Multimodale Interaktion für die Prozessleittechnik !? Leon Urbas

Fakultät ETIT Institut für Automatisierungstechnik, Professur für Prozessleittechnik
19.01.2010
Dr. Engin YEŞİL
Introduction to
Fuzzy Modeling & Control

1998-
Istanbul Technical University
Faculty of Electrical and Electronics Engineering,
Control Engineering Department
– Programs of our Department:
– Control Engineering (Turkish)
– Control Engineering (English) –NEW-
– Control Engineering + Michigan State Uni. –NEW-
PostDoc
(2009-2010)
Fuzzy Modeling & Control (SS 2010)
Wednesday; 4.DS; BAR/213/H
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Dr. Engin YESIL
Room: BAR 277
Tel: 39102
Webpage: www.elk.itu.edu.tr/~yesil
E-mail: engin.yesil@mailbox.tu-dresden.de
Office hours: Anytime (just e-mail before!!!)
19.01.2010 �
Exam (≈20%)
26.01.2010 �
02.02.2010
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FUZZY SETS & FUZZY LOGIC
��
FUZZY SYSTEMS
��
FUZZY PD CONTROLLER
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Evolution of development
1965 Seminal Papers “Fuzzy Sets” by
Prof. Lotfi Zadeh, U.C. Berkeley.
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Why Fuzzy?
fuzzy adj. 1 unclear or confused and lacking details. 2
not clearly thought out or expressed. 3 indistinct,
unclear, distorted or imprecise.
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•
•
•
•
•
•
Cantor: Set Theory at the end of 19th century.
–
(CRISP SETS)
Sanders Peirce (1839-1914):
–
Uncertainty theory
Bertrand Russel
–
(1872-1970):
"All language is vague“
Jan Lukasiewicz (1878-1955):
–
Many-valued logic
Max Black (1909-1988) proto-fuzzy sets
Lotfi
Zadeh Founder of fuzzy sets & logic
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•
•
•
•
•
•
•
•
1975 Fuzzy control was first introduced by E. Mamdani in
“Advances in the linguistic synthesis of fuzzy controllers”
Introduction of Fuzzy Logic in Japan
1980
1985
1990
1995
Empirical Verification of Fuzzy Logic in Europe
Broad Application of Fuzzy Logic in Japan
Broad Application of Fuzzy Logic in Europe
Broad Application of Fuzzy Logic in the U.S.
2000 Fuzzy logic becomes a standard technique for multivariable
control and is also applied in data and sensor signal
analysis.
Application of Fuzzy Logic in business and finance.
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Rice Cooker
Washing Machine
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Automatic gear shift
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Recovery Boiler
Fuzzy Logic Control
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•
•
•
Propositions, Sets and Characteristic Functions
In classical logic a proposition is a statement that is
either true or false.
A proposition is represented by a set, set a collection of
elements that share a common property. These elements
are referred to as members of the set.
A set can be specified by specifying its members, i.e., “a
set A is the set of all elements x in U that have the
property P”. We write
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Examples:
Set of natural numbers smaller than 5:
A = {0,1, 2, 3, 4}
Unit disk in the complex plane:
A = {z | z �
A line in R 2 :
C, |z| ≤
1}
A = {(x, y) | ax + by + c = 0, (x, y, a, b, c) �
R}
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Propositions, Sets and Characteristic Functions
Another way to characterize a set is to use a characteristic
function, function defined by
where we have introduced the truth value
1 for a true
statement, and 0 for a false statement.
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Propositions, Sets and Characteristic Functions
There are three special sets;
1. the universal set (or universe of discourse), U,
which contains all elements
2. the empty set, set �, that contains no elements
3. the singleton set, set � , that contains only one
element.
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Operations on sets
To derive the set representation of a compound proposition,
the set operations corresponding to the logic connectives
“and”, “or” and “not” must be defined.
These operations are called intersection, intersection union and
complement respectively.
Let A and B be two sets defined on the same universe of
discourse, U.
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We
Operations on sets
can then define our three basic set operations as follows:
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EXAMPLE?
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•
Operations on sets
The following formulas give the characteristic functions
resulting from the set operations above.
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Why Fuzzy Sets?
•
•
Classical sets are good for well-defined concepts
(maths, programs, etc.)
Less suitable for representing commonsense
knowledge in terms of vague concepts such as:
– a tall person, slippery road, nice weather, . . .
– want to buy a big car with moderate consumption
– If the temperature is too low, increase heating a lot
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Classical Set Approach
• set of tall people
A = {h | h ≥ 180}
Logical Propositions
• “Engin is tall” . . . true or false?
• Engin’s height:
hEngin = 180.0 μA(180.0) = 1 (true)
hEngin = 179.5 μA(179.5) = 0 (false)
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Fuzzy Set Approach
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Fuzzy Logic Propositions
“Engin is tall” . . . degree of truth
Engin’s height:
hEngin = 180.0 μA (180.0) = 0.6
hEngin = 179.5 μA (179.5) = 0.56
hEngin = 201.0 μA (201.0) = 1
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Subjective and Context Dependent
Tall in
China
Tall in
Europe
Tall in NBA
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Can you give any more example?
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Expensive Cars:
U= {Bugatti, Ferrari, BMW, Porsche, Mercedes, Ford,
FIAT, Rolls-Royce, Lamborghini}
� 1
�
�Bugatti
Expensive
Cars � �
� 0
�
��
FIAT
�
�
1 0.
4 0.
9 0.
7 0.
1 �
� � � �
Ferrari BMW Porsche Mercedes Ford�
�
�
0.
7 0.
65
�
�
Rolls - Royce Lamborghini
��
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Shapes of Membership Functions
•
•
•
A fuzzy set, A, is a set whose characteristic function μA takes values in the interval [0, 1].
In fuzzy logic literature, the characteristic function of a
fuzzy set is always called the membership function of the
fuzzy set.
We interpret the
membership value of
an element to a specific
set as the degree to
which the corresponding
proposition applies.
(a)
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The trapezoidal membership
function:
Trapezoidal membership functions with varying widths and
centers (left), and varying slopes (right).
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The triangular membership
function:
Triangular membership function with varying centers (left)
and slopes (right).
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The Gaussian membership
function:
Gaussian membership functions for different ��:s (left) and
different
x centers (right).
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The sigmoidal
function:
membership
determines the crossover point of the membership function (the
element whose membership value equals 0.5), and � controls the slope
of the function at this point.
Sigmoidal membership functions for different ��:s (left) and
different :s (right).
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The singleton
function:
membership
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Linguistic Variable
Is 27�C high?
27
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Support of a Fuzzy Set
supp(A) = {x | μA(x) > 0}
� support is an ordinary set
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Core (Kernel) of a Fuzzy Set
core(A) = {x | μA(x) = 1}
� core is an ordinary set
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�-cut of a Fuzzy Set
�
A α
= {x | μA(x) ≥α} � A α
is an ordinary set
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Convex and Non-Convex Fuzzy Sets
A fuzzy
set is convex
A �
�
= {x | μA(x) ≥
all its α-cuts are convex sets.
�}
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Non-Convex Fuzzy Set: an Example
High-risk age for car insurance policy.
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Fuzzy Numbers and Singletons
Fuzzy
linear regression:
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Fuzzy Set Operations
•
•
•
In fuzzy logic the classical set operations are extended
to produce an intuitively correct result on fuzzy sets.
This is done through the introduction
S-norms.
of T-norms
It is important to notice that a compound set is
formed by applying the set operations point-wise
all elements on the universe of discourse.
and
over
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INTERSECTION OPERATOR
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UNION OPERATOR
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COMPLEMENT OPERATOR
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Linguistic Modifiers (Hedges)
–
–
–
–
Modify the meaning of a fuzzy set.
For instance, very can change the meaning of the fuzzy
set tall to very
tall. tall
Other common hedges: slightly, more or less, rather,
etc.
Usual approach: powered hedges:
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Fuzzy Logic
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Example?
–
NOT
VERY YOUNG
AND
NOT OLD!
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Fuzzy Set in Multidimensional Domains
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Cylindrical Extension
Projection onto X 1
Projection onto X 2
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Intersection on Cartesian Product Space
• An operation between fuzzy sets are defined in different
domains results in a multi-dimensional fuzzy set.
•
Example: A 1
∩
A 2
on X1 × X2: Folie 51

Fuzzy Relations
•
•
•
Classical relation represents the presence or absence of
interaction between the elements of two or more sets.
With fuzzy relations, the degree of association (correlation)
is represented by membership grades.
An n-dimensional fuzzy relation is a mapping
R : X1 ×X2×X3. . . × Xn → [0, 1]
which assigns membership grades to all n-tuples
(x1, x2, . . . , xn) from the Cartesian product universe.
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Fuzzy Relations: Example
• Example: R : x ≈ y
(“x is approximately equal to y”)
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Relational Composition
• Given fuzzy relation R defined in X ×Y and fuzzy set A
defined in X, derive the corresponding fuzzy set B defined
in Y :
•
max-min composition:
Analogous to evaluating a function.
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crisp function
interval function
fuzzy function
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•
•
•
•
•
Guessing Game:
LEON (a fictitious name) is nearsighted and
colorblind.
When he goes to a local grocery where fruits are
laced on high shelves, he cannot see them very
well.
He can only recognize the size and blurred shape
of the fruits.
He has lived in such a world for some
20 years and now he is a houseman,
and he has some knowledge about the
features of the fruits.
For example, tangerines are round and
relatively small.
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Fruit={tangerine, apple, pineapple, watermelon, strawberry}
Shape={long, round, large}
long
round
large
tangerine apple pineapple watermelon strawberry
0 0 0.3 0 0.8
0.9 1.0 0.3 1.0 0.2
0.2 0.4 0.7 1.0 0.1
Let’s guess a fruit that LEON sees. If he recognizes a fruit that
is “round and big” and if interpret this as
long round large
0 0.7 1.0
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0 0.7 1.0 o
=
0 0 0.3 0 0.8
0.9 1.0 0.3 1.0 0.2
0.2 0.4 0.7 1.0 0.1
tangerine apple pineapple watermelon strawberry
0.7 0.7 0.7 1.0 0.2
From this result, the possibility of watermelon is the
highest and tangerine, apple, and pineapple come next
at an equal possibility.
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If LEON recognizes another fruit as “relatively long,
somewhat round, and not very large” and if we can
interpret his observation as
long round large
0.5 0.5
Answer:
tangerine apple pineapple watermelon strawberry
0.3
0.5 0.5 0.3 0.5 0.5
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