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50WS 19/20 Georg Frey
3. Outline of the Lecture
1. Introduction of Soft Control: definition and limitations, basics of
"smart" systems
2. Knowledge representation and knowledge processing (Symbolic AI)
Application: expert systems
3. Fuzzy systems: Dealing with Fuzzy knowledge
Application: Fuzzy Control
1. Fuzzy-quantities
4. Connective Systems: Neural Networks
Applications: Identification and neural control
5. Genetic algorithms: Stochastic optimization
Application: Optimization
6. Summary & Literature
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Fuzzy Systems
• Core Idea (Natural Model)
Dealing with fuzzy (non-crisp) knowledge
• History
In the mid-1960s Zadeh fuzzy logic
In the mid-1970s Mandani Fuzzy Control
• Application in Automation Engineering
First industrial applications in the early 1980s
Fuzzy controller
• Examples
Drying processes
Gas heater
Fuzzy control of an inverted pendulum
Washing machine (AEG)
Fuzzy control of a hammer drill
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Contents of the 3rd Lecture
1. Classical quantities
1. Definition and essential terms
2. Problems
2. Fuzzy-Quantities
Definition and terms
Operations on quantities and classical connection with the logic
Expansion of operations on fuzzy quantities
3. Summary
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The Classical Concept of Quantity
• A Quantity M is a Summary of wohlbestimmten and
wohlunterschiedenen Objects unserer Anschauung oder unseres
Denkens zu einem Ganzen.
• These objects are elements of so-called M.
• If an object belongs to M, The we write x M, if not, then x M
• Similar Quantities: M1 M2 (x M1 x M2)
• Dissimilar Quantities: M1 M2
• M1 is a Sub-set of quantity M2: M1 M2 (x M1 x M2)
• M1 is a genuine Sub-set of quantity M2: M1 M2, if M1 M2 und
M1 M2
• Blank Quantity:
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Problems in dealing with classical quantities
• Main problem is the binary decision on the affiliation of a quantity (elements are not always well-differentiated)
• Especially critical for continuous measurement (usually given in the Automatic Control)
• Example: for the interval of temperature from 0 ° C to 100 ° C following applies : "temperature is high"
• for T = 60,00°C "the temperature is high" valid
• for T = 59,99°C "the temperature is high" not valid
For use with control based systems, we have to give steps (jumps)
e.g.: R1: If temp. is high, then Heating-systems turns off
R2: if temp. is NOT high, then Heating system turns on
1
0
μ
T/°C60 100
μT=hoch
0
Solution: Fuzzy Quantity
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Affiliation Function (ZGF)
• The affiliation level is 0 or 1
• μ(x) = 1 means, that x completely belongs to Fuzzy-quantity
• μ(x) = 0 means, that x does not belong to Fuzzy-quantity
• Values from 0 to 1 mean that x partly belongs to the fuzzy quantity
• Finally, If G have many Elements discreet representation of ZGF
Indication of the value pairs {x, μ(x)}
• If there are many elements in G or G is a continuum, for example
cont. Measurement parametric representation of ZGF
Functions determined by a few parameters
Advantage: low memory consumption, fine resolution
Disadvantage may be complicated calculation
Function, every element X from a general basic numerical area, has a
G degree of belonging to a fuzzy-quantity, is assigned as μ(x)
(VDI/VDE 3550)
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Parametric Representation (1): step linear
• Indication of the interpolation function
Spezialfall: trapezoide
Funktionen
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Parametric Representation (2): trapezoid or triangular form
For Special case b=c
we obtain, triangular
form ZGF
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Parametric Representation (3): Normalized Gaussian function
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Parametric Representation (4): Sigmoid difference functions
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Parametric Representation (5): generalized bell function
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Parametric Representation (6): LR-Fuzzy-quantity
• Given the parametric presentation of their flanks (separately for right
and left flank)
Between the flanks (m1 <x <m2), μ (x) = 1
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Parametric Representation (7): Singleton (Also discreet)
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Terms for the description of fuzzy quantities
General adaptation of term Quantity
(for two quantities A and B over a basic quantity G)
• Equality of Fuzzy quantities: A = B μA(x) = μB(x) x G
• Blank quantity : μ(x) = 0 x G
• Universal quantity: μU(x) = 1 x G
Further terminologies
• High Normality
• Support
• Core
• -cut
• Fuzzy-subset
• Fuzzy-similarity
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High Normality
• A fuzzy-stock M is normal ,ifH(M) = 1 gilt,
• Otherwise subnormal
The amount of a fuzzy quantity is the maximum value of their affiliation
to function H(M) = max{μM(x) | x G}
Here and normally in practice, only normal fuzzy quantities are considered
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Support
• Synonym: Medium (VDI / VDE 3550), influence width
• English: support
• Calculation:
Let G is the basic quantity and M belongs to G, the support of M
defined as a fuzzy quantity by
supp(M) = {x G | μM(x) > 0}
given
The support of a fuzzy set is the part of the definition frame in which the
affiliation values greater than 0 are accepted
(VDI/VDE 3550)
1
0
μ
xa b c d
supp(M) = {x G | a < x < d}μM
supp(M)
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Core
• Synonyms: Tolerance (VDI/VDE 3550)
• English: core, tolerance
• Calculation:
Let G is the basic quantity and M belongs to G, then core of M is the
is defined as fuzzy quantity
core(M) = {x G | μM(x) = 1}
given
The core of a fuzzy set is the part of the definition frame in which the
affiliation function accepts the value 1
(VDI/VDE 3550)
1
0
μ
xa b c d
core(M) = {x G | b < x < c}μM
core(M)
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-cut
• Synonyms: -Cut (VDI/VDE 3550), -Level
• Englisch: cut
• Calculation:
Let G is the basic quantity and M belongs to G, then the -cut of M
is defined as fuzzy a quantity
-Schnitt(M) = {x G | μM(x) > }
given
Der - cut a fuzzy quantity is the part of the definition frame in which
the affiliation function values greater then 1 are accepted
(VDI/VDE 3550)
1
0
μ
xa b c d
½-Schnitt(M) = {x G | e < x < f}= {x G | (a+b)/2 < x < (d+c)/2 }
μM
½-Schnitt(M)
½
e f
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Basic Quantity
Support
Context: Support , -cut, Core, Basic quantity
• NOTE: basic quantity, support, core and -cut a lot of fuzzy quantities are classical quantities
• Venn-Diagram
-CutCore
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Fuzzy subset
A fuzzy quantity μ1 is called Fuzzy-Subset of a Fuzzy quantity μ2 on
the Basic quantity G (Notation: μ1 μ2 ), is valid if:
μ1(x) μ2(x) x G
1
0
μ
x
μ1
μ2
μ1 μ2
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Fuzzy Similarity
Two fuzzy quantities A and B are fuzzy-similar if
core (A) = core (B) and supp (A) = supp (B)
1
0
μ
x
a b c d
• Two Fuzzy quantities are exactly fuzzy-similar if they only differ in
their forms of left and right flank
• Conclusion 1: Major changes in the description of a fuzzy set
achieved by amendment of support.
• Conclusion 2: It is generally sufficient to use trapezoid or triangular
membership functions.
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Operations of classical set theory and relationship to the logic
• Average of quantities (AND):
x is part of the intersection of M1 and M2
x is part of M1 AND x is part of M2
• Association of quantities (OR):
x is part of the union of M1 and M2
x is part of M1 OR x element of M2
• Complement of quantities (NOT):
x is the element complementary set of M1
x is NOT the element of M1
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Problems with the NOT operator
• Classical:
A AND NOT A = 0
A OR NOT A = 1
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T-standard and S-standard
• T-Standard
Generalization of the logical AND links the membership degrees of
input sizes from the interval [0, 1] into the original size density of 0
to 1 membership degree, with the figure monotonous, associative
and commutative.
• S-Standard (Synonym: t-Conorm)
Generalization of the logical OR links the membership degrees of
input sizes from the interval [0, 1] into the original size density of 0
to 1 membership degree, with the figure monotonous, associative
and commutative.
• Operator pair
If a t-standard,and S-standard are applied together then De-Morgan'
laws are met, and they both together provide a Operator pair.
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Summary and learning of the 3rd Lecture
Know how of elementary notions of classical quantities
Why classical knowledge is problematic to describe quantities of
continuous partial facts
Fuzzy terminologies of quantities and possibilities to display them
Calculation of characteristic values of fuzzy quantities (support,
core, height, cut)
Know how of relationship between quantity and logic
Know how of elementary operators of fuzzy quantities and fuzzy
logic and how they can be applied
