fuzzy fuzzification and defuzzification
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The document provides an overview of fuzzy logic concepts including types of fuzzy systems, membership functions, fuzzy inference, fuzzification and defuzzification methods. It discusses knowledge-based and rule-based fuzzy systems, types of membership functions like triangular, trapezoidal and Gaussian. Examples of fuzzy logic applications in autonomous driving cars and methods for defuzzification like weighted average, centroid, max-membership and centre of sums are also summarized.
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Human
knowledge-based
Rule-based
Fuzzy
IF AND
THEN
distance
speed
acceleration
small
speed is declining
maintain
IF distance perfect AND
speed is declining
THEN increase acceleration
speed [m/s]](https://image.slidesharecdn.com/lect3-fuzzificationanddefuzzification-191204191413/85/fuzzy-fuzzification-and-defuzzification-4-320.jpg)
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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
[ 𝒎 𝟐
]
declining -ve small zero +ve small +ve big +ve big
constant -ve big -ve small zero +ve small +ve big
growing -ve big -ve big -ve small zero +ve small
speed [m/s]](https://image.slidesharecdn.com/lect3-fuzzificationanddefuzzification-191204191413/85/fuzzy-fuzzification-and-defuzzification-10-320.jpg)
![11 of 38
speed [m/s]
Knowledge
Rule base
Distance to next car [ m ]
v.small small perfect big v.big
Speed
Change
[ 𝒎 𝟐
]
declining -ve small zero +ve small +ve big +ve big
constant -ve big -ve small zero +ve small +ve big
growing -ve big -ve big -ve small zero +ve small
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 small AND speed is declining
THEN acceleration zero
Rule 2: IF distance is small AND speed is constant
THEN acceleration negative small
Rule 3: IF distance is perfect AND speed is declining
THEN acceleration positive small
Rule 4: IF distance is perfect AND speed is constant
THEN acceleration zero
max
Take
min](https://image.slidesharecdn.com/lect3-fuzzificationanddefuzzification-191204191413/85/fuzzy-fuzzification-and-defuzzification-11-320.jpg)
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Inductive Reasoning 7
The induction is performed by the entropy
minimization principle, which clusters most
optimally the parameters corresponding to
the output classes [De Luca and Termini,
1972].
Particular General
useful for complex static systems
not useful for dynamic systems
1- subdivide our data set into membership functions
2- determine a threshold line with an entropy
minimization
3- start the segmentation process by moving an
imaginary threshold value x between x1 and x2
4-calculate entropy for each value of x.](https://image.slidesharecdn.com/lect3-fuzzificationanddefuzzification-191204191413/85/fuzzy-fuzzification-and-defuzzification-24-320.jpg)
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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].](https://image.slidesharecdn.com/lect3-fuzzificationanddefuzzification-191204191413/85/fuzzy-fuzzification-and-defuzzification-36-320.jpg)
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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](https://image.slidesharecdn.com/lect3-fuzzificationanddefuzzification-191204191413/85/fuzzy-fuzzification-and-defuzzification-37-320.jpg)
fuzzy fuzzification and defuzzification
- 1. 1 of 38 FUZZY LOGIC Menoufia University Faculty of Electronic Engineering 11/2019
- 2. 2 of 38 Agenda Introduction to Fuzzy01 Fuzzification Methods02 Defuzzification Methods03 References04
- 3. 3 of 38 What Fuzzy Systems? Confused vague blurred
- 4. 4 of 38 Human knowledge-based Rule-based Fuzzy IF AND THEN distance speed acceleration small speed is declining maintain IF distance perfect AND speed is declining THEN increase acceleration speed [m/s]
- 5. 5 of 38 Types of fuzzy systems Fuzzy systems with fuzzifier and defuzzifier pure fuzzy system its inputs and outputs are fuzzy sets (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 not much freedom left to apply different principles in fuzzy
- 6. 6 of 38 a self-parking car in 1983 Nissan has a patent saves fuel F U Z Z Y App. The fuzzy washing machines were the first major consumer products in Japan around 1990 the most advanced subway system on earth in 1987
- 7. 7 of 38 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
- 8. 8 of 38 Classification of fuzzy sets Convex fuzzy set Non-Convex fuzzy set Normal fuzzy set Sub-normal fuzzy set
- 9. 9 of 38 Types of membership function 𝝁 𝒙 = 𝟎 , 𝒙 ≤ 𝒂 𝒙 − 𝒂 𝒃 − 𝒂 , 𝒂 ≤ 𝒙 ≤ 𝒃 𝒄 − 𝒙 𝒄 − 𝒃 , 𝒃 ≤ 𝒙 ≤ 𝒄 𝟎 , 𝒙 ≥ 𝒄 Triangular 𝝁 𝒙 = 𝟎 , 𝒙 ≤ 𝒂 𝒙 − 𝒂 𝒃 − 𝒂 , 𝒂 ≤ 𝒙 ≤ 𝒃 𝟏 , 𝒃 ≤ 𝒙 ≤ 𝒄 𝒄 − 𝒙 𝒄 − 𝒃 , 𝒄 ≤ 𝒙 ≤ 𝒅 𝟎 , 𝒙 ≥ 𝒅 Trapezoidal 𝝁 𝒙 = 𝒆𝒙𝒑 − 𝒙 − 𝒄 𝟐 𝟐𝝈 𝟐 Gaussian
- 10. 10 of 38 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 [ 𝒎 𝟐 ] declining -ve small zero +ve small +ve big +ve big constant -ve big -ve small zero +ve small +ve big growing -ve big -ve big -ve small zero +ve small speed [m/s]
- 11. 11 of 38 speed [m/s] Knowledge Rule base Distance to next car [ m ] v.small small perfect big v.big Speed Change [ 𝒎 𝟐 ] declining -ve small zero +ve small +ve big +ve big constant -ve big -ve small zero +ve small +ve big growing -ve big -ve big -ve small zero +ve small 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 small AND speed is declining THEN acceleration zero Rule 2: IF distance is small AND speed is constant THEN acceleration negative small Rule 3: IF distance is perfect AND speed is declining THEN acceleration positive small Rule 4: IF distance is perfect AND speed is constant THEN acceleration zero max Take min
- 12. 12 of 38 Defuzzification using Weighted average
- 13. 13 of 38 Fuzzification Methods Genetic Algorithm Angular fuzzy sets Rank ordering Inductive Reasoning Intuition Neural Networks Inference
- 14. 14 of 38 Intuition own intelligence and understanding. contextual and semantic knowledge. linguistic truth values. (see Zadeh, 1972). 1
- 15. 15 of 38 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) 𝝁 𝑰 𝑨, 𝑩, 𝑪 = 𝟏 − 𝟏 𝟔𝟎° 𝒎𝒊𝒏(𝑨 − 𝑩, 𝑩 − 𝑪) Right triangle (R) 𝝁 𝑹 𝑨, 𝑩, 𝑪 = 𝟏 − 𝟏 𝟗𝟎° |𝑨 − 𝟗𝟎°| Other triangles (O) 𝝁 𝒐 𝑨, 𝑩, 𝑪 = 𝟏 − 𝒎𝒂𝒙 𝟏 𝟔𝟎° 𝒎𝒊𝒏 𝑨 − 𝑩, 𝑩 − 𝑪 , 𝟏 𝟗𝟎° |𝑨 − 𝟗𝟎°| t Types of triangles:
- 16. 16 of 38 e.g 4-1 Define the triangle for the figure shown in Figure with the three given angles.
- 17. 17 of 38 Rank ordering 3 Preference is determined by pairwise comparisons these determine the ordering of the membership. Suppose 1000 people respond to a questionnaire about 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.”
- 18. 18 of 38 Angular fuzzy sets 4 The linguistic terms 1- Fully anticlockwise (FA) 𝛉 = 𝝅 𝟐 2- Partially anticlockwise (PA) 𝛉 = 𝝅 𝟒 3- No rotation (NR) 𝛉 = 𝟎 4- Partially clockwise (PC) 𝛉 = − 𝝅 𝟒 5- Fully clockwise (FC) 𝛉 = − 𝝅 𝟐 The angular fuzzy set universe angles repeating every 2Π cycles. linguistic values vary with θ on the unit circle membership values μ(θ).
- 19. 19 of 38 𝝁 𝒕 𝐙 = 𝐙 𝐭𝐚𝐧 𝜽 𝐰𝐡𝐞𝐫𝐞 𝒁 = 𝐜𝐨𝐬 𝜽 Angular fuzzy membership function 𝝁 𝒕 𝐙 𝒁 The values for membership functions
- 20. 20 of 38 Neural networks 5 A neural network is a technique that seeks to build an intelligent program using models that simulate the working network of the neurons in the human brain
- 21. 21 of 38 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 process of reproduction, selection, crossover, and mutation among existing organisms. selection cross-over mutationFitness Function Initial Population
- 22. 22 of 38 Inductive Reasoning 7
- 24. 24 of 38 Inductive Reasoning 7 The induction is performed by the entropy minimization principle, which clusters most optimally the parameters corresponding to the output classes [De Luca and Termini, 1972]. Particular General useful for complex static systems not useful for dynamic systems 1- subdivide our data set into membership functions 2- determine a threshold line with an entropy minimization 3- start the segmentation process by moving an imaginary threshold value x between x1 and x2 4-calculate entropy for each value of x.
- 25. 25 of 38 Defuzzification Methods Centre of largest area Mean–max membership Weighted average Maxima Max-membership Centre of sums Centroid method
- 26. 26 of 38 Lambda Cut (𝜆 𝑐𝑢𝑡) Properties of Lambda Cut Sets: 1- 𝑨 ∪ 𝑩 𝝀 = 𝑨 𝝀 ∪ 𝑩 𝝀 2- 𝑨 ∩ 𝑩 𝝀 = 𝑨 𝝀 ∩ 𝑩 𝝀 3- ഥ𝑨 𝝀 ≠ 𝑨 𝝀 𝒆𝒙𝒄𝒆𝒑𝒕 𝒇𝒐𝒓 𝒂 𝒗𝒂𝒍𝒖𝒆 𝒐𝒇 𝝀 = 𝟎. 𝟓 4- 𝑭𝒐𝒓 𝝀 ≤ 𝝁(𝒙), 𝒘𝒉𝒆𝒓𝒆 𝝁(𝒙) 𝒃𝒆𝒕𝒘𝒆𝒆𝒏 𝟎 𝒂𝒏𝒅 𝟏 𝑰𝒇 𝑨 = 𝟎. 𝟗 𝒙 𝟏 + 𝟎. 𝟏 𝒙 𝟐 + 𝟎. 𝟑 𝒙 𝟑 + 𝟎. 𝟐 𝒙 𝟒 + 𝟎. 𝟖 𝒙 𝟓 , 𝒇𝒊𝒏𝒅 𝑨 𝟎.𝟑 𝑨 𝟎.𝟑 = 𝟏 𝒙 𝟏 + 𝟎 𝒙 𝟐 + 𝟏 𝒙 𝟑 + 𝟎 𝒙 𝟒 + 𝟏 𝒙 𝟓
- 27. 27 of 38 Max-membership 1 𝒁∗ This method is given by the expression 𝝁(𝒁∗ ) ≥ 𝝁(𝒁) This method is also referred as height method
- 28. 28 of 38 A railroad company intends to lay a new rail line in a particular part of a county. The whole area through which the new line is passing must be purchased for right-of-way considerations. It is surveyed in three stretches, and the data are collected for analysis. The surveyed data for the road are given by the sets,𝐵1 , 𝐵2 , 𝐵3 , Where the sets are defined on the universe of right-of-way widths, in meters. For the railroad to purchase the land, it must have an assessment of the amount of land to be bought. The three surveys on right-of-way width are ambiguous, however, because some of the land along the proposed railway route is already public domain and will not need to be purchased. Additionally, the original surveys are so old (circa 1860) that some ambiguity exists on boundaries and public right-of-way for old utility lines and old roads. The three fuzzy sets, 𝐵1 , 𝐵2 , 𝐵3 , shown in the following figures, respectively, represent the uncertainty in each survey as to the membership of right-of- way width, in meters, in privately owned land. We now want to aggregate these three survey results to find the single most nearly representative right-of-way width (z) to allow the railroad to make its initial estimate of the right-of-way purchasing cost. We want to find 𝑍∗. Ex. 10 P.112
- 29. 29 of 38 Centroid method 2 also called center of area, center of gravity). it is the most prevalent and physically appealing of all the defuzzification methods 𝒁∗ = 𝝁 𝒁 𝒁 𝒅𝒛 𝝁 𝒁 𝒅𝒛 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 𝞵 Z 𝒁∗ = 𝟒. 𝟗 𝒎
- 30. 30 of 38 Weighted average 3 This method only for symmetrical output membership function. each membership function in the obtained output by its largest membership Value. 𝒁∗ = σ 𝝁(𝒁) 𝒁 σ 𝝁(𝒁) 𝒁∗ = 𝟎. 𝟑 × 𝟐. 𝟓 + 𝟎. 𝟓 × 𝟓 + (𝟏 × 𝟔. 𝟓) 𝟎. 𝟑 + 𝟎. 𝟓 + 𝟏 𝒁∗ = 𝟓. 𝟒𝟏 𝒎 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 𝞵 Z 𝒁∗ = 𝟓. 𝟒𝟏 𝒎
- 31. 31 of 38 Mean–max membership 4 This method is related to max-membership Which needs a single point, while Mean–max membership can be a range. 𝒁∗ = 𝟔 + 𝟕 𝟐 = 𝟔. 𝟓 𝒎 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 𝞵 Z 𝒁∗ = 𝟔. 𝟓 𝒎 𝒁∗ = 𝒂 + 𝒃 𝟐
- 32. 32 of 38 Centre of sums 5 This is one of the most commonly used defuzzification technique. In this method, the overlapping area is counted twice 𝑨 𝟏 = 𝟎. 𝟑 × 𝟎. 𝟓 × 𝟑 + 𝟓 = 𝟏. 𝟐 𝑨 𝟐 = 𝟎. 𝟓 × 𝟎. 𝟓 × 𝟐 + 𝟒 = 𝟏. 𝟓 𝑨 𝟑 = 𝟏 × 𝟎. 𝟓 × 𝟑 + 𝟏 = 𝟐 𝒁∗ = 𝟏. 𝟐 × 𝟐. 𝟓 + 𝟏. 𝟓 × 𝟓 + (𝟐 × 𝟔. 𝟓) 𝟏. 𝟐 + 𝟏. 𝟓 + 𝟐 = 𝟓𝒎 𝒁∗ = σ𝒊=𝟎 𝒏 𝑨𝒊 𝒁𝒊 σ𝒊=𝟎 𝒏 𝑨𝒊 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 𝞵 Z 𝒁∗ = 5 m 𝐴1 𝐴2 𝐴3
- 33. 33 of 38 Center of largest area 6 𝒁∗ = 𝝁 𝑪 𝒎 𝒁 𝒁 𝒅𝒛 𝝁 𝑪 𝒎 𝒁 𝒅𝒛 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 𝞵 Z 𝒁∗ = 4.9 m 𝐴1 𝐴2 If the output fuzzy set has at least two convex subregions, Then 𝑍∗ is calculated using the centroid method. 𝐶 𝑚: is the convex subregion that has the largest area
- 34. 34 of 38 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 𝞵 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)
- 35. 35 of 38 Defuzzification Methods Centre of largest area Mean–max membership Weighted average Maxima Max-membership Centre of sums Centroid method 𝒁∗ = 𝟒. 𝟗 𝒎 𝒁∗ = 𝟓. 𝟒𝟏 𝒎 𝒁∗ = 𝟔. 𝟓 𝒎 𝒁∗ = 𝟓 𝒎 𝒁∗ = 𝟒. 𝟗 𝒎 𝑭𝑶𝑴 𝒁∗ = 𝟔 𝒎 𝑳𝑶𝑴 𝒁∗ = 𝟕 𝒎 𝑴𝑶𝑴 𝒁∗ = 𝟔. 𝟓 𝒎
- 36. 36 of 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].
- 37. 37 of 38 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
- 38. 38 of 38 Thank You Nourhan Selem Salm