Question

VHQ-2, Problem 2: A negative unity feedback system with a forward path transfer function Ka K (Tm8+1) for some scalar gains Ka, K and time constantTm of Write down the system's transfer function. Assuming that K-2, find the values of Ka and Tm so that the system has the fastest response to a step input with no overshoot and a settling time of 0.2 seconds

Answer 1

Given G = K_a K/S (tau_m S + 1) K = 2 t_S = 0.2 sec (consider this to be under 2% band) X (S)----K_a K/S (tau_m S + 1) rightarrow Y (S) Closed loop transfer function, Y (S)/X (S) = G (S)/1 + G (S) + 1 (S) = G (S)/1 + G (S) = K_a K/S (Z_m S + 1)/1 + K_a K/S (Z_m s + 1) \r\n = K_a K /S (tau_m S + 1) + K_a K = K_a K/tau_m S^2 + S + K_a K putting K = 2 Y (S)/X (S) = 2 K_a/tau_m S^2 + S + 2 K_a = 2 K_a/tau_m S^2 + 1/tau_m middot S + 2 K_a/tau_m Comparing denominator with standard 2^nd order equation, S^2 + 2 G w n s + w^_n 2 G w_n = 1/tau_m ---(i) given, setting time, ts = 4/S w_n = 0.2 S w_n = 4/0.2 S w_n = 40/2 S w_n = 20 ---(ii) Putting S w_n = 20 in equation (i), 2 times 20 = 1/tau_0 implies tau_m = 1 /40 S