Defuzzification Convert fuzzy grade to Crisp output *Fuzzy Engineering, Bart Kosko
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
Defuzzification (Cont.) Max-membership principal Also known as height method *Fuzzy Logic with Engineering Applications, Timothy J. Ross
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
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
Defuzzification (Cont.) Center of sums Faster than many defuzzification methods *Fuzzy Logic with Engineering Applications, Timothy J. Ross
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
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
Example: Defuzzification Find an estimate crisp output from the following 3 membership functions *Fuzzy Logic with Engineering Applications, Timothy J. Ross
Example: Defuzzification CENTROID *Fuzzy Logic with Engineering Applications, Timothy J. Ross
Example: Defuzzification Weighted Average *Fuzzy Logic with Engineering Applications, Timothy J. Ross
Example: Defuzzification Mean-Max Z* = (6+7)/2 = 6.5 *Fuzzy Logic with Engineering Applications, Timothy J. Ross
Example: Defuzzification Center of sums *Fuzzy Logic with Engineering Applications, Timothy J. Ross
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
Example: Defuzzification First and Last of maxima *Fuzzy Logic with Engineering Applications, Timothy J. Ross
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
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
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
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
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
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
Designing Antecedent Membership Functions A symmetric Function Design Following the guidelines * Fuzzy Logic: Intelligence, control, and Information, J. Yen and R. Langari, Prentice Hall
Designing Antecedent Membership Functions An asymmetric Function Design Following the guidelines * Fuzzy Logic: Intelligence, control, and Information, J. Yen and R. Langari, Prentice Hall
Example: Furnace Temperature Control Inputs Temperature reading from sensor Furnace Setting Output Power control to motor * Fuzzy Systems Toolbox, M. Beale and H Demuth
MATLAB: Create membership functions - Temp * Fuzzy Systems Toolbox, M. Beale and H Demuth
MATLAB: Create membership functions - Setting * Fuzzy Systems Toolbox, M. Beale and H Demuth
MATLAB: Create membership functions - Power * Fuzzy Systems Toolbox, M. Beale and H Demuth
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
Antecedent Table * Fuzzy Systems Toolbox, M. Beale and H Demuth
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
Consequence Matrix * Fuzzy Systems Toolbox, M. Beale and H Demuth
Evaluating Rules with Function FRULE * Fuzzy Systems Toolbox, M. Beale and H Demuth
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
Example: Fully Automatic Washing Machine * Fuzzy Logic: Intelligence, control, and Information, J. Yen and R. Langari, Prentice Hall
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
Example: Input Membership functions * Fuzzy Logic: Intelligence, control, and Information, J. Yen and R. Langari, Prentice Hall
Example: Output Membership functions * Fuzzy Logic: Intelligence, control, and Information, J. Yen and R. Langari, Prentice Hall
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
Example: Control Surface View (Clipping) * Fuzzy Logic: Intelligence, control, and Information, J. Yen and R. Langari, Prentice Hall
Example: Control Surface View (Scaling) * Fuzzy Logic: Intelligence, control, and Information, J. Yen and R. Langari, Prentice Hall
Example: Control Surface View Clipping Scaling * Fuzzy Logic: Intelligence, control, and Information, J. Yen and R. Langari, Prentice Hall
Example: Rule View (Clipping) * Fuzzy Logic: Intelligence, control, and Information, J. Yen and R. Langari, Prentice Hall
Example: Rule View (Scaling) * Fuzzy Logic: Intelligence, control, and Information, J. Yen and R. Langari, Prentice Hall