Exercise 2.5( Fuzzy Sets): There are many concepts that are used in fuzzy sets that sometimes become useful when studying fuzzy control. The following problems introduce some of the more popular fuzzy set concepts that were not treated earlier in the chapter (a)The"support"of a fuzzy set with membership function u(x)is the(crisp)set of all points x the universe of discourse such that u(x)>0 and the"a-cut?"is the(crisp)set of all points on the universe of e such that u(x)>a. What is the support and 0.5-cut for the fuzzy set shown in Figure 2.6 on page (b)The"height"of a fuzzy set with membership function u() is the highest value that u(x)reaches on the universe of discourse on which it is def ined. A fuzzy set is said to be"normal" if its height is equal to one What is the height of the fuzzy set shown in Figure 2.6 on page 30? Is it normal? Give an example of a fuzzy set that is not normal (c)Afuzzy set with membership function u(x)where the universe of discourse is the set of real numbers is said to be "convexif and only if u(ax,+(1-2)x2)>min u(x),u(*2)) for all x, and x, and all xe[0, 1. Note that just because a fuzy set is said to be convex does not mean that its embership function is a convex function in the usual sense. Prove that the fuzzy set shown in Figure 2.6 on page 30 is convex. Prove that the gaussian membership function is not convex. Give an example(besides the fuzzy set with a gaussian membership function) of a fuzzy set that is not convex (d )A linguistic"hedge"is a modifier to a linguistic value such as"very"or "more or less. When we use linguistic hedges for linguistic values that already have membership functions ve can simply modify these membership It the modified ling page 30. Suppose that we obtain the membership function for"error is very possmall"from the one for "possmall by squaring the membership values(i.e. A,venpossmall=(Apassmall ) ). Sketch the membership function for"error is very possmal l For "error is more or less possmalr"we could use #moreorlesposmal=u, Sketch the membership function for"error is more or less possmall. (a)、0<e(1)< x1<x2∈[D,上时 A1(x)<2(x)
Exercise 2.5 (Fuzzy Sets): There are many concepts that are used in fuzzy sets that sometimes become useful when studying fuzzy control. The following problems introduce some of the more popular fuzzy set concepts that were not treated earlier in the chapter. (a) The “support” of a fuzzy set with membership function ( x) is the (crisp) set of all points x on the universe of discourse such that ( x) 0 and the “ -cut” is the (crisp) set of all points on the universe of discourse such that ( x) . What is the support and 0.5-cut for the fuzzy set shown in Figure 2.6 on page 30? (b) The “height” of a fuzzy set with membership function ( x) is the highest value that ( x) reaches on the universe of discourse on which it is defined. A fuzzy set is said to be “normal” if its height is equal to one. What is the height of the fuzzy set shown in Figure 2.6 on page 30? Is it normal? Give an example of a fuzzy set that is not normal. (c) A fuzzy set with membership function ( x) where the universe of discourse is the set of real numbers is said to be “convex” if and only if (x1 2 1 2 + − (1 , ) x x x ) min ( ) ( ) (2.29) for all 1 x and 2 x and all 0,1 . Note that just because a fuzzy set is said to be convex does not mean that its membership function is a convex function in the usual sense. Prove that the fuzzy set shown in Figure 2.6 on page 30 is convex. Prove that the Gaussian membership function is not convex. Give an example (besides the fuzzy set with a Gaussian membership function) of a fuzzy set that is not convex. (d) A linguistic “hedge” is a modifier to a linguistic value such as “very” or “more or less.” When we use linguistic hedges for linguistic values that already have membership functions, we can simply modify these membership functions so that they represent the modified linguistic values. Consider the membership function in Figure 2.6 on page 30. Suppose that we obtain the membership function for “error is very possmall” from the one for “possmall” by squaring the membership values (i.e. 2 ( ) verypossmall possmall = ,) . Sketch the membership function for “error is very ) possmall.” For “error is more or less possmall” we could use moreorlesspossmall possmall = . Sketch the membership function for “error is more or less possmall.” Exercise 2.5.(Fuzzy Set) (a)、 0 ( ) 2 e t , 3 ( ) 8 8 e t . (b)、1 , yes , (c)、证明：如图所示： 1 2 [0, ]; 4 x x 时 1 2 ( ) ( ) x x
min[1(x)22(x)=1(x) (x1+(1-A)x2)=(Ax+(1-A)x+(1-4)×Q)=(x1+(1-4)×a)a=x2-x1>0 又∵λ∈[0,1 (x1+(1-A)x2)≥(x1) (x1+(1-)x2)≥mn[41(x),2(x) 同理:当x1<x2∈[]时 1(x)>2(x) min[1(x),2(x)]=2(x) (x1+(1-1)x2)=(x2+(1-)x2+xa)=(x2+1×a)2a=x2-x1>0 又∵λ∈[0,1 (x1+(1-A)x2)≥(x2) (x1+(1-A)x2)≥mn[1(x)22(x) (d)、 0.5 0.4
min[ ( ), ( )] ( ); 1 2 1 = x x x 1 2 1 1 1 2 1 + − = + − + − = + − = − ( (1 ) ) ( (1 ) (1 ) ) ( (1 ) ); 0 x x x x x x x 又 [0,1] 1 2 1 + − ( (1 ) ) ( ) x x x 1 2 1 2 + − ( (1 ) ) min[ ( ), ( )] x x x x 同理：当 1 2 [ , ]; 4 2 x x 时 1 2 ( ) ( ) x x min[ ( ), ( )] ( ); 1 2 2 x x x = 1 2 2 2 2 2 1 + − = + − + = + = − ( (1 ) ) ( (1 ) ) ( ); 0 x x x x x x x 又 [0,1] 1 2 2 + − ( (1 ) ) ( ) x x x 1 2 1 2 + − ( (1 ) ) min[ ( ), ( )] x x x x (d)、 2 ( ) veryprossmall prossmall = ：
error is ve 07 0.6 0.4 0.3 0.2 0.1 ror rad Arossmall: serror is more or less nossmal 09 08 0.7 0.5 0.5 0.4 0.3 0.2
moreorlessprossmall prossmall = ：