Fuzzy Logic

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Fuzzy logic ‎Introduction 3‏ ععمعنعأما لإددنط ‎Aleksandar Rakié ‏تور تقد لقن [-و‎ ‎2 

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Contents = Mamdani Fuzzy Inference Fuzzification of the input variables Rule evaluation Aggregation of the rule outputs Defuzzification = Sugeno Fuzzy Inference = Mamdani or Sugeno? 

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Mamdani Fuzzy Inference = The most commonly used fuzzy inference technique is the so-called ‏ری انیت ترا‎ = Jn 1975, Professor Ebrahim Mamdani of London University built one of the first fuzzy systems to control a steam engine and boiler combination. He applied a set of fuzzy rules supplied by experienced human operators. = The Mamdani-style fuzzy inference process is performed in four steps: Fuzzification of the input variables ‏از‎ Is (ol (ali-ic-tite)} 0 eten) Defuzzification. 

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Mamdani Fuzzy Inference We examine a simple two-input one-output problem that includes ۲۱۲66 6۶۰ وز2 7۴۱1 ‎sieee‏ ف قم ذز ول ۱۱ ‎THEN zis C2‏ 2 هلام ول ۳ 3 وا2 7۴1 1ه وذ»ا عا ۱:۱۳ Real-life example for these kinds of rules: Rule: 1 IF project funding is adequate OR project staffing is small THEN _risk 1907 Rule: 2 IF project funding is marginal AND project staffingis large THEN risk irom Rule: 3 IF project funding is inadequate 0 

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Step 1: Fuzzification = The first step is to take the crisp inputs, x1 and y1 (project funding and jroject staffing), and determine the degree to which these inputs belong to each of the appropriate fuzzy 55. 

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Step 2: Rule Evaluation = The second step is to take the fuzzified inputs, Heo O52) - 024, ‏را‎ 07, and apply them to the antecedents of the fuzzy rules. Ifa given fuzzy rule has multiple antecedents, the fuzzy operator (AND or OR) is used to obtain a single number that represents the result of the antecedent evaluation. RECALL: To evaluate the disjunction of the rule antecedents, we use the OR fuzzy operation. Typically, fuzzy expert systems make use of the classical fuzzy operation union: )6 ال تير ال Similarly, in order to evaluate the conjunction of the rule antecedents, we apply the AND fuzzy operation intersection: )وم ,لد)يه] ضام ع ناوي 

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Step 2: Rule Evaluation لاما ۳ 0 ۱۱ مس و —{— Rule1: IF xis A3(0.0) OR yis Bl (0.1) THEN zis C1 (0.1) تا( 7 zis C2(0.2) Rule3: IF xis Al (0.5) THEN zis C3 (0.5) 

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Step 2: Rule Evaluation ™ Now the result of the antecedent evaluationegree of can be applied to the membership functiorMembership 01 16 0550 6. ‏د‎ ‎" 116 ۱۱۵5] 60۲۲۱۳۲۱۵۲۱ ۲۱6۵۱/۱۵۵ ۱5 ۱۵ 1 ry consequent membership function at the level of the antecedent truth. This method is called 0 » ‏2لممعناء وز مماغعدنة متطىءطصعده عط 6ه دمغ عط ععمزك‎ Pent Mel feet eitezayare els lun Oa = However, Clipping is still often preferred because 0 ۱ ‏15هجم‎ 003665, clipping Generates an aggregated output surface that is 625167 ۵ ۰ ۳ ۱ ie at: ea 2 مر ی رز ی كك ‎the original shape of the fuzzy set.‏ 1 ‎consequent is adjusted by multiplying all its‏ ‎It nim It)‏ ا یر ‎antecedent. a‏ سب 0.0 ‎This method, which generally loses less‏ ® 2 ی یزرو yeu scaling 

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Step 3: Aggregation of the Rule Outputs = Aggregation is the process of unification of the outputs 0۲ ۱ ۱۰ ™ We take the membership functions of all rule consequents previously clipped or scaled and combine them into a single fuzzy set. = The input of the aggregation process is the list of clipped or scaled consequent membership functions, and the output is one fuzzy set for each output variable. 8 

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Step 4: Defuzzification The last step in the fuzzy inference process is defuzzification Fuzziness helps us to evaluate the rules, but the final output of a fuzzy system has to be a crisp number. The input for the defuzzification process is the aggregate output fuzzy set and the output is a single number. There are several defuzzification methods, but probably the most popular one is the centroid technique. It finds the point where a vertical line would slice the aggregate set into two equal masses. Mathematically this centre of gravity (COG) can be expressed as: 4 ص99 ‎ns‏ ‏۳7 ‏داز ‏3 

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Step 4: Defuzzification = Centroid defuzzification method finds a point representing the centre of gravity of the aggregated fuzzy set A, on the interval [a, 62 = A reasonable estimate can be obtained by calculating it over a sample of ooints. COG =0+10+ 20) x0.1+ (30+ 40+ 50+ 60) x0.2+ (70+ 804 90+ 100) x0.5 0.1+ 0.1+ 0.14 0.2+ 0.2+ 0.2+0.2+ 0.5+ 0.5+0.5+ 0.5 =67.4 0. 

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Sugeno Fuzzy Inference = Mamdani-style inference, as we have just seen, requires us to find the centroid of a two-dimensional shape by integrating across a continuously varying function. In general, this process is not computationally efficient. ™ Michio Sugeno suggested to use a single spike, a singleton, as the membership function of the rule consequent. = A singleton, or more precisely a fuzzy singleton, is a fuzzy set with a membership function that is unity at a single particular point on the universe of discourse and zero everywhere else. 

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Sugeno Fuzzy Inference Sugeno-style fuzzy inference is very similar to the Mamdani method. Sugeno changed only a rule consequent: instead of a fuzzy set, he used a mathematical function of the input variable. SINC ene ates yt mitre aye ae 0 xisA AND yisB THEN zis Ax, y) للا ‎ies‏ بير 0 إلا همق )ل كعك نامع 5أل 06 مداع لأمنا ده كأع5 22د عرق 8 لمق م 2 :لاع ناتاعءمععر ‎vaeinaulelter!) isleonn‏ 2 0 The most commonly used zero-order Sugeno fuzzy model applies fuzzy rules in the following form: ۴ ‏بر‎ 15 ۸ AND yis B THEN zis k ۱۷۱۶۲۶ 15 3 665181. لمع نوع 5مهء ااه ی ما ‎membership functions are represented by singleton spikes.‏ cy 

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Sugeno Rule Evaluation 01 ۸ 7 ۳ A 7 1: IF xis A3 (0.0) yis Bi (0.1) wa 1 x 0 ‏بر‎ ۲ ۳ : IF xis A2 (0.2) AND yis B2 (0.7) zis k2 (0.2) 41 ob BZ xl xX zis K3 (0.5) ule 3: IF xis Al (0.5) 

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Sugeno Aggregation and Defuzzification re MGA b-a ikea @.0) bd. AL a1 0.) pd.) 5 0,120+ 0,250+ 0 - ‏ا ل‎ aR Tie) 0 6 ص 

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Mamdani or Sugeno? = Mamdani method is widely accepted for capturing expert. knowledge. It allows us to describe the expertise in more intuitive, more human-like manner, However, Mamdani- type fuzzy inference entails a substantial computational burden. ™ On the other hand, Sugeno method is computationally effective and works well with optimization and adaptive techniques, which makes it very attractive in control problems, particularly for dynamic nonlinear systems. 

Fuzzy logic Introduction 3 Fuzzy Inference Aleksandar Rakić rakic@etf.rs Contents  Mamdani Fuzzy Inference • • • •   Fuzzification of the input variables Rule evaluation Aggregation of the rule outputs Defuzzification Sugeno Fuzzy Inference Mamdani or Sugeno? 2 Mamdani Fuzzy Inference  The most commonly used fuzzy inference technique is the so-called Mamdani method.  In 1975, Professor Ebrahim Mamdani of London University built one of the first fuzzy systems to control a steam engine and boiler combination. He applied a set of fuzzy rules supplied by experienced human operators.  The Mamdani-style fuzzy inference process is performed in four steps: 1. Fuzzification of the input variables 2. Rule evaluation (inference) 3. Aggregation of the rule outputs (composition) 4. Defuzzification. 3 Mamdani Fuzzy Inference We examine a simple two-input one-output problem that includes three rules: Rule: 1 IF x is A3 Rule: 2 IF x is A2 Rule: 3 IF x is A1 OR AND y is B1 y is B2 THEN THEN THEN z is C1 z is C2 z is C3 Real-life example for these kinds of rules: Rule: 1 IF project_funding is adequate OR project_staffing is small THEN risk is low Rule: 2 IF project_funding is marginal AND project_staffing is large THEN risk is normal Rule: 3 IF project_funding is inadequate THEN risk is high 4 Step 1: Fuzzification  The first step is to take the crisp inputs, x1 and y1 (project funding and project staffing), and determine the degree to which these inputs belong to each of the appropriate fuzzy sets. Crisp Input x1 1 0.5 0.2 0 A1 A2 x1  (x =A1) =0.5  =0.2 (x =A2) Crisp Input y1 1 B1 0.7 A3 X B2 0.1 0 y1  (y =B1) =0.1  =0.7 Y (y =B2) 5 Step 2: Rule Evaluation   The second step is to take the fuzzified inputs, (x=A1) = 0.5, (x=A2) = 0.2, (y=B1) = 0.1 and (y=B2) = 0.7, and apply them to the antecedents of the fuzzy rules. If a given fuzzy rule has multiple antecedents, the fuzzy operator (AND or OR) is used to obtain a single number that represents the result of the antecedent evaluation. RECALL: To evaluate the disjunction of the rule antecedents, we use the OR fuzzy operation. Typically, fuzzy expert systems make use of the classical fuzzy operation union: AB(x) = max [A(x), B(x)] Similarly, in order to evaluate the conjunction of the rule antecedents, we apply the AND fuzzy operation intersection: AB(x) = min [A(x), B(x)] 6 Step 2: Rule Evaluation 1 1 A3 1 B1 0.1 0.0 0 x1 0 X Rule 1: IF x is A3 (0.0) OR 1 y1 Y y is B1 (0.1) 1 A2 0 x1 y1 Rule 2: IF x is A2 (0.2) AND y is B2 (0.7) 1 0 A1 x1 Rule 3: IF x is A1 (0.5) Z z is C1 (0.1) Y AND 0.2 (min) 0 C1 C2 THEN C3 Z z is C2 (0.2) THEN C2 0 X C3 0 1 0.5 C1 0.5 C2 1 B2 0 0.1 THEN 0.7 0.2 X OR (max) C1 C3 Z z is C3 (0.5) 7 Step 2: Rule Evaluation   Now the result of the antecedent evaluation Degree of Membership can be applied to the membership function 1.0 of the consequent. The most common method is to cut the consequent membership function at the level of the antecedent truth. This method is called clipping (alpha-cut). 0.2 Since the top of the membership function is sliced, the clipped fuzzy set loses some information. 0.0  However, clipping is still often preferred because it involves less complex and faster mathematics, and generates an aggregated output surface that is Degree of easier to defuzzify. Membership While clipping is a frequently used method, 1.0 C2   scaling offers a better approach for preserving the original shape of the fuzzy set.   The original membership function of the rule consequent is adjusted by multiplying all its membership degrees by the truth value of the rule0.2 antecedent. This method, which generally loses less 0.0 information, can be very useful in fuzzy expert systems. clipping Z C2 scaling Z 8 Step 3: Aggregation of the Rule Outputs    Aggregation is the process of unification of the outputs of all rules. We take the membership functions of all rule consequents previously clipped or scaled and combine them into a single fuzzy set. The input of the aggregation process is the list of clipped or scaled consequent membership functions, and the output is one fuzzy set for each output variable. 1 0.1 0 1 C1 1 C2 0.5 C3 0.2 Z z is C1 (0.1) 0 Z z is C2 (0.2) 0 0.5 0.1 Z z is C3 (0.5) 0.2 Z 0  9 Step 4: Defuzzification  The last step in the fuzzy inference process is defuzzification.  Fuzziness helps us to evaluate the rules, but the final output of a fuzzy system has to be a crisp number.  The input for the defuzzification process is the aggregate output fuzzy set and the output is a single number.  There are several defuzzification methods, but probably the most popular one is the centroid technique. It finds the point where a vertical line would slice the aggregate set into two equal masses. Mathematically this centre of gravity (COG) can be expressed as: b  A x xdx COG  ab  A x dx a 10 Step 4: Defuzzification   Centroid defuzzification method finds a point representing the centre of gravity of the aggregated fuzzy set A, on the interval [a, b ]. A reasonable estimate can be obtained by calculating it over a sample of points. Degree of Membership 1.0 0.8 0.6 0.4 0.2 0.0 0 10 20 30 40 50 60 70 67.4 COG  80 90 100 Z (0 10 20) 0.1 (30 40 50 60) 0.2 (70 80 90 100) 0.5 67.4 0.1 0.1 0.1 0.2 0.2 0.2 0.2 0.5 0.5 0.5 0.5 11 Sugeno Fuzzy Inference  Mamdani-style inference, as we have just seen, requires us to find the centroid of a two-dimensional shape by integrating across a continuously varying function. In general, this process is not computationally efficient.  Michio Sugeno suggested to use a single spike, a singleton, as the membership function of the rule consequent.  A singleton, or more precisely a fuzzy singleton, is a fuzzy set with a membership function that is unity at a single particular point on the universe of discourse and zero everywhere else. 12 Sugeno Fuzzy Inference    Sugeno-style fuzzy inference is very similar to the Mamdani method. Sugeno changed only a rule consequent: instead of a fuzzy set, he used a mathematical function of the input variable. The format of the Sugeno-style fuzzy rule is IF x is A AND where:       y is B THEN z is f(x, y) x, y and z are linguistic variables; A and B are fuzzy sets on universe of discourses X and Y, respectively; f (x, y) is a mathematical function. The most commonly used zero-order Sugeno fuzzy model applies fuzzy rules in the following form: IF x is A AND y is B THEN z is k where k is a constant. In this case, the output of each fuzzy rule is constant and all consequent membership functions are represented by singleton spikes. 13 Sugeno Rule Evaluation 1  1 1  0.0  0 x1 x1 X  1 1 B1  B1 A3  A3 0   0 y1  0.1OR 0.1  OR 0.1    ) (max ) (max 0  y1 Y Y k01 k1  IF  xis  OR  Rule 1: Rule 1: IF A3xis(0.0) A3 (0.0) OR y isyis B1B1(0.1) (0.1) 1  1 1 A2  A2 0  0 x1 x1 0.2 X  X0  1  0.7 0.7 0 y1 y1Y  x  2 x (0.2)  AND y  is Rule 2: IFRule is2:AIF AND is A2 (0.2) yisBB2 (0.7) 1 0  1 A1   A1 0 x1 x1  0.5 X  0.5 X   3:  Rule3: Rule IF x isIFAx1is A (0.5) 1 (0.5)  THEN THEN 1  Z  Z  Z  is zis z k 1 k1 (0.1) (0.1) 1  B2  B2AND 0.2 0.2  AND 0.2  1 0.1 0.0 X0      (min) (min) 0 Y  0 k2 k2 Z THEN THEN zis zkis2k2(0.2) (0.2) 1  0.5  1 0.5 0  0 THEN THEN k3Z k3 Z zis zkis3k3(0.5) (0.5) 14 Sugeno Aggregation and Defuzzification 1 1 0.1 0 0.2 k1 Z z is k1 (0.1) 0 k2 Z z is k2 (0.2) 1 1 0.5 0.5 0 0.1 0 k3 Z 0.2 k2 k1 k3 Z  z is k3 (0.5) COG becomes Weighted Average (WA) 0 z1 Z Crisp Output z1 (k1) k1 (k2) k2 (k3) k3 0.120 0.250 0.580 WA   65 (k1)  (k2)  (k3) 0.1 0.2 0.5 15 Mamdani or Sugeno?  Mamdani method is widely accepted for capturing expert knowledge. It allows us to describe the expertise in more intuitive, more human-like manner. However, Mamdanitype fuzzy inference entails a substantial computational burden.  On the other hand, Sugeno method is computationally effective and works well with optimization and adaptive techniques, which makes it very attractive in control problems, particularly for dynamic nonlinear systems. 16