04 - A fuzzy System.ppt

8/10/2019 04 - A fuzzy System.ppt

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Fuzzy Systems

* Fuzzy Systems Toolbox, M. Beale and H Demuth

How can fuzzy systems be used in a world wheremeasurements and actions are expressed as crisp

values?


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Fuzzy Systems (Cont.)

* Fuzzy Systems Toolbox, M. Beale and H Demuth

* Fuzzify crisp inputs to get the fuzzy inputs

* Defuzzify the fuzzy outputs to get crisp outputs


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Fuzzy Systems (Cont.)

90 Degree F.It is too hot!

Turn the fan on highSet the fan at90% speed

Input Fuzzifier Fuzzy System Defuzzifier output


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Fuzzy Set: Vector Representation

Two vectors can represent fuzzy discrete

sets or fuzzy continuous sets, Support Vector (universe vector)

Grade Vector (membership vector]


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Example: Vector Representation

Define the concept of tall over heights

from 5 to 7 feet, using MATLAB

S = [5.00 5.50 6.00 6.50 7.00];

G = [0.00 0.25 0.50 0.75 1.00];

0

0 25

0 5

0 75

1

5 5 5 6 6 5 7

Grade


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Fuzzification

Process of making a crisp quantity fuzzy

Vector representation can be viewed as either

a discrete or an approximation of a continuous

set ( use linear interpolation]

Crisp input

Fuzzy Grade


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Example: Fuzzification

Define fuzzy set near 5

S = [ 0:10];

G = [0.0 0.1 0.3 0.5 0.8 1 0.8 0.5 0.3 0.1 0];

0

0 2

0 4

0 6

0 8

1

0 1 2 3 4 5 6 7 8 9 1 0

Near 50.5

0.9


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Fuzzy Systems

How do you make a machine smart?

Put some FAT in it!

A FAT enough machine can model any

process

A FAT system can always turn inputs to

outputs and turn causes to effects and turn

questions to answer FAT stands for Fuzzy Approximation

Theorem

*Fuzzy Thinking:The new Science of Fuzzy Logic, Bart Kosko


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Fuzzy Systems (cont.)

FAT idea has a simple geometry

Cover a curve with patches!



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Knowledge as rules

Each piece of human knowledge, each rule of

the form IFthis then th tdefines a patch.

All the rules define patches

Try to cover some wiggly curve The better the patches cover the curve, the

smarter the system



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Knowledge as Rules How do you reason?

You want to play golf on Saturday or Sunday

and you dont want to get wet when you play.

Reach it with rules!

If it rains, you get wet!

If you get wet, you cant play golf

If it rains on Saturday and wont rain on Sunday

You play golf on Sunday!



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Artificial Intelligence: AI

Knowledge is rules

Rules are in black-and-white language

Bivalent rules AI has so far, afer over 30 years of research,

not produced smart machines!

Because they cant yet put enough rules in thecomputer (use 100-1000 rules, need >100k}

Throwing more rules at the problem



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Knowledge as Rules

Fuzzy researchers have built hundreds of

smart machines that work!

Yes, we need rules No, we dont need a lot of rules for many

tasks

We need Fuzzy rules



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Knowledge as Rules

Every term in one of our rules is Fuzzy

Every term is vague, hazy, inexact, sloppy

One human rule covers all these casesAI rule covers one precise case



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Fuzzy rule relates fuzzy sets

If X is A, then Y is B

A and B are fuzzy sets and subset of X and Y



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Build a Fuzzy System

3 Steps

Pick the nouns or variables

Example: X be input and Y be output

Let x be temperature and Y be change in motor speed

Cause, effect. Stimulus, response!

Pick the fuzzy sets

Define fuzzy subsets of the nouns X and Y Pick the fuzzy rules

Associate output to the input



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Example: Build a Fuzzy System

Design motor speed controller for air

conditioner

Step 1: assign input and outputvariables

Let X be temperature in Fahrenheit

Let Y be the change in motor speed of the

air conditioner



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conditioner

Step 2: Pick fuzzy sets Define subsets of the noun X and Y

Say 5 fuzzy sets on X

Cold, Cool, Just Right, Warm, and Hot

Say 5 fuzzy sets on Y

Stop, Slow, Medium, Fast, and Blast



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Input Fuzzy set



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Output Fuzzy set



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conditioner

Step 3: Assign a motor speed set toeach temperature set



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Rules If temperature is cold then motor speed stop

If temperature is coolthen motor speed slows

If temperatureis just right then motor speedis medium

If temperature is w rm then motor speed is f st

If temperature is hotthen motor speed bl sts



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Fuzzy Relation



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Fuzzy system with 5 patches



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FAT Theorem

You can always cover a curve with a finite

number of fuzzy patches

Let the cuts overlap

Sloppy rules give big patches

Fine rules give small patches

You pay for precision!



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FAT Theorem (Cont.)

Rough cover of the non linear System



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FAT Theorem (Cont.)

finer cover of the non linear System



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Fuzzy Associative Memory

Which rule fires or activates at which

time?

They all fire all the time They fire in parallel

All rules fire to some degree

Most fire to zero degree The result is a fuzzy weighted average



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Additive Fuzzy System

Stores m fuzzy rules of the form

If X = Ajthen Y = Bj, then computes the output

by defuzzifiy the summed (MAXed) of thepartially fired then-part fuzzy sets Bj

*Fuzzy Engineering, Bart Kosko


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Graphical Technique of

Mamdani (Max-Min] Inference If x1

kis A1kand x2

kis A2kThen Ykis Bk

*Fuzzy Logic with Engineering Applications, Timothy J. Ross


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Mamdani (Max-Min] Inference If x is A Then Y is B

MATLAB

XS = support vector of X;

XG = grade vector of X;

YS = support vector of y;

YG = grade vector of Y;

A = Crisp input;

B = ifthen_min(XS,XG,A,YG);

Where function B= ifthen_min (); calculate

[YG_row,YG_column] = size(YG);

B = Min(fuzzify(XS,XG,A)*ones(1,YG_column),YG);


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Max-Product Inference If x1

kis A1kand x2

kis A2kThen Ykis Bk



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Max-Product Inference If x is A Then Y is B

MATLAB

XS = support vector of X;

XG = grade vector of X;

YS = support vector of y;

YG = grade vector of X;

A = Crisp input;

B = ifthen_prod(XS,XG,A,YG);

Where function B= ifthen_prod (); calculate

B = fuzzify(XS,XG,A) *YG;


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Example: temp. = 65 degree F.

If temperature is just rightthen motor speed is medium

*Fuzzy Thinking, Bart Kosko


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Example: temp. = 63 degree F.

*Fuzzy Thinking, Bart Kosko

If temperature is coolthen motor speed slows

If temperatureis just right then motor speedis medium


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Example: t = 63 degree F. (Cont.)

*Fuzzy Thinkring, Bart Kosko


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Summed (MAXed) of the partially fired then-

part fuzzy sets


OR OUTPUT


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Defuzzify to find the output motor speed



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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 and

physically appealing of all the defuzzification

methods [Sugeno, 1985; Lee, 1990]

Often called

Center of area

Center of gravity



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Max-membership principal

Also known as height method



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Weighted average method

Valid for symmetrical output membership functions


Formed by weightingeach functions in the

output by its respective

maximum membership

value


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Mean-max membership (middle of maxima)

Maximum membership is a plateau


Z* = a + b2


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Center of Largest area

If the output fuzzy set has at least two convex

subregion, defuzzify the largest area using centroid



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First (or last) of maxima

Determine the smallest value of the domain with

maximized membership degree



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Example: Defuzzification

Find an estimate crisp output from the following

3 membership functions



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CENTROID



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Weighted Average



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Center of largest area

Same as the centroid method because the complete

output fuzzy set is convex



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First and Last of maxima



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Defuzzification

Of the seven defuzzification methods presented,

which is the best?

It is context or problem-dependent



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Defuzzification: Criteria

Hellendoorn and Thomas specified 5 criteria

against which 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!



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Defuzzification: Criteria (Cpnt.)

Hellendoorn and Thomas specified 5 criteria

against which to measure the methods

#3 Plausibility

Z* should lie approximatly in the middle of the support region

and have 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

*F L i ith E i i A li ti Ti th J R

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04 - A fuzzy System.ppt