Defuzzification
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Transcript of Defuzzification
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Defuzzification
• Convert fuzzy grade to Crisp output
*Fuzzy Engineering, Bart Kosko
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Defuzzification (Cont.)
• Centroid Method: the most prevalent andphysically appealing of all the defuzzificationmethods [Sugeno, 1985; Lee, 1990]
– Often called• Center of area• Center of gravity
*Fuzzy Logic with Engineering Applications, Timothy J. Ross
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Defuzzification (Cont.)
• Max-membership principal– Also known as height method
*Fuzzy Logic with Engineering Applications, Timothy J. Ross
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Defuzzification (Cont.)
• Weighted average method– Valid for symmetrical output membership functions
*Fuzzy Logic with Engineering Applications, Timothy J. Ross
Formed by weightingeach functions in theoutput by its respectivemaximum membershipvalue
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Defuzzification (Cont.)
• Mean-max membership (middle of maxima)– Maximum membership is a plateau
*Fuzzy Logic with Engineering Applications, Timothy J. Ross
Z* = a + b2
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Defuzzification (Cont.)
• Center of sums– Faster than many defuzzification methods
*Fuzzy Logic with Engineering Applications, Timothy J. Ross
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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
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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
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Example: Defuzzification
• Find an estimate crisp output from the following3 membership functions
*Fuzzy Logic with Engineering Applications, Timothy J. Ross
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Example: Defuzzification
• CENTROID
*Fuzzy Logic with Engineering Applications, Timothy J. Ross
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Example: Defuzzification
• Weighted Average
*Fuzzy Logic with Engineering Applications, Timothy J. Ross
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Example: Defuzzification
• Mean-Max
*Fuzzy Logic with Engineering Applications, Timothy J. Ross
Z* = (6+7)/2 = 6.5
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Example: Defuzzification
• Center of sums
*Fuzzy Logic with Engineering Applications, Timothy J. Ross
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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
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Example: Defuzzification
• First and Last of maxima
*Fuzzy Logic with Engineering Applications, Timothy J. Ross
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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
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Defuzzification: Criteria
• Hellendoorn and Thomas specified 5 criteriaagainst 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
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Defuzzification: Criteria (Cpnt.)
• Hellendoorn and Thomas specified 5 criteriaagainst 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
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Designing Antecedent Membership Functions
• Recommend designer to adopt thefollowing design principles:– Each Membership function overlaps only with
the closest neighboring membershipfunctions;
– For any possible input data, its membershipvalues in all relevant fuzzy sets should sum to 1(or nearly)
* Fuzzy Logic: Intelligence, control, and Information, J. Yen and R. Langari, Prentice Hall
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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
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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
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Designing Antecedent Membership Functions
A symmetric Function Design Following the guidelines
* Fuzzy Logic: Intelligence, control, and Information, J. Yen and R. Langari, Prentice Hall
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Designing Antecedent Membership Functions
An asymmetric Function Design Following the guidelines
* Fuzzy Logic: Intelligence, control, and Information, J. Yen and R. Langari, Prentice Hall
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Example: Furnace Temperature Control
• Inputs– Temperature reading from sensor– Furnace Setting
• Output– Power control to motor
* Fuzzy Systems Toolbox, M. Beale and H Demuth
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MATLAB: Create membership functions - Temp
* Fuzzy Systems Toolbox, M. Beale and H Demuth
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MATLAB: Create membership functions - Setting
* Fuzzy Systems Toolbox, M. Beale and H Demuth
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* Fuzzy Systems Toolbox, M. Beale and H Demuth
MATLAB: Create membership functions - Power
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If - then - Rules
* Fuzzy Systems Toolbox, M. Beale and H Demuth
Fuzzy Rules for Furnace control
Setting
TempLow Medium High
Cold Low Medium High
Cool Low Medium High
Moderate Low Low Low
Warm Low Low Low
Hot low Low Low
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Antecedent Table
* Fuzzy Systems Toolbox, M. Beale and H Demuth
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Antecedent Table
• MATLAB– A = table(1:5,1:3);
• Table generates matrix represents a table of allpossible combinations
* Fuzzy Systems Toolbox, M. Beale and H Demuth
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Consequence Matrix
* Fuzzy Systems Toolbox, M. Beale and H Demuth
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Evaluating Rules with FunctionFRULE
* Fuzzy Systems Toolbox, M. Beale and H Demuth
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Design Guideline (Inference)
* Fuzzy Logic: Intelligence, control, and Information, J. Yen and R. Langari, Prentice Hall
• Recommend—Max-Min (Clipping) Inference method
be used together with the MAXaggregation operator and the MIN ANDmethod
—Max-Product (Scaling) Inferencemethod be used together with the SUMaggregation operator and the PRODUCTAND method
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Example: Fully Automatic Washing Machine
* Fuzzy Logic: Intelligence, control, and Information, J. Yen and R. Langari, Prentice Hall
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Example: Fully Automatic Washing Machine
* Fuzzy Logic: Intelligence, control, and Information, J. Yen and R. Langari, Prentice Hall
• Inputs—Laundry Softness—Laundry Quantity
• Outputs—Washing Cycle
—Washing Time
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Example: Input Membership functions
* Fuzzy Logic: Intelligence, control, and Information, J. Yen and R. Langari, Prentice Hall
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Example: Output Membership functions
* Fuzzy Logic: Intelligence, control, and Information, J. Yen and R. Langari, Prentice Hall
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Example: Fuzzy Rules for Washing Cycle
* Fuzzy Logic: Intelligence, control, and Information, J. Yen and R. Langari, Prentice Hall
Quantity
SoftnessSmall Medium Large
Soft Delicate Light Normal
NormalSoft
Light Normal Normal
NormalHard
Light Normal Strong
Hard Light Normal Strong
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Example: Control Surface View (Clipping)
* Fuzzy Logic: Intelligence, control, and Information, J. Yen and R. Langari, Prentice Hall
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Example: Control Surface View (Scaling)
* Fuzzy Logic: Intelligence, control, and Information, J. Yen and R. Langari, Prentice Hall
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Example: Control Surface View
* Fuzzy Logic: Intelligence, control, and Information, J. Yen and R. Langari, Prentice Hall
ScalingClipping
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Example: Rule View (Clipping)
* Fuzzy Logic: Intelligence, control, and Information, J. Yen and R. Langari, Prentice Hall
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Example: Rule View (Scaling)
* Fuzzy Logic: Intelligence, control, and Information, J. Yen and R. Langari, Prentice Hall