A First Course in FUZZY and NEURAL CONTROL
3.3. COMBINING FUZZY SETS 91 These operations are determined by theircharacteristic functions.Thechar- acteristic functions for ...
92 CHAPTER 3. FUZZY LOGIC FOR CONTROL We say that each of these binary operations isdualto the other, relative to the complement ...
3.3. COMBINING FUZZY SETS 93 The most widely used t-norm, minimum, is also the largest t-norm ñ that is, x◦y≤x∧y for any t-norm◦ ...
94 CHAPTER 3. FUZZY LOGIC FOR CONTROL The following theorem has its roots in the work of the Norwegian mathe- matician Niels Abe ...
3.3. COMBINING FUZZY SETS 95 The ìadditiveî generatorF:[0,1]→[0,∞]for nilpotent t-norms that is com- monly referred to in the li ...
96 CHAPTER 3. FUZZY LOGIC FOR CONTROL Maximumx∨yis the smallest (and most widely used) t-conorm. The largest t-conorm, called th ...
3.3. COMBINING FUZZY SETS 97 Table 3.1 (a). Strict t-norms,a> 0 ,r> 0 Type t-norm generator Algebraic product xy x Hamache ...
98 CHAPTER 3. FUZZY LOGIC FOR CONTROL Table 3.2 (a). Nilpotent t-norms,a> 0 Type t-norm L-generator/Residual Bounded (x+y−1)∨ ...
3.3. COMBINING FUZZY SETS 99 p(S 1 )andp 2 =p(S 2 )of two statementsS 1 andS 2 , then possible values of p 1 &p 2 =p(S 1 &am ...
100 CHAPTER 3. FUZZY LOGIC FOR CONTROL 7.x^0 =1−loga ° a−a^1 −x+1 ¢ ,a> 0 ,a 6 =1 Gˆdel: x^0 =0ifx 6 =0, 00 =1 (Note(x^0 )^0 ...
3.3. COMBINING FUZZY SETS 101 both hold. These are called theDe Morgan laws.They may or may not hold for other pairs of t-norms ...
102 CHAPTER 3. FUZZY LOGIC FOR CONTROL mean √ xy,harmonicmeanxy/(x+y), and median. All averaging operations can be extended to o ...
3.3. COMBINING FUZZY SETS 103 Definition 3.7The ìcompensatory andîoperatoris ≥Yn i=1 xi ¥ 1 −γ≥ 1 − Yn i=1 (1−xi) ¥γ for someγ∈[ ...
104 CHAPTER 3. FUZZY LOGIC FOR CONTROL 3.4 Sensitivityoffunctions........................ The modeling of fuzzy concepts through ...
3.4. SENSITIVITY OF FUNCTIONS 105 Proof. Using the fact thatxand 1 −xhave the same range of values for x∈[0,1], we have the equa ...
106 CHAPTER 3. FUZZY LOGIC FOR CONTROL membership function: ρA(δ)= _ |x−y|≤δ Ø Ø Øαxc−−ya Ø ...
3.4. SENSITIVITY OF FUNCTIONS 107 Example 3.5Here are some examples for logical connectives on[0,1]. Connective Average sensitiv ...
108 CHAPTER 3. FUZZY LOGIC FOR CONTROL so the triangular functionAwith end points(a,0)and(b,0)andhighpoint (c,α)has average sens ...
3.5. COMBINING FUZZY RULES 109 whereA,A,B,ò Bòare fuzzy sets representing fuzzy concepts. The computation of Bòcan be carried ou ...
110 CHAPTER 3. FUZZY LOGIC FOR CONTROL 3.5.1 Productsoffuzzysets .................... The (Cartesian)product of ordinary setsX 1 ...
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