fuzzy fuzzification and defuzzification

it illustrates what is fuzzy logic and the methods of fuzzification and defuzzification

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FUZZY LOGIC
Menoufia University
Faculty of Electronic Engineering
11/2019

2. 2of 38
Agenda
Introduction to Fuzzy01
Fuzzification Methods 02
Defuzzification Methods 03
References 04

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What Fuzzy Systems?
Confused
vague
blurred

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Human
knowledge-based
Rule-based
Fuzzy
IF AND
THEN
distance
speed
acceleration
small
speedis declining
maintain
IF distanceperfect AND
speedis declining
THEN increase acceleration
speed [m/s]

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Types of fuzzy systems
Fuzzy systems with
fuzzifier and defuzzifier
pure fuzzy system
its inputsand outputsare
fuzzysets (natural languages)
in engineering systems the
inputs and outputs are real-
valued variables.
problem
Takagi-Sugeno-Kang
(TSK) fuzzy systems
problem
1-mathematical formula may not
provide a natural framework of
human knowledge.
2-there is notmuch freedomleft to
apply different principles in fuzzy

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aself-parkingcarin1983
Nissanhasapatentsaves
fuel
FUZZY
App.
The fuzzy washing machines
were the first major consumer
products in Japan around
1990
themostadvancedsubway
systemonearthin1987

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Fuzzy Logic
Controller
Sensor
Fuzzification
Fuzzy
Inference
System
to be
controlled
Defuzzification
Membership
function of
input fuzzy set
Rule Base
Membership
function of
output fuzzy set
Feedback

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Classification of fuzzy sets
Convex
fuzzy set
Non-Convex
fuzzy set
Normal
fuzzy set
Sub-normal
fuzzy set

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Types of membership function
��=
�,�≤�
�−�
�−�
,�≤�≤�
�−�
�−�
,�≤�≤�
�,�≥�
Triangular
��=
�,�≤�
�−�
�−�
,�≤�≤�
�,�≤�≤�
�−�
�−�
,�≤�≤�
�,�≥�
Trapezoidal
��=���
−�−�
�
�??????
�
Gaussian

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Autonomous driving car
distance
speed
acceleration
13 m
-2.5 m/s
?
Knowledge
Rule base
Distance to next car [ m ]
v.small small perfect big v.big
Speed
Change
[�
�
]
declining-vesmall zero +vesmall+vebig +vebig
constant -vebig-vesmall zero +vesmall+vebig
growing -vebig -vebig-vesmall zero +vesmall
speed [m/s]

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speed [m/s]
Knowledge
Rule base
Distance to next car [ m ]
v.small small perfect big v.big
Speed
Change
[�
�
]
declining-vesmallzero +vesmall+vebig+vebig
constant-vebig-vesmall zero +vesmall+vebig
growing -vebig-vebig-vesmall zero +vesmall
0.4 0.25
0.4
0.6
0.6
0.75
0.75
0.25
0.25
0.4
0.25
0.6
Rule 1:IF distance is smallAND speed is declining
THENacceleration zero
Rule 2:IF distance issmallAND speed is constant
THEN acceleration negative small
Rule 3:IF distance isperfectAND speed is declining
THEN acceleration positive small
Rule 4:IF distance is perfectAND speed is constant
THEN acceleration zero
max
Take
min

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Defuzzification using Weighted average

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Fuzzification Methods
Genetic Algorithm
Angular fuzzy sets
Rank ordering
Inductive
Reasoning
Intuition Neural Networks
Inference

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Intuition
ownintelligenceand understanding.
contextualand semanticknowledge.
linguistic truth values.
(see Zadeh, 1972).
1

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Inference 2
We wish to deduce a conclusion, given a body of facts and knowledge.
the one we illustrate here relates to our formal knowledge of geometry and geometric shapes
U = {(A,B,C) | A ≥ B ≥ C ≥ 0 ; A + B + C = 180◦}
Isosceles triangle (I) �
??????�,�,�=�−
�
��°
���(�−�,�−�)
Right triangle (R) �
??????�,�,�=�−
�
��°
|�−��°|
Other triangles (O) �
��,�,�=�−���
�
��°
����−�,�−�,
�
��°
|�−��°|t
Types of triangles:

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e.g4-1 Define the triangle for the figure shown in Figure with
the three given angles.

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Rank ordering 3
Preference is determined by pairwise comparisons
these determine the ordering of the membership.
Suppose 1000 peoplerespond to a questionnaireabout
their pairwise preferences among five colors, X = {red,
orange, yellow, green, blue}. Define a fuzzy set as A
on the universe of colors “best color.”

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Angular fuzzy sets 4
The linguistic terms
1-Fully anticlockwise (FA) ??????=
??????
�
2-Partially anticlockwise (PA) ??????=
??????
�
3-No rotation (NR) ??????=�
4-Partially clockwise (PC) ??????=−
??????
�
5-Fully clockwise (FC) ??????=−
??????
�
The angular fuzzy set
universe angles
repeating every 2Πcycles.
linguistic values vary with θ on the unit circle
membership values μ(θ).

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�
��=����??????
??????????????????�??????�=??????��??????
Angular fuzzy membership function
�
��
�
The values for membership functions

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Neural networks 5
Aneuralnetworkisatechnique
thatseekstobuildanintelligent
programusingmodelsthatsimulate
theworkingnetworkofthe
neuronsinthehumanbrain

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Genetic Algorithm 6
Genetic algorithm (GA) uses the concept of Darwin’s theory
of evolution “survival of the fittest.” postulated that the new
classes of living things came into existence through
the processof reproduction, selection, crossover, and
mutationamong existing organisms.
selection cross-over mutation
Fitness
Function
Initial
Population

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Inductive Reasoning 7

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EntropyInduction

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Inductive Reasoning 7
Theinductionisperformedbytheentropy
minimizationprinciple,whichclustersmost
optimallytheparameterscorrespondingto
theoutputclasses[DeLucaandTermini,
1972].
Particular General
usefulfor complex static systems
not useful for dynamicsystems
1-subdivide our dataset into membership functions
2-determine a threshold line with an entropy
minimization
3-startthe segmentationprocess by moving an
imaginary thresholdvalue x between x1 and x2
4-calculate entropy for each value of x.

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Defuzzification Methods
Centre of
largest area
Mean–max
membership
Weighted average
Maxima
Max-membership Centre
of sums
Centroid
method

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Lambda Cut (??????
??????��)
Properties of Lambda Cut Sets:
1-�∪�
�=�
�∪�
�
2-�∩�
�=�
�∩�
�
3-ഥ�
�≠�
���������??????���������=�.�
4-??????�??????�≤�(�),���??????��(�)������������
??????��=
�.�
�
�
+
�.�
�
�
+
�.�
�
�
+
�.�
�
�
+
�.�
�
�
,�����
�.�
�
�.�=
�
�
�
+
�
�
�
+
�
�
�
+
�
�
�
+
�
�
�

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Max-membership 1
�
∗
This method is given by the expression
�(�
∗
)≥�(�)
This method is also referred as height method

28. 28of 38
Arailroadcompanyintendstolayanewraillineina
particularpartofacounty.Thewholeareathroughwhichthenewlineis
passingmustbepurchasedforright-of-wayconsiderations.Itissurveyed
inthreestretches,andthedataarecollectedforanalysis.Thesurveyed
datafortheroadaregivenbythesets,�
1,�
2,�
3,Wherethesetsare
definedontheuniverseofright-of-waywidths,inmeters.Fortherailroad
topurchasetheland,itmusthaveanassessmentoftheamountoflandto
bebought.Thethreesurveysonright-of-waywidthareambiguous,
however,becausesomeofthelandalongtheproposedrailwayrouteis
alreadypublicdomainandwillnotneedtobepurchased.Additionally,the
originalsurveysaresoold(circa1860)thatsomeambiguityexistson
boundariesandpublicright-of-wayforoldutilitylinesandoldroads.The
threefuzzysets,�
1,�
2,�
3,showninthefollowingfigures,respectively,
representtheuncertaintyineachsurveyastothemembershipofright-of-
waywidth,inmeters,inprivatelyownedland.Wenowwanttoaggregate
thesethreesurveyresultstofindthesinglemostnearlyrepresentative
right-of-waywidth(z)toallowtherailroadtomakeitsinitialestimateofthe
right-of-waypurchasingcost.Wewanttofind??????
∗
.
Ex. 10
P.112

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Centroid method 2
also called center of area, center of gravity).
it is the most prevalent and physically
appealing of all the defuzzification methods
�
∗
=
׬����??????
׬���??????
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5 6 7 8
??????
Z
�
∗
=�.��

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Weighted average 3
This method only for symmetrical output
membership function.
each membership function in the obtained output
by its largest membership Value.
�
∗
=
σ�(�)�
σ�(�)
�
∗
=
�.��.�+�.��+(��.�)
�.�+�.�+�
�
∗
=�.���
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5 6 7 8
??????
Z
�
∗
=�.���

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Mean–max membership 4
This method is related to max-membershipWhich
needs a single point, while Mean–max
membershipcan be a range.
�
∗
=
�+�
�
=�.��
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5 6 7 8
??????
Z
�
∗
=�.��
�
∗
=
�+�
�

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Centre of sums 5
This is one of the most commonly used
defuzzification technique. In this method, the
overlapping area is counted twice
�
�=�.��.��+�=�.�
�
�=�.��.��+�=�.�
�
�=��.��+�=�
�
∗
=
�.��.�+�.��+(��.�)
�.�+�.�+�
=��
�
∗
=
σ
�=�
�
�
��
�
σ
�=�
�
�
�
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5 6 7 8
??????
Z
�
∗
=5m
�
1
�
2
�
3

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Center of largest area 6
�
∗
=
׬�
��
���??????
׬�
��
��??????
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5 6 7 8
??????
Z
�
∗
=4.9m
�
1
�
2
If the output fuzzy set has at least two convex subregions,
Then ??????
∗
is calculated using the centroid method.
�
??????: is the convex subregion that has the largest area

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Maxima 7
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5 6 7 8
??????
Z
Z
∗
=6
First of Maxima Method (FOM)
1
2
Z
∗
=7
Last of Maxima Method (LOM)
3
Z
∗
=
6+7
2
=6.5
Mean of Maxima Method (MOM)

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Defuzzification Methods
Centre of
largest area
Mean–max
membership
Weighted average
Maxima
Max-membership Centre
of sums
Centroid
method
�
∗
=�.��
�
∗
=�.���
�
∗
=�.��
�
∗
=��
�
∗
=�.��
????????????��
∗
=��
�??????��
∗
=��
�??????��
∗
=�.��

36. 36of 38
References
[1]L.-X. Wang, A Course in Fuzzy Systems and Control. Prentice Hall PTR, 1997.
[2]S. N. Sivanandam, S. Sumathi, and S. N. Deepa, Introduction to Fuzzy Logic
using MATLAB. Springer, 2006.
[3]T. J. Ross, Fuzzy Logic with Engineering Applications, 2nd ed. Wiley, 2004.
[4]Essam Nabil, “Autonomous driving car,” March,2019, pp. 1–13.[presentation].

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Slide 1 : [1] man thinking [2] working man
Slide 6 : [2]subway
Slide 7 : [1]sensor [2]system to be controlled
Slide 10 : [1]car
Slide 21 : [1]block diagram of genetic algorithm[2]Steps in Genetic Algorithms
Slide 23 : [1]deductive & inductive reasoning[2]entropy [3]ice and water
Sources of images

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Thank You
Nourhan Selem Salm