Fuzzy Systems and Applications

CONTENTS

• History Of Fuzzy Theory• Types of Uncertainty and the Modeling of

Uncertainty • Probability and Uncertainty• Fuzzy Set Theory• Fuzziness versus probability• Fuzzy Logic Control (FLC)

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

Sde 3

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 4

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 5

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 6

“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 11

“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”

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 12

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!

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

Linguistic VariableLinguistic Variable

Slide 13

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 Logic Control (FLC)

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

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

© INFORM 1990-1998 Slide 15

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

© INFORM 1990-1998 Slide 16

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 -

© INFORM 1990-1998 Slide 17

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 -

© INFORM 1990-1998 Slide 18

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 -

© INFORM 1990-1998 Slide 19

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 !

CONCLUSION

• Non-Modeled Based Controller

• Knowledge Based

Thank You for your attention