Exercise 2.7(Fuzzy Logic): There are many concepts that are used in fuzzy logic t useful when studying fuzzy control. The following problems introduce some of the more popular fuzzy logic oncepts that were not treated earlier in the chapter or were treated only briefly (a) The complement(not)of a fuzzy set with a membership function has a membership function given by A()=1-u(). Sketch the complement of the fuzzy set shown in Figure 2.6 on page 30 (b)There are other ways to def ine the"triangular norm"for representing the intersection operation("and")on fuzzy sets, different from the ones introduced in the chapter. Two more are given by defin ing"*as a"bounded zero otherwise). Consider the membership functions shown in Figure 2.9 on page 33. Sketch the membership function for the premiseerror is zero and change-in-error is possmall"when the bounded difference is used to represent this conjunction(premise). Do the same for the case when we use the drastic intersection. Compare these to the case where the minimum operation and the product were used (i.e, plot these also and compare all four) (c) There are other ways to def ine the "triangular co-norm"for representing the union operation(or )on fuz sets,different from the ones introduced in the chapter. Two more are given by defining"e"as a"bounded sum (ie, xe y=min(L,x+y))and"drastic union"(where is xOy when y=0,y when x=0, and one otherwise). Consider the membership functions shown in Figure 2.9 on page 33. Sketch the membership function for"error is zero or change-in-error is possmall"when the bounded sum is used. Do the same for the case when we use the drastic union. Compare these to the case where the maximum operation and the algebraic sum were used(ie, plot these also and compare all four). Exercise 2.7 0.9 0.7 0.4 0.4 0.1 u(x) u(x)
Exercise 2.7 (Fuzzy Logic): There are many concepts that are used in fuzzy logic that sometimes become useful when studying fuzzy control. The following problems introduce some of the more popular fuzzy logic concepts that were not treated earlier in the chapter or were treated only briefly. (a) The complement (“not”) of a fuzzy set with a membership function has a membership function given by ( x) = −1 ( x) . Sketch the complement of the fuzzy set shown in Figure 2.6 on page 30. (b) There are other ways to define the “triangular norm” for representing the intersection operation (“and”) on fuzzy sets, different from the ones introduced in the chapter. Two more are given by defining “*” as a “bounded difference” (i.e.,x*y=max{0,x+y-1}) and “drastic intersection” (where x * y is x when y = 1, y when x = 1 , and zero otherwise). Consider the membership functions shown in Figure 2.9 on page 33. Sketch the membership function for the premise “error is zero and change-in-error is possmall” when the bounded difference is used to represent this conjunction (premise). Do the same for the case when we use the drastic intersection. Compare these to the case where the minimum operation and the product were used (i.e., plot these also and compare all four). (c) There are other ways to define the “triangular co-norm” for representing the union operation (“or”) on fuzzy sets, different from the ones introduced in the chapter. Two more are given by defining “⊕” as a “bounded sum” (i.e., x y x y = + min 1, ) and “drastic union” (where is x y when y = 0, y when x = 0, and one otherwise). Consider the membership functions shown in Figure 2.9 on page 33. Sketch the membership function for “error is zero or change-in-error is possmall” when the bounded sum is used. Do the same for the case when we use the drastic union. Compare these to the case where the maximum operation and the algebraic sum were used (i.e., plot these also and compare all four).Exercise 2.7: (a)
(b) 使用“ minimum”时 08 0.6 0.4 0.2 05 change in error error 0.8 0.6 0.4 0. 0.2 -0.5 change in error error
（b）、 使用“minimum”时 使用“product”时：
使用“ bounded difference”时 08 0.6 0.4 0.2 05 change in error error 使用 drastic intersection”时 0.8 0.6 0.4 0. 0.2 -0.5 change in error error
使用“ bounded difference”时： 使用”drastic intersection”时：
(c)、 使用“ maximum时 08 0.6 0.4 0.2 05 05 change in error error 使用 algebraic sum,时 0.8 0.6 0.4 0. 0.2 change in error error
（c）、 使用“maximum”时： 使用”algebraic sum”时：
使用“ bounded sum时 08 0.6 0.4 0.2 05 05 change in error error 使用“ drastic union时 0.8 0.6 0.4 0. 0.2 -0.5 change in error error
使用“bounded sum”时： 使用“drastic union”时