Answer Happy

Problem 3 For the closed-loop system shown below R(S) Ge(s) 2s+ s +3 K Gp(s) 1 s(s+2) C(s) S 1 TS + 1 the controller and the plant are operating with transfer functions of G(s) and G (s) respectively. R(8) and C(s) are the reference input and the output respectively. If T = 2, a (2) Obtain the open-loop transfer function, Gor(s). Determine the system type (N) of the open-loop system. b (2) Obtain the closed-loop transfer function, Ger(s). Determine the system order (n) of the closed-loop system. c (2) Determine the proportional gain, integral and derivative time constants of the controller. d (2) Use Routh's criterion find the range of K holding the closed-loop system stable. e (2) Plot the root-loci of the closed-loop poles and find the critical value of the gain K at the verge of instability. Is the critical K consistent with the finiding in part (d)? f(1) If the gain K = 10 trace the root-loci and determine the undamped natural frequencies of the system. 8 (2) If the gain K = 10 plot the Bode diagram of the open-loop system. From the Bode diagram at the frequency of w = 1 find the amplitude and the phase of the sinusoidal transfer funcion. •