Fuzzy Logic
History, State of the Art, and
Future Development
History, State of the Art, and
Future Development
Sde 2
Today, Fuzzy Logic Has Today, Fuzzy Logic Has Already Become the Already Become the Standard Technique for Standard Technique for Multi-Variable Control ! Multi-Variable Control !
1965 Seminal Paper “Fuzzy Logic” by Prof. Lotfi Zadeh, Faculty in Electrical Engineering, U.C. Berkeley, Sets the Foundation of the “Fuzzy Set Theory”
1970 First Application of Fuzzy Logic in Control Engineering (Europe)
1975 Introduction of Fuzzy Logic in Japan
1980 Empirical Verification of Fuzzy Logic in Europe
1985 Broad Application of Fuzzy Logic in Japan
1990 Broad Application of Fuzzy Logic in Europe
1995 Broad Application of Fuzzy Logic in the U.S.
2000 Fuzzy Logic Becomes a Standard Technology and Is Also Applied in Data and Sensor Signal Analysis. Application of Fuzzy Logic in Business and Finance. 1965 Seminal Paper “Fuzzy Logic” by Prof. Lotfi Zadeh,
Faculty in Electrical Engineering, U.C. Berkeley, Sets the Foundation of the “Fuzzy Set Theory”
1970 First Application of Fuzzy Logic in Control Engineering (Europe)
1975 Introduction of Fuzzy Logic in Japan
1980 Empirical Verification of Fuzzy Logic in Europe
1985 Broad Application of Fuzzy Logic in Japan
1990 Broad Application of Fuzzy Logic in Europe
1995 Broad Application of Fuzzy Logic in the U.S.
Fuzzy Sets
• Extension of Classical Sets
• Not just a membership value of in the set
and out the set, 1 and 0
The crisp set v.s. the fuzzy set
• The
crisp set
is defined in such a way as to dichotomize(classification)
the individuals in some given universe of discourse into two groups:
members and nonmembers
.
– However, many classification concepts do not exhibit this characteristic.
– For example, the set of tall people, expensive cars, or sunny days.
• A
fuzzy set
can be defined mathematically by assigning to each possible
individual in the universe of discourse a value representing its
grade of
membership
in the fuzzy set.
– For example: a fuzzy set representing our concept of
sunny
might assign
a degree of membership of 1 to a cloud cover of 0%, 0.8 to a cloud
cover of 20%, 0.4 to a cloud cover of 30%, and 0 to a cloud cover of
75%.
Conventional (Boolean) Set Theory:
Conventional (Boolean) Set Theory:
Fuzzy Set Theory
Fuzzy Set Theory
“Strong Fever”
40.1°C
40.1°C
42°C
42°C
41.4°C
41.4°C
39.3°C
39.3°C
38.7°C
38.7°C
37.2°C
37.2°C
38°C
38°C
Fuzzy Set Theory:
Fuzzy Set Theory:
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Application
• Application areas
– Fuzzy Control
• Subway trains
• Cement kilns
Example: Height
• Tall people: say taller than or equal to 1.8m
– 1.8m , 2m, 3m etc member of this set
– 1.0 m, 1.5m or even 1.79999m not a member
• Real systems have measurement uncertainty
1.3 Fuzzy sets: basic types
• A
membership function
:
– A characteristic function: the values assigned to the elements of the
universal set fall within a specified range and indicate the
membership grade of these elements in the set.
– Larger values denote higher degrees of set membership.
• A set defined by membership functions is a
fuzzy set
.
• The most commonly used range of values of membership functions is
the
unit interval
[0,1].
• We think the universal set
X
is always a crisp set.
• Notation:
– The membership function of a fuzzy set
A
is denoted by :
– In the other one, the function is denoted by
A
and has the same form
– In this text, we use the
second notation
.
A
]
1
,
0
[
:
X
A
8 ] 1 , 0 [ :X AMember Functions
• Membership function
– better than listing membership values
• e.g. Tall(x) = {1 if x >= 1.9m ,
– 0 if x <= 1.7m
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Example: Fuzzy Short
• Short(x) = {0 if x >= 1.9m ,
– 1 if x <= 1.7m
Fuzzy Set Operators
• Fuzzy Set:
– Union
– Intersection
– Complement
• Many possible definitions
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Fuzzy Set Union
• Union ( f
A
(x) and f
B
(x) ) =
– max (f
A(x) , f
B(x) )
Fuzzy Set Intersection
• Intersection ( f
A
(x) and f
B
(x) ) =
– min (f
A(x) , f
B(x) )
14
Fuzzy Set Complement
Fuzzy Logic Operators
• Fuzzy Logic:
– NOT (A) = 1 - A
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18
Fuzzy Controllers
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Fuzzification
• Conversion of real input to fuzzy set values
• e.g. Medium ( x ) = {
– 0 if x >= 1.90 or x < 1.70,
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Inference Engine
• Fuzzy rules
– based on fuzzy premises and fuzzy consequences
• e.g.
– If height is Short and weight is Light then feet are
Small
Fuzzification & Inference
Example
• If height is 1.7m and weight is 55kg
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Defuzzification
• Rule base has many rules
– so some of the output fuzzy sets will have
membership value > 0
– Defuzzify to get a real value from the fuzzy
outputs
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Input Fuzzy Sets
Output Fuzzy Sets
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Fuzzy Rules
• If Angle is Zero then output ?
• If Angle is SP then output ?
• If Angle is SN then output ?
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Extended System
• Make use of additional information
– angular velocity:- -5.0 to 5.0 degrees/ second
New Fuzzy Rules
• Make use of old Fuzzy rules for angular velocity
Zero
• If Angle is Zero and
Angular vel is Zero
– then output Zero velocity
• If Angle is SP and
Angular vel is Zero
– then output SN velocity
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Complete Table
• When angular velocity is opposite to the
angle do nothing
– System can correct itself
• If Angle is SP and Angular velocity is SN
– then output ZE velocity
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Example
• Inputs:10 degrees, -3.5 degrees/sec
• Fuzzified Values
• Inference Rules
• Output Fuzzy Sets
Fuzzy Reasoning
• Single rule with single antecedent
Rule
:
if x is A then y is B
Fact
:
x is A’
Conclusion:
y is B’
• Graphic Representation:
Fuzzy Reasoning
• Single rule with multiple antecedent
Rule
:
if x is A and y is B then z is C
Fact:
x is A’ and y is B’
Conclusion:
z is C’
• Graphic Representation:
A
B
T-norm
X
Y
w
A’
B’
C
2Z
C’
Z
X
Y
A’
B’
Fuzzy Reasoning
• Multiple rules with multiple antecedent
Rule 1:
if x is A
1and y is B
1then z is C
1Rule 2:
if x is A
2and y is B
2then z is C
2Fact:
x is A’ and y is B’
Conclusion:
z is C’
Fuzzy Reasoning
• Graphics representation:
A
1B
1A
2B
2T-norm
X
X
Y
Y
w1 w2A’
A’
B’
B’
C
1C
2Z
Z
C’
Z
X
Y
A’
B’
Fuzzy Reasoning: MATLAB
Demo
Other Variants
• Some terminology:
– Degrees of compatibility (match)
– Firing strength
