Topic 2Topic 2

Fuzzy Logic Fuzzy Logic ControlControl

Ming-Feng Yeh 2-2

OutlinesOutlines

Basic concepts of fuzzy set theoryFuzzy relationsFuzzy logic controlGeneral Fuzzy System R.R. Yager and D.P. Filev, Essentials of fuzzy modeling

and control, John Wiley & Sons. Inc., 1994 L.X. Wang, A Course in Fuzzy Systems and Control, Pr

entice-Hall, 1997. K.M. Passino and S. Yurkovich, Fuzzy Control, Addiso

n Wesley, 1998.

Ming-Feng Yeh 2-3

1. Introduction1. Introduction

The concept of a fuzzy subset was originally introduced by L.A. Zadeh in 1965 as a generalization of the idea of an crisp set.

A fuzzy subset whose truth values are drawn from the unit interval [0, 1] rather than the set {0, 1}.

The fuzzy subset has as its underlying logic a multivalued logic.

Ming-Feng Yeh 2-4

Classical Set TheoryClassical Set Theory

Two-valued logic: {0,1}, i.e.,Characteristic function: Intersection:Union:Difference:Complement:

., AaAa

BABA

BA

A

A B

BA

A B

BA

A

BA

B A

A

}1,0{: XA

Ming-Feng Yeh 2-5

Fuzzy Set TheoryFuzzy Set Theory

Fuzzy logic deals with problems that have vagueness, uncertainty, or imprecision, and uses membership functions with values in [0,1].

Membership function:X: universe of discourse

]1,0[: XA

Ming-Feng Yeh 2-6

DefinitionDefinition

Assume X is a set serving as the universe of discourse. A fuzzy subset A of X is a associated with a characteristic function: A(x) or A(x)

Membership function:

The relationship between variables, labels and fuzzy sets.

]1,0[: XA

variable

label

fuzzy set

Temperature

cold cool tepid warm hot

C

A

0

1

20 25

Ming-Feng Yeh 2-7

Fuzzy Set RepresentationFuzzy Set Representation

X: universe of discourse (all the possible values that a variable can assume)

A: a subset of X

Discrete:

Continue:

}.|)(,{or },|)(,{ XxxAxAXxxxA A .)( kkA xxA

.)(X

A xxA

Ming-Feng Yeh 2-8

Membership FunctionsMembership Functions

Four most commonly used membership functions:

a b

monotonic

x

)(x

0

1

x

)(x

0

1

a

b ctriangular

x

)(x

0

1

a b

c dtrapezoidal

x

)(x

0

1

a

b

Gaussian(bell-shaped)

Ming-Feng Yeh 2-9

Fundamental Concepts: 1Fundamental Concepts: 1

Assume A is a fuzzy subset of X

Normal: If these exists at least one element such that A(x)=1. A fuzzy subset that is NOT normal is called subnormal.

Height: The largest membership grade of any element in A. That is, height(A) = max A(x).Crisp sets are special cases of fuzzy sets in which the membership grades are just either zero or one.

Xx

Ming-Feng Yeh 2-10

Fundamental Concepts: 2Fundamental Concepts: 2

Assume A is a fuzzy subset of X

Support of A: all elements of A have nonzero membership grades.

Core of A: all element of A with membership grade one.

} and 0)({)( XxxAxASupp

} and 1)({)( XxxAxACore

Ming-Feng Yeh 2-11

Example 2-1Example 2-1

Assumelet

A is a normal fuzzy subset and B is a subnormal fuzzy subset of X.

Height(A)=1 and Height(B)=0.9

Supp(A)={a,b,c,d} and Supp(B)={a,b,c,d,e}

Core(A)={a} and Core(B)=

},,,,{ edcbaX

}2.0,3.0,1.0,9.0,6.0{

},0,8.0,2.0,3.0,1{

edcbaB

edcbaA

Ming-Feng Yeh 2-12

Fundamental Concepts: 3Fundamental Concepts: 3

Assume A and B are two fuzzy subsets of X

Contain: A is said to be a subset of B,

if

Equal: A and B are said to be equal,

if and

That is,

Null fuzzy subset:

.),()( XxxBxA ,BA

,BA BA .AB

.),()( iff XxxBxABA

.,0)( Xxx

Ming-Feng Yeh 2-13

Operations on Fuzzy SetsOperations on Fuzzy Sets

Assume A and B are two fuzzy subsets of X

Intersection :

Union :

Complement :

)](),(min[)()( xBxAxBxA )](),(max[)()( xBxAxBxA

)(1)( xAxA

BABA

A

)(xA )(xB

x

)(x

1

0

)(xA )(xB

x

)(x

1

0

)(xA

x

)(x

1

0)(xA)()( xBxA )()( xBxA

Ming-Feng Yeh 2-14

Example 2-2Example 2-2

Assume

},,,{ 321 xxxX

321 0.16.03.0)( xxxxA

321 2.08.04.0)( xxxxB

321 0.04.07.0)( xxxxA

321 8.02.06.0)( xxxxB

321 0.18.04.0)()( xxxxBxABA

321 2.06.03.0)()( xxxxBxABA

Ming-Feng Yeh 2-15

Logical Operations: ANDLogical Operations: AND

Fuzzy intersection (AND): The intersection of fuzzy sets A and B, which are defined on the universe of discourse X, is a fuzzy set denoted by , with a membership function asMinimum:Algebraic product:

Triangular norm “ ”:x y = min{x,y} or x y = xyAND: A(x) B(x)

BA}:)(),(min{ Xxxx BABA

}:)()({ Xxxx BABA

Ming-Feng Yeh 2-16

Logical Operations: ORLogical Operations: OR

Fuzzy union (OR): The union of fuzzy sets A and B, which are defined on the universe of discourse X, is a fuzzy set denoted by , with a membership function as

Maximum:Algebraic sum:

Triangular co-norm “ ”:x y = max{x,y} or x y = x+ y – xyOR: A(x) B(x)

BA

}:)(),(max{ Xxxx BABA

}:)()()()({ Xxxxxx BABABA

Ming-Feng Yeh 2-17

-level Set-level Set

Assume A is a fuzzy subset of X, the -level set of A, denoted by A, is the crisp subset of X consisting of all elements in X for which

Any fuzzy subset A of X can be written as

Let

}.,)({ XxxAxA

AA },8,5,2,1{X .80.158.023.011.0 A

}.8{},8,5{

},8,5,2{},8,5,2,1{

0.17.0

3.01.0

AA

AA

83.053.023.03.0 3.0 A