(1, x + y)

  26. Commonly Used Variations Sugeno inferencing Other Norms and co-norms Relationship with Neural Networks Explanation Facilities Teaching a Fuzzy System Tuning a Fuzzy System

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