QUESTION 4 [10 marks] Raphael wants you to design a controller for the following familiar feedback system: C(s) P(s) If the last digit in your uOttawa student ID number is in the set (0, 3, 6, 9), then use 800 P(s): (s+5)(s+20) If the last digit is in the set (1,4,7), then use P(s) 216 (s+3)(s+8) Otherwise, the last digit must be in the set (2,5,8), and you should use P(s): 140 (s+2)(s+20) Raphael has provided the Bode plots for these plants on the following 3 pages. (a) Compute the steady-state tracking error for a unit step input when C(s) = 1. (b) Determine the phase margin when C(s) = 1. (c) Design a lag compensator that accomplishes two objectives: (i) decreases by 50% the steady-state error for a step input, and (ii) increases the phase margin by 10 from the value computed in part (b). Be sure to clearly explain which numbers you have read off the Bode plot at each step. Magnitude (dB) -10 -20 -40 -50 -60 -70 0 -20 -40 -60 -80 -100 -120 -140 Phase (deg) Bode plot for P(s) 800 = (s+5)(s+20) 0.1 10 100 1000 -160 -180 0.1 10 Frequency (rad/s) 100 1000 Phase (deg) Magnitude (dB) 10 -30 -40 50 -60 -70 200288 288 28 -80 0.1 0 100 8 8 8 8 8 8 8 -20 -40 -60 -80 -120 -140 -160 216 Bode plot for P(s) : = (s+3)(s+8) 10 100 1000 -180 0.1 10 Frequency (rad/s) 100 1000 Magnitude (dB) Phase (deg) -30 -40 22088 2 8 8 2 8 0 2 2 8 8 2 -20 -40 -60 -80 -100 -120 -140 -160 -180 Bode plot for P(s): 140 = (s+2)(s+20) 0.1 10 100 1000 0.1 10 Frequency (rad/s) 100 1000