A First Course in FUZZY and NEURAL CONTROL

3.5. COMBINING FUZZY RULES 111 In fuzzy control, the number Wn i=1Ai(x)=Ai^1 (x^1 )∧Ai^2 (x^2 )∧∑∑∑∧Aik(xk) is called thestrengt ...

112 CHAPTER 3. FUZZY LOGIC FOR CONTROL Note that for the set of rules Ri:IfAi 1 andAi 2 and ... andAikthenBi,i=1, 2 ,...,n this ...

3.5. COMBINING FUZZY RULES 113 Example 3.8Take the fuzzy setsAidefinedinEquation3.8andthefunctions f 1 (x)=2+xf 2 (x)=1+x 0 0.2 ...

114 CHAPTER 3. FUZZY LOGIC FOR CONTROL Example 3.9Take the two fuzzy setsA 1 andA 2 of Equation 3.8, and the fuzzy setsC 1 andC ...

3.6. TRUTH TABLES FOR FUZZY LOGIC 115 xyx∧yy^0 x∧y^0 qp 00 010 00 01 000 00 10 011 11 11 100 11 The fact thatqandphave the same ...

116 CHAPTER 3. FUZZY LOGIC FOR CONTROL Observing that the truth values for the expressionsp(x)andq(x)are identical, we can concl ...

3.8. FUZZY RELATIONS 117 Definition 3.11 Afinite set of normal fuzzy subsets{A 1 ,A 2 ,...,An}ofUis a finite fuzzy partitionof a ...

118 CHAPTER 3. FUZZY LOGIC FOR CONTROL Example 3.11IfX={ 2 , 3 , 4 , 6 , 8 }andRis the relation ì(x,y)∈Rif and only ifxdividesy, ...

3.8. FUZZY RELATIONS 119 Example 3.13LetX={x 1 ,x 2 ,x 3 }, R= x 1 x 2 x 3 x 1 x 2 x 3   0. 90. 20. 2 0. 90. 40. 5 1. 00. 61. ...

120 CHAPTER 3. FUZZY LOGIC FOR CONTROL 3.8.2 Orderrelations........................ A partial order≤on a set determines a relati ...

3.9. DEFUZZIFICATION 121 where[a,b]is an interval containing the support ofC. If the support ofCis finite, the computation is z ...

122 CHAPTER 3. FUZZY LOGIC FOR CONTROL 0 y0.5 (^2468101214) y z 0 (1.25) = 7.060 6,heightaboveα=0. 5 3.9.3 Maxcriterionmethod .. ...

3.10. LEVEL CURVES AND ALPHA-CUTS 123 0 y0.5 (^2468101214) y z 0 (1.25) = 6 3.9.5 Middleofmaximamethod.................. Middle ...

124 CHAPTER 3. FUZZY LOGIC FOR CONTROL Anα-cutfor a functionAis the setAαof points for which the value of the function is greate ...

3.10. LEVEL CURVES AND ALPHA-CUTS 125 This procedure is called theextension principle. The extension principle can be viewed as ...

126 CHAPTER 3. FUZZY LOGIC FOR CONTROL Proof.The theorem follows immediately from the equalities below. f(Aα)={f(u):A(x)≥α} = {y ...

3.11. UNIVERSAL APPROXIMATION 127 as theStone-Weierstrass theorem, provides the most general framework for designing function ap ...

128 CHAPTER 3. FUZZY LOGIC FOR CONTROL Theorem 3.8 (Stone-Weierstrass theorem)Let(X,d)be a compact met- ric space. LetH⊆C(X)such ...

3.12. EXERCISES AND PROJECTS 129 ii.(A∩B)(x)=A(x)B(x) iii.(A∩B)(x)=max{A(x)+B(x)− 1 , 0 } (b) union: i.(A∪B)(x)=A(x)∨B(x) ii.(A∪ ...

130 CHAPTER 3. FUZZY LOGIC FOR CONTROL Show that any t-conorm∗satisfies the following. (a)x∗y≥x∨yfor everyx,y∈[0,1] (b) x∗1=1f ...