View
228
Download
0
Category
Preview:
Citation preview
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
1/108
1
159.320 FUZZY LOGICFUZZY LOGIC
22
Introduction, Inference System, Fuzzy Control
System, Examples
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
2/108
menu
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
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
3/108
menu
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
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
4/108
menu
Fuzzy LogicFuzzy Logic
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
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
5/108
menu
Fuzzy LogicFuzzy Logic
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!
$%at is Fuzzy Logic&
It is a+le to simultaneously handle numerical data and linguistic
-no,ledge!
A techni2ue that facilitates the control of a complicated system ,ithout
-no,ledge of its mathematical description!
FL is in fact. a precise pro'lem(solving met%odologya precise pro'lem(solving met%odology!
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
6/108
menu
Fuzzy LogicFuzzy Logic
Professor Lotfi )# *ade%
In +-.. Lotfi )# *ade% of the 6niersity of California at &er-eley pu+lished
7Fuzzy Sets.7 ,hich laid out the mathematics of fuzzy set theory and. +y
e$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;
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
7/108menu
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$ploit his patents in the field!
2 years after its conception
http://en.wikipedia.org/wiki/Fuzzy_control_system
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
8/108
eeting Lotfi in :ermanyeeting Lotfi in :ermany
y Fuzzy Logic('ased;esearc%es
• Do+ot Eaigation
– Dealtime pathplanning/y+rid Fuzzy AG0
• (achine Hision – Dealtime colouro+ect
recognition
– Colour correction
– Fuzzy Colour ContrastFusion
– Fuzzyenetic Colour
Contrast Fusion
th Fuzzy "ays /
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
9/108menu
eeting Prof# 7ama5a4a in
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
10/108menu
Fuzzy LogicFuzzy Logic
Fuzzy Logic is one of the most tal-eda+out technologies to hit the em+edded
control field in recent years! It has already transformed many product
mar-ets in Mapan and orea. and has +egun to attract a ,idespread follo,ing
In the 6nited States! Industry ,atchers predict that fuzzy technology is on its
,ay to +ecoming a multi+illiondollar +usiness!
Introduction of FL in t%e Engineering 4orld =+>s?,
Fuzzy Logic ena+les lo, cost microcontrollers to perform functions traditionally
performed +y more po,erful e$pensie machines ena+ling lo,er cost products
to e$ecute adanced features!
Intel Corporation@s Em'edded icrocomputer "ivision Fuzzy Logic Aperation
http:;;,,,!intel!com;design;mcsJ;designe$;
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
11/108
Sample )pplicationsSample )pplications
In t%e city of Sendai in
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
12/108
Sample )pplicationsSample )pplications
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!
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
13/108
Sample )pplicationsSample )pplications
/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!
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
14/108menu
Control SystemsControl Systems
• Conentional Control s! Fuzzy Control
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
15/108menu
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
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
16/108menu
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$t!
"ra4'ac5s
Loo5(up ta'le
Sample
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
17/108menu
Conventional Control vs# FuzzyConventional Control vs# Fuzzy
%he usual alternatie to loo-up ta+les is to hae the controller e$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$. and ,or-ing them out in realtime may +e
more than an afforda+le controller /or machine0 can manage!
"o4nside
at%ematical formula
θ θ θ θ θ sinsin)cos(cos)sin(2
2
2
2
mgl l l t
ml l xt
m =
∂
∂−
+
∂
∂
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
18/108menu
Conventional Control vs# FuzzyConventional Control vs# Fuzzy
It may +e difficult or impossi+le to derie a ,or-a+le mathematical model in the
first place. ma-ing +oth ta+ular and formula+ased methods impractical!
"o4nside of at%ematical modeling approac%
%hough an automotie engineer might understand the general relationship
+et,een say. ignition timing. air flo,. fuel mi$ 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!
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
19/108menu
Conventional Control vs# FuzzyConventional Control vs# Fuzzy
$%y use Fuzzy Logic&
FL can ma-e deelopment and implementation much simpler!
It needs no intricate mathematical models. only a practical understanding of the
oerall system +ehaiour!
FL mechanisms can result to %ig%er accuracy%ig%er accuracy and smoot%er controlsmoot%er control as ,ell!
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
20/108menu
Fuzzy Logic ExplainedFuzzy Logic ExplainedFuzzy Set T%eory
Is a man ,hose height is @O 111;
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
21/108menu
Fuzzy Logic ExplainedFuzzy Logic Explained
Fuzzy Set T%eory
Is a man ,hose height is @O 111;
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
22/108
menu
Fuzzy Inference ProcessFuzzy Inference Process
• What are the steps inoled in creating a
Fuzzy Control System?
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
23/108
menu
Fuzzy Inference ProcessFuzzy Inference Process
Fuzzification ;ule
Evaluation"efuzzification
e!g! thetae!g! theta e!g! forcee!g! force
Fuzzification0 %ranslate input into truth alues;ule Evaluation0 Compute output truth alues
"efuzzification0 %ransfer truth alues into output
Fuzzy Inference ProcessFuzzy Inference Process
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
24/108
menu
A'stacle )voidance Pro'lemA'stacle )voidance Pro'lem
o'stacle
/o+s$. 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
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
25/108
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);
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
26/108
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
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
27/108
menu
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!
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
28/108
menu
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!
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
29/108
menu
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 $.5!"!0.25 !0
0.25
=
=
Crisp Input:
Fzero/5!
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
30/108
menu
Sample CalculationsSample CalculationsCrisp Input:
Fzero/5!
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
31/108
menu
LeftTrapezoid
LeftPSlope Q 5
DightPSlope Q 1 ; /A &0
CAS# 1: T U a
(em+ership Halue Q 1
CAS#
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
32/108
menu
Trapezoidal em'ers%ip FunctionsTrapezoidal em'ers%ip Functions
;ig%tTrapezoid
LeftPSlope Q 1 ; /& A0
DightPSlope Q 5
CAS# 1: T UQ a
(em+ership Halue Q 5
CAS#
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
33/108
menu
Trapezoidal em'ers%ip FunctionsTrapezoidal em'ers%ip Functions
;egular Trapezoid
LeftPSlope Q 1 ; /& A0
DightPSlope Q 1 ; /C "0
CAS# 1: T UQ a *r T VQ d (em+ership Halue Q 5
CAS#
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
34/108
Inputs are applied to a set of ift%en control rules!
2# ;ule Evaluation
e!g! IF temperature is ery hot. T/ED set fan speed ery high!
Fuzzy ControlFuzzy Control%i22erent stages o2 Fuzzy control %i22erent stages o2 Fuzzy control
F C t lF C t l
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
35/108
menu
i2 (inpt! is mem"ership #nction!$ 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
Fuzzy ControlFuzzy Control%i22erent stages o2 Fuzzy control %i22erent stages o2 Fuzzy control
F C t lF C t l
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
36/108
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
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
37/108
menu
Fuzzy ControlFuzzy Control
[ ] [ ]
[ ] [ ]
[ ] [ ]
[ ] [ ]
[ ] [ ]
"
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
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
38/108
menu
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
Fuzzy ControlFuzzy Control%i22erent stages o2 Fuzzy control %i22erent stages o2 Fuzzy control
F C t lF C t l
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
39/108
menu
Fuzzy ControlFuzzy Control
" 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!
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
40/108
menu
ExampleExample
• LetOs loo- at the Inerted 'endulum
'ro+lem
S f StS f St
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
41/108
menu
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
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
42/108
menu
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$periments: mounted a ,ine
glass containing ,ater or een a lie mouse to the top of the pendulum0!
I t d P d l P 'lIn erted Pend l m Pro'lem
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
43/108
Inverted Pendulum Pro'lemInverted Pendulum Pro'lem
%he solution uses a secondorder differential e2uation thatdescri+es cart motion as a function of pole position and elocity:
Conventional mat%ematical solution
θ θ θ θ θ sinsin)cos(cos)sin(2
2
2
2
mgl l l t
ml l xt
m =
∂∂
−
+
∂∂
I t d P d l P 'lInverted Pendulum Pro'lem
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
44/108
menu
Inverted Pendulum Pro'lemInverted Pendulum Pro'lem
Sensed values0
T – position of o+ect ,ith respect to the horizontal a$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
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
45/108
"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
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
46/108
menu
Inverted Pendulum Pro'lemInverted Pendulum Pro'lem
origin
Parameters for a Fuzzy SystemParameters for a Fuzzy System
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
47/108
menu
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
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
48/108
menu
Fuzzy SetsFuzzy Sets
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!
Fuzzy SetsFuzzy Sets
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
49/108
menu
Fuzzy SetsFuzzy Sets
%he angular velocityangular velocity could +e descri+ed as:
1! Falling to the Left /E0!
N! Falling to the Dight /'0!
Fuzzy ;ulesFuzzy ;ules
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
50/108
menu
Fuzzy ;ulesFuzzy ;ules
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
Fuzzy ;ulesFuzzy ;ules
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
51/108
menu
Fuzzy ;ulesFuzzy ;ules
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
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
52/108
menu
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
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
53/108
menu
)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
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
54/108
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#
em'ers%ip Functionsem'ers%ip Functions
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
55/108
em'ers%ip Functionsem'ers%ip Functionsem'ers%ip Functions for t%e Cart KelocityKelocity
Trapezoid Vertices
Left Trapezoid Regular Rigt
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
em'ers%ip Functionsem'ers%ip Functions
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
56/108
em'ers%ip Functionsem'ers%ip Functions
em'ers%ip Functions for t%e Pole )ngle)ngle
Trapezoid Vertices
Left Trapezoid Regular Rigt
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
em'ers%ip Functionsem'ers%ip Functions
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
57/108
em'ers%ip Functionsem'ers%ip Functions
em'ers%ip Functions for t%e !room )ngular Kelocity)ngular Kelocity
Trapezoid Vertices
Left Trapezoid Regular Rigt
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
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
58/108
menu
• *&S%ACL# AH*I"AEC# F688B
L#C%6D#!doc
• Lec< Fuzzy Logic Fuzzy#2ns!doc
Fuzzy ;uleFuzzy ;ule
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
59/108
menu
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
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
60/108
menu
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!
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
61/108
menu
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
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
62/108
menu
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
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
63/108
menu
amdani FISamdani FIS
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$QK@5)rdfre2Q@55)r,fre2Q@55)rleadQ
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
64/108
menu
amdani FISamdani FIS
amdani FISamdani FIS
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
65/108
menu
amdani FISamdani FIS
Flo4 of Fuzzy InferenceFlo4 of Fuzzy Inference
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
66/108
Flo4 of Fuzzy InferenceFlo4 of Fuzzy Inference
In this figure. the flo, proceeds up from the inputs in the lo,er left. then
across each ro,. or rule. and then do,n the rule outputs to finish in the lo,er
right! %his compact flo, sho,s eerything at once. from linguistic aria+le
fuzzification all the ,ay through defuzzification of the aggregate output!
amdani FISamdani FIS
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
67/108
menu
amdani FISamdani FIS
http:;;en!,i-ipedia!org;,i-i;FuzzyPsystem
(a$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
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
68/108
menu
Sample0 amdaniSample0 amdani
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
Sample0 amdaniSample0 amdani
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
69/108
menu
Sample0 amdaniSample0 amdani
;ule !ase =;!?0
Classical Inference Engine0 max(min(C:max(min(C:
Let us consider a sample input to the system. Mo 2, 2.Q size, 4eig%tQ
e2erence: :enetic fuzzy systems 'y Ascar CordOn, Francisco /errera, Fran5 /offmann
Sample0 amdaniSample0 amdani
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
70/108
menu
Sample0 amdaniSample0 amdani
;ule !ase0
Classical Inference Engine0 max(min(C:max(min(C:
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:
e2erence: :enetic fuzzy systems 'y Ascar CordOn, Francisco /errera, Fran5 /offmann
Sample0 amdaniSample0 amdani
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
71/108
menu
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 >>
e2erence: :enetic fuzzy systems 'y Ascar CordOn, Francisco /errera, Fran5 /offmann
Sample0 amdaniSample0 amdani
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
72/108
menu
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:
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
73/108
Sample0 amdaniSample0 amdani
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
74/108
menu
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:?
e2erence: :enetic fuzzy systems 'y Ascar CordOn, Francisco /errera, Fran5 /offmann
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
Sample0 amdaniSample0 amdani
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
75/108
menu
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:?
e2erence: :enetic fuzzy systems 'y Ascar CordOn, Francisco /errera, Fran5 /offmann
Sample0 amdaniSample0 amdani
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
76/108
menu
Sample0 amdaniSample0 amdani
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
e2erence: :enetic fuzzy systems 'y Ascar CordOn, Francisco /errera, Fran5 /offmann
T%e final crisp output is 7T%e final crisp output is 7oo #1-8 #1-8
Sample0 amdaniSample0 amdani
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
77/108
menu
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
e2erence: :enetic fuzzy systems 'y Ascar CordOn, Francisco /errera, Fran5 /offmann
amdani FISamdani FIS
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
78/108
menu
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
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
79/108
menu
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$actly the same! %he main difference +et,een (amdani and Sugeno is that theSugeno output mem+ership functions are either linear or constant!
Sugeno FISSugeno FIS
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
80/108
menu
Suge o Sg
FIS0 Sugeno vs# amdaniFIS0 Sugeno vs# amdani
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
81/108
menu
gg
)dvantages of t%e amdani et%od
It is intuitie!
It has ,idespread acceptance!
It is ,ell suited to human input!
)dvantages of t%e Sugeno et%od
It is computationally efficient!
It can +e used to model any inference system in ,hich the
output mem+ership functions are either linear or constant!
It ,or-s ,ell ,ith linear techni2ues /e!g!. 'I" control0!
It ,or-s ,ell ,ith optimization and adaptie techni2ues!
It has guaranteed continuity of the output surface!
It is ,ell suited to mathematical analysis!
Fuzzy SetsFuzzy Sets
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
82/108
menu
yy
Fuzzy Set !Fuzzy Set )
#$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?
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
83/108
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 &
#$ample:
Fuzzy RFuzzy Randand
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
84/108
menu
yy
*ade% ( min= 6 " =x?, 6 7 =x??
%his techni2ue. named after the inentor of fuzzy set
%heoryY it simply computes the 3and4 +y ta-ing the minimum of the t,o /or
more0 mem+ership alues! %his is the most common definition of the fuzzy
3and4!
Product ( 6 " =x? U 6 7 =x?
%his techni2ue computes the fuzzy 3and4 +y multiplying the t,o mem+ership
alues!
%here are many ,ays to compute 3and4! %he t,o most common are:
Fuzzy RFuzzy Randand
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
85/108
menu
yy!ot% tec%niGues %ave t%e follo4ing t4o properties0
T=,? T=a,? T=,a?
T=a,+? T=+,a? a
*ne of the nice things a+out +oth definitions is that they also can +e used to
compute the &oolean 3and4! %he fuzzy Rand is an e$tension of the !oolean
Rand to num+ers that are not ust 5 or 1. +ut +et,een 5 and 1!
Fuzzy Ror
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
86/108
menu
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 !
#$ample:
Fuzzy RFuzzy Ror or
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
87/108
menu
yy
T%e fuzzy Ror is an extension of t%e !oolean Ror to
num'ers t%at are not 6ust or +, 'ut 'et4een and +#
Fuzzy RFuzzy Ror or
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
88/108
menu
yy
*ade%ma$/ 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
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
89/108
yy
Lu5asie4icz "is6unction
min/1. A /$0 X ) /$00
*ther ,ays to compute Fuzzy 3or 4:
( ! ) in("! ) x y x yσ = +
/amac%er "is6unction "is6unction
2
( ! ) "
x y xy
x y xyσ
+ −
= −
)()("
)()(2)()(
x x
x x x x
B A
B A B A
µ µ
µ µ µ µ
−
−+
Fuzzy RFuzzy Ror or
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
90/108
yy*ther ,ays to compute Fuzzy 3or 4:
Einstein "is6unction
)()(")()(
x x x x
B A
B A
µ µ µ µ
+ +
( ! )"
x y x y
xyσ
+=
+
Fuzzy RFuzzy Rnotnot
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
91/108
menu
y
DAT =)? + ( )
Eegation of )
#$ample:
Fuzzy set )
Fuzzy Set operationsFuzzy Set operations
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
92/108
menu
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
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
93/108
menu
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$ample. the fuzzy 3and4 is used to com+ine the mem+ership functions to compute the rule strength!
+
ConseGuenceConseGuence
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
94/108
menu
Clipping the output mem+ership function at the rule strength!
IF T/ED;uleStrengt%
Input "istri'ution Autput
"istri'ution
Mo 7o
xo yo
and
and
In this e$ample. the fuzzy 3and4 is used to com+ine the mem+ership functions to compute the rule strength!
2
ConseGuenceConseGuence1
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
95/108
menu
%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$ample of this!
IF T/ED;uleStrengt%
Input "istri'ution Autput
"istri'ution
Mo 7o
xo yo
and
and
%he output
mem+ershipfunctions on the
right hand side of
the figure are
com+ined using the
fuzzy 3or 4 to o+tain
the outputdistri+ution sho,n
on the lo,er right
corner of the figure!
"efuzzification of Autput"i t i' ti
.
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
96/108
"istri'ution
In many instances. it is desired to come up ,ith a single crisp output from aFIS! For e$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$ample outcome of this
computation is sho,n in the figure at the
right!
a?
"efuzzification of Autput"i t i' ti
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
97/108
"istri'ution
ean of maximum %his techni2ue ta-es the output distri+utionfound in the preious section and finds its mean of ma$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
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
98/108
menu
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
em'ers%ip Functionsem'ers%ip Functions
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
99/108
menu
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
em'ers%ip Functionsem'ers%ip Functions
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
100/108
menu
T%e :aussian function
%he symmetric aussian function depends on t,o parameters and c as gien +yσ
em'ers%ip Functionsem'ers%ip Functions
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
101/108
menu
T%e Trapezoidal function
or
%he trapezoidal cure is a function of a ector. $. and depends on four scalarparameters a. +. c. and d. as gien +y
%he parameters a and d locate the 7feet7 of the trapezoid and the parameters + and c
locate the 7shoulders!7
/edges/edges http:;;+log!peltarion!com;
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
102/108
'.")( x A
µ
%.")( x A µ
2)( x A
µ
')( x A µ
*)( x A µ
2
"
)( x A µ
5.0)(0i6 )(22 ≤≤ x x A A µ µ
) little) little
Slig%tlySlig%tly
KeryKery
ExtremelyExtremely
Kery veryKery very
Some4%atSome4%at
IndeedIndeed
/edge EffectAperator
")(0.5i6 ))("(2" 2 ≤
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
103/108
Sigma N!@=
c @
Fuzzy Logic )pplicationsFuzzy Logic )pplications
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
104/108
menu
•Inverted Pendulum – Analyze the simulation /see implementation0
– Hariations of settings:
• /N$N. @$@ FA((S0
• Shape of mem+ership functions
• Fuzzy *utput alues settings. time delay
– 6tilisation of #ulerOs (ethod in the simulation
/see pdf0
Fuzzy Logic )pplicationsFuzzy Logic )pplications
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
105/108
menu
•;o'ot Davigation – "eriation of e2uations for motion
– Hariations of settings:
• FA((
• Shape of mem+ership functions
• Fuzzy *utput alues settings. time delay
– Startup codes
;eferences;eferences
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
106/108
menu
• enetic fuzzy systems +y *scar Cord]n. Francisco errera. Fran-
offmann• Eeural Eet,or- and Fuzzy Logic Applications in C;CXX /Wiley 'rofessional
Computing0 +y Stephen Welstead
• Fuzzy Logic ,ith #ngineering Applications +y %imothy Doss
• Fuzzy Sets and 'attern Decognition +y &enamin napp
Euler et%odEuler et%od
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
107/108
menu
• In mathematics and computational science. the Euler met%od.
named after Leonhard #uler. is a numerical procedure forsoling ordinary differential e2uations /*"#s0 ,ith a gieninitial alue! It is the most +asic -ind of e$plicit method fornumerical integration for ordinary differential e2uations!
Sorce: http:**www.answers.com*topic*eler+metho'+!,cat-technology
Euler et%odEuler et%od
8/17/2019 Lec2011 - 3 - AI - Fuzzy Logic
108/108
• %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
Recommended