Defuzzification

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


• Max-membership principal– Also known as height method



• Weighted average method– Valid for symmetrical output membership functions


Formed by weightingeach functions in theoutput by its respectivemaximum membershipvalue


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


Z* = a + b2


• Center of sums– Faster than many defuzzification methods



• Center of Largest area– If the output fuzzy set has at least two convex

subregion, defuzzify the largest area using centroid



• First (or last) of maxima– Determine the smallest value of the domain with

maximized membership degree


Example: Defuzzification

• Find an estimate crisp output from the following3 membership functions



• CENTROID



• Weighted Average



• Mean-Max


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


• Center of sums



• Center of largest area– Same as the centroid method because the complete

output fuzzy set is convex



• First and Last of maxima


Defuzzification

• Of the seven defuzzification methods presented,which is the best?

– It is context or problem-dependent


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!


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


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


A Membership Function Design that violates the second principle



A Membership Function Design that violates both principle



A symmetric Function Design Following the guidelines



An asymmetric Function Design Following the guidelines


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


MATLAB: Create membership functions - Setting



MATLAB: Create membership functions - Power

If - then - Rules


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


Antecedent Table

• MATLAB– A = table(1:5,1:3);

• Table generates matrix represents a table of allpossible combinations


Consequence Matrix


Evaluating Rules with FunctionFRULE


Design Guideline (Inference)


• 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


Example: Fully Automatic Washing Machine


• Inputs—Laundry Softness—Laundry Quantity

• Outputs—Washing Cycle

—Washing Time

Example: Input Membership functions


Example: Output Membership functions


Example: Fuzzy Rules for Washing Cycle


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)


Example: Control Surface View (Scaling)


Example: Control Surface View


ScalingClipping

Example: Rule View (Clipping)


Example: Rule View (Scaling)


Defuzzification

Engineering

fuzzy logic

fuzzy engineering

engineering applications

fuzzy grade

defuzzification centroid

fuzzy systems toolbox

defuzzification methods

completeoutput fuzzy