Fuzzy inferencesystem2024 in engineering control

control

1. 1
State of Libya
Al-Asmarya Islamic University
Faculty of Engineering
Electrical and Computer Engineering Department
Fuzzy Inferences System
Student:Mustafa EnwiderStudent:Ali Al-sharani
Student:Hisham Ali Student:Waleed Aleyan
Presentation of each of:

2. Fuzzy Inference System
.
Important Inference systems:
Mamdani
Sugeno
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3. Mamdani method
Themostcommonlyusedfuzzyinferencetechniqueistheso-called
Mamdanimethod.In1975,ProfessorEbrahimMamdaniof
LondonUniversitybuiltoneofthefirstfuzzysystemstocontrola
steamengineandboilercombination.Heappliedasetoffuzzy
rulessuppliedbyexperiencedhumanoperators.
.
lfuzzification of the input variables
lrule evaluation
laggregation of the rule outputs
ldefuzzification.
The Mamdani-style fuzzy inference process is performed in four steps:
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We examine a simple two-input one-output problem that
includes three rules:

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Step 1: Fuzzification
Thefirststepistotakethecrispinputs,x1andy1
(projectfundingandprojectstaffing),anddetermine
thedegreetowhichtheseinputsbelongtoeachofthe
appropriatefuzzysets.SothatitisaProcesswhichmakesacrisp
quantityfuzzyCrispInput
0.1
0.7
1
0
y1
B1 B2
Y
CrispInput
0.2
0.5
1
0
A1 A2 A3
x1
x1 X
(x =A1) = 0.5
(x =A2) = 0.2
(y =B1) = 0.1
(y =B2) = 0.7
A1 =inadequate,A2=marginal,A3=adequate 
A(x=A3)=0.0
B1=small,B2=large

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Step 2: Rule Evaluation
Thesecondstepistotakethefuzzifiedinputs,
m
(x=A1)
=0.5,m
(x=A2)
=0.2,m
(y=B1)
=0.1andm
(y=B2)
=
0.7,andapplythemtotheantecedentsofthefuzzy
rules.Ifagivenfuzzyrulehasmultipleantecedents,
thefuzzyoperator(ANDorOR)isusedtoobtaina
singlenumberthatrepresentstheresultofthe
antecedentevaluation.Thisnumber(thetruthvalue)
isthenappliedtotheconsequentmembership
function.

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Toevaluatethedisjunctionoftheruleantecedents,
weusetheORfuzzyoperation.Typically,fuzzy
expertsystemsmakeuseoftheclassicalfuzzy
operationunion:

A

B
(x) = max [
A
(x), 
B
(x)]
Similarly, in order to evaluate the conjunction of the
rule antecedents, we apply the AND fuzzy operation
intersection:

A

B
(x) = min [
A
(x), 
B
(x)]

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1
0
X
1
y1
0
Y
0.0
x1
0
0.1
C1
1
C2
Z
1
0
X
0.2
0
0.2
C1
1
C2
Z
A2
x1
Rule 3:IFx isA1 (0.5)
A1
1
0
X
0
1
Zx1
THEN
C1 C2
1
y1
B2
0
Y
0.7
B1
0.1
C3
C3
C30.5 0.5
OR
(max)
AND
(min)
OR THENRule 1:IFx isA3 (0.0)
AND THENRule 2:IFx isA2 (0.2)
yisB1 (0.1) zisC1 (0.1)
yisB2 (0.7) zisC2 (0.2)
zisC3 (0.5)
Mamdani-style rule evaluation

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Nowtheresultoftheantecedentevaluationcanbe
appliedtothemembershipfunctionofthe
consequent.
nThemostcommonmethodofcorrelatingtherule
consequentwiththetruthvalueoftherule
antecedentistocuttheconsequentmembership
functionattheleveloftheantecedenttruth.This
methodiscalledclipping.Sincethetopofthe
membershipfunctionissliced,theclippedfuzzyset
losessomeinformation.However,clippingisstill
oftenpreferredbecauseitinvolveslesscomplexand
fastermathematics,andgeneratesanaggregated
outputsurfacethatiseasiertodefuzzify.

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nWhileclippingisafrequentlyusedmethod,scaling
offersabetterapproachforpreservingtheoriginal
shapeofthefuzzyset.Theoriginalmembership
functionoftheruleconsequentisadjustedby
multiplyingallitsmembershipdegreesbythetruth
valueoftheruleantecedent.Thismethod,which
generallyloseslessinformation,canbeveryuseful
infuzzyexpertsystems.

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Clipped and scaled membership functions1.0
0.0
0.2
Z Z
C2
1.0
0.0
0.2
C2
Degree of
Membership
Degree of
Membership

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Step 3: Aggregation of the rule outputs
Aggregationistheprocessofunificationofthe
outputsofallrules.Wetakethemembership
functionsofallruleconsequentspreviouslyclippedor
scaledandcombinethemintoasinglefuzzyset.
Theinputoftheaggregationprocessisthelistof
clippedorscaledconsequentmembershipfunctions,
andtheoutputisonefuzzysetforeachoutput
variable.

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0.1
1
C1
zisC1 (0.1)
C2
0
0.2
1
zisC2 (0.2)
0
0.5
1
zisC3 (0.5)
ZZZ
0.2
Z0

C3
0.5
0.1
Aggregation of the rule outputs

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Step 4: Defuzzification
Thelaststepinthefuzzyinferenceprocessis
defuzzification.Fuzzinesshelpsustoevaluatethe
rules,butthefinaloutputofafuzzysystemhastobe
acrispnumber.Theinputforthedefuzzification
processistheaggregateoutputfuzzysetandthe
outputisasinglenumber.

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nThereareseveraldefuzzificationmethods,but
probablythemostpopularoneisthecentroid
technique.Itfindsthepointwhereaverticalline
wouldslicetheaggregatesetintotwoequalmasses.
Mathematicallythiscentreofgravity(COG)can
beexpressedas:




b
a
A
b
a
A
COG
xxdx
xdx

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nCentroid defuzzification method finds a point
representing the centre of gravity of the fuzzy set, A,
on the interval, ab.
nAreasonableestimatecanbeobtainedbycalculating
itoverasampleofpoints.1.0
0.0
0.2
0.4
0.6
0.8
160170180190200
a b
210
A
150
X

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Centre of gravity (COG):4.67
5.05.05.05.02.02.02.02.01.01.01.0
5.0)100908070(2.0)60504030(1.0)20100(



COG
1.0
0.0
0.2
0.4
0.6
0.8
0 2030405010 70809010060
Z
Degreeof
Membership
67.4

18. Examples on Mamadani Method:
Ex.1
Ex.2

19. Steps Manual:

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Example 2:

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nMamdani-styleinference,aswehavejustseen,
requiresustofindthecentroidofatwo-dimensional
shapebyintegratingacrossacontinuouslyvarying
function.Ingeneral,thisprocessisnot
computationallyefficient.
nMichioSugenosuggestedtouseasinglespike,a
singleton,asthemembershipfunctionoftherule
consequent.Asingleton,,ormorepreciselyafuzzy
singleton,isafuzzysetwithamembership
functionthatisunityatasingleparticularpointon
theuniverseofdiscourseandzeroeverywhereelse.
Sugenofuzzy inference

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Sugeno-stylefuzzyinferenceisverysimilartothe
Mamdanimethod.Sugenochangedonlyarule
consequent.Insteadofafuzzyset,heuseda
mathematicalfunctionoftheinputvariable.The
formatoftheSugeno-stylefuzzyruleis
IF x is A
AND y is B
THEN z is f (x, y)
where x, y and z are linguistic variables; A and B are
fuzzy sets on universe of discourses X and Y,
respectively; and f (x, y) is a mathematical function.

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nMamdanimethodiswidelyacceptedforcapturing
expertknowledge.Itallowsustodescribethe
expertiseinmoreintuitive,morehuman-like
manner.However,Mamdani-typefuzzyinference
entailsasubstantialcomputationalburden.
nOntheotherhand,Sugenomethodiscomputationally
effectiveandworkswellwithoptimisationand
adaptivetechniques,whichmakesitveryattractivein
controlproblems,particularlyfordynamicnonlinear
systems.
How to make a decision on which method
to apply –Mamdanior Sugeno?