Answer Happy

r y C(s) G(s) Figure 4

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(OH) a= 0B 1+ sin() 1 - sin(on) -19 con

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