Fuzzy Logic Theory

8/3/2019 Fuzzy Logic Theory

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Institute of Intelligent Power Electronics IPEPage 1

Fuzzy Logic Theory

Dr. Xiao-Zhi Gao

Department of Electrical Engineering

Helsinki University of Technology

[email protected]


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Outline

Fuzzy set theory

Basic concepts in fuzzy sets

Operations on fuzzy sets

Fuzzy rules and reasoning

Fuzzy inference systems

Mamdani fuzzy systems

Sugeno fuzzy systems


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Primer of Fuzzy Sets

From classical crisp sets to fuzzy sets

Change of range of membership functions

Notions of fuzzy sets

Representation, support, cut, convexity ...

Fuzzy set operations Intersection, union, complement, etc.

Share almostthe same mathematicsfoundamentals with those of classical sets


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Basics of Fuzzy Sets Classical sets

two-valuemembership functions

An element either belongs to or does not belong

to a given classical set (sharp boundary) Fuzzy sets

Extend the degrees of membership:

An element partiallybelongs to and partiallydoes not belong to a given fuzzy set

Smooth and Gradual boundary

Membership functions Assignment of membership functions is subjective

{ } ]1,0[1,0


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Father of Fuzzy Logic: Lotfi A. Zadeh

Lotfi Asker Zadeh is a Professor in the GraduateSchool, Computer Science Division, Departmentof EECS, University of California, Berkeley. In

addition, he is serving as the Director of BISC(Berkeley Initiative in Soft Computing).

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

http://www-bisc.cs.berkeley.edu/


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Four Typical Membership Functions

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

1.2

Memb

ershipGrades

(a) Triangular MF

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

1.2

Memb

ershipGrades

(b) Trapezoidal MF

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

1.2

Mem

bershipGrades

(c) Gaussian MF

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

1.2

Mem

bershipGrades

(d) Generalized Bell MF


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Supports

Core

Crossover points

Height

Fuzzy singleton

A fuzzy singleton is a crispset

A fuzzy set that has only one element

x0

Definitions in Fuzzy Sets

(x0) = 1

(x) > 0

(x) =1

(x) = 0.5

max A(x)[


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Definitions in Fuzzy Sets

FuzzySingleton


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Representations of

Membership Functions

Discrete Fuzzy Sets Universe of discourse is discrete

Continuous Fuzzy Sets Universe of discourse is continuous


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Discrete and Continuous

Fuzzy Sets

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

1.2

X = Number of Children

MembershipGrades

(a) MF on a Discrete Universe

0 10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1

1.2

X = Age

Mem

bershipGrades

(b) MF on a Continuous Universe


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Cuts of Fuzzy Sets

(a constant) cut of a fuzzy set is a

crispset

Strong cuts

{ } = )(xxA Aa

{ } >= )(xxA Aa

support(A) = A0

core(A) =A1


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Resolution Principle

A fuzzy membership function can be

expressed in terms of the characteristicfunctions of its cuts

An example of decomposing a fuzzy setwith discrete objects

=

3

3.0,2

2.0,1

1.0A

],min[max)( AxA =

R l i P i i l


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33.0

22.0

11.0

3)3.0,2.0,1.0max(

2)2.0,1.0max(

11.0

3

3.03.0

32.0

22.02.0

3

1.0

2

1.0

1

1.01.0

3

13

1

2

13

1

2

1

1

1

3.0

2.0

1.0

3.0

2.0

1.0

++=

++=

=

+=

++=

=

+=

++=

A

A

A

A

A

A

A


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0 10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1

1.2Resolution Principle

x

Mem

bershipGrades


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Extension Principle

Extending crispdomains of mathematicalexpressions to fuzzydomains

Suppose fis a function mapping from Xto Y, and A is a fuzzy set:

we have image of A under f:n

nAAA

xx

xx

xxA )()()(

2

2

1

1 +++= L

)()(

)()(

)()()(

2

2

1

1

n

nAAA

xfx

xfx

xfxAfB +++== L


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Extension Principle

An example of extension principle

2

3.0

1

9.0

0

8.0

1

4.0

2

1.0+++

+

=A

3)( 2 =xxf

1

3.0

2

9.0

3

8.0

1

)3.01.0(

2

)9.04.0(

3

8.0

13.0

29.0

38.0

24.0

11.0

+

+

=

+

+

=

+

+

+

+=B

E t i P i i l


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Extension Principle


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Convex Fuzzy Sets Convexity of fuzzy sets

where and are two arbitrary elements

in A, and is an arbitrary constantx1 x2

C d N F


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Convex and Nonconvex Fuzzy


0

0.2

0.4

0.6

0.8

1

1.2

MembershipG

rades

(a) Two Convex Fuzzy Sets

0

0.2

0.4

0.6

0.8

1

1.2

MembershipGrades

(b) A Nonconvex Fuzzy Set


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Operations on Fuzzy Sets

0

0.2

0.4

0.6

0.8

1

1.2(a) Fuzzy Sets A and B

A B

0

0.2

0.4

0.6

0.8

1

1.2(b) Fuzzy Set "not A"

0

0.2

0.4

0.6

0.8

1

1.2(c) Fuzzy Set "A OR B"

0

0.2

0.4

0.6

0.8

1

1.2(d) Fuzzy Set "A AND B"

A

A IBA U B

and B


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Operations on Fuzzy Sets Subset

Equality

andA B B A


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Concept of A B

0

0.2

0.4

0.6

0.8

1

1.2A Is Contained in B

Memb

ershipGrades

B

A

Li i ti V i bl


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Linguistic Variables

Modeling of human thinking Numerical values are not sufficient

Linguistic variables exist in real world

e.g, chatting with a stranger on the phone

Estimation of your partners age:

(40? probability of 40? About middle-aged?) Linguistic variables are characterized by linguistic

values (Age:[Young, Old, Very Old, etc.])

Linguistic values are described by their fuzzymembership functions


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Membership Functions For Linguistic Values

Young, Middle Aged, and Old

0 10 20 30 40 50 60 70 80 900

0.2

0.4

0.6

0.8

1

1.2

X = Age

MembershipGrades

Young Middle Aged Old

Age:

LinguisticVariable

Young, Middle

Aged, and Old:Linguistic Values

Membership Functions For Primary


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Membership Functions For Primary

and Composite Linguistic Values

0 10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1

X = age

Members

hipGrades

(a) Primary Linguistic Values

OldYoung

0 10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1

X = age

M

embershipGrades

(b) Composite Linguistic Values

Not Young and Not Old

More or Less Old

Extremely Old

Young but

Not Too Young

Extremely Old

= Old4


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Fuzzy IF-THEN Rules

A fuzzy IF-THEN rule is expressed as

IF xis A THEN yis B

A and B are fuzzy membership functions

A fuzzy IF-THEN rule associatesfuzzyinput and output membership functions

Premise Part Consequent Part


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Fuzzy Reasoning

Fuzzy reasoning derives conclusionsbased on givenfuzzy IF-THEN rules

and knownfacts

An example:

Given a fuzzy rule: IF bath is very hot THENadd a lot of cold water

Known fact: bath is a little hot

Conclusion: how much cold water should beadded?


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Fuzzy Reasoning

Premise 1: IF X is A THEN Y is B

Premise 2: IF X is

Conclusion: THEN Y is

First, measure the similaritybetween A and

Second, projectthis similarity to B

There are a few composition operations

used in fuzzy reasoning procedure Max-Minis most widely employed

AB

A

Fuzzy Reasoning for Single Rule


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Fuzzy Reasoning for Single Rulewith Single Antecedent

IntersectionIF X is A THEN Y is B

W

Known Fact: X is A



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with Multiple Antecedents

w=min(w1, w2)or

w=w1*w2

IF X is A and Y is B THEN Z is C

Known Facts: X is Aand Y is B

Fuzzy Reasoning For Multiple Rules


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Fuzzy Reasoning For Multiple Rules

with Multiple Antecedents

Aggregation

If X is A1 and Y is B1 Then Z is C1If X is A2 and Y is B2 Then Z is C2

Known Facts:

X is A

Y is B


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Fuzzy Logic SystemsFuzzification

Interface

Inference

Engine

Fuzzy Rule

Base

Defuzzification

InterfaceCrisp

Input

Crisp

Output

Rules

Fuzzy

Input

Fuzzy

Output

Mamdani and Sugeno Fuzzy Inference Systems


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Fuzzification

Converts a crispvalue (from sensors,devices, measurement meters, etc.) into

a fuzzyvalue

A crisp value = A fuzzy singleton (no

fuzziness is introduced)

x0: Crisp Value

Fuzzy singleton

A(x)

A(x)

x0

1


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Mamdani Fuzzy Inference Systems

Mamdani fuzzy inference systems

IF x is A THEN y is B Both premise and consequent parts

consist of fuzzy membership functions

Max-Mincomposition can be applied infuzzy reasoning procedure



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If X is A1 and Y is B1 THEN Z is C1

If X is A2 and Y is B2 THEN Z is C2

x and y are two input values

Defuzzification Methods


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D f ifi ti M th d


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Centroid of Area (COA)

where is a crisp output

Bisector of Area (BOA)

where is a crisp output

ZCOA*

=

A (xi)xii=1

n

A (x i)i=1

n

A (xi)i=1

M

= A (x j)j=M+1

n

ZBisec t* = xM

*

COAZ



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Mean of Maximum (MOM)

where reaches maximal values of

MOM could generatate wrong discreteoutputs in certain cases

Different defuzzification methods mayhave similarperformances [Lee 88]

ZMOM* =

xi*

i

N

N

A (xi*) A (x)

Mamdani F Model An E ample


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Mamdani Fuzzy Model: An Example

A Two-Input and Single-Output Mamdanifuzzy logic system with four rules

IF X is small and Y is small THEN Z is large negative

IF X is small and Y is large THEN Z is small negative

IF X is large and Y is small THEN Z is small positive

IF X is large and Y is large THEN Z is large positive

Fuzzy Input and Output


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y p p


-5 -4 -3 -2 -1 0 1 2 3 4 50

0.5

1

X

MembershipGrades small large

-5 -4 -3 -2 -1 0 1 2 3 4 50

0.5

1

Y

Mem

bershipGrades small large

-5 -4 -3 -2 -1 0 1 2 3 4 50

0.5

1

Z

MembershipGrades large negative small negative small positive large positive

Mamdani Input/Output Surface


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Mamdani Input/Output Surface

-5

0

5

-5

0

5

-3

-2

-1

0

1

2

3

XY

Z

NonlinearInput/Output

Surface

Sugeno Fuzzy Inference Systems


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Sugeno fuzzy inference systems

IF x is A THEN y = f(x)

Only premisepart employs fuzzymembership functions

Consequent output is a functionof inputvariables

First order = linear consequent part

Higher orders = nonlinear consequent part



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If X is A1 and Y is B1

THEN Z = p1x+q1y+c1

If X is A2 and Y is B2THEN Z = p2x+q2y+c2

First Order Sugeno Inference System

Z: Output

S F I f S t


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Advantages:

Nodefuzzification method is needed Easy to analyze

linearwith respect to consequent parameters

(only first order consequent part case)

Disadvantages

Difficult to interpret practical meanings ofconsequent part

Sugeno Fuzzy Model: An Example


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Sugeno Fuzzy Model: An Example

A Two-Input and Single-Output Sugeno

fuzzy logic system with four rules

IF X is small and Y is small THEN z = -x+y+1

IF X is small and Y is large THEN z = -y+3 IF X is large and Y is small THEN z = -x+3

IF X is large and Y is large THEN z = -x+y+2

Fuzzy Input Membership Functions


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Fuzzy Input Membership Functions

-5 -4 -3 -2 -1 0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

1.2

X

M

embershipGrades

Small Large

-5 -4 -3 -2 -1 0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

1.2

Y

MembershipGrades

Small Large

Sugeno Input/Output Surface


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Sugeno Input/Output Surface

-5

0

5

-5

0

5

-2

0

2

4

6

8

10

XY

Z NonlinearInput/Output

Surface

Diagram of A Fuzzy Logic System


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Diagram of A Fuzzy Logic System

Conclusions


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Conclusions

Background knowledge of fuzzy sets isintroduced

Concept of fuzzy sets is basically anaturalgeneralization of classical sets

Fuzzy sets have some characteristicsdifferent from classical sets

Fuzzy inference systems are used tomodel human thinking

Conclusions


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Two fuzzy logic systems: Mamdani andSugeno types

Engineering potential is more importantthan pure theoretical research

Applications of fuzzy logic and fusion

with neural networks will be discussedlater

Fuzzy control and fuzzy signal filtering Fuzzy neural networks

Fuzzy Logic Theory

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