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Unformatted text preview: MAE143B Homework Set 4 - Solutions Question 1 FPE-N Question 6-25: Suppose that G ( s ) = 25( s + 1) s ( s + 2)( s 2 + 2 s + 16) , in the unity feedback system in the figure. Use matlabs margin command to calculate the gain + _ G ( s ) r t y t margin and phase margin for G ( s ) and, on the basis of the Bode plots, conclude which margin would provide more useful information to the control designer for this system. We use the following matlab commands to draw the Bode plot with margins indicated in the figure below. >> sys=tf(25 * [1 1],conv([1 0],conv([1 2],[1 2 16]))) Transfer function: 25 s + 25--------------------------- s4 + 4 s3 + 20 s2 + 32 s >> margin(sys) 100 80 60 40 20 20 Magnitude (dB) 10 1 10 10 1 10 2 270 225 180 135 90 45 Phase (deg) Bode Diagram Gm = 3.91 dB (at 4.22 rad/sec) , Pm = 101 deg (at 1.08 rad/sec) Frequency (rad/sec) From the figure, it is clear that the small gain margin is a real problem. Indeed, the bode plot indicates that the system frequency response comes and stays quite close to the -1 point. So a small modeling error, which is what the margins are meant to capture, could lead to instability of the unity feedback system. The huge phase margin is rather illusory, since the gain remains very close to 0dB well after the gain crossover point. The corresponding Nyquist diagram below shows better the proximity to the-1 point and the failure of the phase margin to capture this. 0.8 0.6 0.4 0.2 0.2 0.4 1 0.5 0.5 1 1.5 Nyquist Diagram Real Axis Imaginary Axis Question 2 FPE-N Question 6-37: The Nyquist diagrams for two stable, open-loop systems are sketched in the figure below. The proposed operating gain is indicated as K , and arrows indicate increasing frequency. In each case give a rough estimate of the following quantities for the closed-loop (unity feedback) system. (a) Phase margin. The phase margin at the nominal gain K is the angle associated with the intersection of the Nyquist diagram and a circle of radius 1 /K centered at the origin. For System (a), this is roughly 25 . For System (b), it is roughly 60 ....
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- Fall '09