AI
Fuzzy Systems
History, State of the Art, and
Future Development
Today, Fuzzy Logic Has
Already Become the
Standard Technique for
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.
Sde 2
Types of Uncertainty and the
Modeling of Uncertainty
Stochastic Uncertainty:
 The Probability of Hitting the Target Is 0.8
Lexical Uncertainty:
 "Tall Men", "Hot Days", or "Stable Currencies"
 We Will Probably Have a Successful Business Year.
 The Experience of Expert A Shows That B Is Likely to
Occur. However, Expert C Is Convinced This Is Not True.
Most Words and Evaluations We Use in Our Daily Reasoning Are
Not Clearly Defined in a Mathematical Manner. This Allows
Humans to Reason on an Abstract Level!
Slide 3
Probability and Uncertainty
“... a person suffering from hepatitis shows in
60% of all cases a strong fever, in 45% of all
cases yellowish colored skin, and in 30% of all
cases suffers from nausea ...”
Stochastics and Fuzzy Logic
Complement Each Other !
Slide 4
Fuzzy Set Theory
Conventional (Boolean) Set Theory:
38.7°C
38°C
40.1°C
39.3°C
41.4°C
Fuzzy Set Theory:
42°C
“Strong Fever”
37.2°C
38.7°C
38°C
40.1°C
39.3°C
“More-or-Less” Rather Than “Either-Or” !
41.4°C
42°C
“Strong Fever”
37.2°C
Slide 5
Fuzzy Sets...
Representing
crisp and
fuzzy sets as
subsets of a
domain
(universe) U".
Fuzziness versus probability
Probability
density
function for
throwing a
dice and the
membership
functions of
the concepts
"Small"
number,
"Medium",
"Big".
Conceptualising in fuzzy
terms...
One
representation
for the fuzzy
number "about
600".
Conceptualising in fuzzy
terms...
Representing
truthfulness
(certainty) of
events as
fuzzy sets over
the [0,1]
domain.
Strong Fever Revisited
Conventional (Boolean) Set Theory:
38.7°C
38°C
40.1°C
39.3°C
41.4°C
Fuzzy Set Theory:
42°C
“Strong Fever”
37.2°C
38.7°C
38°C
40.1°C
39.3°C
41.4°C
42°C
“Strong Fever”
37.2°C
Slide 10
 Why fuzzy?
As Zadeh said, the term is concrete, immediate and
descriptive; we all know what it means. However,
many people in the West were repelled by the word
fuzzy , because it is usually used in a negative sense.
 Why logic?
Fuzziness rests on fuzzy set theory, and fuzzy logic
is just a small part of that theory.
Range of logical values in Boolean and fuzzy logic
0
01
0 1
(a) Boolean Logic.
1
0 0
0.2
0.4
0.6
0.8
(b) Multi-valued Logic.
1 1
 The classical example in fuzzy sets is tall men.
The elements of the fuzzy set “tall men” are all
men, but their degrees of membership depend on
their height.
Fuzzy Set Definitions
Discrete Definition:
µSF(35°C) = 0
µSF(38°C) = 0.1
µSF(41°C) = 0.9
µSF(36°C) = 0
µSF(39°C) = 0.35
µSF(42°C) = 1
µSF(37°C) = 0
µSF(40°C) = 0.65
µSF(43°C) = 1
Continuous Definition:
µ(x)
No More Artificial Thresholds!
1
0
36°C
37°C
38°C
39°C
40°C
41°C
42°C
Slide 14
Representation of hedges in fuzzy logic
Hedge
Mathematical
Expression
A little
[ A( x)] 1.3
Slightly
[ A(x)] 1.7
Very
[ A (x)] 2
Extremely
[ A(x) ] 3
Graphical Representation
Representation of hedges in fuzzy logic (continued)
Hedge
Very very
Mathematical
Expression
[A ( x)]4
More or less
A ( x)
Somewhat
A ( x)
2 [A (x )]2
Indeed
if 0  A  0.5
1  2 [1  A ( x)]2
if 0.5 < A  1
Graphical Representation
Linguistic Variable
...Terms, Degree of Membership, Membership Function, Base Variable...
µ(x)
1
strong fever
low temp normal raised temperature
… pretty much raised …
A Linguistic Variable
Defines a Concept of Our
Everyday Language!
... but just slightly strong …
0
36°C
37°C
38°C
39°C
40°C
41°C
42°C
Slide 17
Fuzzy Sets
• Formal definition:
A fuzzy set A in X is expressed as a set of
ordered pairs:
A  {( x ,  A ( x ))| x  X }
Fuzzy set
Membership
function
(MF)
Universe or
universe of discourse
A fuzzy set is totally characterized by a
membership function (MF).
Cantor’s sets
Not A
B
A
AA
Complement
A
B
Intersection
Containment
AA
B
Union
Operations of fuzzy sets
 ( x)
 ( x)
1
1
B
A
A
0
1
x
Complement
x
 ( x)
1
x
A
0
Containment
x
 ( x)
1
AB
0
1
B
1
Not A
0
0
x
AB
0
Intersection
AB
0
x
1
x
0
AB
Union
x
 Complement
Crisp Sets: Who does not belong to the set?
Fuzzy Sets: How much do elements not belong to
the set?
The complement of a set is an opposite of this set.
For example, if we have the set of tall men, its
complement is the set of NOT tall men. When we
remove the tall men set from the universe of
discourse, we obtain the complement. If A is the
fuzzy set, its complement A can be found as
follows:
A(x) = 1  A(x)
 Containment
Crisp Sets: Which sets belong to which other sets?
Fuzzy Sets: Which sets belong to other sets?
Similar to a Chinese box, a set can contain other
sets. The smaller set is called the subset. For
example, the set of tall men contains all tall men;
very tall men is a subset of tall men. However, the
tall men set is just a subset of the set of men. In
crisp sets, all elements of a subset entirely belong to
a larger set. In fuzzy sets, however, each element
can belong less to the subset than to the larger set.
Elements of the fuzzy subset have smaller
memberships in it than in the larger set.
 Intersection
Crisp Sets: Which element belongs to both sets?
Fuzzy Sets: How much of the element is in both
sets?
In classical set theory, an intersection between two
sets contains the elements shared by these sets. For
example, the intersection of the set of tall men and
the set of fat men is the area where these sets
overlap. In fuzzy sets, an element may partly
belong to both sets with different memberships. A
fuzzy intersection is the lower membership in both
sets of each element. The fuzzy intersection of two
fuzzy sets A and B on universe of discourse X:
AB(x) = min [A (x), B (x)] = A (x)  B(x),
where xX
 Union
Crisp Sets: Which element belongs to either set?
Fuzzy Sets: How much of the element is in either
set?
The union of two crisp sets consists of every element
that falls into either set. For example, the union of
tall men and fat men contains all men who are tall
OR fat. In fuzzy sets, the union is the reverse of the
intersection. That is, the union is the largest
membership value of the element in either set. The
fuzzy operation for forming the union of two fuzzy
sets A and B on universe X can be given as:
AB(x) = max [A (x), B(x)] = A (x)  B(x),
where xX
• Subset:
Set-Theoretic Operations
A  B  A  B
• Complement:
A  X  A  A ( x )  1  A ( x )
• Union:
C  A  B  c ( x )  max( A ( x ), B ( x ))  A ( x ) B ( x )
• Intersection:
C  A  B  c ( x )  min( A ( x ), B ( x ))  A ( x ) B ( x )
What is the difference between classical and
fuzzy rules?
A classical IF-THEN rule uses binary logic, for
example,
Rule: 1
Rule: 2
IF
speed is > 100
IF
speed is < 40
THEN stopping_distance is long THEN stopping_distance is short
The variable speed can have any numerical value
between 0 and 220 km/h, but the linguistic variable
stopping_distance can take either value long or short.
In other words, classical rules are expressed in the
black-and-white language of Boolean logic.
We can also represent the stopping distance rules in a
fuzzy form:
Rule: 1
Rule: 2
IF
speed is fast
IF
speed is slow
THEN stopping_distance is long THEN stopping_distance is short
In fuzzy rules, the linguistic variable speed also has
the range (the universe of discourse) between 0 and
220 km/h, but this range includes fuzzy sets, such as
slow, medium and fast. The universe of discourse of
the linguistic variable stopping_distance can be
between 0 and 300 m and may include such fuzzy
sets as short, medium and long.
Basic Elements of a
Fuzzy Logic System
Fuzzy Logic Defines
the Control Strategy
on a Linguistic Level!
Fuzzification, Fuzzy Inference, Defuzzification:
Measured Variables
(Linguistic Values)
2. Fuzzy-Inference
Command Variables
(Linguistic Values)
Linguistic
Level
Numerical
Level
3. Defuzzification
1. Fuzzification
Measured Variables
(Numerical Values)
Plant
Command Variables
(Numerical Values)
Basic Elements of a
Fuzzy Logic System
Control Loop of the Fuzzy Logic Controlled Container Crane:
Angle, Distance
(Numerical Values)
2. Fuzzy-Inference
Closing the Loop
With Words !
Power
(Linguistic Variable)
Linguistic
Level
Numerical
Level
3. Defuzzification
1. Fuzzification
Angle, Distance
(Numerical Values)
Container Crane
Power
(Numerical Values)
Types of Fuzzy Controllers:
- Direct Controller The Outputs of the Fuzzy Logic System Are the Command Variables of the Plant:
Command
Variables
IF temp=low
AND P=high
THEN A=med
Plant
IF ...
Fuzzification
Inference Defuzzification
Measured Variables
Fuzzy Rules Output
Absolute Values !
Types of Fuzzy Controllers:
- Supervisory Control Fuzzy Logic Controller Outputs Set Values for Underlying PID Controllers:
IF temp=low
AND P=high
THEN A=med
Set Values
PID
PID
IF ...
Fuzzification
Inference
Defuzzification
Plant
PID
Measured Variables
Human Operator
Type Control !
Types of Fuzzy Controllers:
- PID Adaptation Fuzzy Logic Controller Adapts the P, I, and D Parameter of a Conventional PID Controller:
Set Point Variable
IF temp=low
AND P=high
THEN A=med
P
I
D
IF ...
Fuzzification
Inference
Command Variable
PID
Plant
Defuzzification
Measured Variable
The Fuzzy Logic System
Analyzes the Performance of the
PID Controller and Optimizes It !