Exercise 9.1. Derive the compact formula for the fuzzy systems with fuzzy rule base (7.1), Zadeh inference engine (7.31), singleton fuzzifier (8.1), and center average defuzzifier (8.18) Attachment Ru) : IF a is A and.. and n is A THEN y is B' * Zadeh Inference Engine: In Zadeh inference engine, we use: (i) individual- rule based inference with intersection combination (7.22), (ii) Zadeh implica- tion (5.25), and (iii) min for all the t-norm operators. Specifically, from (7.22), (7.20), (5.25), and (7.13) we obtain ' (y) = min{sup min[nA, (X), na (min( Al ( 1), , l-M (zn), (y) ), l-1 xeU (7.31) Singleton fuzzifier: The singleton fuzzifier maps a real-valued point x* E U into a fuzzy singleton A' in U, which has membership value 1 at x and 0 at all other points in U; that is, if x x (x) 0 otherwise 8.2.2 Center Average Defuzzifier Because the fuzzy set B' is the union or intersection of M fuzzy sets, a good approx- imation of (8.15) is the weighted average of the centers of the M fuzzy sets, with the weights equal the heights of the corresponding fuzzy sets. Specifically, let g' be the center of the l'th fuzzy set and wi be its height, the center average defuzzifier determines y* as (8.18) WI Exercise 9.1. Derive the compact formula for the fuzzy systems with fuzzy rule base (7.1), Zadeh inference engine (7.31), singleton fuzzifier (8.1), and center average defuzzifier (8.18) Attachment Ru) : IF a is A and.. and n is A THEN y is B' * Zadeh Inference Engine: In Zadeh inference engine, we use: (i) individual- rule based inference with intersection combination (7.22), (ii) Zadeh implica- tion (5.25), and (iii) min for all the t-norm operators. Specifically, from (7.22), (7.20), (5.25), and (7.13) we obtain ' (y) = min{sup min[nA, (X), na (min( Al ( 1), , l-M (zn), (y) ), l-1 xeU (7.31) Singleton fuzzifier: The singleton fuzzifier maps a real-valued point x* E U into a fuzzy singleton A' in U, which has membership value 1 at x and 0 at all other points in U; that is, if x x (x) 0 otherwise 8.2.2 Center Average Defuzzifier Because the fuzzy set B' is the union or intersection of M fuzzy sets, a good approx- imation of (8.15) is the weighted average of the centers of the M fuzzy sets, with the weights equal the heights of the corresponding fuzzy sets. Specifically, let g' be the center of the l'th fuzzy set and wi be its height, the center average defuzzifier determines y* as (8.18) WI