topic 2 fuzzy logic control. ming-feng yeh2-2 outlines basic concepts of fuzzy set theory fuzzy...
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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