AI

Fuzzy Systems

History, State of the Art, and Future DevelopmentHistory, State of the Art, and Future Development

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1965 Seminal Paper “Fuzzy Logic” by Prof. Lotfi Zadeh, Faculty in Electrical Engineering, U.C. Berkeley, Sets the Foundation of the “Fuzzy Set Theory”

1970 First Application of Fuzzy Logic in Control Engineering (Europe)

1975 Introduction of Fuzzy Logic in Japan

1980 Empirical Verification of Fuzzy Logic in Europe

1985 Broad Application of Fuzzy Logic in Japan

1990 Broad Application of Fuzzy Logic in Europe

1995 Broad Application of Fuzzy Logic in the U.S.

2000 Fuzzy Logic Becomes a Standard Technology and Is Also Applied in Data and Sensor Signal Analysis. Application of Fuzzy Logic in Business and Finance.

1965 Seminal Paper “Fuzzy Logic” by Prof. Lotfi Zadeh, Faculty in Electrical Engineering, U.C. Berkeley, Sets the Foundation of the “Fuzzy Set Theory”

1970 First Application of Fuzzy Logic in Control Engineering (Europe)

1975 Introduction of Fuzzy Logic in Japan

1980 Empirical Verification of Fuzzy Logic in Europe

1985 Broad Application of Fuzzy Logic in Japan

1990 Broad Application of Fuzzy Logic in Europe

1995 Broad Application of Fuzzy Logic in the U.S.

2000 Fuzzy Logic Becomes a Standard Technology and Is Also Applied in Data and Sensor Signal Analysis. Application of Fuzzy Logic in Business and Finance.

Today, Fuzzy Logic Has Today, Fuzzy Logic Has Already Become the Already Become the Standard Technique for Standard Technique for Multi-Variable Control !Multi-Variable Control !

Stochastic Uncertainty:

The Probability of Hitting the Target Is 0.8

Lexical Uncertainty:

"Tall Men", "Hot Days", or "Stable Currencies"

We Will Probably Have a Successful Business Year.

The Experience of Expert A Shows That B Is Likely to Occur. However, Expert C Is Convinced This Is Not True.

Stochastic Uncertainty:

The Probability of Hitting the Target Is 0.8

Lexical Uncertainty:

"Tall Men", "Hot Days", or "Stable Currencies"

We Will Probably Have a Successful Business Year.

The Experience of Expert A Shows That B Is Likely to Occur. However, Expert C Is Convinced This Is Not True.

Types of Uncertainty and the Modeling of Uncertainty Types of Uncertainty and the Modeling of Uncertainty

Slide 3

Most Words and Evaluations We Use in Our Daily Reasoning Are Most Words and Evaluations We Use in Our Daily Reasoning Are Not Clearly Defined in a Mathematical Manner. This Allows Not Clearly Defined in a Mathematical Manner. This Allows Humans to Reason on an Abstract Level!Humans to Reason on an Abstract Level!

“... a person suffering from hepatitis shows in 60% of all cases a strong fever, in 45% of all cases yellowish colored skin, and in 30% of all cases suffers from nausea ...”

“... a person suffering from hepatitis shows in 60% of all cases a strong fever, in 45% of all cases yellowish colored skin, and in 30% of all cases suffers from nausea ...”

Probability and UncertaintyProbability and Uncertainty

Slide 4

Stochastics and Fuzzy Logic Stochastics and Fuzzy Logic Complement Each Other !Complement Each Other !

Conventional (Boolean) Set Theory:Conventional (Boolean) Set Theory:

Fuzzy Set TheoryFuzzy Set Theory

Slide 5

“Strong Fever”

40.1°C40.1°C

42°C42°C

41.4°C41.4°C

39.3°C39.3°C

38.7°C38.7°C

37.2°C37.2°C

38°C38°C

Fuzzy Set Theory:Fuzzy Set Theory:

40.1°C40.1°C

42°C42°C

41.4°C41.4°C

39.3°C39.3°C

38.7°C

37.2°C

38°C

““More-or-Less” Rather Than “Either-Or” !More-or-Less” Rather Than “Either-Or” !

“Strong Fever”

Fuzzy Sets...

Representing crisp and fuzzy sets as subsets of a domain (universe) U".

Fuzziness versus probability

Probability density function for throwing a dice and the membership functions of the concepts "Small" number, "Medium", "Big".

Conceptualising in fuzzy terms...

One representation for the fuzzy number "about 600".

Conceptualising in fuzzy terms...

Representing truthfulness (certainty) of events as fuzzy sets over the [0,1] domain.

Conventional (Boolean) Set Theory:Conventional (Boolean) Set Theory:

Strong Fever RevisitedStrong Fever Revisited

Slide 10

“Strong Fever”

40.1°C40.1°C

42°C42°C

41.4°C41.4°C

39.3°C39.3°C

38.7°C38.7°C

37.2°C37.2°C

38°C38°C

Fuzzy Set Theory:Fuzzy Set Theory:

40.1°C40.1°C

42°C42°C

41.4°C41.4°C

39.3°C39.3°C

38.7°C

37.2°C

38°C

“Strong Fever”

As Zadeh said, the term is concrete, immediate and As Zadeh said, the term is concrete, immediate and descriptive; we all know what it means. descriptive; we all know what it means. However, many people in the West However, many people in the West were repelled by the word fuzzywere repelled by the word fuzzy , , because it is usually used in a negative sense.because it is usually used in a negative sense.

Fuzziness rests on fuzzy set theory, and fuzzy logic Fuzziness rests on fuzzy set theory, and fuzzy logic is just a small part of that theory. is just a small part of that theory.

Why fuzzy?Why fuzzy?

Why logic?Why logic?

Range of logical values in Boolean and fuzzy logicRange of logical values in Boolean and fuzzy logic

(a) Boolean Logic. (b) Multi-valued Logic0 1 10 0.2 0.4 0.6 0.8 1001 10

.

The classical example in fuzzy sets is tall men. The classical example in fuzzy sets is tall men. The elements of the fuzzy set “tall men” are all The elements of the fuzzy set “tall men” are all men, but their degrees of membership depend on men, but their degrees of membership depend on their height.their height.

Discrete Definition:

µSF

(35°C) = 0 µSF

(38°C) = 0.1 µSF

(41°C) = 0.9

µSF

(36°C) = 0 µSF

(39°C) = 0.35 µSF

(42°C) = 1

µSF

(37°C) = 0 µSF

(40°C) = 0.65 µSF

(43°C) = 1

Discrete Definition:

µSF

(35°C) = 0 µSF

(38°C) = 0.1 µSF

(41°C) = 0.9

µSF

(36°C) = 0 µSF

(39°C) = 0.35 µSF

(42°C) = 1

µSF

(37°C) = 0 µSF

(40°C) = 0.65 µSF

(43°C) = 1

Fuzzy Set DefinitionsFuzzy Set Definitions

Slide 14

Continuous Definition:Continuous Definition:

39°C 40°C 41°C 42°C38°C37°C36°C

1

0

µ(x)No More Artificial Thresholds!No More Artificial Thresholds!

Representation of hedges in fuzzy logicRepresentation of hedges in fuzzy logic

Hedge MathematicalExpression

A little

Slightly

Very

Extremely

Graphical Representation

[A(x)]1.3

[A(x)]1.7

[A(x)]2

[A(x)]3

Representation of hedges in fuzzy logic (continued)Representation of hedges in fuzzy logic (continued)

Hedge MathematicalExpression Graphical Representation

Very very

More or less

Indeed

Somewhat

2 [A(x )]2

A(x)

A(x)

if 0 A 0.5

if 0.5 < A 1

1 2 [1 A(x)]2

[A(x)]4

...Terms, Degree of Membership, Membership Function, Base Variable......Terms, Degree of Membership, Membership Function, Base Variable...

Linguistic VariableLinguistic Variable

Slide 17

39°C 40°C 41°C 42°C38°C37°C36°C

1

0

µ(x)low temp normal raised temperature strong fever

… pretty much raised … … pretty much raised …

... but just slightly strong … ... but just slightly strong …

A Linguistic Variable A Linguistic Variable Defines a Concept of Our Defines a Concept of Our Everyday Language!Everyday Language!

Fuzzy Sets• Formal definition:

A fuzzy set A in X is expressed as a set of ordered pairs:

A x x x XA {( , ( ))| }

Universe oruniverse of discourse

Fuzzy setMembership

function(MF)

A fuzzy set is totally characterized by aA fuzzy set is totally characterized by amembership function (MF).membership function (MF).

Cantor’s setsCantor’s sets

Intersection Union

Complement

NotA

A

Containment

AA

B

BA AA B

Operations of fuzzy setsOperations of fuzzy sets

Complement

0x

1

(x)

0x

1

Containment

0x

1

0x

1

AB

NotA

A

Intersection

0x

1

0x

AB

Union0

1

ABAB

0x

1

0x

1

B

A

B

A

(x)

(x) (x)

ComplementComplementCrisp Sets:Crisp Sets: Who does not belong to the set?Who does not belong to the set?Fuzzy Sets:Fuzzy Sets: How much do elements not belong toHow much do elements not belong tothe set?the set?The complement of a set is an opposite of this set.The complement of a set is an opposite of this set.For example, if we have the set of tall men, itsFor example, if we have the set of tall men, itscomplement is the set of NOT tall men. When wecomplement is the set of NOT tall men. When weremove the tall men set from the universe ofremove the tall men set from the universe ofdiscourse, we obtain the complement. If A is thediscourse, we obtain the complement. If A is thefuzzy set, its complement fuzzy set, its complement A can be found asA can be found asfollows:follows:

AA((xx) 1 ) 1 AA((xx))

ContainmentContainmentCrisp Sets:Crisp Sets: Which sets belong to which other sets? Which sets belong to which other sets?Fuzzy Sets:Fuzzy Sets: Which sets belong to other sets? Which sets belong to other sets?Similar to a Chinese box, a set can contain otherSimilar to a Chinese box, a set can contain othersets. The smaller set is called the sets. The smaller set is called the subsetsubset. For. Forexample, the set of tall men contains all tall men;example, the set of tall men contains all tall men;very tall men is a subset of tall men. However, thevery tall men is a subset of tall men. However, thetall men set is just a subset of the set of men. Intall men set is just a subset of the set of men. Incrisp sets, all elements of a subset entirely belong tocrisp sets, all elements of a subset entirely belong toa larger set. In fuzzy sets, however, each elementa larger set. In fuzzy sets, however, each elementcan belong less to the subset than to the larger set.can belong less to the subset than to the larger set.Elements of the fuzzy subset have smallerElements of the fuzzy subset have smallermemberships in it than in the larger set.memberships in it than in the larger set.

IntersectionIntersection Crisp Sets:Crisp Sets: Which element belongs to both sets? Which element belongs to both sets?Fuzzy Sets:Fuzzy Sets: How much of the element is in both How much of the element is in both sets?sets?In classical set theory, an intersection between two In classical set theory, an intersection between two sets contains the elements shared by these sets. For sets contains the elements shared by these sets. For example, the intersection of the set of tall men and example, the intersection of the set of tall men and the set of fat men is the area where these sets the set of fat men is the area where these sets overlap. In fuzzy sets, an element may partly overlap. In fuzzy sets, an element may partly belong to both sets with different memberships. A belong to both sets with different memberships. A fuzzy intersection is the lower membership in both fuzzy intersection is the lower membership in both sets of each element. The fuzzy intersection of two sets of each element. The fuzzy intersection of two fuzzy sets A and B on universe of discourse X:fuzzy sets A and B on universe of discourse X:AABB(x) = min [(x) = min [AA (x), (x), BB (x)] = (x)] = AA (x) (x) BB(x)(x),,where xwhere xXX

UnionUnion Crisp Sets:Crisp Sets: Which element belongs to either set? Which element belongs to either set?Fuzzy Sets:Fuzzy Sets: How much of the element is in either How much of the element is in either set?set?The union of two crisp sets consists of every elementThe union of two crisp sets consists of every elementthat falls into either set. For example, the union ofthat falls into either set. For example, the union oftall men and fat men contains all men who are talltall men and fat men contains all men who are tallOR OR fat. In fuzzy sets, the union is the reverse of thefat. In fuzzy sets, the union is the reverse of theintersection. That is, the union is the largestintersection. That is, the union is the largestmembership value of the element in either set. Themembership value of the element in either set. Thefuzzy operation for forming the union of two fuzzyfuzzy operation for forming the union of two fuzzysets A and B on universe X can be given as:sets A and B on universe X can be given as:

AABB(x) = max [(x) = max [AA (x), (x), BB(x)] = (x)] = AA (x) (x) BB(x)(x),,where xwhere xXX

Set-Theoretic Operations• Subset:

• Complement:

• Union:

• Intersection:

A B A B

C A B x x x x xc A B A B ( ) max( ( ), ( )) ( ) ( )

C A B x x x x xc A B A B ( ) min( ( ), ( )) ( ) ( )

A X A x xA A ( ) ( )1

What is the difference between classical andWhat is the difference between classical andfuzzy rules?fuzzy rules?

Rule: 1Rule: 1 Rule: 2Rule: 2IF IF speed is > 100speed is > 100 IF IF speed is < 40 speed is < 40 THEN THEN stopping_distance is longstopping_distance is long THEN stopping_distance is shortTHEN stopping_distance is short

The variable speed can have any numerical valueThe variable speed can have any numerical valuebetween 0 and 220 km/h, but the linguistic variablebetween 0 and 220 km/h, but the linguistic variablestopping_distance can take either value long stopping_distance can take either value long or short.or short.In other words, classical rules are expressed in theIn other words, classical rules are expressed in theblack-and-white language of Boolean logic.black-and-white language of Boolean logic.

A classical IF-THEN rule uses binary logic, for A classical IF-THEN rule uses binary logic, for example,example,

We can also represent the stopping distance rules in aWe can also represent the stopping distance rules in afuzzy form:fuzzy form:

Rule: 1Rule: 1 Rule: 2Rule: 2IF speed is fastIF speed is fast IF IF speed is slowspeed is slowTHEN stopping_distance is longTHEN stopping_distance is long THEN stopping_distance is shortTHEN stopping_distance is short

In fuzzy rules, the linguistic variable speed also hasIn fuzzy rules, the linguistic variable speed also hasthe range (the universe of discourse) between 0 andthe range (the universe of discourse) between 0 and220 km/h, but this range includes fuzzy sets, such as220 km/h, but this range includes fuzzy sets, such asslow, medium and slow, medium and fast. The universe of discourse offast. The universe of discourse ofthe linguistic variable stopping_distance can bethe linguistic variable stopping_distance can bebetween 0 and 300 m and may include such fuzzybetween 0 and 300 m and may include such fuzzysets as short, medium andsets as short, medium and long.long.

Fuzzification, Fuzzy Inference, Defuzzification:Fuzzification, Fuzzy Inference, Defuzzification:

Basic Elements of a Fuzzy Logic SystemBasic Elements of a Fuzzy Logic System

LinguisticLevel

NumericalLevel

Measured Variables

Measured Variables

(Numerical Values)

(Linguistic Values)2. Fuzzy-Inference Command Variables

3. Defuzzification

Plant

1. Fuzzification

(Linguistic Values)

Command Variables(Numerical Values)

Fuzzy Logic Defines Fuzzy Logic Defines the Control Strategy on the Control Strategy on a Linguistic Level!a Linguistic Level!

Control Loop of the Fuzzy Logic Controlled Container Crane:Control Loop of the Fuzzy Logic Controlled Container Crane:

Basic Elements of a Fuzzy Logic SystemBasic Elements of a Fuzzy Logic System

LinguisticLevel

NumericalLevel

Angle, Distance

Angle, Distance

(Numerical Values)

(Numerical Values)2. Fuzzy-Inference

Power

Power

(Numerical Values)

(Linguistic Variable)

3. Defuzzification

Container Crane

1. Fuzzification

Closing the Loop Closing the Loop With Words !With Words !

Types of Fuzzy Controllers:- Direct Controller - Types of Fuzzy Controllers:- Direct Controller -

The Outputs of the Fuzzy Logic System Are the Command Variables of the Plant: The Outputs of the Fuzzy Logic System Are the Command Variables of the Plant:

Fuzzification Inference Defuzzification

IF temp=lowAND P=highTHEN A=med

IF ...

Variables

Measured Variables

Plant

Command

Fuzzy Rules Output Fuzzy Rules Output Absolute Values ! Absolute Values !

Types of Fuzzy Controllers:- Supervisory Control - Types of Fuzzy Controllers:- Supervisory Control -

Fuzzy Logic Controller Outputs Set Values for Underlying PID Controllers: Fuzzy Logic Controller Outputs Set Values for Underlying PID Controllers:

Fuzzification Inference Defuzzification

IF temp=lowAND P=highTHEN A=med

IF ...

Set Values

Measured Variables

Plant

PID

PID

PID

Human Operator Human Operator Type Control ! Type Control !

Types of Fuzzy Controllers:- PID Adaptation - Types of Fuzzy Controllers:- PID Adaptation -

Fuzzy Logic Controller Adapts the P, I, and D Parameter of a Conventional PID Controller:Fuzzy Logic Controller Adapts the P, I, and D Parameter of a Conventional PID Controller:

Fuzzification Inference Defuzzification

IF temp=lowAND P=highTHEN A=med

IF ...

P

Measured Variable

PlantPID

ID

Set Point Variable

Command Variable

The Fuzzy Logic System The Fuzzy Logic System Analyzes the Performance of the Analyzes the Performance of the PID Controller and Optimizes It !PID Controller and Optimizes It !