A process consists of an integrating element operating in parallel with a first-order element (Fig. E6.6). U(s) Y(s) Figure E6.6 (a) What is the order of the overall transfer function, G(s) = Y(s)/U(s)? (b) What is the gain of G(s)? (c) What are the poles of G(s)? Where are they located in the complex s-plane? (d) What are the zeros of G(s)? Where are they located? Under what condition(s) will one or more of the zeros be located in the right-half s-plane? (e) Under what conditions, will this process exhibit a righthalf plane zero? (f) For any input change, what functions of time (response modes) will be included in the response, y(t)? (g) Is the output bounded for any bounded input change, for example, u(t) = M?

Problem 6.6A process consists of an integrating element operating in parallel with a first-order element(Fig. E6.6). U(s) Y(s) Figure E6.6(a) What is the order of the overall transfer function, G(s) = Y(s)/U(s)(b) What is the gain of G(s)(c) What are the poles of G(s) Where are they located in the complex s-plane(d) What are the zeros of G(s) Where are they located Under what condition(s) will one ormore of the zeros be located in the right-half s-plane(e) Under what conditions, will this process exhibit a right half plane zero(f) For any input change, what functions of time (response modes) will be included in theresponse, y(t)(g) Is the output bounded for any bounded input change, for example, u(t) = M Step-by-step solution Step 1 of 9 ^(a)Refer to Figure E6.6 in the textbook for the closed-loop transfer function. K 1 K2 Y (s) = s U(s) + s + 1U(s) K1 K2 Y (s) = ( s + s + 1 U(s) Y (s) K K = s1+ s + 1 U(s) K (s+1)+K s Y (s)= 1 2 U(s) s(s+1) Y (s) K1(s+1)+K 2 U(s) = (s +s) (K +K )s+K Y (s)= 1 2 1 U(s) (s +s) Y (s)Substitute G(s) for U(s) in the above equation. (K +K )s+K G(s) = 1 2 1 ……….(1) (s +s)From equation (1), the maximum exponent on s in denominator is 2.\nTherefore, the order of overall transfer function is .