Fuzzy systems

Calculate the

degree of matching

Fuzzy inference

engine

Defuzzification module

Fuzzy rule base

General scheme of a fuzzy systemGeneral scheme of a fuzzy system

Linguistic rules: IF x = A THEN y = B

A is the rule antecedent, B is the rule consequent Example: „IF traffic is heavy in this direction THEN keep the green light longer”

Fuzzy rule base relation R containing two fuzzy rules A1→ B1, A2→ B2 (R1, R2)

If x = A then y = B "fuzzy point" A×BIf x = Ai then y = Bi i = 1,...,r "fuzzy graph"

Fuzzy rule = fuzzy relation (Ri)

Fuzzy rule base = fuzzy relation (R), is the union (s-norm) of the fuzzy rule relations Ri:

Fuzzy rule base relation:

)),((max),( ),(1

),( yxyx yxR

r

iyxR i

))(),(min(),( )()(),( yxyx yBxAyxR iii

more dimensional case:

))(),(),...,(min(),,...,,( )()(1)(21),,...,,( ,1,121yxxyxxx yBnxAxAnyxxxR ininini

Linguistic variables and their Linguistic variables and their representationsrepresentations

• Linguistic variable (linguistic term) defined by Zadeh:

"By a linguistic variable we mean a variable whose values are words or sentences in a natural or artificial language. For example, Age is a linguistic variable if its values are linguistic rather than numerical, i.e., young, not young, very young, quite young, old, not very old and not very young, etc., rather than 20, 21, 22, 23, ..."

Frame of Cognition (fuzzy partition)Frame of Cognition (fuzzy partition)

• Partition A={Ai} "covers" the universe X; i.e. each element of this universe is assigned to at least one granule with a nonzero degree of membership. Thus:

> 0 denotes the level of "coverage" of X. )(],,1[ xAniXx i

The fuzzy partition (frame of cognition) -covers the universe of discourse X

Ruspini-partitionRuspini-partition

sup(supp(Ai(x)))=inf(core(Ai+1(x)))

sup(core(Ai(x)))= inf(supp(Ai+1(x)))

Boolean partitionBoolean partition

• A induced by the fuzzy partition:

)))(((sup))((),(],1[

xAxAifxAx ini

ii

Specificity of fuzzy partitionsSpecificity of fuzzy partitions

• Fuzzy partition A* is more specific than A if all the elements of A* are more specific (e.g. in terms of their specificity measure) than the elements of A. Then, the number of elements of A* is greater than the number of linguistic terms in A.

• For instance, the fuzzy partition: A = { Negative, Zero, Positive} is less specific than the fuzzy partition A* containing seven items:

Large Positive Middle,Positive Small,Positive

Zero,

Small,Negative Middle,Negative Large, Negative

A*

Specificity of fuzzy partitionsSpecificity of fuzzy partitions

Fuzzy Partition A containing three linguistic terms

Fuzzy Partition A* containing seven linguistic terms

FuzzyFuzzy inferenceinference mechanismmechanism (Mamdani) (Mamdani)

• If x1 = A1,i and x2 = A2,i and...and xn = An,i then y = Bi

)}}(),({min{max,, jAjX

xij xxw

ijj

The weighting factor wji characterizes, how far the input xj corresponds to the rule antecedent fuzzy set Aj,i in one

dimension

},,,min{ ,,2,1 iniii wwww

The weighting factor wi

characterizes, how far the input x fulfils to the

antecedents of the rule Ri.

ConclusionConclusion

))(,min()( ywyii Biy

The conclusion of rule Ri for a given x observation is yi

The whole inferenceThe whole inference

Compositional Rule of InferenceCompositional Rule of Inference

Takagi-Sugeno methodTakagi-Sugeno method

If x1 = A1,i and x2 = A2,i and...and xn = An,i then yi = fi(x1,x2,...,xn)

where wi is the weighting factor, the level of the firing of the rule

Ri, similarly to the Mamdani method

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r

inii

r

ii

r

iii

w

xxxfw

w

ywy

1

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Defuzzification methodsDefuzzification methodsCenter of Gravity Method (COG)Center of Gravity Method (COG)

Yy

y

Yy

y

COGdyy

dyyy

y)(

)(

Defuzzification methodsDefuzzification methodsCenter of Center of SumsSums Method (CO Method (COSS))

r

i Yy

y

r

i Yy

y

COS

dyy

dyyy

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i

i

1

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Defuzzification methodsDefuzzification methodsMean of Maxima Method (MOM)Mean of Maxima Method (MOM)

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MAXy

MAXy

MOMdy

ydy

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FuFuzzy systems: zzy systems: an examplean example

Fuzzy systems operate on fuzzy rules:

IF temperature is COLD THEN motor_speed is LOWIF temperature is WARM THEN motor_speed is MEDIUMIF temperature is HOT THEN motor_speed is HIGH

TEMPERATURE MOTOR_SPEED

Inference mechanismInference mechanism ((MamdaniMamdani))

Temperature = 55 Motor Speed

Motor Speed = 43.6

RULE 1

RULE 2

RULE 3