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Fuzzy logic

Nourhan Selem Salm, profile picture
Nourhan Selem Salm

This document provides an introduction to fuzzy logic and fuzzy systems. It discusses classical set theory versus fuzzy set theory and membership functions. Types of fuzzy membership functions like triangular, trapezoidal, and Gaussian are shown. The key components of a fuzzy logic controller including fuzzification, fuzzy inference system, and defuzzification are described. Several defuzzification methods such as mean of maxima, centroid, and approximate centroid are explained. Examples of fuzzy applications in areas like washing machines and autonomous vehicles are presented. The document also discusses building fuzzy systems using MATLAB/Simulink and at the command line. Finally, it briefly introduces PID fuzzy controllers.

Engineering
Related topics:
Fuzzy Logic Insights
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FUZZY LOGIC Menoufia University Faculty of Electronic Engineering 4/2020
2 of 57 References08 07 Fuzzy on Simulink 05 Fuzzy at the Cmd line 06 PID – Fuzzy controller Agenda Introduction to Fuzzy01 Fuzzification & Defuzzification 02 Fuzzy application03 04 FIS tool
3 of 57 0 0.5 1 0 1 2 3 4 5 6 7 8 9 10 𝞵(x) x Classical control theory 1 0 On off Yes No
4 of 57 Classical set theory 𝐴 = 0.1,0.3,0.5 & 𝐵 = {0.2,0.3,0.5,0.7} 𝒖𝒏𝒊𝒐𝒏 ∶ 𝐴 ∪ 𝐵 = {0.1,0.2,0.3,0.5,0.7} 𝒊𝒏𝒕𝒆𝒓𝒔𝒆𝒄𝒕𝒊𝒐𝒏 ∶ 𝐴 ∩ 𝐵 = {0.3,0.5} 𝒅𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒄𝒆 ∶ 𝐴 − 𝐵 = {0.1} 𝒄𝒐𝒎𝒑𝒍𝒆𝒎𝒆𝒏𝒕: ҧ𝐴 = 0.9,0.7,0.5 𝒄𝒂𝒓𝒕𝒆𝒔𝒊𝒂𝒏 𝒑𝒓𝒐𝒅𝒖𝒄𝒕 ∶ 𝐴 × 𝐵 𝒅𝒆𝒎𝒐𝒓𝒈𝒆𝒏′ 𝒔 𝒍𝒂𝒘 ∶ 𝐴 ∩ 𝐵 ′ = 𝐴′ ∪ 𝐵′ 0.1 0.3 0.5 A 0.7 0.2 B
5 of 57 Crisp set Vs Fuzzy set
6 of 57 What Fuzzy Systems? Confused vague blurred
7 of 57 Fuzzy he wrote that to handle biological systems "we need a radically different kind of mathematics, the mathematics of fuzzy or cloudy quantities which are not describable in terms of probability distributions" 1962 1965 Classical control Is a 160 m person is tall ? True Possibly True
8 of 57 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
9 of 57 speed [m/s] 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
10 of 57 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
11 of 57 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
12 of 57 Defuzzification Methods Centre of largest area Mean–max membership Maxima (MOM) Max-membership Centre of sums Centroid method Approx. Centroid method
13 of 57 Mean of Maxima (MOM) 1 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∗ = 𝒂 + 𝒃 𝟐 𝒁∗ = 𝟔 + 𝟕 𝟐 = 𝟔. 𝟓 𝒎
14 of 57 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 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∗ = 𝟎. 𝟑 × (𝟏 + 𝟐 + 𝟑) + 𝟎. 𝟓 × (𝟒 + 𝟓) + 𝟏 × (𝟔 + 𝟕) (𝟎. 𝟑 × 𝟑) + (𝟎. 𝟓 × 𝟐) + (𝟏 × 𝟐) 𝒁∗ = 𝟑. 𝟑𝟑 𝒎 𝒁∗ = σ 𝝁(𝒁) 𝒁 σ 𝝁(𝒁)
15 of 57 The approximate COA 3 𝒁∗ = σ 𝝁(𝒁) 𝑪 σ 𝝁(𝒁) 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∗ = 𝟎. 𝟑 × 𝟐. 𝟓 + 𝟎. 𝟓 × 𝟓 + (𝟏 × 𝟔. 𝟓) 𝟎. 𝟑 + 𝟎. 𝟓 + 𝟏 = 𝟓. 𝟒𝟏 𝒎
16 of 57 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]
17 of 57 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
18 of 57 Defuzzification using approximate COA
19 of 57 Washing Machine
20 of 57 Washing Machine 40 30 ?0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 50 60 70 80 90 100 𝞵(weight) Weight (g) v.Light light Heavy V.heavy 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 50 60 70 80 90 100 μ(Dirtiness) Dirtiness (%) Almost Clean Dirty Soiled Filthy 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 50 60 70 80 90 100 μ(detergent) Detergent (%) v.Light little Much V.Much Maximum Knowledge Rule base Weight [ Kg ] V.Light Light Heavy V.Heavy Dirtiness Almost Clean V.Little Little Much Much Dirty Little Little Much V.Much Soiled Much Much V.Much Maximum Filthy V.Much Much V.Much Maximum weight dirtiness amount of detergent output
21 of 57 heavy dirty Much heavy soiled V.Much 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 50 60 70 80 90 100 μ(Dirtiness) Dirtiness (%) Almost Clean Dirty Soiled Filthy 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 50 60 70 80 90 100 𝞵(weight) Weight (g) v.Light light Heavy V.heavy Light dirty little Light soiled Much 0.4 0.4 0.40.8 0.80.6 0.60.2 0.20.2 0.6 0.2 Little 0.4 Much 0.6 V.Much 0.2 IF weight is light(0.4) AND dirtiness is dirty(0.8) THEN detergent is little(0.4) IF weight is light(0.4) AND dirtiness is soiled(0.2) THEN detergent is Much(0.2) IF weight is heavy(0.6) AND dirtiness is dirty(0.8) THEN detergent is Much(0.6) IF weight is heavy(0.6) AND dirtiness is soiled(0.2) THEN detergent is V.Much(0.2)
22 of 57 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 50 60 70 80 90 100μ(detergent) detergent v.Light little Much V.Much Maximum 𝒁∗ = 𝟒𝟎 + 𝟔𝟎 𝟐 = 𝟓𝟎% 𝒁∗ = 𝒂 + 𝒃 𝟐 𝒁∗ = 𝟎. 𝟒 × 𝟐𝟓 + 𝟎. 𝟔 × 𝟓𝟎 + (𝟎. 𝟐 × 𝟕𝟓) 𝟎. 𝟒 + 𝟎. 𝟔 + 𝟎. 𝟐 𝒁∗ = 𝟒𝟓. 𝟖𝟑 % 𝒁∗ = σ 𝝁(𝒁) 𝒁 σ 𝝁(𝒁) 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 50 60 70 80 90 100 μ(detergent) Detergent (%) v.Light little Much V.Much Maximum approximate COA MOM (Mean of Maxima )
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24 of 57 Inputs Output
25 of 57 Service = 3 Food = 8 Rule 1 : IF Service is poor OR Food is rancid THEN Tip is cheap Rule 2 : IF Service is good THEN Tip is average Rule 3 : IF Service is excellent OR Food is delicious THEN Tip is generous 0.125 0.4 0 0 0.5 0.5 0.4 0.125
26 of 57 Build Fuzzy using Fuzzy Logic Designer
27 of 57
28 of 57 Build Fuzzy at the Command Line
29 of 57 Generate new fuzzy01 Add the first input (service)02 Add its membership functions03
30 of 57 04
31 of 57 Add the rules to the FIS05 Rule 1 : IF Service is poor OR Food is rancid THEN Tip is cheap Rule 2 : IF Service is good THEN Tip is average Rule 3 : IF Service is excellent OR Food is delicious THEN Tip is generous 1 - Index of membership function for first input 5 - Fuzzy operator (1 for AND, 2 for OR) 2 - Index of membership function for second input 3 - Index of membership function for output 4 - Rule weight
32 of 57 Evaluate fuzzy06
33 of 57 Plotting07
34 of 57
35 of 57 PID Fuzzy Controller System
36 of 57 𝐈𝐧𝐩𝐮𝐭: 𝐞 𝐤 𝐨𝐮𝐭𝐩𝐮𝐭: 𝐮 𝐤 𝐑𝐮𝐥𝐞: 𝐢𝐟 𝐞 𝐤 𝐢𝐬 … 𝐭𝐡𝐞𝐧 𝐮 𝐤 𝐢𝐬 … P- like FLC1
37 of 57 𝐈𝐧𝐩𝐮𝐭: 𝐞 𝐤 , ∆ 𝒆(𝒌) 𝐨𝐮𝐭𝐩𝐮𝐭: ∆ 𝐮 𝐤 𝐑𝐮𝐥𝐞: 𝐢𝐟 𝐞 𝐤 𝐢𝐬 … 𝐚𝐧𝐝 ∆ 𝒆 𝒌 𝒊𝒔 … 𝐭𝐡𝐞𝐧 ∆𝐮 𝐤 𝐢𝐬 … PI- like FLC2
38 of 57 𝐈𝐧𝐩𝐮𝐭: 𝐞 𝐤 , ∆ 𝒆 𝒌 𝐨𝐮𝐭𝐩𝐮𝐭: 𝐮 𝐤 𝐑𝐮𝐥𝐞: 𝐢𝐟 𝐞 𝐤 𝐢𝐬 … 𝐚𝐧𝐝 ∆ 𝒆 𝒌 𝒊𝒔 … 𝐭𝐡𝐞𝐧 𝐮 𝐤 𝐢𝐬 … PD- like FLC3
39 of 57 𝐈𝐧𝐩𝐮𝐭: 𝐞 𝐤 , ∆ 𝒆 𝒌 , Σ 𝒆(𝒌) 𝐨𝐮𝐭𝐩𝐮𝐭: 𝐮 𝐤 𝐑𝐮𝐥𝐞: 𝐢𝐟 𝐞 𝐤 𝐢𝐬 … 𝐚𝐧𝐝 ∆ 𝒆 𝒌 𝒊𝒔 … 𝐚𝐧𝐝 Σ𝒆 𝒌 𝒊𝒔 … 𝐭𝐡𝐞𝐧 𝐮 𝐤 𝐢𝐬 … PID- like FLC3
40 of 57 𝐈𝐧𝐩𝐮𝐭: 𝐞 𝐤 𝐨𝐮𝐭𝐩𝐮𝐭: 𝐮 𝐤 𝐈𝐧𝐩𝐮𝐭: 𝐞 𝐤 , ∆ 𝒆(𝒌) 𝐨𝐮𝐭𝐩𝐮𝐭: 𝐮 𝐤 𝐈𝐧𝐩𝐮𝐭: 𝐞 𝐤 , ∆ 𝒆(𝒌) 𝐨𝐮𝐭𝐩𝐮𝐭: ∆ 𝐮 𝐤 𝐈𝐧𝐩𝐮𝐭: 𝐞 𝐤 , ∆ 𝒆 𝒌 , Σ 𝒆(𝒌) 𝐨𝐮𝐭𝐩𝐮𝐭: 𝐮 𝐤
41 of 57 Consider a system model is describe by: 𝒚 (𝒌) = 𝟎. 𝟔 × 𝒚 (𝒌 − 𝟏) + 𝒖(𝒌 − 𝟏) PI-Like FLC is designed to regulate this system around a set point of R=2. Five fuzzy sets are used to represent the linguistic variables NB, NS, Z, PS and PB for the controller both input and output variables. Triangular membership functions are used to represent these fuzzy sets and defined on the normalized domain [-1,1] as shown in Fig. 1. The suggested rule-base is depicted in table. If the measured parameters are obtained as y(k-1)=1.5 and u(k-1)=0.5,find the controller output signal taking into account the actual domain of the controller variables is [-2, 2]. Knowledge Rule base e(k) NB NS Z PS PB ∆ 𝒆(𝒌) NB NB NB NB NS Z NS NB NB NS Z PS Z NB NS Z PS PB PS NS Z PS PB PB PB Z PS PB PB PB
42 of 5742 of 24 𝒚 (𝒌) = 𝟎. 𝟔 × 𝒚 (𝒌 − 𝟏) + 𝒖(𝒌 − 𝟏) 𝒂𝒄𝒕𝒖𝒂𝒍 𝒅𝒐𝒎𝒂𝒊𝒏 ∈ −𝟐, 𝟐 𝒕𝒉𝒆 𝒄𝒐𝒏𝒕𝒓𝒐𝒍𝒍𝒆𝒓 𝒐𝒖𝒕𝒑𝒖𝒕 𝒔𝒊𝒈𝒏𝒂𝒍 find 𝒏𝒐𝒓𝒎𝒂𝒍𝒊𝒛𝒆𝒅 𝒅𝒐𝒎𝒂𝒊𝒏 ∈ [−𝟏, 𝟏] 𝑷𝑰 − 𝑳𝒊𝒌𝒆 𝑭𝑳𝑪 𝒚 𝒌 = 𝟎. 𝟔 × 𝒚 𝒌 − 𝟏 + 𝒖 𝒌 − 𝟏 = 𝟎. 𝟔 × 𝟏. 𝟓 + 𝟎. 𝟓 = 𝟏. 𝟒 𝒆(𝒌) = 𝑹(𝒌) − 𝒚(𝒌) = 𝟐 − 𝟏. 𝟒 = 𝟎. 𝟔 𝜟𝒆 (𝒌) = 𝒆 (𝒌) − 𝒆 (𝒌 − 𝟏) = 𝑹 𝒌 − 𝒚 𝒌 − 𝑹 𝒌 − 𝒚 𝒌 − 𝟏 = 𝒚 𝒌 − 𝟏 − 𝒚 𝒌 = 𝟏. 𝟓 − 𝟏. 𝟒 = 𝟎. 𝟏 𝑻𝒉𝒆 𝒂𝒄𝒕𝒖𝒂𝒍 𝒔𝒚𝒔𝒕𝒆𝒎 𝒐𝒖𝒕𝒑𝒖𝒕 𝒇𝒐𝒓 𝒕𝒉𝒆 𝒎𝒆𝒂𝒔𝒖𝒓𝒆𝒅 𝒗𝒂𝒍𝒖𝒆𝒔: 𝒚 𝒌 − 𝟏 = 𝟏. 𝟓 𝒖 𝒌 − 𝟏 = 𝟎. 𝟓
43 of 5743 of 24 𝒚 (𝒌) = 𝟎. 𝟔 × 𝒚 (𝒌 − 𝟏) + 𝒖(𝒌 − 𝟏) 𝒚 𝒌 = 𝟏. 𝟒 𝒂𝒄𝒕𝒖𝒂𝒍 𝒅𝒐𝒎𝒂𝒊𝒏 ∈ −𝟐, 𝟐 𝒕𝒉𝒆 𝒄𝒐𝒏𝒕𝒓𝒐𝒍𝒍𝒆𝒓 𝒐𝒖𝒕𝒑𝒖𝒕 𝒔𝒊𝒈𝒏𝒂𝒍 find 𝒆 𝒌 = 𝟎. 𝟔 𝒏𝒐𝒓𝒎𝒂𝒍𝒊𝒛𝒆𝒅 𝒅𝒐𝒎𝒂𝒊𝒏 ∈ [−𝟏, 𝟏] 𝑷𝑰 − 𝑳𝒊𝒌𝒆 𝑭𝑳𝑪 ∆𝒆 𝒌 = 𝟎. 𝟏 𝑻𝒉𝒆 𝒏𝒐𝒓𝒎𝒂𝒍𝒊𝒛𝒆𝒅 𝒊𝒏𝒑𝒖𝒕 𝒗𝒂𝒍𝒖𝒆𝒔 𝒇𝒐𝒓 𝑭𝑳𝑪 𝒘𝒊𝒍𝒍 𝒃𝒆: 𝑮 𝒆 = 𝟏 𝟐 = 𝟎. 𝟓 , 𝑮∆𝒆 = 𝟏 𝟐 = 𝟎. 𝟓 , 𝑮∆𝒖 = 𝟐 𝟏 = 𝟐 𝑻𝒐 𝒐𝒃𝒕𝒂𝒊𝒏 𝒔𝒄𝒂𝒍𝒊𝒏𝒈 𝒇𝒂𝒄𝒕𝒐𝒓𝒔: 𝒆 𝒏 𝒌 = 𝑮 𝒆 × 𝒆 𝒌 = 𝟎. 𝟓 × 𝟎. 𝟔 = 𝟎. 𝟑 ∆𝒆 𝒏(𝒌) = 𝑮∆𝒆 × ∆𝒆 𝒌 = 𝟎. 𝟓 × 𝟎. 𝟏 = 𝟎. 𝟎𝟓
44 of 57 𝒇𝒖𝒛𝒛𝒊𝒇𝒊𝒄𝒂𝒕𝒊𝒐𝒏 ∶ 𝝁 𝒁_𝒆 𝒏(𝒌) = 𝟏 𝟑 − 𝟎. 𝟑 𝟏 𝟑 − 𝟎 = 𝟎. 𝟏 𝝁 𝑷𝑺_𝒆 𝒏(𝒌) = 𝟎. 𝟑 − 𝟎 𝟏 𝟑 − 𝟎 = 𝟎. 𝟗 𝝁 𝒁_∆𝒆 𝒏(𝒌) = 𝟏 𝟑 − 𝟎. 𝟎𝟓 𝟏 𝟑 − 𝟎 = 𝟎. 𝟖𝟓 𝝁 𝑷𝑺_𝒆 𝒏(𝒌) = 𝟎. 𝟎𝟓 − 𝟎 𝟏 𝟑 − 𝟎 = 𝟎. 𝟏𝟓
45 of 57 PS Z PS PS PS PB Z Z Z Z PS PS 0.1 0.1 0.10.85 0.850.9 0.90.15 0.150.1 0.85 0.15 IF e (k) is Z (0.1) AND ∆𝒆(𝒌) is Z (0.85) THEN ∆u(𝒌) is Z (0.1) IF e (k) is Z (0.1) AND ∆𝒆(𝒌) is PS (0.15) THEN ∆u(𝒌) is PS (0.1) IF e (k) is PS (0.9) AND ∆𝒆(𝒌) is Z (0.85) THEN ∆u(𝒌) is PS (0.85) IF e (k) is PS (0.9) AND ∆𝒆(𝒌) is PS (0.15) THEN ∆u(𝒌) is PB (0.15) Knowledge Rule base e(k) NB NS Z PS PB ∆ 𝒆(𝒌) NB NB NB NB NS Z NS NB NB NS Z PS Z NB NS Z PS PB PS NS Z PS PB PB PB Z PS PB PB PB Z 0.1 PS 0.85 PB 0.15
46 of 57 ∆𝒖 𝒏 𝒌 = 𝟎. 𝟏 × 𝟎 + 𝟎. 𝟖𝟓 × 𝟏 𝟑 + (𝟎. 𝟏𝟓 × 𝟐 𝟑 ) 𝟎. 𝟏 + 𝟎. 𝟖𝟓 + 𝟎. 𝟏𝟓 ∆𝒖 𝒏 𝒌 = 𝟎. 𝟑𝟓 ∆𝒖 𝒏 𝒌 = σ 𝝁(∆𝒖 𝒏) 𝑪 σ 𝝁(∆𝒖 𝒏) approximate COA 𝒅𝒆𝒇𝒖𝒛𝒛𝒊𝒇𝒊𝒄𝒂𝒕𝒊𝒐𝒏 ∶ ∆𝒖 𝒌 = 𝑮∆𝒖 × ∆𝒖 𝒏 𝒌 = 𝟐 × 𝟎. 𝟑𝟓 = 𝟎. 𝟕 𝒖 𝒌 = 𝒖 𝒌 − 𝟏 + ∆𝒖 𝒌 = 𝟎. 𝟓 + 𝟎. 𝟕 = 𝟏. 𝟐 𝑻𝒉𝒆 𝒂𝒄𝒕𝒖𝒂𝒍 𝒄𝒐𝒏𝒕𝒓𝒐𝒍 𝒐𝒖𝒕𝒑𝒖𝒕 𝜟𝒖 𝒌 𝒐𝒇 𝑭𝑳𝑪: 𝑻𝒉𝒆 𝒄𝒐𝒏𝒕𝒓𝒐𝒍 𝒔𝒊𝒈𝒏𝒂𝒍 𝒖 𝒌 𝒘𝒊𝒍𝒍 𝒃𝒆:
Build Fuzzy using Simulink
48 of 57 Tank Level Control System 5 cm ? 0 0.25 0.5 0.75 1 1.25 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 𝞵(Level) Liquid level (cm) low okay high 0 0.5 1 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 𝞵(valvecontrol) Valve control signal (%/s) close fast no change open fast Rule 1 : IF level is okay THEN valve is no change Rule 2 : IF level is low THEN valve is open fast Rule 3 : IF level is high THEN valve is close fast Liquid level Valve control signal
49 of 57 𝑻𝒂𝒏𝒌 𝑺𝒚𝒔𝒕𝒆𝒎 𝑺𝒊𝒎𝒖𝒍𝒂𝒕𝒊𝒐𝒏
50 of 57 𝑺𝒖𝒃𝒔𝒚𝒔𝒕𝒆𝒎 𝑽𝒂𝒍𝒗𝒆 𝑺𝒖𝒃𝒔𝒚𝒔𝒕𝒆𝒎 𝑻𝒂𝒏𝒌 ሶℎ = 1 𝐴 𝑞𝑖𝑛 − 𝑎 𝐴 2𝑔ℎ 𝑨 = 𝟎. 𝟒 𝒎 𝟐 𝒂 = 𝟎. 𝟎𝟏𝟐 𝒎 𝟐 𝒒𝒊,𝒎𝒂𝒙 = 𝟐𝟎 𝒍/𝒔 𝐴: 𝑐𝑟𝑜𝑠𝑠 − 𝑠𝑒𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑎𝑛𝑘 𝑎: 𝑐𝑟𝑜𝑠𝑠 − 𝑠𝑒𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑖𝑝𝑒
51 of 57 𝒍𝒊𝒒𝒖𝒊𝒅 𝒍𝒆𝒗𝒆𝒍 𝒗𝒂𝒍𝒗𝒆 𝒄𝒐𝒏𝒕𝒓𝒐𝒍 𝒗𝒂𝒍𝒗𝒆 𝒐𝒑𝒆𝒏𝒊𝒏𝒈
52 of 57 𝑻𝒂𝒏𝒌 𝑺𝒚𝒔𝒕𝒆𝒎 𝑺𝒊𝒎𝒖𝒍𝒂𝒕𝒊𝒐𝒏 With reference
53 of 57 𝒍𝒊𝒒𝒖𝒊𝒅 𝒍𝒆𝒗𝒆𝒍 𝒗𝒂𝒍𝒗𝒆 𝒄𝒐𝒏𝒕𝒓𝒐𝒍 𝒗𝒂𝒍𝒗𝒆 𝒐𝒑𝒆𝒏𝒊𝒏𝒈
54 of 57 With referenceWithout reference
55 of 57 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]. [5] Essam Nabil, “Fuzzy logic control system applications,” March,2019, pp. 1-30 .[presentation]. [6] Essam Nabil, “Tipping problem,” March,2019, pp. 1–18.[presentation].
56 of 57 References [7] “Build Fuzzy Systems Using Fuzzy Logic Designer - MATLAB & Simulink.” [Online]. Available: https://www.mathworks.com/help/fuzzy/building-systems-with-fuzzy-logic-toolbox- software.html. [Accessed: 22-Nov-2019]. [8] “Build Fuzzy Systems at the Command Line - MATLAB & Simulink.” [Online]. Available: https://www.mathworks.com/help/fuzzy/working-from-the-command-line.html. [Accessed: 22-Nov-2019] [9] Essam Nabil, “Tank control system” March,2019, pp. 1–18.[presentation]. [10] Essam Nabil, “PID - Like Fuzzy Logic control” March,2019, pp. 1–39.[presentation].
57 of 57 Thank You Nourhan Selem Salm FUZZY

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Fuzzy logic

  • 1. FUZZY LOGIC Menoufia University Faculty of Electronic Engineering 4/2020
  • 2. 2 of 57 References08 07 Fuzzy on Simulink 05 Fuzzy at the Cmd line 06 PID – Fuzzy controller Agenda Introduction to Fuzzy01 Fuzzification & Defuzzification 02 Fuzzy application03 04 FIS tool
  • 3. 3 of 57 0 0.5 1 0 1 2 3 4 5 6 7 8 9 10 𝞵(x) x Classical control theory 1 0 On off Yes No
  • 4. 4 of 57 Classical set theory 𝐴 = 0.1,0.3,0.5 & 𝐵 = {0.2,0.3,0.5,0.7} 𝒖𝒏𝒊𝒐𝒏 ∶ 𝐴 ∪ 𝐵 = {0.1,0.2,0.3,0.5,0.7} 𝒊𝒏𝒕𝒆𝒓𝒔𝒆𝒄𝒕𝒊𝒐𝒏 ∶ 𝐴 ∩ 𝐵 = {0.3,0.5} 𝒅𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒄𝒆 ∶ 𝐴 − 𝐵 = {0.1} 𝒄𝒐𝒎𝒑𝒍𝒆𝒎𝒆𝒏𝒕: ҧ𝐴 = 0.9,0.7,0.5 𝒄𝒂𝒓𝒕𝒆𝒔𝒊𝒂𝒏 𝒑𝒓𝒐𝒅𝒖𝒄𝒕 ∶ 𝐴 × 𝐵 𝒅𝒆𝒎𝒐𝒓𝒈𝒆𝒏′ 𝒔 𝒍𝒂𝒘 ∶ 𝐴 ∩ 𝐵 ′ = 𝐴′ ∪ 𝐵′ 0.1 0.3 0.5 A 0.7 0.2 B
  • 5. 5 of 57 Crisp set Vs Fuzzy set
  • 6. 6 of 57 What Fuzzy Systems? Confused vague blurred
  • 7. 7 of 57 Fuzzy he wrote that to handle biological systems "we need a radically different kind of mathematics, the mathematics of fuzzy or cloudy quantities which are not describable in terms of probability distributions" 1962 1965 Classical control Is a 160 m person is tall ? True Possibly True
  • 8. 8 of 57 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
  • 9. 9 of 57 speed [m/s] 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
  • 10. 10 of 57 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
  • 11. 11 of 57 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
  • 12. 12 of 57 Defuzzification Methods Centre of largest area Mean–max membership Maxima (MOM) Max-membership Centre of sums Centroid method Approx. Centroid method
  • 13. 13 of 57 Mean of Maxima (MOM) 1 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∗ = 𝒂 + 𝒃 𝟐 𝒁∗ = 𝟔 + 𝟕 𝟐 = 𝟔. 𝟓 𝒎
  • 14. 14 of 57 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 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∗ = 𝟎. 𝟑 × (𝟏 + 𝟐 + 𝟑) + 𝟎. 𝟓 × (𝟒 + 𝟓) + 𝟏 × (𝟔 + 𝟕) (𝟎. 𝟑 × 𝟑) + (𝟎. 𝟓 × 𝟐) + (𝟏 × 𝟐) 𝒁∗ = 𝟑. 𝟑𝟑 𝒎 𝒁∗ = σ 𝝁(𝒁) 𝒁 σ 𝝁(𝒁)
  • 15. 15 of 57 The approximate COA 3 𝒁∗ = σ 𝝁(𝒁) 𝑪 σ 𝝁(𝒁) 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∗ = 𝟎. 𝟑 × 𝟐. 𝟓 + 𝟎. 𝟓 × 𝟓 + (𝟏 × 𝟔. 𝟓) 𝟎. 𝟑 + 𝟎. 𝟓 + 𝟏 = 𝟓. 𝟒𝟏 𝒎
  • 16. 16 of 57 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]
  • 17. 17 of 57 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
  • 18. 18 of 57 Defuzzification using approximate COA
  • 19. 19 of 57 Washing Machine
  • 20. 20 of 57 Washing Machine 40 30 ?0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 50 60 70 80 90 100 𝞵(weight) Weight (g) v.Light light Heavy V.heavy 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 50 60 70 80 90 100 μ(Dirtiness) Dirtiness (%) Almost Clean Dirty Soiled Filthy 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 50 60 70 80 90 100 μ(detergent) Detergent (%) v.Light little Much V.Much Maximum Knowledge Rule base Weight [ Kg ] V.Light Light Heavy V.Heavy Dirtiness Almost Clean V.Little Little Much Much Dirty Little Little Much V.Much Soiled Much Much V.Much Maximum Filthy V.Much Much V.Much Maximum weight dirtiness amount of detergent output
  • 21. 21 of 57 heavy dirty Much heavy soiled V.Much 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 50 60 70 80 90 100 μ(Dirtiness) Dirtiness (%) Almost Clean Dirty Soiled Filthy 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 50 60 70 80 90 100 𝞵(weight) Weight (g) v.Light light Heavy V.heavy Light dirty little Light soiled Much 0.4 0.4 0.40.8 0.80.6 0.60.2 0.20.2 0.6 0.2 Little 0.4 Much 0.6 V.Much 0.2 IF weight is light(0.4) AND dirtiness is dirty(0.8) THEN detergent is little(0.4) IF weight is light(0.4) AND dirtiness is soiled(0.2) THEN detergent is Much(0.2) IF weight is heavy(0.6) AND dirtiness is dirty(0.8) THEN detergent is Much(0.6) IF weight is heavy(0.6) AND dirtiness is soiled(0.2) THEN detergent is V.Much(0.2)
  • 22. 22 of 57 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 50 60 70 80 90 100μ(detergent) detergent v.Light little Much V.Much Maximum 𝒁∗ = 𝟒𝟎 + 𝟔𝟎 𝟐 = 𝟓𝟎% 𝒁∗ = 𝒂 + 𝒃 𝟐 𝒁∗ = 𝟎. 𝟒 × 𝟐𝟓 + 𝟎. 𝟔 × 𝟓𝟎 + (𝟎. 𝟐 × 𝟕𝟓) 𝟎. 𝟒 + 𝟎. 𝟔 + 𝟎. 𝟐 𝒁∗ = 𝟒𝟓. 𝟖𝟑 % 𝒁∗ = σ 𝝁(𝒁) 𝒁 σ 𝝁(𝒁) 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 50 60 70 80 90 100 μ(detergent) Detergent (%) v.Light little Much V.Much Maximum approximate COA MOM (Mean of Maxima )
  • 25. 25 of 57 Service = 3 Food = 8 Rule 1 : IF Service is poor OR Food is rancid THEN Tip is cheap Rule 2 : IF Service is good THEN Tip is average Rule 3 : IF Service is excellent OR Food is delicious THEN Tip is generous 0.125 0.4 0 0 0.5 0.5 0.4 0.125
  • 26. 26 of 57 Build Fuzzy using Fuzzy Logic Designer
  • 28. 28 of 57 Build Fuzzy at the Command Line
  • 29. 29 of 57 Generate new fuzzy01 Add the first input (service)02 Add its membership functions03
  • 31. 31 of 57 Add the rules to the FIS05 Rule 1 : IF Service is poor OR Food is rancid THEN Tip is cheap Rule 2 : IF Service is good THEN Tip is average Rule 3 : IF Service is excellent OR Food is delicious THEN Tip is generous 1 - Index of membership function for first input 5 - Fuzzy operator (1 for AND, 2 for OR) 2 - Index of membership function for second input 3 - Index of membership function for output 4 - Rule weight
  • 32. 32 of 57 Evaluate fuzzy06
  • 35. 35 of 57 PID Fuzzy Controller System
  • 36. 36 of 57 𝐈𝐧𝐩𝐮𝐭: 𝐞 𝐤 𝐨𝐮𝐭𝐩𝐮𝐭: 𝐮 𝐤 𝐑𝐮𝐥𝐞: 𝐢𝐟 𝐞 𝐤 𝐢𝐬 … 𝐭𝐡𝐞𝐧 𝐮 𝐤 𝐢𝐬 … P- like FLC1
  • 37. 37 of 57 𝐈𝐧𝐩𝐮𝐭: 𝐞 𝐤 , ∆ 𝒆(𝒌) 𝐨𝐮𝐭𝐩𝐮𝐭: ∆ 𝐮 𝐤 𝐑𝐮𝐥𝐞: 𝐢𝐟 𝐞 𝐤 𝐢𝐬 … 𝐚𝐧𝐝 ∆ 𝒆 𝒌 𝒊𝒔 … 𝐭𝐡𝐞𝐧 ∆𝐮 𝐤 𝐢𝐬 … PI- like FLC2
  • 38. 38 of 57 𝐈𝐧𝐩𝐮𝐭: 𝐞 𝐤 , ∆ 𝒆 𝒌 𝐨𝐮𝐭𝐩𝐮𝐭: 𝐮 𝐤 𝐑𝐮𝐥𝐞: 𝐢𝐟 𝐞 𝐤 𝐢𝐬 … 𝐚𝐧𝐝 ∆ 𝒆 𝒌 𝒊𝒔 … 𝐭𝐡𝐞𝐧 𝐮 𝐤 𝐢𝐬 … PD- like FLC3
  • 39. 39 of 57 𝐈𝐧𝐩𝐮𝐭: 𝐞 𝐤 , ∆ 𝒆 𝒌 , Σ 𝒆(𝒌) 𝐨𝐮𝐭𝐩𝐮𝐭: 𝐮 𝐤 𝐑𝐮𝐥𝐞: 𝐢𝐟 𝐞 𝐤 𝐢𝐬 … 𝐚𝐧𝐝 ∆ 𝒆 𝒌 𝒊𝒔 … 𝐚𝐧𝐝 Σ𝒆 𝒌 𝒊𝒔 … 𝐭𝐡𝐞𝐧 𝐮 𝐤 𝐢𝐬 … PID- like FLC3
  • 40. 40 of 57 𝐈𝐧𝐩𝐮𝐭: 𝐞 𝐤 𝐨𝐮𝐭𝐩𝐮𝐭: 𝐮 𝐤 𝐈𝐧𝐩𝐮𝐭: 𝐞 𝐤 , ∆ 𝒆(𝒌) 𝐨𝐮𝐭𝐩𝐮𝐭: 𝐮 𝐤 𝐈𝐧𝐩𝐮𝐭: 𝐞 𝐤 , ∆ 𝒆(𝒌) 𝐨𝐮𝐭𝐩𝐮𝐭: ∆ 𝐮 𝐤 𝐈𝐧𝐩𝐮𝐭: 𝐞 𝐤 , ∆ 𝒆 𝒌 , Σ 𝒆(𝒌) 𝐨𝐮𝐭𝐩𝐮𝐭: 𝐮 𝐤
  • 41. 41 of 57 Consider a system model is describe by: 𝒚 (𝒌) = 𝟎. 𝟔 × 𝒚 (𝒌 − 𝟏) + 𝒖(𝒌 − 𝟏) PI-Like FLC is designed to regulate this system around a set point of R=2. Five fuzzy sets are used to represent the linguistic variables NB, NS, Z, PS and PB for the controller both input and output variables. Triangular membership functions are used to represent these fuzzy sets and defined on the normalized domain [-1,1] as shown in Fig. 1. The suggested rule-base is depicted in table. If the measured parameters are obtained as y(k-1)=1.5 and u(k-1)=0.5,find the controller output signal taking into account the actual domain of the controller variables is [-2, 2]. Knowledge Rule base e(k) NB NS Z PS PB ∆ 𝒆(𝒌) NB NB NB NB NS Z NS NB NB NS Z PS Z NB NS Z PS PB PS NS Z PS PB PB PB Z PS PB PB PB
  • 42. 42 of 5742 of 24 𝒚 (𝒌) = 𝟎. 𝟔 × 𝒚 (𝒌 − 𝟏) + 𝒖(𝒌 − 𝟏) 𝒂𝒄𝒕𝒖𝒂𝒍 𝒅𝒐𝒎𝒂𝒊𝒏 ∈ −𝟐, 𝟐 𝒕𝒉𝒆 𝒄𝒐𝒏𝒕𝒓𝒐𝒍𝒍𝒆𝒓 𝒐𝒖𝒕𝒑𝒖𝒕 𝒔𝒊𝒈𝒏𝒂𝒍 find 𝒏𝒐𝒓𝒎𝒂𝒍𝒊𝒛𝒆𝒅 𝒅𝒐𝒎𝒂𝒊𝒏 ∈ [−𝟏, 𝟏] 𝑷𝑰 − 𝑳𝒊𝒌𝒆 𝑭𝑳𝑪 𝒚 𝒌 = 𝟎. 𝟔 × 𝒚 𝒌 − 𝟏 + 𝒖 𝒌 − 𝟏 = 𝟎. 𝟔 × 𝟏. 𝟓 + 𝟎. 𝟓 = 𝟏. 𝟒 𝒆(𝒌) = 𝑹(𝒌) − 𝒚(𝒌) = 𝟐 − 𝟏. 𝟒 = 𝟎. 𝟔 𝜟𝒆 (𝒌) = 𝒆 (𝒌) − 𝒆 (𝒌 − 𝟏) = 𝑹 𝒌 − 𝒚 𝒌 − 𝑹 𝒌 − 𝒚 𝒌 − 𝟏 = 𝒚 𝒌 − 𝟏 − 𝒚 𝒌 = 𝟏. 𝟓 − 𝟏. 𝟒 = 𝟎. 𝟏 𝑻𝒉𝒆 𝒂𝒄𝒕𝒖𝒂𝒍 𝒔𝒚𝒔𝒕𝒆𝒎 𝒐𝒖𝒕𝒑𝒖𝒕 𝒇𝒐𝒓 𝒕𝒉𝒆 𝒎𝒆𝒂𝒔𝒖𝒓𝒆𝒅 𝒗𝒂𝒍𝒖𝒆𝒔: 𝒚 𝒌 − 𝟏 = 𝟏. 𝟓 𝒖 𝒌 − 𝟏 = 𝟎. 𝟓
  • 43. 43 of 5743 of 24 𝒚 (𝒌) = 𝟎. 𝟔 × 𝒚 (𝒌 − 𝟏) + 𝒖(𝒌 − 𝟏) 𝒚 𝒌 = 𝟏. 𝟒 𝒂𝒄𝒕𝒖𝒂𝒍 𝒅𝒐𝒎𝒂𝒊𝒏 ∈ −𝟐, 𝟐 𝒕𝒉𝒆 𝒄𝒐𝒏𝒕𝒓𝒐𝒍𝒍𝒆𝒓 𝒐𝒖𝒕𝒑𝒖𝒕 𝒔𝒊𝒈𝒏𝒂𝒍 find 𝒆 𝒌 = 𝟎. 𝟔 𝒏𝒐𝒓𝒎𝒂𝒍𝒊𝒛𝒆𝒅 𝒅𝒐𝒎𝒂𝒊𝒏 ∈ [−𝟏, 𝟏] 𝑷𝑰 − 𝑳𝒊𝒌𝒆 𝑭𝑳𝑪 ∆𝒆 𝒌 = 𝟎. 𝟏 𝑻𝒉𝒆 𝒏𝒐𝒓𝒎𝒂𝒍𝒊𝒛𝒆𝒅 𝒊𝒏𝒑𝒖𝒕 𝒗𝒂𝒍𝒖𝒆𝒔 𝒇𝒐𝒓 𝑭𝑳𝑪 𝒘𝒊𝒍𝒍 𝒃𝒆: 𝑮 𝒆 = 𝟏 𝟐 = 𝟎. 𝟓 , 𝑮∆𝒆 = 𝟏 𝟐 = 𝟎. 𝟓 , 𝑮∆𝒖 = 𝟐 𝟏 = 𝟐 𝑻𝒐 𝒐𝒃𝒕𝒂𝒊𝒏 𝒔𝒄𝒂𝒍𝒊𝒏𝒈 𝒇𝒂𝒄𝒕𝒐𝒓𝒔: 𝒆 𝒏 𝒌 = 𝑮 𝒆 × 𝒆 𝒌 = 𝟎. 𝟓 × 𝟎. 𝟔 = 𝟎. 𝟑 ∆𝒆 𝒏(𝒌) = 𝑮∆𝒆 × ∆𝒆 𝒌 = 𝟎. 𝟓 × 𝟎. 𝟏 = 𝟎. 𝟎𝟓
  • 44. 44 of 57 𝒇𝒖𝒛𝒛𝒊𝒇𝒊𝒄𝒂𝒕𝒊𝒐𝒏 ∶ 𝝁 𝒁_𝒆 𝒏(𝒌) = 𝟏 𝟑 − 𝟎. 𝟑 𝟏 𝟑 − 𝟎 = 𝟎. 𝟏 𝝁 𝑷𝑺_𝒆 𝒏(𝒌) = 𝟎. 𝟑 − 𝟎 𝟏 𝟑 − 𝟎 = 𝟎. 𝟗 𝝁 𝒁_∆𝒆 𝒏(𝒌) = 𝟏 𝟑 − 𝟎. 𝟎𝟓 𝟏 𝟑 − 𝟎 = 𝟎. 𝟖𝟓 𝝁 𝑷𝑺_𝒆 𝒏(𝒌) = 𝟎. 𝟎𝟓 − 𝟎 𝟏 𝟑 − 𝟎 = 𝟎. 𝟏𝟓
  • 45. 45 of 57 PS Z PS PS PS PB Z Z Z Z PS PS 0.1 0.1 0.10.85 0.850.9 0.90.15 0.150.1 0.85 0.15 IF e (k) is Z (0.1) AND ∆𝒆(𝒌) is Z (0.85) THEN ∆u(𝒌) is Z (0.1) IF e (k) is Z (0.1) AND ∆𝒆(𝒌) is PS (0.15) THEN ∆u(𝒌) is PS (0.1) IF e (k) is PS (0.9) AND ∆𝒆(𝒌) is Z (0.85) THEN ∆u(𝒌) is PS (0.85) IF e (k) is PS (0.9) AND ∆𝒆(𝒌) is PS (0.15) THEN ∆u(𝒌) is PB (0.15) Knowledge Rule base e(k) NB NS Z PS PB ∆ 𝒆(𝒌) NB NB NB NB NS Z NS NB NB NS Z PS Z NB NS Z PS PB PS NS Z PS PB PB PB Z PS PB PB PB Z 0.1 PS 0.85 PB 0.15
  • 46. 46 of 57 ∆𝒖 𝒏 𝒌 = 𝟎. 𝟏 × 𝟎 + 𝟎. 𝟖𝟓 × 𝟏 𝟑 + (𝟎. 𝟏𝟓 × 𝟐 𝟑 ) 𝟎. 𝟏 + 𝟎. 𝟖𝟓 + 𝟎. 𝟏𝟓 ∆𝒖 𝒏 𝒌 = 𝟎. 𝟑𝟓 ∆𝒖 𝒏 𝒌 = σ 𝝁(∆𝒖 𝒏) 𝑪 σ 𝝁(∆𝒖 𝒏) approximate COA 𝒅𝒆𝒇𝒖𝒛𝒛𝒊𝒇𝒊𝒄𝒂𝒕𝒊𝒐𝒏 ∶ ∆𝒖 𝒌 = 𝑮∆𝒖 × ∆𝒖 𝒏 𝒌 = 𝟐 × 𝟎. 𝟑𝟓 = 𝟎. 𝟕 𝒖 𝒌 = 𝒖 𝒌 − 𝟏 + ∆𝒖 𝒌 = 𝟎. 𝟓 + 𝟎. 𝟕 = 𝟏. 𝟐 𝑻𝒉𝒆 𝒂𝒄𝒕𝒖𝒂𝒍 𝒄𝒐𝒏𝒕𝒓𝒐𝒍 𝒐𝒖𝒕𝒑𝒖𝒕 𝜟𝒖 𝒌 𝒐𝒇 𝑭𝑳𝑪: 𝑻𝒉𝒆 𝒄𝒐𝒏𝒕𝒓𝒐𝒍 𝒔𝒊𝒈𝒏𝒂𝒍 𝒖 𝒌 𝒘𝒊𝒍𝒍 𝒃𝒆:
  • 48. 48 of 57 Tank Level Control System 5 cm ? 0 0.25 0.5 0.75 1 1.25 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 𝞵(Level) Liquid level (cm) low okay high 0 0.5 1 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 𝞵(valvecontrol) Valve control signal (%/s) close fast no change open fast Rule 1 : IF level is okay THEN valve is no change Rule 2 : IF level is low THEN valve is open fast Rule 3 : IF level is high THEN valve is close fast Liquid level Valve control signal
  • 49. 49 of 57 𝑻𝒂𝒏𝒌 𝑺𝒚𝒔𝒕𝒆𝒎 𝑺𝒊𝒎𝒖𝒍𝒂𝒕𝒊𝒐𝒏
  • 50. 50 of 57 𝑺𝒖𝒃𝒔𝒚𝒔𝒕𝒆𝒎 𝑽𝒂𝒍𝒗𝒆 𝑺𝒖𝒃𝒔𝒚𝒔𝒕𝒆𝒎 𝑻𝒂𝒏𝒌 ሶℎ = 1 𝐴 𝑞𝑖𝑛 − 𝑎 𝐴 2𝑔ℎ 𝑨 = 𝟎. 𝟒 𝒎 𝟐 𝒂 = 𝟎. 𝟎𝟏𝟐 𝒎 𝟐 𝒒𝒊,𝒎𝒂𝒙 = 𝟐𝟎 𝒍/𝒔 𝐴: 𝑐𝑟𝑜𝑠𝑠 − 𝑠𝑒𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑎𝑛𝑘 𝑎: 𝑐𝑟𝑜𝑠𝑠 − 𝑠𝑒𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑖𝑝𝑒
  • 51. 51 of 57 𝒍𝒊𝒒𝒖𝒊𝒅 𝒍𝒆𝒗𝒆𝒍 𝒗𝒂𝒍𝒗𝒆 𝒄𝒐𝒏𝒕𝒓𝒐𝒍 𝒗𝒂𝒍𝒗𝒆 𝒐𝒑𝒆𝒏𝒊𝒏𝒈
  • 52. 52 of 57 𝑻𝒂𝒏𝒌 𝑺𝒚𝒔𝒕𝒆𝒎 𝑺𝒊𝒎𝒖𝒍𝒂𝒕𝒊𝒐𝒏 With reference
  • 53. 53 of 57 𝒍𝒊𝒒𝒖𝒊𝒅 𝒍𝒆𝒗𝒆𝒍 𝒗𝒂𝒍𝒗𝒆 𝒄𝒐𝒏𝒕𝒓𝒐𝒍 𝒗𝒂𝒍𝒗𝒆 𝒐𝒑𝒆𝒏𝒊𝒏𝒈
  • 54. 54 of 57 With referenceWithout reference
  • 55. 55 of 57 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]. [5] Essam Nabil, “Fuzzy logic control system applications,” March,2019, pp. 1-30 .[presentation]. [6] Essam Nabil, “Tipping problem,” March,2019, pp. 1–18.[presentation].
  • 56. 56 of 57 References [7] “Build Fuzzy Systems Using Fuzzy Logic Designer - MATLAB & Simulink.” [Online]. Available: https://www.mathworks.com/help/fuzzy/building-systems-with-fuzzy-logic-toolbox- software.html. [Accessed: 22-Nov-2019]. [8] “Build Fuzzy Systems at the Command Line - MATLAB & Simulink.” [Online]. Available: https://www.mathworks.com/help/fuzzy/working-from-the-command-line.html. [Accessed: 22-Nov-2019] [9] Essam Nabil, “Tank control system” March,2019, pp. 1–18.[presentation]. [10] Essam Nabil, “PID - Like Fuzzy Logic control” March,2019, pp. 1–39.[presentation].
  • 57. 57 of 57 Thank You Nourhan Selem Salm FUZZY