Fuzzy System

• Based on intuition and judgment• No need for a mathematical model• Provides a smooth transition between

members and nonmembers• Relatively simple, fast and adaptive • Less sensitive to system fluctuations• Can implement design objectives,

(difficult to express mathematically), in linguistic or descriptive rules.

Applications DomainApplications Domain• Fuzzy Logic• Fuzzy Control

– Neuro-Fuzzy System– Intelligent Control– Hybrid Control

• Fuzzy Pattern Recognition• Fuzzy Modeling

Some Interesting Some Interesting ApplicationsApplications

• Ride smoothness control• Camcorder auto-focus and jiggle

control• Braking systems• Copier quality control• Rice cooker temperature control• High performance drives• Air-conditioning systems

Conventional or Conventional or crispcrisp sets sets are binary. An element are binary. An element either belongs to the set or either belongs to the set or doesn't. doesn't.

{True, false}{True, false}{1, 0}{1, 0}

Universe (X)

Subset A

Subset B

Subset C

Crisp Set/SubsetCrisp Set/Subset1A

1B1C

0A

9

9 9.5

10e.g. On a scale of

one to 10, how good was the

dive?

• Examples of fuzzy measures include close, heavy, light, big, small, smart, fast, slow, hot, cold, tall and short.

Fuzzy IndicatorsFuzzy Indicators• Can you distinguish between Can you distinguish between

American and French person?American and French person?• Some Rules:Some Rules:

– IfIf speaks English speaks English thenthen American American– IfIf speaks French speaks French thenthen French French– IfIf loves perfume loves perfume thenthen French French– IfIf loves outdoors loves outdoors thenthen American American– IfIf good cook good cook thenthen French French– IfIf plays baseball plays baseball thenthen American American

Fuzzy IndicatorsFuzzy Indicators• Rules may give contradictory indicators{good cook, loves outdoors, speaks French}• The right answer is a question of a degree

of association• Fuzzy logic resolves these conflicting

indicators– Membership of the person in the French set is

0.9– Membership of the person in the American set

is 0.1

• Fuzzy Probability• Probability deals with

uncertainty and likelihood• Fuzzy logic deals with

ambiguity and vagueness

• Fuzzy Probability

• Example #1– Billy has ten toes. The

probability Billy has nine toes is zero. The fuzzy membership of Billy in the set of people with nine toes, however, is nonzero.

Example #2

– A bottle of liquid has a probability of ½ of being rat poison and ½ of being pure water.

– A second bottle’s contents, in the fuzzy set of liquids containing lots of rat poison, is ½.

– The meaning of ½ for the two bottles clearly differs significantly and would impact your choice should you be dying of thirst.

– 50% probability means 50% chance that the water is clean.

– 50% fuzzy membership means that the water has poison.

(cite: Bezdek)

#1

#2

• Crisp membership functions are either one or zero.

• e.g. Numbers greater than 10.

A ={x | x>10}

1

x10

A(x)

• The set, B, of numbers near to 2 can be represented by a membership function

|2|)( xB ex

0 1 2 3

x

B(x)

• A fuzzy set, A, is said to be a subset of B if

• e.g. B = far and A=very far.• For example...

)()( xx BA

)()( 2 xx BA

Fuzzy SetsFuzzy Sets

Tall Short

Fuzzy SetsFuzzy Sets

Tall ShortTall or Short?Tall or Short?

Fuzzy Fuzzy MeasuresMeasures

4’

5’

6’

7’

Very Tall

Tall

Medium

Short

Very Short

Membership FunctionMembership FunctionVery TallTallMediumShortVery Short

4’ 5’ 6’ 7’

1.0

vttmsvs ,,,,

Membership FunctionMembership Function

Short Medium Tall

0,0,0,1,0 0,5.0,5.0,0,0

Very TallTallMediumShortVery Short

4’ 5’ 6’ 7’

1.0

Fuzzy Logic OperationsFuzzy Logic Operations

)](),([max )( A xxx BBA

Fuzzy union operation or fuzzy OR

Fuzzy Logic OperationsFuzzy Logic Operations Fuzzy intersection operation or fuzzy AND

)](),([min )( A xxx BBA

Fuzzy Logic OperationsFuzzy Logic Operations Complement operation

)(1)( xx AA

Fuzzy Logic OperationsFuzzy Logic Operations



Fuzzy union operation or fuzzy OR

Fuzzy intersection operation or fuzzy AND

)(1)( xx AA

Complement operation

0 1 2 3

x

AB(x)

0 1 2 3 x

B(x)

0 1 2 3 x

A(x)1


0 1 2 3 x

B(x)

0 1 2 3 x

A(x)1

A+B (x)

0 1 2 3 x


• Fuzzifier converts a crisp input into a fuzzy variable.

• Definition of the membership functions must– reflect the designer's knowledge– provide smooth transition between

member and nonmembers of a fuzzy set– be simple to calculate

• Typical shapes of the membership function are Gaussian, trapezoidal and triangular.

• Assume we want to evaluate the health of a person based on his height and weight.

• The input variables are the crisp numbers of the person’s heightheight and weight.

• Fuzzification is a process by which the numbers are changed into linguistic words

Fuzzification of Height

4’ 5’ 6’ 7’

Very Short Short Medium Tall Very Tall

VS = very shortS = ShortM = Mediumetc.

1.0

Fuzzification of Weight

Very HeavyHeavyMediumSlimVery Slim

150lb 200lb 250lb 300lb

1.0

VS = very slimS = SlimM = Mediumetc.

• Rules reflect expert’s decisions.• Rules are tabulated as fuzzy words• Rules can be grouped in subsets• Rules can be redundant• Rules can be adjusted to match desired

results

• Rules are tabulated as fuzzy words– Healthy (H)– Somewhat healthy (SH)– Less Healthy (LH)– Unhealthy (U)

• Rule function f f{ U, LH, SH, H}

f

f{ U, LH, SH, H}

Weight

Height

VerySlim Slim Mediu

m Heavy VeryHeavy

VeryShort H SH LH U U

Short SH H SH LH UMediu

m LH H H LH U

Tall U SH H SH UVeryTall U LH H SH LH

• For a given person, compute the membership of his/her weight and height

• Example:– Assume that a person’s height is 6’ 1”– Assume that the person’s weight is 140 lbs

Very Short Short Medium Tall Very Tall

4’ 5’ 6’ 7’

1.0

0.7

0.3

height ={ VS, S, M, T , VT}

height={ 0 0 0.7 0.3 0 }

Very Slim Slim Medium Heavy Very Heavy

weight ={ VS, S, M, H , VH}

Weight={ 0.8 0.2 0 0 0 }

150lb 200lb 250lb 300lb

1.0

0.8

0.2

Weight

Height

Very Slim Slim Mediu

m Heavy VeryHeavy

VeryShort H SH LH U U

Short SH H SH LH UMediu

m H LH U

Tall H SH UVeryTall U LH H SH LH

LHSHHLHUVeryTall

USHH0.3

ULHH0.7

ULHSHHSHShort

UULHSHHVeryShort

VeryHeavyHeavyMediu

m0.20.8

Height

Weight

SHU

HLH

Weight

Height

0.8 0.2Mediu

m(0)

Heavy(0)

V.Heavy(0)

V.Short(0) 0 0 0 0 0

Short(0) 0 0 0 0 0

0.7 0.7 0.2 0 0 0

0.3 0.3 0.2 0 0 0

V.Tall(0) 0 0 0 0 0

f = {U,LH,SH,H}f = {0.3,0.7,0.2,0.2}

Weight

Height

0.8 0.2V.Short(

0) 0 0

Short(0) 0 0

0.7 0.7 0.20.3 0.3 0.2

V.Tall(0) 0 0

Weight

Height

0.8 0.2V.Short(

0) H SH

Short(0) SH H0.7 LH H0.3 U SH

V.Tall(0) U LH

f = { U, LH, SH, H}f = { 0.3, 0.7, 0.2, 0.2}

• Use the fuzzified rules to compute the final decision.

• Two methods are often used. - Maximum Method(not often

used) - Centroid

• Fuzzy set with the largest membership value is selected.

• Fuzzy decision:f={U, LH, SH, H}f={0.3, 0.7, 0.2, 0.2}

• Final Decision(FD)=Less Healthy• If two decisions have same membership

max, use the average of the two.

..........

DDD

FDu

u

u LH LH

LH

0.44292.02.07.03.0

0.82.00.62.00.47.00.20.3

FD

•Crisp Decision Index (D) = 0.4429

• Assume that we need to evaluate student applicants based on their GPA and GRE scores.

• Let us assume that the decision should be Excellent (E), Very Good(VG), Good(G), Fair(F) or Poor(P)

• An expert will associate the decision to the GPA and GRE score. They are then Tabulated.

• Assume that we need to evaluate student applicants based on their GPA and GRE scores.

• For simplicity, let us have three categories for each score [High (H), Medium (M), and Low(L)]

• Let us assume that the decision should be Excellent (E), Very Good (VG), Good (G),

Fair (F) or Poor (P)• An expert will associate the decisions to

the GPA and GRE score. They are then Tabulated.

Excellent = 95-100%Very Good = 90 - 94% Good = 80 - 89% Fair = 70 - 79% Poor = 0 - 69%

Issue94 is VG, but is also very close to Excellent

• Fuzzifier converts a crisp input into a fuzzy variable.

• Definition of the membership functions must – Reflects the designer’s knowledge– Provides smooth transition between member

and nonmembers of a fuzzy set– Simple to calculate• Typical shapes of the membership function are

Gaussian, trapezoidal and triangular.

GRE = {L , M ,

H }

GRE

GPA = {L , M ,

H }

GPA

• Assume a student with GRE=900 and GPA=3.6

• The decisions on the classification of the applicant are – Excellent– Very good– Etc.

GRE=900

GRE = {L = 0.8 , M = 0.2 , H = 0}

GRE

GPA=3.6

GPA = {L = 0 , M = 0.4 , H = 0.6}

GPA

GRE = {L = 0.8 , M = 0.2 , H = 0}GPA = {L = 0 , M = 0.4 , H = 0.6}

Fn

0.6

0.4

0.2

•Converting the output fuzzy variable into a unique number

•Two defuzzifier methods are often used. –Maximum Method (not often used)

–Centroid

• Fuzzy set with the largest membership value is selected.

• Fuzzy decision: Fn = {P, F, G,VG, E}Fn = {0.6, 0.4, 0.2, 0.2, 0}

• Final Decision (FD) = Poor Student• If two decisions have same

membership max, use the average of the two.

..........

VGEfn

FDVGE

VGE

if

if

Final Decision (FD) = Fair Student

706.04.02.02.0

606.0704.0802.0902.01000

FD

Feedback (error, change in error)

Reference

FuzzyFuzzyControllerController

SystemSystemu

Input

Output

CELN MN SN ZE SP MP LP

LN LN LN LN LN MN SN SNMN LN LN LN MN SN ZE ZESN LN LN MN SN ZE ZE SP

E ZE LN MN SN ZE SP MP LPSP SN ZE ZE SP MP LP LPMP ZE ZE SP MP LP LP LPLP SP SP MP LP LP LP LP

• Requires mathematical models• Nonlinear processes are linearized• Poor Performance when model

deviates• Difficult to tune for unknown

dynamics• Poor performance for widely varying

operation

• Poor performance in noisy environments

• Inclusion of some design objectives can be challenging (e.g. comfort of ride and safety)

• Engineers are comfortable with the classical control design.

– Well-established technologies.– Verifiable overall system stability. – System’s reliability can be

evaluated.

• Stability and reliability studies are based on linearized models

• Most systems do not behave linearly

• Most systems do drift

• Neural and Fuzzy Control.• Based on intuitions and judgments.• Relatively simple, fast and

adaptable.• Can implement design objectives.

Difficult to express mathematically; in linguistic or descriptive rules.

• No need for mathematical model .• Less sensitive to system fluctuations. • Design objectives difficult to express

mathematically can be incorporated in a fuzzy controller by linguistic rules.

• Implementation is simple and straight forward.

System Inputs

Execution Layer ….FLC FLC FLC

Supervisor Layer

MLFC Outputs

-3 -2 -1 0 1 2 3

LN MN SN ZE SP MP LP

0

1

0 1 3 6-1-3-60

1ZE SP MP LPSNMNLNE

CE

Fuzzy System

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