Answer Happy

Consider again the generic feedback system shown in Figure 4, where G(s) is the plant's transfer function assumed to be stable and its Bode diagram is given in Figure 5 and C(s) is the controller's transfer function. Use the provided Bode plot to answer the following questions. (a) Let C(s)= 1 (unitary static gain), deduce: i) the crossover frequency of the loop transfer function. [2 marks] ii) the phase margin of the loop transfer function. [2 marks] (b) Let C(s)= k (proportional controller): i) find the largest crossover frequency which can be achieved with this type of controller while maintaining a phase margin of at least 30. [3 marks] ii) find the value of k for which the crossover frequency found in question (b)-i) is attained. [3 marks] (c) Let C(s)= kH(s), where k is a static gain, and H(s) is a phase advance compensator in the form: Bs+a H as+B. =

i) find values of k, a and b such that the cross-over frequency of the resulting loop transfer function is 30 rad/sec and the phase margin is at least 30. [10 marks) ii) draw the Bode diagram of the so-obtained controller C(s). You can either use Matlab to have the exact Bode plot or you can draw the line approximation of the Bode diagram. (5 marks] Hint: use the following tuning rules to design the phase advance compensator. In order to tune H(s) to attain desired wn (0 dB frequency) and on (maximum phase advance) choose: sin(on) 1+ sin(n) Q= 0), B 1+ sin(on) 1-sin(n) --04

20 10 0 - 10 -20 Magnitude (dB) -30 -40 -50 -60 -70 -80 -30 -60 -90 Phase (deg) -120 - 150 -180 -210 102 10 10 10° Frequency (rad/s) 10 Figure 5