Fuzzy Control
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Transcript of Fuzzy Control
Fuzzy Control
Lecture 2 Fuzzy Set
Basil HamedElectrical Engineering Islamic University of Gaza
Content Crisp Sets Fuzzy Sets Set-Theoretic Operations Fuzzy Relations
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Introduction
Fuzzy set theory provides a means for representing uncertainties.
Natural Language is vague and imprecise.
Fuzzy set theory uses Linguistic variables, rather than quantitative variables to represent imprecise concepts.
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Fuzzy Logic
Fuzzy Logic is suitable toVery complex modelsJudgmentalReasoningPerceptionDecision making
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Crisp Set and Fuzzy Set
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Information World
Crisp set has a unique membership function
A(x) = 1 x A 0 x A
A(x) {0, 1}
Fuzzy Set can have an infinite number of membership functions
A [0,1]
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Fuzziness
Examples:
A number is close to 5
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Fuzziness
Examples:
He/she is tall
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Classical Sets
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CLASSICAL SETSDefine a universe of discourse, X, as a collection of objects all having the same characteristics. The individual elements in the universe X will be denoted as x. The features of the elements in X can be discrete, or continuous valued quantities on the real line. Examples of elements of various universes might be as follows:
the clock speeds of computer CPUs;the operating currents of an electronic motor;the operating temperature of a heat pump;the integers 1 to 10.
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Operations on Classical Sets
Union:A B = {x | x A or x B}
Intersection:A B = {x | x A and x B}
Complement:A’ = {x | x A, x X}
X – Universal SetSet Difference:
A | B = {x | x A and x B} Set difference is also denoted by A - B
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Union of sets A and B (logical or).
Intersection of sets A and B (logical and)
Operations on Classical Sets
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Operations on Classical Sets
Complement of set A.
Difference operation A|B.
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Properties of Classical Sets
A B = B AA B = B AA (B C) = (A B) CA (B C) = (A B) C
A (B C) = (A B) (A C)A (B C) = (A B) (A C)
A A = AA A = A
A X = XA X = AA = AA =
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Mapping of Classical Sets to Functions
Mapping is an important concept in relating set-theoretic forms to function-theoretic representations of information. In its most general form it can be used to map elements or subsets in one universe of discourse to elements or sets in another universe.
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Fuzzy Sets
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A fuzzy set, is a set containing elements that have varying degrees of membership in the set.
Elements in a fuzzy set, because their membership need not be complete, can also be members of other fuzzy sets on the same universe.
Elements of a fuzzy set are mapped to a universe of membership values using a function-theoretic form.
Fuzzy Sets
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An object has a numeric “degree of membership” Normally, between 0 and 1 (inclusive)
0 membership means the object is not in the set 1 membership means the object is fully inside the set In between means the object is partially in the set
Fuzzy Set Theory
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If U is a collection of objects denoted generically by x, then a fuzzy set A in U is defined as a set of ordered pairs:
membershipfunction
U : universe of discourse.
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Fuzzy Sets
Characteristic function X, indicating the belongingness of x to the set A
X(x) = 1 x A 0 x A
or called membership
Hence,A B XA B(x)
= XA(x) XB(x)= max(XA(x),XB(x))
Note: Some books use + for , but still it is not ordinary addition!
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Fuzzy Sets
A B XA B(x)= XA(x) XB(x)= min(XA(x),XB(x))
A’ XA’(x) = 1 – XA(x)
A’’ = A
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Fuzzy Set Operations
A B(x) = A(x) B(x) = max(A(x), B(x))
A B(x) = A(x) B(x) = min(A(x), B(x))
A’(x) = 1 - A(x)
De Morgan’s Law also holds: (A B)’ = A’ B’ (A B)’ = A’ B’
But, in generalA A’ A A’
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Union of fuzzy sets A and B∼
.
Intersection of fuzzy sets A and B∼
.
Fuzzy Set Operations
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Complement of fuzzy set A∼
.
Fuzzy Set Operations
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Operations
A B
A B A B ADr Basil Hamed 25
A A’ = X A A’ = Ø
Excluded middle axioms for crisp sets. (a) Crisp set A and its complement; (b) crisp A ∪ A = X (axiom of excluded middle); and (c) crisp A ∩ A = Ø (axiom of contradiction).
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A A’ A A’
Excluded middle axioms for fuzzy sets are not valid. (a) Fuzzy set A∼
and its complement; (b) fuzzy A ∪ A∼ = X (axiom of excluded middle); and (c) fuzzy A ∩ A = Ø (axiom of contradiction).
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Set-Theoretic Operations
A B
A B
A
A B
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Examples of Fuzzy Set Operations
Fuzzy union (): the union of two fuzzy sets is the maximum (MAX) of each element from two sets.E.g. A = {1.0, 0.20, 0.75} B = {0.2, 0.45, 0.50} A B = {MAX(1.0, 0.2), MAX(0.20, 0.45),
MAX(0.75, 0.50)} = {1.0, 0.45, 0.75}
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Examples of Fuzzy Set Operations
Fuzzy intersection (): the intersection of two fuzzy sets is just the MIN of each element from the two sets.E.g. A B = {MIN(1.0, 0.2), MIN(0.20,
0.45), MIN(0.75, 0.50)} = {0.2, 0.20, 0.50}
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Examples of Fuzzy Set OperationsA = {1/a, 0.3/b, 0.2/c 0.8/d, 0/e}B = {0.6/a, 0.9/b, 0.1/c, 0.3/d, 0.2/e}Complement: = {0/a, 0.7/b, 0.8/c 0.2/d, 1/e}Union:A B = {1/a, 0.9/b, 0.2/c, 0.8/d, 0.2/e}Intersection:A B = {0.6/a, 0.3/b, 0.1/c, 0.3/d, 0/e} Dr Basil Hamed 31
Properties of Fuzzy Sets
A B = B AA B = B AA (B C) = (A B) CA (B C) = (A B) C
A (B C) = (A B) (A C)A (B C) = (A B) (A C)
A A = A A A = AA X = X A X = AA = A A =
If A B C, then A C
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Fuzzy Sets
Note (x) [0,1] not {0,1} like Crisp set
A = {A(x1) / x1 + A(x2) / x2 + …} = { A(xi) / xi}Note: ‘+’ add
‘/ ’ divide
Only for representing element and its membership.
Also some books use (x) for Crisp Sets too.
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Example (Discrete Universe)
{1, 2,3,4,5,6,7,8}U # courses a student may take in a semester.
(1,0.1) (2,0.3) (3,0.8) (4,1)(5,0.9) (6,0.5) (7,0.2) (8,0.1)
A
appropriate # courses taken
0.5
1
02 4 6 8
x : # courses
( )A x
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Example (Discrete Universe)
{1, 2,3,4,5,6,7,8}U # courses a student may take in a semester.
(1,0.1) (2,0.3) (3,0.8) (4,1)(5,0.9) (6,0.5) (7,0.2) (8,0.1)
A
appropriate # courses taken
Alternative Representation:
1 2 3 40.1/ 0.3 / 0.8 / 1.0 / 0.9 / 0.5 / 0.2 / 0.1/5 6 7 8A
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Example (Continuous Universe)
possible ages
U : the set of positive real numbers
( , ( ))BB x x x U
4
1( )501
5
B xx
about 50 years old
00.20.40.60.8
11.2
0 20 40 60 80 100
x : age
( )B x
4505
11 xR
B x
Alternative Representation:
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Alternative Notation
( , ( ))AA x x x U
U : discrete universe
U : continuous universe
( ) /i
A i ix U
A x x
( ) /AUA x x
Note that and integral signs stand for the union of membership grades; “ / ” stands for a marker and does not imply division.
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Fuzzy DisjunctionAB max(A, B)AB = C "Quality C is the disjunction of Quality A and B"
0
1
0.375
A
0
1
0.75
B
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Fuzzy ConjunctionAB min(A, B)AB = C "Quality C is the conjunction of Quality A and B"
0
1
0.375
A
0
1
0.75
B
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Example: Fuzzy Conjunction
Calculate AB given that A is .4 and B is 20
0
1A
0
1B
.1 .2 .3 .4 .5 .6 .7 .8 .9 1 5 10 15 20 25 30 35 40
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Example: Fuzzy Conjunction
Calculate AB given that A is .4 and B is 20
0
1A
0
1B
.1 .2 .3 .4 .5 .6 .7 .8 .9 1 5 10 15 20 25 30 35 40
• Determine degrees of membership:
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Example: Fuzzy Conjunction
Calculate AB given that A is .4 and B is 20
0
1A
0
1B
.1 .2 .3 .4 .5 .6 .7 .8 .9 1 5 10 15 20 25 30 35 40
• Determine degrees of membership:• A = 0.7
0.7
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Example: Fuzzy Conjunction
Calculate AB given that A is .4 and B is 20
0
1A
0
1B
.1 .2 .3 .4 .5 .6 .7 .8 .9 1 5 10 15 20 25 30 35 40
• Determine degrees of membership:• A = 0.7 B = 0.9
0.70.9
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Example: Fuzzy Conjunction
Calculate AB given that A is .4 and B is 20
0
1A
0
1B
.1 .2 .3 .4 .5 .6 .7 .8 .9 1 5 10 15 20 25 30 35 40
• Determine degrees of membership:• A = 0.7 B = 0.9
• Apply Fuzzy AND• AB = min(A, B) = 0.7
0.70.9
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Generalized Union/Intersection
Generalized Union Or called triangular norm.
Generalized Intersection
t-norm
t-conorm Or called s-norm.
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T-norms and S-norms
And/OR definitions are called T-norms (S-norms) Duals of one another A definition of one defines the other implicitly
Many different ones have been proposed Min/Max, Product/Bounded-Sum, etc. Tons of theoretical literature We will not go into this.
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Examples: T-Norm & T-Conorm
Minimum/Maximum:
Lukasiewicz:
( , ) min( , )T a b a b a b
( , ) max( , )S a b a b a b
( , ) max( 1,0) ( , )T a b a b LAND a b
( , ) min( ,1) ( , )S a b a b LOR a b
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Classical Logic &Fuzzy LogicHypothesis : Engineers are mathematicians. Logical thinkers do not believe in magic. Mathematicians are logical thinkers.Conclusion : Engineers do not believe in magic.Let us decompose this information into individual propositionsP: a person is an engineerQ: a person is a mathematicianR: a person is a logical thinkerS: a person believes in magicThe statements can now be expressed as algebraic propositions as((PQ)(RS)(QR))(PS)
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Fuzzy Relations
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Aa1
a2
a3
a4
Bb1
b2
b3
b4
b5
Crisp Relation (R)
R A B Dr Basil Hamed 50
Aa1
a2
a3
a4
Bb1
b2
b3
b4
b5
Crisp Relation (R)R A B
1 1 1 3 2 5
3 1 3 4 4 2
( , ), ( , ), ( , )( , ), ( , ), ( , )a b a b a b
Ra b a b a b
1 0 1 0 00 0 0 0 11 0 0 1 00 1 0 0 0
RM
1 1a Rb 1 3a Rb 2 5a Rb
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Crisp Relations
Example:
If X = {1,2,3} Y = {a,b,c}R = { (1 a),(1 c),(2 a),(2 b),(3 b),(3 c) }
a b c1 1 0 1
R = 2 1 1 03 0 1 1
Using a diagram to represent the relation
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The Real-Life Relation 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
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Fuzzy RelationsTriples showing connection between two sets:
(a,b,#): a is related to b with degree #
Fuzzy relations are set themselves
Fuzzy relations can be expressed as matrices
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Fuzzy Relations MatricesExample: Color-Ripeness relation for tomatoesR1(x, y) unripe semi ripe ripe
green 1 0.5 0
yellow 0.3 1 0.4
Red 0 0.2 1
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CompositionLet R be a relation that relates, or maps, elements from universe X to universe Y, and let S be a relation that relates, or maps, elements from universe Y to universe Z.
A useful question we seek to answer is whether we can find a relation, T, that relates the same elements in universe X that R contains to the same elements in universe Z that S contains. It turns out that we can find such a relation using an operation known as composition.
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CompositionIf R is a fuzzy relation on the space X x Y S is a fuzzy relation on the space Y x ZThen, fuzzy composition is T = R SThere are two common forms of the composition operation: 1. Fuzzy max-min composition
T(xz) = (R(xy) s(yz))
2. Fuzzy max-production compositionT(xz) = (R(xy) s(yz))
Note: R S S R multiplication
y Y
y Y
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A fuzzy relation defined on X an Z.
Max-Min Composition
X Y ZR: fuzzy relation defined on X and Y.
S: fuzzy relation defined on Y and Z.R 。 S: the composition of R and S.
( , ) max min ( , ), ( , )R S y R Sx z x y y z
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Example
1 0.1 0.2 0.0 1.02 0.3 0.3 0.0 0.23 0.8 0.9 1.0 0.4
R a b c d0.9 0.0 0.30.2 1.0 0.80.8 0.0 0.70.4 0.2 0.3
Sabcd
1 0.4 0.2 0.32 0.3 0.3 0.33 0.8 0.9 0.8
R S
0.1 0.2 0.0 1.00.9 0.2 0.8 0.4min0.1 0.2 0.0 0.4max
( , ) max min ( , ), ( , )S R v R Sx y x v v y
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Max-Product Composition
( , ) max ( , ) ( , )R S v R Sx y x v v y
A fuzzy relation defined on X an Z.
X Y ZR: fuzzy relation defined on X and Y.
S: fuzzy relation defined on Y and Z.R。 S: the composition of R and S.
.
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Example
1 0.1 0.2 0.0 1.02 0.3 0.3 0.0 0.23 0.8 0.9 1.0 0.4
R a b c d0.9 0.0 0.30.2 1.0 0.80.8 0.0 0.70.4 0.2 0.3
Sabcd
0.1 0.2 0.0 1.00.9 0.2 0.8 0.4Product
max .09 .04 0.0 0.4
R S
1 0.4 0.2 0.32 0.27 0.3 0.243 0.8 0.9 0.7
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Properties of Fuzzy Relations
Example: y1 y2 z1 z2 z3
R = x1 0.7 0.5 S = y1 0.9 0.6 0.2x2 0.8 0.4 y2 0.1 0.7 0.5
z1 z2 z3Using max-min, T = x1 0.7 0.6 0.5
x2 0.8 0.6 0.4
z1 z2 z3Using max-product, T = x1 0.63 0.42 0.25
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Example 3.8 (Page 59)
Suppose we are interested in understanding the speed control of the DC shunt motor under no-load condition, as shown.
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Example 3.8Initially, the series resistance Rse in should be kept in the cut-in position for the following reasons:1. The back electromagnetic force, given by Eb = kNφ, where k is a constant of proportionality, N is the motor speed, and φ is the flux (which is proportional to input voltage, V ), is equal to zero because the motor speed is equal to zero initially.2. We have V = Eb + Ia(Ra + Rse), therefore Ia = (V − Eb)/(Ra + Rse), where Ia is the armature current and Ra is the armature resistance. Since Eb is equal to zero initially, the armature current will be Ia = V/(Ra + Rse), which is going to be quite large initially and may destroy the armature.
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Example 3.8Let Rse be a fuzzy set representing a number of possible values for series resistance, say sn values, given as
and let Ia be a fuzzy set having a number of possible values of the armature current, say m values, given as
The fuzzy sets Rse and Ia can be related through a fuzzy relation, say R, which would allow for the establishment of various degrees of relationship between pairs of resistance and current.Dr Basil Hamed 65
Example 3.8Let N be another fuzzy set having numerous values for the motor speed, say v values, given as
Now, we can determine another fuzzy relation, say S, to relate current to motor speed, that is, Ia to N.Using the operation of composition, we could then compute a relation, say T, to be used to relate series resistance to motor speed, that is, Rse to N.
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Example 3.8The operations needed to develop these relations are as follows – two fuzzy Cartesian products and one composition:
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Example 3.8Suppose the membership functions for both series resistance Rse and armature current Ia are given in terms of percentages of their respective rated values, that is,
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Example 3.8
The following relation then result from use of the Cartesian product to determine R:
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Example 3.8Cartesian product to determine S:
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Example 3.8
The following relation results from a max–min composition for T:
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HW 1 2.4, 2.5,2.7, 2.11, 3.2, 3.4, 3.8 Due 6/ 10/ 2013Good Luck
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