Question

PROBLEM2 The unit step response of two systems, T(s) and T,(s), are shown in Figure P2.1 below T,(s) T(s) 0 0102 0.3 Tefseconds 08 07 0.8 0. Figure P2.1: Unit step responses of linear second-order control systems T(s) and T,(s) Assuming that T(s) and T,(6) are linear second-order prototypical control systems, do the following (a) Determine the closed-loop transfer function 7(s). (b) Determine the closed-loop transfer function T,(s). (7 points (3 points)

Answer 1

For T_2 (s), c (infinity) = steady state value = 5 = system gain (k) c (t_p) = 7.6 = maximum value at t = t_p (peak time) % M_p = e^-E pi/squareroot 1 - E^2 times 100 % (For a II^nd order system) = e (E_p) - c (infinity)/c(infinity) times 100 % = 52 % Therefore e^-E pi/squareroot 1 - E^2 = 0.52 - E pi/squareroot 1 - E^2 = l_n (0.52) rightarrow E^2 pi^2 = [l_n (0.52)]^2 (1 - E^2) E^2 (pi^2 + (ln (0.52))^2) = (ln (0.52))^2 rightarrow E = 0.203 k_p = peak time = pi/w_d = 0.1 sec rightarrow w_d = pi/0.1 = 10 pi rad/sec but w_d = w_n squareroot 1 - E^2 rightarrow w_n = w_d/squareroot 1 - E^2 = 32 rad/sec Therefore T_2 (s) = k . w_n^2/s^2 + 2 {w_n} + w_n^2 (standard second order system) T_2 (s) = 5 . (1029)/s^2 + 4185 + 1029 For T_1 (s)