Lec2011 - 3 - AI - Fuzzy Logic

8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic

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159.320 FUZZY LOGICFUZZY LOGIC

22

Introduction, Inference System, Fuzzy Control

System, Examples


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Fuzzy LogicFuzzy Logic

• Introduction – What is Fuzzy Logic?

– Applications of Fuzzy Logic – Classical Control System s! Fuzzy Control

• "eeloping a Fuzzy Control System

• #$amples• %heory of Fuzzy Sets

• Fuzzy Inference Systems


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TopicsTopics

• Introduction

• &asic Algorithm

• Control Systems

• Sample Computations

• Inerted 'endulum

• Fuzzy Inference Systems – (amdani %ype

– Sugeno %ype• Fuzzy Sets ) *perators

• "efuzzification

• (em+ership Functions

Control Systems

Inverted Pendulum

Computations

Sugeno

amdani

!asics

Fuzzy Sets

"efuzzification

em# Fcns


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A computational paradigm that is +ased on ho, humans thin-

Fuzzy Logic loo-s at the ,orld in imprecise terms. in much the same ,ay

that our +rain ta-es in information /e!g! temperature is hot. speed is slo,0.

then responds ,ith precise actions!

$%at is Fuzzy Logic&

%he human +rain can reason ,ith uncertainties. agueness. and

udgments! Computers can only manipulate precise aluations! Fuzzy logic

is an attempt to com+ine the t,o techni2ues!

3Fuzzy4 – a misnomer. has resulted in the mista-en suspicion that FL is

someho, less e$acting than traditional logic


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Fuzzy logic differs from classical logic in that statements are no longer

+lac- or ,hite. true or false. on or off!In traditional logic an o+ect ta-es on a alue of either zero or one!

In fuzzy logic. a statement can assume any real alue +et,een 5 and 1.

representing the degree to ,hich an element +elongs to a gien set!

$%at is Fuzzy Logic&

It is a+le to simultaneously handle numerical data and linguistic

-no,ledge!

A techni2ue that facilitates the control of a complicated system ,ithout

-no,ledge of its mathematical description!

FL is in fact. a precise pro'lem(solving met%odologya precise pro'lem(solving met%odology!


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Professor Lotfi )# *ade%

In +-.. Lotfi )# *ade% of the 6niersity of California at &er-eley pu+lished

7Fuzzy Sets.7 ,hich laid out the mathematics of fuzzy set theory and. +y

e$tension. fuzzy logic! 8adeh had o+sered that conentional computer logic

couldn9t manipulate data that represented su+ectie or ague ideas. so he created

fuzzy logic to allo, computers to determine the distinctions among data ,ith

shades of gray. similar to the process of human reasoning!

/istory of Fuzzy Logic

Source0 )ugust 1, 23

/Computer,orld0http:;;,,,!computer,orld!com;ne,s;


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Pioneering 4or5sPioneering 4or5s

• Interest in fuzzy systems ,as spar-ed +y Sei6i 7asuno'u and So6i iyamoto of /itac%i. ,ho in +8. proided simulations thatdemonstrated the superiority of fuzzy control systems for the Sendairail4ay! %heir ideas ,ere adopted. and fuzzy systems ,ere used tocontrol accelerating and +ra-ing ,hen the line opened in +89!

• Also in +89. during an international meeting of fuzzy researchers in%o-yo. Ta5es%i 7ama5a4a demonstrated the use of fuzzy control.through a set of simple dedicated fuzzy logic chips. in an 7 invertedpendulum7 e$periment! %his is a classic control pro+lem. in ,hich aehicle tries to -eep a pole mounted on its top +y a hinge upright +ymoing +ac- and forth!

• *+serers ,ere impressed ,ith this demonstration. as ,ell as latere$periments +y 7ama5a4a in ,hich he mounted a ,ine glasscontaining ,ater or een a lie mouse to the top of the pendulum!%he system maintained sta+ility in +oth cases! Bama-a,a eentually,ent on to organize his o,n fuzzysystems research la+ to help

e$ploit his patents in the field!

2 years after its conception

http://en.wikipedia.org/wiki/Fuzzy_control_system


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eeting Lotfi in :ermanyeeting Lotfi in :ermany

y Fuzzy Logic('ased;esearc%es

• Do+ot Eaigation

– Dealtime pathplanning/y+rid Fuzzy AG0

• (achine Hision – Dealtime colouro+ect

recognition

– Colour correction

– Fuzzy Colour ContrastFusion

– Fuzzyenetic Colour

Contrast Fusion

th Fuzzy "ays /


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eeting Prof# 7ama5a4a in


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Fuzzy Logic is one of the most tal-eda+out technologies to hit the em+edded

control field in recent years! It has already transformed many product

mar-ets in Mapan and orea. and has +egun to attract a ,idespread follo,ing

In the 6nited States! Industry ,atchers predict that fuzzy technology is on its

,ay to +ecoming a multi+illiondollar +usiness!

Introduction of FL in t%e Engineering 4orld =+>s?,

Fuzzy Logic ena+les lo, cost microcontrollers to perform functions traditionally

performed +y more po,erful e$pensie machines ena+ling lo,er cost products

to e$ecute adanced features!

Intel Corporation@s Em'edded icrocomputer "ivision Fuzzy Logic Aperation

http:;;,,,!intel!com;design;mcsJ;designe$;


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Sample )pplicationsSample )pplications

In t%e city of Sendai in


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D)S) has studied fuzzy control for automated space

doc5ing: simulations sho, that a fuzzy control system cangreatly reduce fuel consumption

Canon deeloped an auto(focusing camera that uses a

chargecoupled deice /CC"0 to measure the clarity of the

image in si$ regions of its field of ie, and use the informationproided to determine if the image is in focus! It also trac-s the

rate of change of lens moement during focusing. and controls

its speed to preent oershoot!

http:;;en!,i-ipedia!org;,i-i;FuzzyPsystem

%he camera9s fuzzy control system uses +2 inputs: J to o+tain the current claritydata proided +y the CC" and J to measure the rate of change of lens moement!

%he output is the position of the lens! %he fuzzy control system uses +1 rules and

re2uires +#+ 5ilo'ytes of memory!


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/aier ESL(T2+ Top Load $as%er

Miele WT94 Front !oad "ll#in#$ne Washer / %ryer

"&' !!()(* Front !oad Washer

+anussi +WF(4,*W Front !oad Washer

!' W%(4(-( Front !oad Washer

For 4as%ing mac%ines, Fuzzy Logic control is almost

'ecoming a standard feature

:E $P;!++$/ Top Load $as%er

*thers: Samsung. %oshi+a. Eational. (atsushita. etc!

fuzzy controllers to load,eight. fa+ricmi$. and dirt sensors and automatically set the

,ash cycle for the +est use of po,er. ,ater. and detergent!


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Control SystemsControl Systems

• Conentional Control s! Fuzzy Control


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Control Systems in :eneralControl Systems in :eneral

%he aim of any control system is to produce a set of

desired outputs for a given set of inputs!

A household thermostat ta-es a

temperature input and sends a controlsignal to a furnace!

A car engine controller responds to aria+les such as engine position.

manifold pressure and cylinder temperature to regulate fuel flo, and

spar- timing!

Samples

A'6ective

Cran-shaft /red0. pistons /gray0 intheir cylinders /+lue0. and fly,heel

/+lac-0

Image0 %ttp0en#4i5ipedia#org4i5iCran5s%aft


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Conventional Control vs# FuzzyConventional Control vs# Fuzzy

In the simplest case. a controller ta-es its cues from a loo-up ta+le. ,hich tells,hat output to produce for eery input or com+ination of inputs!

%he ta+le might tell the controller.

3IF temperature is 8., T/ED increase furnace fan speed to 1 ;P#4

%he pro+lem ,ith the ta+ular approach is that the ta'le can get very longta'le can get very long.

especially in situations ,here there are many inputs or outputs! And that. inturn. may reGuire more memory t%an t%e controller can %andlemay reGuire more memory t%an t%e controller can %andle. or more

than is costeffectie!

%a+ular control mechanisms may also gie a 'umpy, uneven response'umpy, uneven response. as the

controller umps from one ta+le+ased alue to the ne$t!

"ra4'ac5s

Loo5(up ta'le

Sample


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%he usual alternatie to loo-up ta+les is to hae the controller e$ecute a

mathematical formula – a set of control eGuations t%at express t%e output

as a function of t%e input!

Ideally. these e2uations represent an accurate model of the system+ehaiour!

For example:

%he formulas can +e ery comple$. and ,or-ing them out in realtime may +e

more than an afforda+le controller /or machine0 can manage!

"o4nside

at%ematical formula

θ θ θ θ θ sinsin)cos(cos)sin(2

2

2

2

mgl l l t

ml l xt

m =

∂

∂−

+

∂

∂


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It may +e difficult or impossi+le to derie a ,or-a+le mathematical model in the

first place. ma-ing +oth ta+ular and formula+ased methods impractical!

"o4nside of at%ematical modeling approac%

%hough an automotie engineer might understand the general relationship

+et,een say. ignition timing. air flo,. fuel mi$ and engine D'(. the e$act maththat underlies those interactions may +e completely o+scure!

$%y use Fuzzy Logic&

FL oercomes the disadantages of +oth ta+le+ased and formula+asedcontrol!

Fuzzy has no un4ieldy memory reGuirementsno un4ieldy memory reGuirements of loo-up ta+les.

and no %eavy num'er(crunc%ing demandsno %eavy num'er(crunc%ing demands of formula+ased solutions!


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$%y use Fuzzy Logic&

FL can ma-e deelopment and implementation much simpler!

It needs no intricate mathematical models. only a practical understanding of the

oerall system +ehaiour!

FL mechanisms can result to %ig%er accuracy%ig%er accuracy and smoot%er controlsmoot%er control as ,ell!


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Fuzzy Logic ExplainedFuzzy Logic ExplainedFuzzy Set T%eory

Is a man ,hose height is @O 111;


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Fuzzy Logic ExplainedFuzzy Logic Explained

Fuzzy Set T%eory

Is a man ,hose height is @O 111;


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Fuzzy Inference ProcessFuzzy Inference Process

• What are the steps inoled in creating a

Fuzzy Control System?


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Fuzzification ;ule

Evaluation"efuzzification

e!g! thetae!g! theta e!g! forcee!g! force

Fuzzification0 %ranslate input into truth alues;ule Evaluation0 Compute output truth alues

"efuzzification0 %ransfer truth alues into output



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A'stacle )voidance Pro'lemA'stacle )voidance Pro'lem

o'stacle

/o+s$. o+sy0

θ

/$.y0

Can you descri+e ho, the ro+ot

should turn +ased on the position

and angle of the o+stacle?

;o'ot Davigation

"emonstration

*+stacle Aoidance ) %arget 'ursuit


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Another example:Another example: Fuzzy Sets for Robot NavigationFuzzy Sets for Robot Navigation

Angle and "istanceAngle and "istance

Su' ranges for angles J distances overlapSu' ranges for angles J distances overlap**

S)LL

E"IB

L);:E

DE);

F);

KE;7 F);


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Fuzzy SystemsFuzzy Systems forfor A'stacle )voidanceA'stacle )voidance

NEAR FAR VERY FAR

SMALL Very Sharp Sharp Turn Med Turn

MEDIUM Sharp Turn Med Turn Mild Turn

LARGE Med Turn Mild Turn Zero Turn

earest $stacle 0%istance and "ngle1

NEAR FAR VERY FAR

SMALL Very Slow Slow Speed Fast Fast

MEDIUM Slow Speed Fast Speed Very Fast

LARGE Fast Speed Very Fast Top Speed

e#g# If the Distance from the *+stacle is NEAR and

the Angle from the *+stacle is SMALL

T%en turn Very Sharply !

Fuzzy System 1 =Steering?Fuzzy System 1 =Steering?

Fuzzy System 3 =Speed )d6ustment?Fuzzy System 3 =Speed )d6ustment?

e.g. If the Distance from the *+stacle is NEAR and

the Angle from the *+stacle is SMALL

T%en moe Very Slowly.

Kision System

)ngle)ngle

SpeedSpeed

http://var/www/apps/conversion/tmp/scratch_6/C:%5CCore%5CResearch%5CConferences%5C9th%20Fuzzy%20Days%202006%5CUnmanned%2030.7%20-%20Scaled%20+%20Waypoint%20-%20Dynamic%20Obstacles%20-%20NZ%5CUnmanned%2030.7%20-%20Scaled%20+%20Waypoint%20-%20Dynamic%20Obstacles%20-%20NZ%5CUnmanned.exe


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Fuzzy ControlFuzzy Control%i22erent stages o2 Fuzzy control %i22erent stages o2 Fuzzy control

Input aria+les are assigned degrees of mem+ership in arious classes

+# Fuzzification

e!g! A temperature input might +e graded according to its degree of coldness.

coolness. ,armth or heat!

$e 4ill see a complete$e 4ill see a complete

example of t%e stepsexample of t%e steps

involved later#involved later#

%he purpose of fuzzificationfuzzification is to map t%e inputs from a set of sensors /or

features of those sensors0 to values from to ++ using a set of input

mem+ership functions!


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FuzzificationFuzzification

5!5 5!@5!@ 1!51!5

1!5

5!5

DE:)TIKEDE:)TIKE PASITIKEPASITIKE*E;A*E;A

Fuzzy Sets Q R Eegatie. 8ero. 'ositie

Fuzzi2ication &3ampleFuzzi2ication &3ample

Crisp Input: x #2.x #2.$%at is t%e degree of$%at is t%e degree of

mem'ers%ip of x in eac%mem'ers%ip of x in eac%

of t%e Fuzzy Sets&of t%e Fuzzy Sets&

Assuming that ,e are using trapezoidal mem+ership functions!


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Sample CalculationsSample Calculationsx 0.25=

ZE

x a d xF (0.25) ax in !"! !0

# a d c

− − = ÷ ÷− −

0.25 ( ") " 0.25ax in !"! ! 0

0.25 ( ") " 0.25

− − −= ÷ ÷− − − −

( )( )ax in ".$%!"!" !0

"

=

=

&

0.25 ( 0.5) ' 0.25F (0.25) ax in !"! ! 0

0.5 ( 0.5) ' 0.25

− − −= ÷ ÷− − −

( )( )ax in 0.%5!"!5.5 !0

0.%5

=

=

0.25 ( ') 0.5 0.25F (0.25) ax in !"! ! 0

2.5 ( ') 0.5 ( 0.5)

− − −= ÷ ÷− − − − −

( )( )ax in $.5!"!0.25 !0

0.25

=

=

Crisp Input:

Fzero/5!


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Sample CalculationsSample CalculationsCrisp Input:

Fzero/5!


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LeftTrapezoid

LeftPSlope Q 5

DightPSlope Q 1 ; /A &0

CAS# 1: T U a

(em+ership Halue Q 1

CAS#


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Trapezoidal em'ers%ip FunctionsTrapezoidal em'ers%ip Functions

;ig%tTrapezoid

LeftPSlope Q 1 ; /& A0

DightPSlope Q 5

CAS# 1: T UQ a

(em+ership Halue Q 5

CAS#


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Trapezoidal em'ers%ip FunctionsTrapezoidal em'ers%ip Functions

;egular Trapezoid

LeftPSlope Q 1 ; /& A0

DightPSlope Q 1 ; /C "0

CAS# 1: T UQ a *r T VQ d (em+ership Halue Q 5

CAS#


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Inputs are applied to a set of ift%en control rules!

2# ;ule Evaluation

e!g! IF temperature is ery hot. T/ED set fan speed ery high!


F C t lF C t l


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i2 (inpt! is mem"ership #nction!$ and/or (inpt% is mem"ership

#nction%$ and/or &.

then (otpt is otpt mem"ership #nction$.

For e$ample. one could ma-e up a rule that says:

i2 temperatre is high and hmi'ity is high then room is hot.

Fuzzy rulesFuzzy rules are al4ays 4ritten in t%e follo4ing form0


F C t lF C t l


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Inputs are applied to a set of ift%en control rules!

2# ;ule Evaluation

e!g! IF temperature is ery hot. T/ED set fan speed ery high!

%he results of arious rules are smme' together to generate a set of 3fuzzy

outputs4!

Fuzzy ControlFuzzy Control

S S

S ZE &S

&S &S &

%i22erent stages o2 Fuzzy control %i22erent stages o2 Fuzzy control

E 8# '

E

8#

'

F)F)

*utputsELQ@

ESQ

8#Q5

'SQ

'LQ@!5

W1 W4 W7

W2 W5 W8

W3 W6 W9

x

y

F C t lF C t l


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[ ] [ ]

[ ] [ ]

[ ] [ ]

[ ] [ ]

[ ] [ ]

"

2 ZE

' &

* ZE

5 ZE ZE

+ in F (0.25)!F ( 0.25) in 0.25!0.%5 0.25

+ in F (0.25)!F ( 0.25) in 0.25!" 0.25

+ in F (0.25)!F ( 0.25) in 0.25!0.25 0.25

+ in F (0.25)!F ( 0.25) in "!0.%5 0.%5

+ in F (0.25)!F ( 0.25) in "!" "

+

= − = =

= − = =

= − = =

= − = =

= − = =

[ ] [ ]

[ ] [ ]

[ ] [ ]

[ ] [ ]

$ ZE &

% &

, & ZE

- & &

in F (0.25)!F ( 0.25) in "!0.25 0.25

+ in F (0.25)!F ( 0.25) in 0.%5!0.%5 0.%5

+ in F (0.25)!F ( 0.25) in 0.%5!" 0.%5

+ in F (0.25)!F ( 0.25) in 0.%5!0.25 0.25

= − = =

= − = =

= − = =

= − = =

ule &5aluation &3ampleule &5aluation &3ample

Assuming that ,e are using the conunction operator /AE"0 in theantecedents of the rules. ,e calculate the rule firing strengt% $n!

S S

S ZE &S

&S &S &

E 8# '

E

8#

'

F)F)

x

y

W1 W4 W7

W2 W5 W8

W3 W6 W9

F C t lF C t l


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Fuzzy outputs are com+ined into discrete alues needed to drie the control

mechanism

$e 4ill see a complete$e 4ill see a complete

example of t%e stepsexample of t%e steps

involved later#involved later#

1# "efuzzification

/e!g! A cooling fan0


F C t lF C t l


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" 2 ' * 5 $ % , -

-

i

i "

(+ + S + &S + S + ZE + &S + S + &S + &)/T&/T

+

=

× + × + × + × + × + × + × + × + ×=

∑

( )

( )

0.25 ( 5) 0.25 2.5 0.25 2.5 0.%5 2.5 " 0 0.25 2.5 0.%5 2.5 0.%5 2.5 0.25 5

0.25 0.25 0.25 0.%5 " 0.25 0.%5 0.%5 0.25

× − + × + × + × + × + × + × + × + ×=

+ + + + + + + +

Q 1!


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ExampleExample

• LetOs loo- at the Inerted 'endulum

'ro+lem

S f StS f St


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Summary of StepsSummary of Steps

1! determining a set of fuzzy rules

N! com+ining the fuzzified inputs according to the fuzzy rules to esta+lish

a rule strength.

=! finding the conse2uence of the rule +y com+ining the rule strength andthe output mem+ership function /if itOs a mamdani FIS0.

@! com+ining the conse2uences to get an output distri+ution. and

J! defuzzifying the output distri+ution /this step applies only if a crisp

output /class0 is needed0!

To compute t%e output of t%is FIS given t%e inputs, one must go t%roug% six

steps0

I t d P d l P 'lI t d P d l P 'l


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Inverted Pendulum Pro'lemInverted Pendulum Pro'lem

A pole ,ith a ,eight on top is mounted on a motordrien cart! %he pole

can s,ing freely. and the cart must moe +ac- and forth to -eep it ertical!

A controller monitors the angle and motion of the pole and directs the cartto e$ecute the necessary +alancing moements!

) Classic test case in em'edded control

) :limpse at /istory0 International Conference in %o-yo /1>K0 Ta5es%i 7ama5a4a

demonstrated the use of fuzzy control. through a set of simple dedicated fuzzy logic

chips. in an 7inverted penduluminverted pendulum7 e$periment! /Later e$periments: mounted a ,ine

glass containing ,ater or een a lie mouse to the top of the pendulum0!

I t d P d l P 'lIn erted Pend l m Pro'lem


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%he solution uses a secondorder differential e2uation thatdescri+es cart motion as a function of pole position and elocity:

Conventional mat%ematical solution

θ θ θ θ θ sinsin)cos(cos)sin(2

2

2

2

mgl l l t

ml l xt

m =

∂∂

−

+

∂∂

I t d P d l P 'lInverted Pendulum Pro'lem


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Sensed values0

T – position of o+ect ,ith respect to the horizontal a$is

θ angle of pole relatie to the ertical a$is

"erived values0

TO Helocity along the $a$is

θO Angular elocity

Input varia'les0 sensed and deried alues

Controller output0 FF – force to +e applied to the cart

" i d I t K l"erived Input Kalues


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"erived Input Kalues"erived Input Kalues

t%eta time t%eta@

< 1

15 < >

N5 N

A sample calculation of some of the deried Halues: angular

elocity /thetaO0

We can derie ne, input alues for our Fuzzy Control System

using 'hysics e2uations!

In erted Pend l m Pro'lemInverted Pendulum Pro'lem


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origin

Parameters for a Fuzzy SystemParameters for a Fuzzy System


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Parameters for a Fuzzy SystemParameters for a Fuzzy System

1! specify the fuzzy sets to +e associated ,ith each aria+le!

N! specify the shape of the mem+ership functions!

Ance you %ave determined t%e appropriate inputsinputs andoutputsoutputs for your application, t%ere are t%ree steps to

designing t%e parameters for a fuzzy system0

Fuzzy SetsFuzzy Sets


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We might +egin designing a fuzzy system +y su+diiding the t,o input

aria+les /pole angle and angular elocity0 into mem+ership sets!

%he angleangle could +e descri+ed as:

1! Inclined to the Left /E0!

N! Inclined to the Dight /'0!



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%he angular velocityangular velocity could +e descri+ed as:

1! Falling to the Left /E0!

N! Falling to the Dight /'0!

Fuzzy ;ulesFuzzy ;ules


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Fuzzy rule +ase and the corresponding FA(( for the velocity and

position ectors of the inerted pendulum+alancing pro+lem

1! IF cart is on the left AE" cart is going left %#E largely push cart to the right

J! IF cart is centered AE" cart is going right %#E slightly push cart to the left

K! IF cart is on the right AE" cart is going left %#E donOt push cart

>! IF cart is on the right AE" cart is not moing %#E push cart to the left

! IF cart is on the right AE" cart is going right %#E largely push cart to the left



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Fuzzy rule +ase and the corresponding FA(( for the angle

and angular velocity ectors of the inerted pendulum+alancing pro+lem

1! IF pole is leaning to the left AE" pole is dropping to the left %#E largely

push cart to the left

the cart

! and so on. and so forth

!

!

Position vs KelocityPosition vs Kelocity


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Position vs# KelocityPosition vs# Kelocity

If t%e cart is too near t%e end of t%e pat%, t%en regardless of t%e

state of t%e 'room angle pus% t%e cart to4ards t%e ot%er end#

1 x 1 F)

)ngle vs )ngular Kelocity)ngle vs )ngular Kelocity


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)ngle vs# )ngular Kelocity)ngle vs# )ngular Kelocity

If t%e 'room angle is too 'ig or c%anging too Guic5ly, t%en

regardless of t%e location of t%e cart on t%e cart pat%, pus% t%ecart to4ards t%e direction it is leaning to#

1 x 1 F)

em'ers%ip Functionsem'ers%ip Functions


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em'ers%ip Functionsem'ers%ip Functionsem'ers%ip Functions for t%e Cart PositionPosition

xx

Trapezoid Vertices

Left Trapezoid Regular Rigt

1 2 1 ".5 1 "

3 1 " 3 1 0.5 3 1 2

4 1 0 4 1 0.5 4 1 0

1 0 1 ".5 1 0

Ta5e note of t%e

position of t%e

origin#

Ta5e note of t%e

position of t%e

origin#



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em'ers%ip Functionsem'ers%ip Functionsem'ers%ip Functions for t%e Cart KelocityKelocity

Trapezoid Vertices


1 ' 1 ".5 1 0

3 1 0 3 1 0.5 3 1 '

4 1 0 4 1 0.5 4 1 0

1 0 1 ".5 1 0



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em'ers%ip Functions for t%e Pole )ngle)ngle

Trapezoid Vertices


1 0." 1 0." 1 0

3 1 0 3 1 0.0' 3 1 0."

4 1 0 4 1 0.0' 4 1 0

1 0 1 0." 1 0



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em'ers%ip Functions for t%e !room )ngular Kelocity)ngular Kelocity

Trapezoid Vertices


1 0." 1 0."5 1 0

3 1 0 3 1 0.0' 3 1 0."

4 1 0 4 1 0.0' 4 1 0

1 0 1 0."5 1 0


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• *&S%ACL# AH*I"AEC# F688B

L#C%6D#!doc

• Lec< Fuzzy Logic Fuzzy#2ns!doc

Fuzzy ;uleFuzzy ;ule


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Fuzzy ;uleFuzzy ;ule

FDE);/"istance0 Q degree of

mem+ership of the gien distance

in the Fuzzy Set E#AD

If "istance is E#AD and Angle is S(ALL T%en %urn Sharp Left!

FS)LL/Angle0 Q degree of

mem+ership of the gien angle

in the Fuzzy Set S(ALL

Could +e aconstant or

another

(F

amdani Inference Systemamdani Inference System


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amdani Inference Systemamdani Inference System

IF T/ED;uleStrengt%

Input "istri'utionAutput

"istri'ution

Mo 7o

xo yo

and

and

T4o input, t4o rule amdani FIS 4it% crisp inputs

Fuzzy rules are a collection of linguistic statements that descri+e ho, the FIS should

ma-e a decision regarding classifying an input or controlling an output!


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ore detailsNore detailsN

• Fuzzy Inference Systems =FIS?Fuzzy Inference Systems =FIS?• Fuzzy ;ulesFuzzy ;ules• Fuzzy Com'ination AperatorsFuzzy Com'ination Aperators• em'ers%ip functionsem'ers%ip functions

Fuzzy InferenceFuzzy Inference


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Fuzzy InferenceFuzzy Inference

Fuzzy inference is the process of formulating the mapping from a gien input to

an output using fuzzy logic! %he mapping then proides a +asis from ,hichdecisions can +e made. or patterns discerned! %he process of fuzzy inference

inoles all of the pieces that are descri+ed in the preious sections: (em+ership

Functions. Logical *perations. and If%hen Dules!

amdani FISamdani FIS


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amdani(type inference. e$pects the output mem+ership functions to +e fuzzy

sets! After the aggregation process. there is a fuzzy set for each output aria+le thatneeds defuzzification!

http:;;,,,!math,or-s!com;access;helpdes-;help;tool+o$;fuzzy;inde$!html?;access;helpdes-;help;tool+o$;fuzzy;+pK>lJP1!html)http:;;,,,!math,or-s!com;cgi+in;te$is;,e+inator;search;?

d+Q(SS)pro$Qpage)rorderQK@5)rpro$QK@5)rdfre2Q@55)r,fre2Q@55)rleadQ


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Flo4 of Fuzzy InferenceFlo4 of Fuzzy Inference


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Flo4 of Fuzzy InferenceFlo4 of Fuzzy Inference

In this figure. the flo, proceeds up from the inputs in the lo,er left. then

across each ro,. or rule. and then do,n the rule outputs to finish in the lo,er

right! %his compact flo, sho,s eerything at once. from linguistic aria+le

fuzzification all the ,ay through defuzzification of the aggregate output!



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http:;;en!,i-ipedia!org;,i-i;FuzzyPsystem

(a$min inferencing and centroid defuzzification for a system ,ith input aria+les 7$7.

7y7. and 7z7 and an output aria+le 7n7! Eote that 7mu7 is standard fuzzylogic

nomenclature for 7truth alue7:

Sample0 amdaniSample0 amdani


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Consider a pro+lem ,ith < input aria+les. size and 4eig%t. and one output aria+le.

Guality, ,ith the follo,ing lingustic term sets associated0

Classical Inference Engine0 max(min(C:max(min(C:

%he semantics of these linguistic terms are defined 'y triangular(s%aped

mem'ers%ip functions0

e2erence: :enetic fuzzy systems 'y Ascar CordOn, Francisco /errera, Fran5 /offmann



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;ule !ase =;!?0


Let us consider a sample input to the system. Mo 2, 2.Q size, 4eig%tQ




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;ule !ase0


Let us consider a sample input to the system. Mo 2, 2.Q#

T%is input is matc%ed against t%e rule antecedents in order to determine t%e rule(rule(

firing strengt%firing strengt% hhi i of eac% rule ;i in t%e ;ule !ase#

6sing a minimum %norm as conunctie. the follo,ing results are o+tained:




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Sample0 amdaniSample0 amdaniClassical Inference Engine0 max(min(C:max(min(C:

;ule(firing strengt%;ule(firing strengt% %i of eac% rule ;i in t%e ;ule !ase0

%he inference system applies the%he inference system applies the compositional rule of inferencecompositional rule of inference to o+tain theto o+tain the

inferred fuzzy sets !inferred fuzzy sets !i >>




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Sample0 amdaniSample0 amdani;ule(firing strengt%;ule(firing strengt% %i of eac% rule ;i in t%e ;ule !ase0

Application of Application of compositional rule of inferencecompositional rule of inference to o+tain theto o+tain the inferred fuzzy sets !inferred fuzzy sets !i >>

raphical representation:raphical representation:


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Sample0 amdaniSample0 amdaniraphical representation:raphical representation:

%he final output is calculated +y defuzzification using%he final output is calculated +y defuzzification using Centre of :ravity =C:?Centre of :ravity =C:?


Aggregation of the = indiidual output fuzzy sets +y means of the Aggregation of the = indiidual output fuzzy sets +y means of the maximum t(conormmaximum t(conorm



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Sample0 amdaniSample0 amdaniraphical representation:raphical representation:

%he final output is calculated +y defuzzification using%he final output is calculated +y defuzzification using Centre of :ravity =C:?Centre of :ravity =C:?




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ode )(F)TI 4it% ax ( Centre of :ravity =C:? strategyode )(F)TI 4it% ax ( Centre of :ravity =C:? strategy

RRFirst )ggregate, T%en InferFirst )ggregate, T%en Infer


T%e final crisp output is 7T%e final crisp output is 7oo #1-8 #1-8



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Sample0 amdaniSa p e a da)lternatively, =RFirst Infer, T%en )ggregate?)lternatively, =RFirst Infer, T%en )ggregate?

ode !(FIT) 4it% aximum Kalue 4eig%ted 'y t%e matc%ing strategyode !(FIT) 4it% aximum Kalue 4eig%ted 'y t%e matc%ing strategy




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amdani FIS

• It is possi+le. and in many cases much more efficient. to use asingle spi5e as the output mem'ers%ip function rather than a

distri+uted fuzzy set!

• %his type of output is sometimes -no,n as a singleton output

mem'ers%ip function. and it can +e thought of as a predefuzzified

fuzzy set!

• It enhances the efficiency of the defuzzification process +ecause it

greatly simplifies the computation re2uired +y the more general(amdani method. ,hich finds the centroid of a 2(" function!

• Dather than integrating across the t,odimensional function to find

the centroid. you use the 4eig%ted average of a fe4 data points!

ABTPBT E!E;S/IP FBDCTIAD

Sugeno FISSugeno FIS


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Sugeno FISg

A typical rule in a Sugeno fuzzy model has the form:

If Input 1 Q $ and Input < Q y. then *utput is z ax 'y c

For a zero(order Sugeno model. the output leel zz is a constant /aQ+ Q50!

Sugeno FIS is similar to the (amdani method in many respects! %he first t,o parts of

the fuzzy inference process. fuzzifying the inputs and applying the fuzzy operator. are

e$actly the same! %he main difference +et,een (amdani and Sugeno is that theSugeno output mem+ership functions are either linear or constant!

Sugeno FISSugeno FIS


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Suge o Sg

FIS0 Sugeno vs# amdaniFIS0 Sugeno vs# amdani


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gg

)dvantages of t%e amdani et%od

It is intuitie!

It has ,idespread acceptance!

It is ,ell suited to human input!

)dvantages of t%e Sugeno et%od

It is computationally efficient!

It can +e used to model any inference system in ,hich the

output mem+ership functions are either linear or constant!

It ,or-s ,ell ,ith linear techni2ues /e!g!. 'I" control0!

It ,or-s ,ell ,ith optimization and adaptie techni2ues!

It has guaranteed continuity of the output surface!

It is ,ell suited to mathematical analysis!



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yy

Fuzzy Set !Fuzzy Set )

#$amples:

We ,ill use the follo,ing fuzzy sets in e$plaining the different fuzzy

operators that follo,s ne$t!

Fuzzy com'inations =T(norms?


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y = ?In ma-ing a fuzzy rule. ,e use the concept of 3and4. 3or 4. and sometimes

3not4! %he sections +elo, descri+e the most common definitions of these

3fuzzy com+ination4 operators! Fuzzy com+inations are also referred to as3T(norms4!

Fuzzy Rand

%he fuzzy 3and4 is ,ritten as:

,here A is read as 3the mem+ership in class A4 and ) is read as 3themem+ership in class &4!

Intersection of A and &

#$ample:

Fuzzy RFuzzy Randand


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yy

*ade% ( min= 6 " =x?, 6 7 =x??

%his techni2ue. named after the inentor of fuzzy set

%heoryY it simply computes the 3and4 +y ta-ing the minimum of the t,o /or

more0 mem+ership alues! %his is the most common definition of the fuzzy

3and4!

Product ( 6 " =x? U 6 7 =x?

%his techni2ue computes the fuzzy 3and4 +y multiplying the t,o mem+ership

alues!

%here are many ,ays to compute 3and4! %he t,o most common are:

Fuzzy RFuzzy Randand


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yy!ot% tec%niGues %ave t%e follo4ing t4o properties0

T=,? T=a,? T=,a?

T=a,+? T=+,a? a

*ne of the nice things a+out +oth definitions is that they also can +e used to

compute the &oolean 3and4! %he fuzzy Rand is an e$tension of the !oolean

Rand to num+ers that are not ust 5 or 1. +ut +et,een 5 and 1!

Fuzzy Ror


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y

Fuzzy Ror

%he fuzzy 3or4 is ,ritten as:

,here A is read as 3the mem+ership in class A4 and ) is read as 3themem+ership in class &4!

6nion of ) and !

#$ample:

Fuzzy RFuzzy Ror or


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yy

T%e fuzzy Ror is an extension of t%e !oolean Ror to

num'ers t%at are not 6ust or +, 'ut 'et4een and +#

Fuzzy RFuzzy Ror or


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yy

*ade%ma$/ A /$0. ) /$00

it simply computes the 3or4 +y ta-ing the maximum of the t,o /or more0

mem+ership alues! %his is the most common definition of the fuzzy 3or4!

Product

/ A /$0 X ) /$00 – / A /$0 G ) /$00

%his techni2ue uses the difference +et,een the sum of the t,o /or more0

mem+ership alues and the product of their mem+ership alues!

%here are many ,ays to compute 3or 4! %he t,o most common are:

Similar to the fuzzy 3and4. +oth definitions of the fuzzy 3or4 also can +e used to

compute the &oolean 3or4!

( ! ) ax( ! ) x y x yσ =

( ! ) x y x y xyσ = + −

Fuzzy RFuzzy Ror or


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yy

Lu5asie4icz "is6unction

min/1. A /$0 X ) /$00

*ther ,ays to compute Fuzzy 3or 4:

( ! ) in("! ) x y x yσ = +

/amac%er "is6unction "is6unction

2

( ! ) "

x y xy

x y xyσ

+ −

= −

)()("

)()(2)()(

x x

x x x x

B A

B A B A

µ µ

µ µ µ µ

−

−+

Fuzzy RFuzzy Ror or


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yy*ther ,ays to compute Fuzzy 3or 4:

Einstein "is6unction

)()(")()(

x x x x

B A

B A

µ µ µ µ

+ +

( ! )"

x y x y

xyσ

+=

+

Fuzzy RFuzzy Rnotnot


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y

DAT =)? + ( )

Eegation of )

#$ample:

Fuzzy set )

Fuzzy Set operationsFuzzy Set operations


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y p

Fuzzy logic is a supersetsuperset of conventional=!oolean? logic

All other operations on classical sets also hold for fuzzy sets. e$cept for

the e$cluded middle la,s!

0≠∩

≠∪

A A

X A A

ConseGuenceConseGuence


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T%e conseGuence of a fuzzy rule is computed using t4o steps0

Computing the rule strength +y com+ining the fuzzified inputs

using the fuzzy com'ination process

IF T/ED;uleStrengt%

Input "istri'ution Autput

"istri'ution

Mo 7o

xo yo

and

and

In this e$ample. the fuzzy 3and4 is used to com+ine the mem+ership functions to compute the rule strength!

+

ConseGuenceConseGuence


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Clipping the output mem+ership function at the rule strength!

IF T/ED;uleStrengt%


"istri'ution

Mo 7o

xo yo

and

and

In this e$ample. the fuzzy 3and4 is used to com+ine the mem+ership functions to compute the rule strength!

2

ConseGuenceConseGuence1


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%he outputs of all of the fuzzy rules must no, +e com+ined to o+tain one

fuzzy output distri+ution! %his is usually. +ut not al,ays. done +y using the

fuzzy 3or4! %he figure +elo, sho,s an e$ample of this!

IF T/ED;uleStrengt%


"istri'ution

Mo 7o

xo yo

and

and

%he output

mem+ershipfunctions on the

right hand side of

the figure are

com+ined using the

fuzzy 3or 4 to o+tain

the outputdistri+ution sho,n

on the lo,er right

corner of the figure!

"efuzzification of Autput"i t i' ti

.


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"istri'ution

In many instances. it is desired to come up ,ith a single crisp output from aFIS! For e$ample. if one ,as trying to classify a letter dra,n +y hand on a

dra,ing ta+let. ultimately the FIS ,ould hae to come up ,ith a crisp num+er

to tell the computer ,hich letter ,as dra,n! %his crisp num+er is o+tained in a

process -no,n as defuzzification!

%here are t,o common techni2ues for defuzzifying:

Center of mass %his techni2ue ta-es the output distri+ution found inthe preious slide and finds its center of mass to come up ,ith one crisp

num+er! %his is computed as follo,s:

,here z is the center of mass and Zc is

the mem+ership in class c at alue 8 !

An e$ample outcome of this

computation is sho,n in the figure at the

right!

a?

"efuzzification of Autput"i t i' ti


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"istri'ution

ean of maximum %his techni2ue ta-es the output distri+utionfound in the preious section and finds its mean of ma$ima to come up ,ith

one crisp num+er! %his is computed as follo,s:

'?

,here z is the mean of ma$imum. * 6 is the point at ,hich

the mem+ership function is ma$imum. and l is the num+er

of times the output distri+ution reaches the ma$imum leel!

An e$ample outcome of this computation is sho,n the

figure at the right!

amdani FIS 4it% a Fuzzy Inputamdani FIS 4it% a Fuzzy Input


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IF T/ED;uleStrengt%

and

and

) t4o Input, t4o rule amdani FIS 4it% a fuzzy input

sho,s a modification

of the (amdani FIS,here the input y5 is

fuzzy. not

crisp!

%his can +e used to

model inaccuracies in

the measurement!

For e$ample. ,e may

+e measuring theoutput of a pressure

sensor! #en ,ith the

e$act same pressure

applied. the sensor is

measured to hae

slightly different

oltages!%he fuzzy input

mem+ership

function models this

uncertainty!

%he input fuzzy function is com+ined ,ith the rule input mem+ership function +y using the fuzzy

3and4



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sigmf/$.[a c\0. as gien in the follo,ing e2uation +y f/$.a.c0 is a mapping on a ector

$. and depends on t,o parameters a and c!

T%e Sigmoidal function



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T%e :aussian function

%he symmetric aussian function depends on t,o parameters and c as gien +yσ



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T%e Trapezoidal function

or

%he trapezoidal cure is a function of a ector. $. and depends on four scalarparameters a. +. c. and d. as gien +y

%he parameters a and d locate the 7feet7 of the trapezoid and the parameters + and c

locate the 7shoulders!7

/edges/edges http:;;+log!peltarion!com;


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'.")( x A

µ

%.")( x A µ

2)( x A

µ

')( x A µ

*)( x A µ

2

"

)( x A µ

5.0)(0i6 )(22 ≤≤ x x A A µ µ

) little) little

Slig%tlySlig%tly

KeryKery

ExtremelyExtremely

Kery veryKery very

Some4%atSome4%at

IndeedIndeed

/edge EffectAperator

")(0.5i6 ))("(2" 2 ≤


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Sigma N!@=

c @

Fuzzy Logic )pplicationsFuzzy Logic )pplications


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•Inverted Pendulum – Analyze the simulation /see implementation0

– Hariations of settings:

• /N$N. @$@ FA((S0

• Shape of mem+ership functions

• Fuzzy *utput alues settings. time delay

– 6tilisation of #ulerOs (ethod in the simulation

/see pdf0

Fuzzy Logic )pplicationsFuzzy Logic )pplications


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•;o'ot Davigation – "eriation of e2uations for motion

– Hariations of settings:

• FA((

• Shape of mem+ership functions

• Fuzzy *utput alues settings. time delay

– Startup codes

;eferences;eferences


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• enetic fuzzy systems +y *scar Cord]n. Francisco errera. Fran-

offmann• Eeural Eet,or- and Fuzzy Logic Applications in C;CXX /Wiley 'rofessional

Computing0 +y Stephen Welstead

• Fuzzy Logic ,ith #ngineering Applications +y %imothy Doss

• Fuzzy Sets and 'attern Decognition +y &enamin napp

Euler et%odEuler et%od


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• In mathematics and computational science. the Euler met%od.

named after Leonhard #uler. is a numerical procedure forsoling ordinary differential e2uations /*"#s0 ,ith a gieninitial alue! It is the most +asic -ind of e$plicit method fornumerical integration for ordinary differential e2uations!

Sorce: http:**www.answers.com*topic*eler+metho'+!,cat-technology

Euler et%odEuler et%od


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• %he idea is that ,hile the cure is initially un-no,n. its starting point. ,hich,e denote +y A5. is -no,n /see the picture on top right0! %hen. from thedifferential e2uation. the slope to the cure at A5 can +e computed. andso. the tangent line!

• %a-e a small step along that tangent line up to a point A1! If ,e pretendthat A1 is still on the cure. the same reasoning as for the point A5 a+oecan +e used! After seeral steps. a polygonal cure A5 A1 A

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Lec2011 - 3 - AI - Fuzzy Logic