Fuzzy Logic Theory
Transcript of Fuzzy Logic Theory
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Fuzzy Logic Theory
Dr. Xiao-Zhi Gao
Department of Electrical Engineering
Helsinki University of Technology
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Outline
Fuzzy set theory
Basic concepts in fuzzy sets
Operations on fuzzy sets
Fuzzy rules and reasoning
Fuzzy inference systems
Mamdani fuzzy systems
Sugeno fuzzy systems
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Primer of Fuzzy Sets
From classical crisp sets to fuzzy sets
Change of range of membership functions
Notions of fuzzy sets
Representation, support, cut, convexity ...
Fuzzy set operations Intersection, union, complement, etc.
Share almostthe same mathematicsfoundamentals with those of classical sets
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Basics of Fuzzy Sets Classical sets
two-valuemembership functions
An element either belongs to or does not belong
to a given classical set (sharp boundary) Fuzzy sets
Extend the degrees of membership:
An element partiallybelongs to and partiallydoes not belong to a given fuzzy set
Smooth and Gradual boundary
Membership functions Assignment of membership functions is subjective
{ } ]1,0[1,0
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Father of Fuzzy Logic: Lotfi A. Zadeh
Lotfi Asker Zadeh is a Professor in the GraduateSchool, Computer Science Division, Departmentof EECS, University of California, Berkeley. In
addition, he is serving as the Director of BISC(Berkeley Initiative in Soft Computing).
http://en.wikipedia.org/wiki/Lotfi_Asker_Zadeh
http://www-bisc.cs.berkeley.edu/
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Four Typical Membership Functions
0 20 40 60 80 1000
0.2
0.4
0.6
0.8
1
1.2
Memb
ershipGrades
(a) Triangular MF
0 20 40 60 80 1000
0.2
0.4
0.6
0.8
1
1.2
Memb
ershipGrades
(b) Trapezoidal MF
0 20 40 60 80 1000
0.2
0.4
0.6
0.8
1
1.2
Mem
bershipGrades
(c) Gaussian MF
0 20 40 60 80 1000
0.2
0.4
0.6
0.8
1
1.2
Mem
bershipGrades
(d) Generalized Bell MF
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Supports
Core
Crossover points
Height
Fuzzy singleton
A fuzzy singleton is a crispset
A fuzzy set that has only one element
x0
Definitions in Fuzzy Sets
(x0) = 1
(x) > 0
(x) =1
(x) = 0.5
max A(x)[
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Definitions in Fuzzy Sets
FuzzySingleton
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Representations of
Membership Functions
Discrete Fuzzy Sets Universe of discourse is discrete
Continuous Fuzzy Sets Universe of discourse is continuous
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Discrete and Continuous
Fuzzy Sets
0 1 2 3 4 5 60
0.2
0.4
0.6
0.8
1
1.2
X = Number of Children
MembershipGrades
(a) MF on a Discrete Universe
0 10 20 30 40 50 60 70 80 90 1000
0.2
0.4
0.6
0.8
1
1.2
X = Age
Mem
bershipGrades
(b) MF on a Continuous Universe
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Cuts of Fuzzy Sets
(a constant) cut of a fuzzy set is a
crispset
Strong cuts
{ } = )(xxA Aa
{ } >= )(xxA Aa
support(A) = A0
core(A) =A1
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Resolution Principle
A fuzzy membership function can be
expressed in terms of the characteristicfunctions of its cuts
An example of decomposing a fuzzy setwith discrete objects
=
3
3.0,2
2.0,1
1.0A
],min[max)( AxA =
R l i P i i l
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Resolution Principle
33.0
22.0
11.0
3)3.0,2.0,1.0max(
2)2.0,1.0max(
11.0
3
3.03.0
32.0
22.02.0
3
1.0
2
1.0
1
1.01.0
3
13
1
2
13
1
2
1
1
1
3.0
2.0
1.0
3.0
2.0
1.0
++=
++=
=
+=
++=
=
+=
++=
A
A
A
A
A
A
A
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Resolution Principle
0 10 20 30 40 50 60 70 80 90 1000
0.2
0.4
0.6
0.8
1
1.2Resolution Principle
x
Mem
bershipGrades
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Extension Principle
Extending crispdomains of mathematicalexpressions to fuzzydomains
Suppose fis a function mapping from Xto Y, and A is a fuzzy set:
we have image of A under f:n
nAAA
xx
xx
xxA )()()(
2
2
1
1 +++= L
)()(
)()(
)()()(
2
2
1
1
n
nAAA
xfx
xfx
xfxAfB +++== L
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Extension Principle
An example of extension principle
2
3.0
1
9.0
0
8.0
1
4.0
2
1.0+++
+
=A
3)( 2 =xxf
1
3.0
2
9.0
3
8.0
1
)3.01.0(
2
)9.04.0(
3
8.0
13.0
29.0
38.0
24.0
11.0
+
+
=
+
+
=
+
+
+
+=B
E t i P i i l
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Extension Principle
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Convex Fuzzy Sets Convexity of fuzzy sets
where and are two arbitrary elements
in A, and is an arbitrary constantx1 x2
C d N F
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Convex and Nonconvex Fuzzy
Membership Functions
0
0.2
0.4
0.6
0.8
1
1.2
MembershipG
rades
(a) Two Convex Fuzzy Sets
0
0.2
0.4
0.6
0.8
1
1.2
MembershipGrades
(b) A Nonconvex Fuzzy Set
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Operations on Fuzzy Sets
0
0.2
0.4
0.6
0.8
1
1.2(a) Fuzzy Sets A and B
A B
0
0.2
0.4
0.6
0.8
1
1.2(b) Fuzzy Set "not A"
0
0.2
0.4
0.6
0.8
1
1.2(c) Fuzzy Set "A OR B"
0
0.2
0.4
0.6
0.8
1
1.2(d) Fuzzy Set "A AND B"
A
A IBA U B
and B
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Operations on Fuzzy Sets Subset
Equality
andA B B A
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Concept of A B
0
0.2
0.4
0.6
0.8
1
1.2A Is Contained in B
Memb
ershipGrades
B
A
Li i ti V i bl
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Linguistic Variables
Modeling of human thinking Numerical values are not sufficient
Linguistic variables exist in real world
e.g, chatting with a stranger on the phone
Estimation of your partners age:
(40? probability of 40? About middle-aged?) Linguistic variables are characterized by linguistic
values (Age:[Young, Old, Very Old, etc.])
Linguistic values are described by their fuzzymembership functions
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Membership Functions For Linguistic Values
Young, Middle Aged, and Old
0 10 20 30 40 50 60 70 80 900
0.2
0.4
0.6
0.8
1
1.2
X = Age
MembershipGrades
Young Middle Aged Old
Age:
LinguisticVariable
Young, Middle
Aged, and Old:Linguistic Values
Membership Functions For Primary
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Membership Functions For Primary
and Composite Linguistic Values
0 10 20 30 40 50 60 70 80 90 1000
0.2
0.4
0.6
0.8
1
X = age
Members
hipGrades
(a) Primary Linguistic Values
OldYoung
0 10 20 30 40 50 60 70 80 90 1000
0.2
0.4
0.6
0.8
1
X = age
M
embershipGrades
(b) Composite Linguistic Values
Not Young and Not Old
More or Less Old
Extremely Old
Young but
Not Too Young
Extremely Old
= Old4
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Fuzzy IF-THEN Rules
A fuzzy IF-THEN rule is expressed as
IF xis A THEN yis B
A and B are fuzzy membership functions
A fuzzy IF-THEN rule associatesfuzzyinput and output membership functions
Premise Part Consequent Part
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Fuzzy Reasoning
Fuzzy reasoning derives conclusionsbased on givenfuzzy IF-THEN rules
and knownfacts
An example:
Given a fuzzy rule: IF bath is very hot THENadd a lot of cold water
Known fact: bath is a little hot
Conclusion: how much cold water should beadded?
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Fuzzy Reasoning
Premise 1: IF X is A THEN Y is B
Premise 2: IF X is
Conclusion: THEN Y is
First, measure the similaritybetween A and
Second, projectthis similarity to B
There are a few composition operations
used in fuzzy reasoning procedure Max-Minis most widely employed
AB
A
Fuzzy Reasoning for Single Rule
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Fuzzy Reasoning for Single Rulewith Single Antecedent
IntersectionIF X is A THEN Y is B
W
Known Fact: X is A
Fuzzy Reasoning for Single Rule
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Fuzzy Reasoning for Single Rule
with Multiple Antecedents
w=min(w1, w2)or
w=w1*w2
IF X is A and Y is B THEN Z is C
Known Facts: X is Aand Y is B
Fuzzy Reasoning For Multiple Rules
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Fuzzy Reasoning For Multiple Rules
with Multiple Antecedents
Aggregation
If X is A1 and Y is B1 Then Z is C1If X is A2 and Y is B2 Then Z is C2
Known Facts:
X is A
Y is B
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Fuzzy Logic SystemsFuzzification
Interface
Inference
Engine
Fuzzy Rule
Base
Defuzzification
InterfaceCrisp
Input
Crisp
Output
Rules
Fuzzy
Input
Fuzzy
Output
Mamdani and Sugeno Fuzzy Inference Systems
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Fuzzification
Converts a crispvalue (from sensors,devices, measurement meters, etc.) into
a fuzzyvalue
A crisp value = A fuzzy singleton (no
fuzziness is introduced)
x0: Crisp Value
Fuzzy singleton
A(x)
A(x)
x0
1
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Mamdani Fuzzy Inference Systems
Mamdani fuzzy inference systems
IF x is A THEN y is B Both premise and consequent parts
consist of fuzzy membership functions
Max-Mincomposition can be applied infuzzy reasoning procedure
Mamdani Fuzzy Inference Systems
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Mamdani Fuzzy Inference Systems
If X is A1 and Y is B1 THEN Z is C1
If X is A2 and Y is B2 THEN Z is C2
x and y are two input values
Defuzzification Methods
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Defuzzification Methods
D f ifi ti M th d
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Defuzzification Methods
Centroid of Area (COA)
where is a crisp output
Bisector of Area (BOA)
where is a crisp output
ZCOA*
=
A (xi)xii=1
n
A (x i)i=1
n
A (xi)i=1
M
= A (x j)j=M+1
n
ZBisec t* = xM
*
COAZ
Defuzzification Methods
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Defuzzification Methods
Mean of Maximum (MOM)
where reaches maximal values of
MOM could generatate wrong discreteoutputs in certain cases
Different defuzzification methods mayhave similarperformances [Lee 88]
ZMOM* =
xi*
i
N
N
A (xi*) A (x)
Mamdani F Model An E ample
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Mamdani Fuzzy Model: An Example
A Two-Input and Single-Output Mamdanifuzzy logic system with four rules
IF X is small and Y is small THEN Z is large negative
IF X is small and Y is large THEN Z is small negative
IF X is large and Y is small THEN Z is small positive
IF X is large and Y is large THEN Z is large positive
Fuzzy Input and Output
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y p p
Membership Functions
-5 -4 -3 -2 -1 0 1 2 3 4 50
0.5
1
X
MembershipGrades small large
-5 -4 -3 -2 -1 0 1 2 3 4 50
0.5
1
Y
Mem
bershipGrades small large
-5 -4 -3 -2 -1 0 1 2 3 4 50
0.5
1
Z
MembershipGrades large negative small negative small positive large positive
Mamdani Input/Output Surface
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Mamdani Input/Output Surface
-5
0
5
-5
0
5
-3
-2
-1
0
1
2
3
XY
Z
NonlinearInput/Output
Surface
Sugeno Fuzzy Inference Systems
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Sugeno Fuzzy Inference Systems
Sugeno fuzzy inference systems
IF x is A THEN y = f(x)
Only premisepart employs fuzzymembership functions
Consequent output is a functionof inputvariables
First order = linear consequent part
Higher orders = nonlinear consequent part
Sugeno Fuzzy Inference Systems
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Sugeno Fuzzy Inference Systems
If X is A1 and Y is B1
THEN Z = p1x+q1y+c1
If X is A2 and Y is B2THEN Z = p2x+q2y+c2
First Order Sugeno Inference System
Z: Output
S F I f S t
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Sugeno Fuzzy Inference Systems
Advantages:
Nodefuzzification method is needed Easy to analyze
linearwith respect to consequent parameters
(only first order consequent part case)
Disadvantages
Difficult to interpret practical meanings ofconsequent part
Sugeno Fuzzy Model: An Example
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Sugeno Fuzzy Model: An Example
A Two-Input and Single-Output Sugeno
fuzzy logic system with four rules
IF X is small and Y is small THEN z = -x+y+1
IF X is small and Y is large THEN z = -y+3 IF X is large and Y is small THEN z = -x+3
IF X is large and Y is large THEN z = -x+y+2
Fuzzy Input Membership Functions
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Fuzzy Input Membership Functions
-5 -4 -3 -2 -1 0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
1.2
X
M
embershipGrades
Small Large
-5 -4 -3 -2 -1 0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
1.2
Y
MembershipGrades
Small Large
Sugeno Input/Output Surface
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Sugeno Input/Output Surface
-5
0
5
-5
0
5
-2
0
2
4
6
8
10
XY
Z NonlinearInput/Output
Surface
Diagram of A Fuzzy Logic System
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Diagram of A Fuzzy Logic System
Conclusions
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Conclusions
Background knowledge of fuzzy sets isintroduced
Concept of fuzzy sets is basically anaturalgeneralization of classical sets
Fuzzy sets have some characteristicsdifferent from classical sets
Fuzzy inference systems are used tomodel human thinking
Conclusions
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Two fuzzy logic systems: Mamdani andSugeno types
Engineering potential is more importantthan pure theoretical research
Applications of fuzzy logic and fusion
with neural networks will be discussedlater
Fuzzy control and fuzzy signal filtering Fuzzy neural networks