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Fuzzy Sets & Systems

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Outline

• Introduction• Basic definitions and terminology• Set-theoretic operations• MF formulation and parameterization• Extension principle• Fuzzy relations• Fuzzy if-then rules• Compositional rule of inference• Fuzzy reasoning• Fuzzy inference systems• Fuzzy modeling

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Fuzzy Sets

•Sets with fuzzy boundaries

A = Set of tall people

Heights5’10’’

1.0

Crisp set A

Membershipfunction

Heights5’10’’ 6’2’’

.5

.9

Fuzzy set A1.0

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Membership Functions (MFs)

• Characteristics of MFs:– Subjective measures– Not probability functions

MFs

Heights5’10’’

.5

.8

.1

“tall” in Asia

“tall” in the US

“tall” in NBA

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Fuzzy Sets

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

A x x x XA= ∈{( , ( ))| }µ

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).

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Fuzzy Sets with Discrete Universes

• Fuzzy set C = “desirable city to live in”X = {SF, Boston, LA} (discrete and nonordered)C = {(SF, 0.9), (Boston, 0.8), (LA, 0.6)}

• Fuzzy set A = “sensible number of children”X = {0, 1, 2, 3, 4, 5, 6} (discrete universe)A = {(0, .1), (1, .3), (2, .7), (3, 1), (4, .6), (5, .2), (6, .1)}

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Fuzzy Sets with Cont. Universes

• Fuzzy set B = “about 50 years old”X = Set of positive real numbers (continuous)

B = {(x, µB(x)) | x in X}

µ B xx

( ) =+ −

1

150

10

2

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Alternative Notation

• A fuzzy set A can be alternatively denoted as follows:

A x xAx X

i i

i

=∈∑ µ ( ) /

A x xA

X

= ∫ µ ( ) /

X is discrete

X is continuous

Note that ΣΣΣΣ and integral signs stand for the union of membership grades; “/” stands for a marker and does not imply division.

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Fuzzy Partition

• Fuzzy partitions formed by the linguistic values “young”, “middle aged”, and “old”:

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More Definitions

SupportCoreNormalityCrossover pointsFuzzy singleton

α-cut, strong α-cut

ConvexityFuzzy numbersBandwidthSymmetricityOpen left or right,

closed

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MF Terminology

MF

X

.5

1

0Core

Crossover points

Support

α α α α - cut

αααα

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Set-Theoretic Operations

• Subset:

• Complement:

• Union:

• Intersection:

A B A B⊆ ⇔ ≤µ µ

C A B x x x x xc A B A B= ∪ ⇔ = = ∨µ µ µ µ µ( ) max( ( ), ( )) ( ) ( )

C A B x x x x xc A B A B= ∩ ⇔ = = ∧µ µ µ µ µ( ) min( ( ), ( )) ( ) ( )

A X A x xA A= − ⇔ = −µ µ( ) ( )1

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Set-Theoretic Operations

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MF Formulation

Triangular MF: trimf x a b cx a

b a

c x

c b( ; , , ) max min , ,=

−−

−−

0

Trapezoidal MF: trapmf x a b c dx a

b a

d x

d c( ; , , , ) max min , , ,=

−−

−−

1 0

Generalized bell MF: gbellmf x a b cx c

b

b( ; , , ) =+

−1

1

2

Gaussian MF: gaussmf x a b c e

x c

( ; , , ) =−

−

1

2

2

σ

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MF Formulation

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MF Formulation

Sigmoidal MF: sigmf x a b ce a x c( ; , , ) ( )=

+ − −

1

1

Extensions:

Abs. differenceof two sig. MF

Productof two sig. MF

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MF Formulation

L-R MF:

L R x c

Fc x

x c

Fx c

x c

L

R

( ; , , )

,

,

α βα

β

=

−

<

−

≥

Example:F x xL ( ) max( , )= −0 1 2 F x xR ( ) exp( )= − 3

c=65

a=60

b=10

c=25

a=10

b=40

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Extension Principle

A is a fuzzy set on X :

A x x x x x xA A A n n= + + +µ µ µ( ) / ( ) / ( ) /1 1 2 2 L

The image of A under f( ) is a fuzzy set B:

B x y x y x yB B B n n= + + +µ µ µ( ) / ( ) / ( ) /1 1 2 2 L

where yi = f(xi), i = 1 to n.

If f( ) is a many-to-one mapping, then

µ µBx f y

Ay x( ) max ( )( )

== −1

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Fuzzy Relations

•A fuzzy relation R is a 2D MF:

•Examples:– x is close to y (x and y are numbers)– x depends on y (x and y are events)– x and y look alike (x, and y are persons or objects)– If x is large, then y is small (x is an observed reading and Y

is a corresponding action)

R x y x y x y X YR= ∈ ×{(( , ), ( , ))| ( , ) }µ

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Max-Min Composition

• The max-min composition of two fuzzy relations R1 (defined on Xand Y) and R2 (defined on Y and Z) is

• Properties:– Associativity:– Distributivity over union:

– Week distributivity over intersection:

– Monotonicity:

µ µ µR Ry

R Rx z x y y z1 2 1 2o ( , ) [ ( , ) ( , )]= ∨ ∧

R S T R S R To U o U o( ) ( ) ( )=

R S T R S To o o o( ) ( )=

R S T R S R To I o I o( ) ( ) ( )⊆

S T R S R T⊆ ⇒ ⊆( ) ( )o o

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Linguistic Variables

• A numerical variables takes numerical values:Age = 65

• A linguistic variables takes linguistic values:Age is old

• A linguistic values is a fuzzy set.• All linguistic values form a term set:

T(age) = {young, not young, very young, ...middle aged, not middle aged, ...old, not old, very old, more or less old, ...not very yound and not very old, ...}

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Linguistic Variables (Terms)

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Fuzzy If-Then Rules

• General format:– If x is A then y is B

• Examples:– If pressure is high, then volume is small.– If the road is slippery, then driving is dangerous.– If a tomato is red, then it is ripe.– If the speed is high, then apply the brake a little.

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Fuzzy Reasoning

• Single rule with single antecedentRule: if x is A then y is BFact: x is A’Conclusion: y is B’

• Graphic Representation:

A

X

w

A’ B

Y

x is A’

B’

Y

A’

Xy is B’

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Fuzzy Reasoning

• Single rule with multiple antecedentRule: if x is A and y is B then z is CFact: x is A’ and y is B’Conclusion: z is C’

• Graphic Representation:

A B T-norm

X Y

w

A’ B’ C2

Z

C’

ZX Y

A’ B’

x is A’ y is B’ z is C’

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Fuzzy Reasoning

• Multiple rules with multiple antecedentRule 1: if x is A1 and y is B1 then z is C1

Rule 2: if x is A2 and y is B2 then z is C2

Fact: x is A’ and y is B’Conclusion: z is C’

• Graphic Representation: (next slide)

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Fuzzy Reasoning

• Graphics representation:

A1 B1

A2 B2

T-norm

X

X

Y

Y

w1

w2

A’

A’ B’

B’ C1

C2

Z

Z

C’Z

X Y

A’ B’

x is A’ y is B’ z is C’

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Fuzzy Inference Systems (FIS)

• Also known as– Fuzzy models– Fuzzy associate memories (FAM)– Fuzzy controllers

Rule base(Fuzzy rules)

Data base(MFs)

Fuzzy reasoning

input output

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How to Construct a FIS for a Specific Application?

• Incorporate human expertise about the target system: it is called the domain knowledge (linguistic data!)

• Use conventional system identification techniques for fuzzy modeling when input-output data of a target system are available (numerical data)

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General Guidelines about Fuzzy Modeling

A. Identification of the surface structure

• Select relevant input-output variables• Choose a specific type of FIS• Determine the number of linguistic terms associated with

each input & output variables (for a Sugeno model, determine the order of consequent equations)

Part A describes the behavior of the target system by means of linguistic terms

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General Guidelines about Fuzzy Modeling

B.Identification of deep structure

• Choose an appropriate family of parameterized MF’s

• Interview human experts familiar with the target systems to determine the parameters of the MF’s used in the rule base

• Refine the parameters of the MF’s using regression & optimization techniques (best performance for a plant in control!)

• + (ii): assumes the availability of human experts

• (iii): assumes the availability of the desired input-output data set