Defuzzification. Convert fuzzy grade to Crisp output. *Fuzzy Engineering, Bart Kosko

Transcription

1 Defuzzification Convert fuzzy grade to Crisp output *Fuzzy Engineering, Bart Kosko

2 Defuzzification (Cont.) Centroid Method: the most prevalent and physically appealing of all the defuzzification methods [Sugeno, 1985; Lee, 1990] Often called Center of area Center of gravity *Fuzzy Logic with Engineering Applications, Timothy J. Ross

3 Defuzzification (Cont.) Max-membership principal Also known as height method *Fuzzy Logic with Engineering Applications, Timothy J. Ross

4 Defuzzification (Cont.) Weighted average method Valid for symmetrical output membership functions Formed by weighting each functions in the output by its respective maximum membership value *Fuzzy Logic with Engineering Applications, Timothy J. Ross

5 Defuzzification (Cont.) Mean-max membership (middle of maxima) Maximum membership is a plateau Z* = a + b 2 *Fuzzy Logic with Engineering Applications, Timothy J. Ross

6 Defuzzification (Cont.) Center of sums Faster than many defuzzification methods *Fuzzy Logic with Engineering Applications, Timothy J. Ross

7 Defuzzification (Cont.) Center of Largest area If the output fuzzy set has at least two convex subregion, defuzzify the largest area using centroid *Fuzzy Logic with Engineering Applications, Timothy J. Ross

8 Defuzzification (Cont.) First (or last) of maxima Determine the smallest value of the domain with maximized membership degree *Fuzzy Logic with Engineering Applications, Timothy J. Ross

9 Example: Defuzzification Find an estimate crisp output from the following 3 membership functions *Fuzzy Logic with Engineering Applications, Timothy J. Ross

10 Example: Defuzzification CENTROID *Fuzzy Logic with Engineering Applications, Timothy J. Ross

11 Example: Defuzzification Weighted Average *Fuzzy Logic with Engineering Applications, Timothy J. Ross

12 Example: Defuzzification Mean-Max Z* = (6+7)/2 = 6.5 *Fuzzy Logic with Engineering Applications, Timothy J. Ross

13 Example: Defuzzification Center of sums *Fuzzy Logic with Engineering Applications, Timothy J. Ross

14 Example: Defuzzification Center of largest area Same as the centroid method because the complete output fuzzy set is convex *Fuzzy Logic with Engineering Applications, Timothy J. Ross

15 Example: Defuzzification First and Last of maxima *Fuzzy Logic with Engineering Applications, Timothy J. Ross

16 Defuzzification Of the seven defuzzification methods presented, which is the best? It is context or problem-dependent *Fuzzy Logic with Engineering Applications, Timothy J. Ross

17 Defuzzification: Criteria Hellendoorn and Thomas specified 5 criteria against whnic to measure the methods #1 Continuity Small change in the input should not produce the large change in the output #2 Disambiguity Defuzzification method should always result in a unique value, I.e. no ambiguity Not satisfied by the center of largest area! *Fuzzy Logic with Engineering Applications, Timothy J. Ross

18 Defuzzification: Criteria (Cpnt.) Hellendoorn and Thomas specified 5 criteria against whnic to measure the methods #3 Plausibility Z* should lie approximatly in the middle of the support region and hve high degree of membership #4 Computational simplicity Centroid and center of sum required complex computation! #5 Constitutes the difference between centroid, weighted average and center of sum Problem-dependent, keep computation simplicity *Fuzzy Logic with Engineering Applications, Timothy J. Ross

19 Designing Antecedent Membership Functions Recommend designer to adopt the following design principles: Each Membership function overlaps only with the closest neighboring membership functions; For any possible input data, its membership values in all relevant fuzzy sets should sum to 1 (or nearly) * Fuzzy Logic: Intelligence, control, and Information, J. Yen and R. Langari, Prentice Hall

20 Designing Antecedent Membership Functions A Membership Function Design that violates the second principle * Fuzzy Logic: Intelligence, control, and Information, J. Yen and R. Langari, Prentice Hall

21 Designing Antecedent Membership Functions A Membership Function Design that violates both principle * Fuzzy Logic: Intelligence, control, and Information, J. Yen and R. Langari, Prentice Hall

22 Designing Antecedent Membership Functions A symmetric Function Design Following the guidelines * Fuzzy Logic: Intelligence, control, and Information, J. Yen and R. Langari, Prentice Hall

23 Designing Antecedent Membership Functions An asymmetric Function Design Following the guidelines * Fuzzy Logic: Intelligence, control, and Information, J. Yen and R. Langari, Prentice Hall

24 Example: Furnace Temperature Control Inputs Temperature reading from sensor Furnace Setting Output Power control to motor * Fuzzy Systems Toolbox, M. Beale and H Demuth

25 MATLAB: Create membership functions - Temp * Fuzzy Systems Toolbox, M. Beale and H Demuth

26 MATLAB: Create membership functions - Setting * Fuzzy Systems Toolbox, M. Beale and H Demuth

27 MATLAB: Create membership functions - Power * Fuzzy Systems Toolbox, M. Beale and H Demuth

28 If - then - Rules Fuzzy Rules for Furnace control Temp Setting Low Medium High Cold Low Medium High Cool Low Medium High Moderate Low Low Low Warm Low Low Low Hot low Low Low * Fuzzy Systems Toolbox, M. Beale and H Demuth

29 Antecedent Table * Fuzzy Systems Toolbox, M. Beale and H Demuth

30 Antecedent Table MATLAB A = table(1:5,1:3); Table generates matrix represents a table of all possible combinations * Fuzzy Systems Toolbox, M. Beale and H Demuth

31 Consequence Matrix * Fuzzy Systems Toolbox, M. Beale and H Demuth

32 Evaluating Rules with Function FRULE * Fuzzy Systems Toolbox, M. Beale and H Demuth

33 Design Guideline (Inference) Recommend Max-Min (Clipping) Inference method be used together with the MAX aggregation operator and the MIN AND method Max-Product (Scaling) Inference method be used together with the SUM aggregation operator and the PRODUCT AND method * Fuzzy Logic: Intelligence, control, and Information, J. Yen and R. Langari, Prentice Hall

34 Example: Fully Automatic Washing Machine * Fuzzy Logic: Intelligence, control, and Information, J. Yen and R. Langari, Prentice Hall

35 Example: Fully Automatic Washing Machine Inputs Laundry Softness Laundry Quantity Outputs Washing Cycle Washing Time * Fuzzy Logic: Intelligence, control, and Information, J. Yen and R. Langari, Prentice Hall

36 Example: Input Membership functions * Fuzzy Logic: Intelligence, control, and Information, J. Yen and R. Langari, Prentice Hall

37 Example: Output Membership functions * Fuzzy Logic: Intelligence, control, and Information, J. Yen and R. Langari, Prentice Hall

38 Example: Fuzzy Rules for Washing Cycle Quantity Softness Small Medium Large Soft Delicate Light Normal Normal Soft Normal Hard Light Normal Normal Light Normal Strong Hard Light Normal Strong * Fuzzy Logic: Intelligence, control, and Information, J. Yen and R. Langari, Prentice Hall

39 Example: Control Surface View (Clipping) * Fuzzy Logic: Intelligence, control, and Information, J. Yen and R. Langari, Prentice Hall

40 Example: Control Surface View (Scaling) * Fuzzy Logic: Intelligence, control, and Information, J. Yen and R. Langari, Prentice Hall

41 Example: Control Surface View Clipping Scaling * Fuzzy Logic: Intelligence, control, and Information, J. Yen and R. Langari, Prentice Hall

42 Example: Rule View (Clipping) * Fuzzy Logic: Intelligence, control, and Information, J. Yen and R. Langari, Prentice Hall

43 Example: Rule View (Scaling) * Fuzzy Logic: Intelligence, control, and Information, J. Yen and R. Langari, Prentice Hall