Islamic Azad University Central Tehran Branch Fuzzy Controliauctb.ac.ir/Files/وب سایت اساتید/fuzzy-lecture.pdf · Fuzzy Control Dr. Shahram Javadi Assistant Professor

Fuzzy Control

Dr. Shahram Javadi Assistant Professor of Electrical Engineering dept. [email protected] Subject: FUZZY-CONTROL

Islamic Azad University Central Tehran Branch

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mailto:[email protected]

Text Book


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1- Introduction An Historical Perspective Limitations of Fuzzy Systems Uncertainty and Information Fuzzy Sets and Membership Chance versus Fuzziness

2- Classical Sets and Fuzzy Sets Classical Sets Operations on Classical Sets Properties of Classical (Crisp) Sets Fuzzy Sets

3- Classical Relations and Fuzzy Relations Crisp Relations Composition Fuzzy Relations Fuzzy Cartesian Product and Composition

4- Properties of Membership Functions, Fuzzification, and Defuzzification

Features of the Membership Function Fuzzification Defuzzification to Crisp Sets λ-cuts for Fuzzy Relations Defuzzification to Scalars

Syllabus Outline

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5- Logic and Fuzzy Systems

Classical Logic Fuzzy Logic Approximate Reasoning Natural Language Linguistic Hedges Fuzzy (Rule-Based) Systems

6- Development of Membership Functions 7- Fuzzy Arithmetic and the Extension Principle

Extension Principle Crisp Functions, Mapping, and Relations Functions of Fuzzy Sets – Extension Principle Fuzzy Transform (Mapping) Fuzzy Arithmetic Approximate Methods of Extension (Vertex Method, DSW Algorithm, Restricted DSW algorithm)

8- Fuzzy Control Systems

Control System Design Problem Control (Decision) Surface Fuzzy Control

Syllabus Outline (cont.)


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1- INTRODUCTION

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HISTORICAL PERSPECTIVE

Newtonian mechanics (considered no uncertainty)

Statistical mechanics described by a probability theory

(uncertainty)

The leading theory in quantifying uncertainty in scientific models from the late nineteenth century until the late twentieth century had been

probability theory! The gradual evolution of the expression of uncertainty using probability theory was challenged, first in 1937 by Max Black, with his studies in

Vagueness!

With introduction of fuzzy sets by Lotfi Zadeh in 1965, it had a profound influence on the thinking about uncertainty because it challenged not only probability theory as the sole representation for uncertainty, but the very foundations upon which probability theory was based: classical binary (two-valued) logic.

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Probability theory dominated the mathematics of uncertainty for over five centuries (1500s-2000s)

The twentieth century saw the first developments of alternatives to probability theory and to classical Aristotelian logic as paradigms to address more kinds of uncertainty than just the random kind

Jan Lukasiewicz developed a multivalued, discrete logic (1930). In the 1960’s Arthur Dempster developed a theory of evidence In 1965 Lotfi Zadeh introduced his seminal idea in a continuous-valued

logic that he called fuzzy set theory. In the 1970s Glenn Shafer extended Dempster’s work to produce a

complete theory of evidence dealing with information from more than one source

Later in the 1980s other investigators showed a strong relationship between evidence theory, probability theory, and possibility theory with the use of what was called fuzzy measures

HISTORICAL PERSPECTIVE (cont.)

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Uncertainty can be manifested in many forms:

Fuzzy (not sharp, unclear, imprecise, approximate)

Vague (not specific, amorphous)

Ambiguous (too many choices, contradictory)

Ignorance (dissonant, not knowing something)

Natural variability (conflicting, random, chaotic, unpredictable)

Uncertainty

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Uncertainty can be manifested in many forms:

‘‘I shall return soon’’ is vague not known to be associated with any unit of time (seconds, hours, days)

‘‘I shall return in a few minutes’’ is fuzzy associated with an uncertainty that is at least known to be on the order of minutes.

‘‘I shall return within 2 minutes of 6pm’’ involves an uncertainty which has a quantifiable imprecision

Uncertainty

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Fuzzy Logic : An Idea

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Operates Independently of Probability

Probability theory measures how likely the proposition is to be correct.

Fuzzy logic measures the degree of correctness to which the proposition is correct.

Probability vs. Fuzzy Logic

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The important distinction between probabilistic information and fuzzy logic is that there is no uncertainty about the fullness of the glass but rather about the degree to which it matches the category ‘full' .

Many terms/fuzzy sets, such as 'tall,' 'rich,' 'famous‘ or ‘full,' are valid only to a certain degree when applied to a particular individual or situation.

Fuzzy logic tries to measure that degree and to allow computers to manipulate such information.

Probability vs. Fuzzy Logic

What is fuzzy?

A dictionary definition 1. Of or resembling fuzz. 2. Not clear; indistinct: a fuzzy recollection of past

events. 3. Not coherent; confused: a fuzzy plan of action. 4. Covered with fuzz.

And so what is a Fuzzy Set? a not clear Set?

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Why Use Fuzzy Logic?

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Fuzzy logic is conceptually easy to understand. Fuzzy logic is flexible. Fuzzy logic is tolerant of imprecise data. Fuzzy logic can model nonlinear functions of arbitrary

complexity. Fuzzy logic can be built on top of the experience of experts. Fuzzy logic can be blended with conventional control

techniques. Fuzzy logic is based on natural language.

Fuzzy logic keeps you from bogging down in unnecessary detail. It’s all a matter of perspective. Life is complicated enough already.

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Fuzzy Sets

Proposed by Ladeh Zadeh in 1965 , "Fuzzy sets," Information and Control, vol. 8, pp. 338--353, 1965.

A generalization of set theory that allows partial membership in a set. Membership is a real number with a range [0, 1]

Membership functions are commonly triangular or Gaussian because ease of computation.

Utility comes from overlapping membership functions – a value can belong to more than one set

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Definition

As the complexity of a system increases, it becomes more difficult and eventually impossible to make a precise statement about its behavior.

Provides a very precise approach for dealing with uncertainty which grows out of the complexity of human behavior.

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Precise versus fuzzy statement

John is tall John if 180Cm. If John is on the basket ball team:

He is 210Cm. It is cold outside. In the winter

It is -10° C outside. In the summer:

It is 40° C outside. In the summer in northern Canada

It is 20° C outside.

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Range of logical values in Boolean and fuzzy logic

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

Degree of MembershipFuzzy

MarkJohnTom

Bob

Bill

11100

1.001.000.980.820.78

Peter

Steven

MikeDavid

ChrisCrisp

1

0000

0.240.150.060.010.00

Name Height, cm

205198181

167

155152

158

172179

208

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150 210170 180 190 200160Height, cm

Degree ofMembership

Tall Men

150 210180 190 200

1.0

0.0

0.2

0.4

0.6

0.8

160

Degree ofMembership

170

1.0

0.0

0.2

0.4

0.6

0.8

Height, cm

Fuzzy Sets

Crisp Sets

Crisp and fuzzy sets of “tall men”

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Fuzzy Control

Dr. Shahram Javadi Assistant Professor of Electrical Engineering dept. [email protected]


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2- Classical Sets and Fuzzy Sets

Let X be the universe of discourse and its elements be denoted as x. In

the classical set theory, crisp set A of X is defined as function fA(x) called the characteristic function of A

fA(x): X → {0, 1}, where

∉∈

=AxAx

xf A if0, if 1,

)(

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Classical set

χA : X → {0, 1}.

A fuzzy set is a set with fuzzy boundaries

In the fuzzy theory, fuzzy set A of universe X is defined by function µA(x) called the membership function of set A

µA(x): X → [0, 1], where µA(x) = 1 if x is totally in A; µA(x) = 0 if x is not in A; 0 < µA(x) < 1 if x is partly in A.

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Fuzzy set

µA(x): X → [0, 1]

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Classic set versus Fuzzy set

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Discrete Form of a Fuzzy Set

Continuous Form of a Fuzzy Set

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Operations on Classical set

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Classical Union

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Classical Intersection

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Classical Complement

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Classical Difference

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Properties of Classical (Crisp) Sets

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Properties of Classical (Crisp) Sets

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Operations in function-theoretic terms

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Operations on Fuzzy set

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∩

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Fuzzy Control



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3- Classical Relations and Fuzzy Relations

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CARTESIAN PRODUCT

A = {0, 1} B = {a, b, c}

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CRISP RELATIONS

A subset of the Cartesian product A1 × A2 ×・・・×Ar is called an r-aray relation over A1,A2, . . . ,Ar

The most common case is for r = 2;

The Cartesian product of two universes X and Y is determined as

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Sagittal diagram of an unconstrained relation

Relation Matrix

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Constrained Relation

Let X = {1, 2} and Y = {a, b}

Universal Relation and identity relation

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Continuous Relation

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Operations on Crisp Relations

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Composition

There are two common forms of the composition operation: max–min composition

max–product composition

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Example

max–min composition:

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FUZZY RELATIONS

The ‘‘strength’’ of the relation between ordered pairs of the two universes is not measured with the characteristic function, but rather with a membership function expressing various ‘‘degrees’’ of strength of the relation on the unit interval.

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Operations On Fuzzy relations

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Properties of Fuzzy relations Just as for crisp relations, the properties of commutativity, associativity, distributivity, involution, and idempotency all hold for fuzzy relations. Moreover, de morgan’s principles hold for fuzzy relations just as they do for crisp (classical) relations, and the null relation, O, and the complete relation, E, are analogous to the null set and the whole set in set-theoretic form, respectively

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Fuzzy Cartesian Product and Composition

(C) 2001 by Yu Hen Hu 15

Fuzzy Composition Example

Let the two relations R and S be, respectively:

The goal is to compute RoS using both Max-min and Max-product composition rules.

R y1

y2

y3

S z1

z2

x1

0.4 0.6 0 y1

0.5 0.8

x2

0.9 1 0.1 y2

0.1 1

y1

0 0.6


MAX-MIN Composition RoS = max{min(0.4,0.5), min(0.6, 0.1), min(0, 0)} = max{ 0.4, 0.1, 0} = 0.4 max{min(0.4,0.8), min(0.6, 1), min(0, 0.6)} = max{ 0.4, 0.6, 0} = 0.6 max{min(0.9,0.5), min(1, 0.1), min(0.1, 0)} = max{ 0.5, 0.1, 0} = 0.5 max{min(0.9,0.8), min(1, 1), min(0.1, 0.6)} = max{ 0.8, 1, 0.1} = 1

=

15.06.04.0

6.0011.08.05.0

1.019.006.04.0


MAX-PRODUCT Composition

RoS =

=

145.06.02.0

6.0011.08.05.0

1.019.006.04.0

max{0.40.5, 0.60.1, 00} = max{0.2,0.06,0} = 0.2

max{0.40.8, 0.61, 00.6} = max{0.32, 0.6, 0} = 0.6

max{0.90.5, 10.1, 0.10} = max{0.45, 0.1, 0} = 0.45

max{0.90.8, 11, 0.10.6} = max{0.72, 1, 0.06} = 1

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Example

fuzzy max–min composition

fuzzy max–product composition

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Crisp Equivalence Relation

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Crisp Tolerance Relation

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Fuzzy tolerance and equivalence relations

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Other forms of the composition operation

Fuzzy Control



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4- Membership Functions, Fuzzification, and

Defuzzification

FEATURES OF THE MEMBERSHIP FUNCTION

The core comprises those elements x of the universe such that

The support comprises those elements x of the universe such that

the boundaries comprise those elements x of the universe such that

۳

VARIOUS FORMS

The most common forms of membership functions are those that are normal and convex

Many operations on fuzzy sets, like extension principle and the union operator both can produce subnormal or non-convex fuzzy sets

Membership functions can be symmetrical or asymmetrical

ordinary membership functions generalized membership functions interval-valued membership function type-2 fuzzy set

۴

Interval-valued membership function

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Type-2 fuzzy set

۶

The membership function of a general type-2 fuzzy set is three-dimensional

Type-2 fuzzy sets and systems generalize (type-1) fuzzy sets and systems so that more uncertainty can be handled

http://upload.wikimedia.org/wikipedia/en/9/93/FOU_for_IT2_FS.jpg

Type-2 fuzzy set

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The membership function of a general type-2 fuzzy set is three-dimensional

Type-2 fuzzy sets and systems generalize (type-1) fuzzy sets and systems so that more uncertainty can be handled

http://upload.wikimedia.org/wikipedia/en/a/a4/MF_of_general_T2_FS.jpg

FUZZIFICATION

۸

Fuzzification is the process of making a crisp quantity fuzzy

In the real world, hardware such as a digital voltmeter generates crisp data, but these data are subject to experimental error

FUZZIFICATION

۹

Membership function representing imprecision in ‘‘crisp voltage reading.’’

FUZZIFICATION

۱۰

Comparisons of fuzzy sets and crisp or fuzzy readings: (a) fuzzy set and crisp reading; (b) fuzzy set and fuzzy reading

DEFUZZIFICATION TO CRISP SETS

۱۱

Any particular fuzzy set A can be transformed into an infinite number of λ-cut sets, because there are an infinite number of values λ on the interval [0 1]

DEFUZZIFICATION

۱۲

Any particular fuzzy set A can be transformed into an infinite number of λ-cut sets, because there are an infinite number of values λ on the interval [0 1]

A discrete fuzzy setA∼

DEFUZZIFICATION - λ cut

۱۳

Lambda-cut sets for λ = 1, 0.9, 0.6, 0.3, 0+, 0.

λ-cut re-scales the memberships to 1 or 0

The properties of λ-cut:

1. (A ∪ B)λ = Aλ ∪ Bλ 2. (A ∩ B)λ = Aλ ∩ Bλ 3. (A’)λ ≠ (Aλ)’ except for x = 0.5 4. Aα ⊆ Aλ ∀ λ < α and 0 < α < 1 A0 = X

Core = A1

Support = A0+

Boundaries = [A0 + A1]


0.6 0.3

λ-cuts for fuzzy relations 1 0.8 0 0.1 0.2

0.8 1 0.4 0 0.9

0 0.4 1 0 0

0.1 0 0 1 0.5

0.2 0.9 0 0.5 1

R =


We can define λ-cut for relations similar to the one for sets

Rλ = {(x y) | µR(x y) > λ}

1 1 φ

1 φ 1

1

R1 =

1 1 φ

1 φ 1

1 1

R0.9 =

R0 = E


λ-cuts on relations have the following properties:

(R ∪ S)λ = Rλ ∪ Sλ

(R ∩ S)λ = Rλ ∩ Sλ

(R’)λ ≠ (Rλ)’

Rα < Rλ ∀ λ ≤ α and 0 ≤ α ≤ 1


DEFUZZIFICATION TO SCALARS

۱۸

Defuzzification is the conversion of a fuzzy quantity to a precise quantity, just as fuzzification is the conversion of a precise quantity to a fuzzy quantity

The output of a fuzzy process can be the logical union of two or more fuzzy membership functions defined on the universe of discourse of the output variable

DEFUZZIFICATION TO SCALARS

۱۹

DEFUZZIFICATION TO SCALARS METHODES

۲۰

1. Max membership principle

2. Centroid method

3. Weighted average method

4. Mean max membership

5. Center of sums

6. Center of largest area

7. First (or last) of maxima

1. Max membership principle

۲۱

2. Centroid method

۲۲

3. Weighted average method

۲۳

Unfortunately it is usually restricted to symmetrical output membership functions.

4. Mean max membership (middle-of-maxima)

۲۴

Defuzzification Methods

Example1: A railroad company intends to lay a new rail line in a particular part of a county. The whole area through which the new line is passing must be purchased for right-of-way considerations. It is surveyed in three stretches, and the data are collected for analysis. The surveyed data for the road are given by the sets , where the sets are defined on the universe of right-of-way widths, in meters. For the railroad to purchase the land, it must have an assessment of the amount of land to be bought. The three surveys on the right-of-way width are ambiguous , however, because some of the land along the proposed railway route is already public domain and will not need to be purchased. Additionally, the original surveys are so old (circa 1860) that some ambiguity exists on the boundaries and public right-of-way for old utility lines and old roads. The three fuzzy sets , shown in the figures below, represent the uncertainty in each survey as to the membership of the right-of-way width, in meters, in privately owned land.

We now want to aggregate these three survey results to find the single most nearly representative right-of-way width (z) to allow the railroad to make its initial estimate

3~2~1~BandB,B

3~2~1~BandB,B


Fuzzy set B1: public right-of-way width (z) for survey 1

Fuzzy set B2: public right-of-way width (z) for survey 2 Fuzzy set B3: public right-of-way width (z) for survey 3


( ) ( )

( )

meters

dzzdzdzzdzdzzdzdzz

zdzzzdzzdzzzdzzdzzdzzzdzz

dzz

zdzzz

B

B

9.4

)8()5()5(.2

3)3(.3.

85)5(.2

3)3(.)3(.

)(

)(

1

0

6.3

1

4

6.3

5.5

4

6

5.5

7

6

8

7

1

0

6.3

1

4

6.3

5.5

4

6

5.5

7

6

8

7

*

~

~

=

−++−++

−

++÷

−++−++

−

++

=•

=

∫ ∫ ∫ ∫ ∫ ∫ ∫

∫ ∫ ∫ ∫ ∫ ∫ ∫

∫∫

µ

µ

Centroid method:

The centroid method for finding z*.


Weighted-Average Method: ( ) ( ) ( ) metersz 41.5

15.3.5.6155.5.23.* =

++×+×+×

=

Mean-Max Method: meters5.62/)76( =+


Example2: Many products, such as tar, petroleum jelly, and petroleum, are extracted from crude oil. In a newly drilled oil well, three sets of oil samples are taken and tested for their viscosity. The results are given in the form of the three fuzzy sets B1, B2, and B3, all defined on a universe of normalized viscosity, we want to find the most nearly representative viscosity value for all three oil samples, and hence find z* for the three fuzzy viscosity sets.




According to the centroid method,

( ) ( ) ( )

( ) ( )

( ) ( ) ( )

( ) ( )5.2

5.5.25.5.

3)1(67.267.

5.5.25.5.

3167.2)67(.

)(

)(

5

3

3

33.2

33.2

2

2

8.1

8.1

5.1

5.1

0

5

3

3

33.2

33.2

2

2

8.1

8.1

5.1

5.1

0

*

~

~

=

−+−+

−+−+−+÷

−+−+

−+−+−+

==

∫∫∫∫∫∫

∫∫∫∫∫∫

∫∫

dzzdzz

dzzdzzdzzdzz

zdzzzdzz

zdzzzdzzzdzzzdzz

dzz

zdzzz

B

B

µ

µ


The centroid value obtained, z*, is shown in the figure below:


According to the weighted average method:

( ) ( ) ( ) 25.2111

31215.11* =++

×+×+×=z

5. Center of sums

۳۵

This is faster than many defuzzification methods that are presently in use, and the method is not restricted to symmetric membership functions. This process involves the algebraic sum of individual output fuzzy sets, sayC1 and C2, instead of their union.

Intersecting areas are added twice

The method also involves finding the centroids of the individual membership functions

is the distance to the centroid of each of the respective membership functions

5. Center of sums

۳۶

6. Center of largest Area

۳۷

If the output fuzzy set has at least two convex subregions, then the center of gravity of the convex fuzzy subregion with the largest area is used to obtain the defuzzified value z∗ of the output

Where Cm is the convex subregion that has the largest area making up Ck

6. Center of largest Area

۳۸


۳۹

This method uses the overall output or union of all individual output fuzzy sets Ck to determine the smallest value of the domain with maximized membership degree in Ck.


۴۰

The supremum (sup) is the least upper bound and the infimum (inf) is the greatest lower bound

Other Methods for Defuzzifications:

۴۱ 1: Defuzzification: criteria and classification, from the journal Fuzzy Sets and Systems, Van Leekwijck and Kerre, Vol. 108 (1999), pp. 159-178 [1]

Fuzzy Control



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5- Logic and Fuzzy Systems

1- CLASSICAL LOGIC

۳

Now let P and Q be two simple propositions on the same universe of discourse that can be combined using the following five logical connectives

CLASSICAL LOGIC

۴

Now define sets A and B from universe X

A propositional calculus will exist for the case where proposition P measures the truth of the statement that an element, x, from the universe X is contained in set A and the truth of the statement Q that this element, x, is contained in set B,

CLASSICAL LOGIC

۵

where truth is measured in terms of the truth value

Logical connectives

۶

Logical connectives

۷

The logical implication is analogous to the set-theoretic form!

Logical connectives

۸

Graphically, this implication and the analogous set operation are represented by the Venn diagram

Graphical analog of the classical implication operation; gray area is where implication holds

Logical connectives

۹

Logical connectives

۱۰

Implication operation involving two different universes of discourse: P is a proposition described by set A, which is defined on universe X Q is a proposition described by set B, which is defined on universe Y Then the implication P →Q can be represented in set-theoretic terms by the relation R, where R is defined by

Logical connectives

۱۱

IF A, THEN B

Logical connectives

۱۲

Logical connectives

۱۳

IF A, THEN B

Logical connectives

۱۴

IF A, THEN B ELSE C

Logical connectives

۱۵

IF A, THEN B ELSE C

Tautologies

۱۶

Compound Propositions that are always True !

Example: If A is the set of all prime numbers (A1 = 1, A2 = 2, A3 = 3, A4 = 5,. . .) on the real line universe, X, Then the proposition: ‘‘Ai is not divisible by 6’’ is a tautology.

Tautologies

۱۷

Some common tautologies:

Tautologies

۱۸

Proof:

Contradictions

۱۹

Compound Propositions that are always False !

Example: If A is the set of all prime numbers (A1 = 1, A2 = 2, A3 = 3, A4 = 5,. . .) on the real line universe, X, Then the proposition: ‘‘Ai is a multiple of 4’’ is a contradiction.

Exclusive OR

۲۰

The exclusive or is of interest because it arises in many situations involving natural language and human reasoning.

For example, when you are going to travel by plane or boat to some destination, the implication is that you can travel by air or sea, but not both, i.e., one or the other.

Exclusive OR

۲۱

Exclusive NOR

۲۲

Deductive Inferences

۲۳

Suppose we have a rule of the form IF A, THEN B Can we deduce, in rule form, IF A, THEN B? YES

2- FUZZY LOGIC

۲۴

Many interesting paradoxes over the ages!

Barber of Seville: In the small Spanish town of Seville, there is a rule that all and only those men who do not shave themselves are shaved by the barber. Who shaves the barber?

Ancient Greece Does the liar from Crete lie when he claims, ‘‘All Cretians are liars?’’

‘‘I lie.’’: The statement can not be both true and false

2- FUZZY LOGIC

۲۵

Multivalued Logic:

FUZZY LOGIC

۲۶

Example Suppose we are evaluating a new invention to determine its commercial potential. We will use two metrics to make our decisions regarding the innovation of the idea. Our metrics are the ‘‘uniqueness’’ of the invention, denoted by a universe of novelty scales, X = {1, 2, 3, 4}, and the ‘‘market size’’ of the invention’s commercial market, denoted on a universe of scaled market sizes, Y = {1, 2, 3, 4, 5, 6}. In both universes the lowest numbers are the ‘‘highest uniqueness’’ and the ‘‘largest market,’’ respectively. A new invention in your group, say a compressible liquid of very useful temperature and viscosity conditions, has just received scores of ‘‘medium uniqueness,’’ denoted by fuzzy set A∼ , and ‘‘medium market size,’’ denoted fuzzy set B∼ . We wish to determine the implication of such a result, i.e., IF A∼ THENB∼ We assign the invention the following fuzzy sets to represent its ratings:

FUZZY LOGIC

۲۷

IF A, THEN B

۲۸

APPROXIMATE REASONING

۲۹

OTHER FORMS OF THE IMPLICATION OPERATION

۳۰

NATURAL LANGUAGE

۳۱

LINGUISTIC HEDGES

۳۲

LINGUISTIC HEDGES

۳۳

LINGUISTIC HEDGES

Fuzzy concentration

Fuzzy dilation

۳۴

LINGUISTIC HEDGES

preference table

۳۵

LINGUISTIC HEDGES

Example: ‘‘plus very minus very small’’ should be interpreted as

plus (very (minus (very (small))))

Parentheses may be used to change the precedence order and ambiguities may be resolved by the use of association-to-the-right.

۳۶

LINGUISTIC HEDGES

Example:

۳۷

LINGUISTIC HEDGES

Example: α = ‘‘not very small and not very, very large’’

۳۸

FUZZY (RULE-BASED) SYSTEMS

IF premise (antecedent), THEN conclusion (consequent)

In the field of artificial intelligence (machine intelligence) there are various ways to represent knowledge. Perhaps the most common way to represent human knowledge is to form it into natural language expressions of the type

deductive form.

This form of knowledge representation, characterized as shallow knowledge, is quite appropriate in the context of linguistics because it expresses human empirical and heuristic knowledge in our own language of communication. It does not, however, capture the deeper forms of knowledge usually associated with intuition, structure, function.

۳۹


۴۰


۴۱

GRAPHICAL TECHNIQUES OF INFERENCE

Three common methods of deductive inference for fuzzy systems based on linguistic rules: (1)Mamdani systems,

(2) Sugeno models, (3) Tsukamoto models

۴۲


(1)Mamdani systems:

۴۳

GRAPHICAL TECHNIQUES OF INFERENCE Case 1: Using Max-Min Composition

Truncated

membership functions for each rule

۴۴

GRAPHICAL TECHNIQUES OF INFERENCE Case 2: Using Max-Prod Composition

Scaled membership functions for each

rule

۴۵


(2): Sugeno method, or the TSK method (Takagi, Sugeno, and Kang) [Takagi and Sugeno, 1985; Sugeno and Kang, 1988],

where z = f (x, y) is a crisp function in the consequent.

Usually f (x, y) is a polynomial function in the inputs x and y, but it can be any general function as long as it describes the output of the system within the fuzzy region specified in the antecedent of the rule to which it is applied.

When f (x, y) is a constant the inference system is called a zero-order Sugeno model, which is a special case of the Mamdani system.

۴۶


When f (x, y) is a linear function of x and y, the inference

first-order Sugeno model:

۴۷


Example: first-order Sugeno model

۴۸


(3): Tsukamoto Method [1979] In this method the consequent of each fuzzy rule is represented by a fuzzy set with a monotonic membership function,

۴۹


(3): Tsukamoto Method [1979] Advantage: Since each rule infers a crisp output, the Tsukamoto model’s aggregation of the overall output also avoids the time-consuming process of defuzzification. Disadvantage: Because of the special nature of the output membership functions required by the method, it is not as useful as a general approach, and must be employed in specific situations.

۵۰

GRAPHICAL TECHNIQUES OF INFERENCE (3): Tsukamoto Method [1979]

Example:

Fuzzy Inference

۵۱

Contents

Fuzzy Inference • Fuzzification of the input variables • Rule evaluation • Aggregation of the rule outputs • Defuzzification

Mamdani Sugeno

۵۲

Fuzzy Inference

The most commonly used fuzzy inference technique is the so-called Mamdani method.

In 1975, Professor Ebrahim Mamdani of London University built one of the first fuzzy systems to control a steam engine and boiler combination. He applied a set of fuzzy rules supplied by experienced human operators.

۵۳

Mamdani Fuzzy Inference

The Mamdani-style fuzzy inference process is performed in four steps:

1. Fuzzification of the input variables

2. Rule evaluation (inference)

3. Aggregation of the rule outputs (composition)

4. Defuzzification.

۵۴

Mamdani Fuzzy Inference We examine a simple two-input one-output problem that includes three rules: Rule: 1 IF x is A3 OR y is B1 THEN z is C1 Rule: 2 IF x is A2 AND y is B2 THEN z is C2 Rule: 3 IF x is A1 THEN z is C3

Real-life example for these kinds of rules: Rule: 1 IF project_funding is adequate OR project_staffing is small THEN risk is low Rule: 2 IF project_funding is marginal AND project_staffing is large THEN risk is normal Rule: 3 IF project_funding is inadequate THEN risk is high

۵۵

Mamdani Fuzzy Inference We examine a simple two-input one-output problem that includes three rules: Rule: 1 IF x is A3 OR y is B1 THEN z is C1 Rule: 2 IF x is A2 AND y is B2 THEN z is C2 Rule: 3 IF x is A1 THEN z is C3

Real-life examle for these kinds of rules: Rule: 1 IF project_funding is adequate OR project_staffing is small THEN risk is low Rule: 2 IF project_funding is marginal AND project_staffing is large THEN risk is normal Rule: 3 IF project_funding is inadequate THEN risk is high

۵۶

Step 1: Fuzzification

The first step is to take the crisp inputs, x1 and y1 (project funding and project staffing), and determine the degree to which these inputs belong to each of the appropriate fuzzy sets.

Crisp Inputy1

0.1

0.71

0 y1

B1 B2

Y

Crisp Input

0.20.5

1

0

A1 A2 A3

x1

x1 Xµ (x = A1) = 0.5µ (x = A2) = 0.2

µ (y = B1) = 0.1µ (y = B2) = 0.7

57

Step 2: Rule Evaluation

The second step is to take the fuzzified inputs, µ(x=A1) = 0.5, µ(x=A2) = 0.2, µ(y=B1) = 0.1 and µ(y=B2) = 0.7,

and apply them to the antecedents of the fuzzy rules. If a given fuzzy rule has multiple antecedents, the fuzzy operator

(AND or OR) is used to obtain a single number that represents the result of the antecedent evaluation.

RECALL: To evaluate the disjunction of the rule antecedents, we use

the OR fuzzy operation. Typically, fuzzy expert systems make use of the classical fuzzy operation union:

µA∪B(x) = max [µA(x), µB(x)] Similarly, in order to evaluate the conjunction of the rule antecedents,

we apply the AND fuzzy operation intersection: µA∩B(x) = min [µA(x), µB(x)]

۵۸


A31

0 X

1

y10 Y0.0

x1 0

0.1C1

1C2

Z

1

0 X

0.2

0

0.2 C11

C2

Z

A2

x1

Rule 3:

A11

0 X 0

1

Zx1

THEN

C1 C2

1

y1

B2

0 Y

0.7

B10.1

C3

C3

C30.5 0.5

OR(max)

AND(min)

OR THENRule 1:

AND THENRule 2:

IF x is A3 (0.0) y is B1 (0.1) z is C1 (0.1)

IF x is A2 (0.2) y is B2 (0.7) z is C2 (0.2)

IF x is A1 (0.5) z is C3 (0.5)

۵۹


Now the result of the antecedent evaluation can be applied to the membership function of the consequent.

There are two main methods for doing so: Clipping Scaling

Degree ofMembership1.0

0.0

0.2

Z

Degree ofMembership

Z

C2

1.0

0.0

0.2

C2

clipping scaling 60

Step 2: Rule Evaluation The most common method of correlating the rule consequent with

the truth value of the rule antecedent is to cut the consequent membership function at the level of the antecedent truth. This method is called clipping (alpha-cut).

Since the top of the membership function is sliced, the clipped fuzzy set loses some information.

However, clipping is still often preferred because it involves less complex and faster mathematics, and generates an aggregated output surface that is easier to defuzzify.

While clipping is a frequently used method, scaling offers a better approach for preserving the original shape of the fuzzy set.

The original membership function of the rule consequent is adjusted by multiplying all its membership degrees by the truth value of the rule antecedent.

This method, which generally loses less information, can be very useful in fuzzy expert systems.

۶۱

Step 3: Aggregation of the rule outputs

Aggregation is the process of unification of the outputs of all rules.

We take the membership functions of all rule consequents previously clipped or scaled and combine them into a single fuzzy set.

The input of the aggregation process is the list of clipped or scaled consequent membership functions, and the output is one fuzzy set for each output variable.

00.1

1C1

Cz is 1 (0.1)

C2

00.2

1

Cz is 2 (0.2)

0

0.5

1

Cz is 3 (0.5)ZZZ

0.2

Z0

∑

C30.50.1

62

Step 4: Defuzzification The last step in the fuzzy inference process is defuzzification.

Fuzziness helps us to evaluate the rules, but the final output of a fuzzy system has to be a crisp number.

The input for the defuzzification process is the aggregate output fuzzy set and the output is a single number.

There are several defuzzification methods, but probably the most popular one is the centroid technique. It finds the point where a vertical line would slice the aggregate set into two equal masses. Mathematically this centre of gravity (COG) can be expressed as:

( )

( )∫

∫

µ

µ

= b

aA

b

aA

dxx

dxxx

COG

63

Centroid defuzzification method finds a point representing the centre of gravity of the aggregated fuzzy set A, on the interval [a, b ].

A reasonable estimate can be obtained by calculating it over a sample of points.

Step 4: Defuzzification

64

Sugeno Fuzzy Inference

Mamdani-style inference, as we have just seen, requires us to find the centroid of a two-dimensional shape by integrating across a continuously varying function. In general, this process is not computationally efficient.

Michio Sugeno suggested to use a single spike, a singleton, as the membership function of the rule consequent.

A singleton, or more precisely a fuzzy singleton, is a fuzzy set with a membership function that is unity at a single particular point on the universe of discourse and zero everywhere else.

۶۵

۶۶

Sugeno Fuzzy Inference

The most commonly used zero-order Sugeno fuzzy model applies fuzzy rules in the following form:

IF x is A AND y is B THEN z is k

where k is a constant.

In this case, the output of each fuzzy rule is constant. All consequent membership functions are represented by

singleton spikes.

۶۷

Sugeno Rule Evaluation

A31

0 X

1

y10 Y

0.0

x1 0

0.1

1

Z

1

0 X

0.2

0

0.2

1

Z

A2

x1

IF x is A1 (0.5) z is k3 (0.5)Rule 3:

A11

0 X 0

1

Zx1

THEN

1

y1

B2

0 Y

0.7

B10.1

0.5 0.5

OR(max)

AND(min)

OR y is B1 (0.1) THEN z is k1 (0.1)Rule 1:

IF x is A2 (0.2) AND y is B2 (0.7) THEN z is k2 (0.2)Rule 2:

k1

k2

k3

IF x is A3 (0.0)

A31

0 X

1

y10 Y

0.0

x1 0

0.1

1

Z

1

0 X

0.2

0

0.2

1

Z

A2

x1

IF x is A1 (0.5) z is k3 (0.5)Rule 3:

A11

0 X 0

1

Zx1

THEN

1

y1

B2

0 Y

0.7

B10.1

0.5 0.5

OR(max)

AND(min)

OR y is B1 (0.1) THEN z is k1 (0.1)Rule 1:

IF x is A2 (0.2) AND y is B2 (0.7) THEN z is k2 (0.2)Rule 2:

k1

k2

k3

IF x is A3 (0.0)

۶۸

Sugeno Aggregation of the Rule Outputs

z is k1 (0.1) z is k2 (0.2) z is k3 (0.5) ∑0

1

0.1Z 0

0.5

1

Z00.2

1

Zk1 k2 k3 0

1

0.1Zk1 k2 k3

0.20.5

۶۹

Sugeno Defuzzification

655.02.01.0

805.0502.0201.0)3()2()1(

3)3(2)2(1)1(=

++×+×+×

=µ+µ+µ

×µ+×µ+×µ=

kkkkkkkkkWA

Weighted Average (WA)

0 Z

Crisp Outputz1

z1

70

Mamdani or Sugeno?

Mamdani method is widely accepted for capturing expert knowledge. It allows us to describe the expertise in more intuitive, more human-like manner. However, Mamdani-type fuzzy inference entails a substantial computational burden.

On the other hand, Sugeno method is computationally effective and works well with optimization and adaptive techniques, which makes it very attractive in control problems, particularly for dynamic nonlinear systems.

۷۱

The most commonly used fuzzy inference technique is the so-called Mamdani method. In 1975, Professor Ebrahim Mamdani of London University built one of the first fuzzy systems to control a steam engine and boiler combination. He applied a set of fuzzy rules supplied by experienced human operators.

Fuzzy inference

7/6/2006 73

m

s

M

RL

VL

S

RS

L

VS

S

M

MVS S

L

M

S

The square FAM representation

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The rule table Rule m s ρ n Rule m s ρ n Rule m s ρ n

1 VS S L VS 10 VS S M S 19 VS S H VL

2 S S L VS 11 S S M VS 20 S S

S

3 M S L VS 12 M S M VS 21 M S

4 VS M L VS 13 VS M M RS 22 VS M H M

M

M

M

5 S M L VS 14 S M M S 23 S M

6 M M L VS 15 M M M VS 24 M M

7 VS L L S 16 VS L M M 25 VS L H

H

H

H

H

H

RL

8 S L

L

L S 17 S L M RS 26 S L

9 M L L VS 18 M L M S 27 M L H RS

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Rule Base 1 1. If (utilisation_factor is L) then (number_of_spares is S)2. If (utilisation_factor is M) then (number_of_spares is M)3. If (utilisation_factor is H) then (number_of_spares is L)

4. If (mean_delay is VS) and (number_of_servers is S) then (number_of_spares is VL)5. If (mean_delay is S) and (number_of_servers is S) then (number_of_spares is L)6. If (mean_delay is M) and (number_of_servers is S) then (number_of_spares is M)

7. If (mean_delay is VS) and (number_of_servers is M) then (number_of_spares is RL)8. If (mean_delay is S) and (number_of_servers is M) then (number_of_spares is RS)9. If (mean_delay is M) and (number_of_servers is M) then (number_of_spares is S)

10. If (mean_delay is VS) and (number_of_servers is L) then ( number_of_spares is M)11. If (mean_delay is S) and (number_of_servers is L) then ( number_of_spares is S)12. If (mean_delay is M) and (number_of_servers is L) then ( number_of_spares is VS)

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Gui

VS VS VSVS VS VS

VS VS VSVL L M

HS

VS VS VSVS VS VS

VS VS VSM

VS VS VSVS VS VSS S VSL

s

LVS S

m

MH

ρρ

VS VS VSLVS S

S

m

VS VS VSM

S S VSL

s

S VS VSMVS S

m

VS S

m

S

RS S VSM

M RS SL

s

S

M M SM

RL M RSL

s

M

M

M

M

Cube FAM of Rule Base 2

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Gui

00.2

0.40.6

0.81

0

0.2

0.4

0.6

0.2

0.3

0.4

0.5

0.6

number_of_serversmean_delay

Three-dimensional plots for Rule Base 1

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Gui


00.2

0.40.6

0.81

0

0.2

0.4

0.6

0.2

0.3

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0.6

utilisation_factormean_delay

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Gui

00.2

0.40.6

0.81

0

0.2

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0.15

0.2

0.25

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0.35



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Gui

VS VS VS

VS VS VS

VS VS VS

VS VS VS

VS VS VS

VS VS VS

VS VS VS

VS VS VS

VS VS VS

VS VS VS

VS VS VS

VS VS VS

VS VS VS

S S VS

S S VS

VL L M

VL RL RS

M M S

RL M RS

L M RS

HS

M

RL

L

RS

s

LVS Sm

MM

Hρρ

VS VS VS

VS VS VS

VS VS VS

S S VS

S S VS

LVS S

S

M

M

RL

L

RS

m

s

S VS VS

S VS VS

RS S VS

M RS S

M RS S

MMVS Sm

VS Sm

S

M

RL

L

RS

s

S

M

M

RL

L

RS

s

Cube FAM of Rule Base 3

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Gui

00.2

0.40.6

0.81

0

0.2

0.4

0.6

0.15

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0.35



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0.5


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Gui

Tuning fuzzy systems 1. Review model input and output variables, and if

required redefine their ranges.

2. Review the fuzzy sets, and if required define additional sets on the universe of discourse. The use of wide fuzzy sets may cause the fuzzy system to perform roughly.

3. Provide sufficient overlap between neighboring sets. It is suggested that triangle-to-triangle and trapezoid-to-triangle fuzzy sets should overlap between 25% to 50% of their bases.

7/6/2006 85 A

Gui

4. Review the existing rules, and if required add new rules to the rule base.

5. Examine the rule base for opportunities to write hedge rules to capture the pathological behaviour of the system.

6. Adjust the rule execution weights. Most fuzzy logic tools allow control of the importance of rules by changing a weight multiplier.

7. Revise shapes of the fuzzy sets. In most cases, fuzzy systems are highly tolerant of a shape approximation.

Documents

Islamic Azad University Central Tehran Branch Fuzzy Controliauctb.ac.ir/Files/وب سایت اساتید/fuzzy-lecture.pdf · Fuzzy Control Dr. Shahram Javadi Assistant Professor