Defuzzification

• Convert fuzzy grade to Crisp output

*Fuzzy Engineering, Bart Kosko

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

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

*Fuzzy Logic with Engineering Applications, Timothy J. Ross

Formed by weightingeach functions in theoutput by its respectivemaximum membershipvalue

Defuzzification (Cont.)

• Mean-max membership (middle of maxima)– Maximum membership is a plateau

*Fuzzy Logic with Engineering Applications, Timothy J. Ross

Z* = a + b2

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 following3 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

*Fuzzy Logic with Engineering Applications, Timothy J. Ross

Z* = (6+7)/2 = 6.5

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 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

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

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

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

* Fuzzy Systems Toolbox, M. Beale and H Demuth

MATLAB: Create membership functions - Power

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

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 allpossible combinations

* Fuzzy Systems Toolbox, M. Beale and H Demuth

Consequence Matrix

* Fuzzy Systems Toolbox, M. Beale and H Demuth

Evaluating Rules with FunctionFRULE

* Fuzzy Systems Toolbox, M. Beale and H Demuth

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

Example: Fully Automatic Washing Machine

* 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

• Inputs—Laundry Softness—Laundry Quantity

• Outputs—Washing Cycle

—Washing Time

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

* 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

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

* Fuzzy Logic: Intelligence, control, and Information, J. Yen and R. Langari, Prentice Hall

ScalingClipping

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