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Unformatted text preview: HOMEWORK 5 SOLUTION 1. (10 pts) For the following process and the controller transfer functions, determine if the closedloop system exhibits stability using the Bode Stability Criterion, g p ( s ) = 2 s + 1 3 s 3 + 2 s 2 + 4 s + 1 ; g c ( s ) = 0 . 1 1 + 1 s ¶ If the system is stable, determine the Gain Margin and the Phase Margin. Show your work clearly. Solution: Based on the Bode diagram, the closedloop system is stable.10050 50 Magnitude (dB) 102 101 10 10 1 10 22251801359045 Phase (deg) Bode Diagram Gm = 18.2 dB (at 1.31 rad/sec) , Pm = 85.1 deg (at 0.0971 rad/sec) Frequency (rad/sec) Matlab code: s=tf(’s’); % process transfer function % G_p=(2*s+1)/(3*s^3+2*s^2+4*s+1); % controller % G_c=0.1*(1+1/s); % open loop transfer function % G_ol=G_c*G_p; % get gain margin and phase margin % margin(G_ol); 2. (10 pts) The following transfer function is given for a process: g p ( s ) = 1 2 s 3 + 6 s 2 + 3 s + 1 If a P controller is used with a gain of 10, show, using Nyquist Stability Criterion, that the system is unstable. What value of the gain would make this system stable? Show, again, using the Nyquist Stability Criterion. Solution: Based on the Bode diagrams, the closedloop system is unstable with a gain of k c = 10, since the Nyquist contour of G ol encircles the critical point ( 1 , 0). A gain of k c = 8 would make the system stable. k c = 10:42 2 4 6 8 10 12108642 2 4 6 8 10 Nyquist Diagram Real Axis Imaginary Axis 2 k c = 8:42 2 4 6 8 108642 2 4 6 8 Nyquist Diagram Real Axis Imaginary Axis zoom in: 31.0151.011.00510.9950.010.005 0.005 0.01 0.015 Nyquist Diagram Real Axis Imaginary Axis k c = 5: 421 1 2 3 4 5 654321 1 2 3 4 5 Nyquist Diagram Real Axis Imaginary Axis Matlab code: s=tf(’s’); % process transfer function % G_p=1/(2*s^3+6*s^2+3*s+1); % controller %...
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This note was uploaded on 04/01/2011 for the course ECH 157 taught by Professor Palagozu during the Spring '08 term at UC Davis.
 Spring '08
 PALAGOZU

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