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homework5_f10_sol

homework5_f10_sol - HOMEWORK 5 SOLUTION 1(10 pts For the...

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HOMEWORK 5 SOLUTION 1. (10 pts) For the following process and the controller transfer functions, determine if the closed-loop 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 closed-loop system is stable. -100 -50 0 50 Magnitude (dB) 10 -2 10 -1 10 0 10 1 10 2 -225 -180 -135 -90 -45 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);
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% 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 closed-loop 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: -4 -2 0 2 4 6 8 10 12 -10 -8 -6 -4 -2 0 2 4 6 8 10 Nyquist Diagram Real Axis Imaginary Axis 2
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k c = 8: -4 -2 0 2 4 6 8 10 -8 -6 -4 -2 0 2 4 6 8 Nyquist Diagram Real Axis Imaginary Axis zoom in: 3
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-1.015 -1.01 -1.005 -1 -0.995 -0.01 -0.005 0 0.005 0.01 0.015 Nyquist Diagram Real Axis Imaginary Axis k c = 5: 4
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-2 -1 0 1 2 3 4 5 6 -5 -4 -3 -2 -1 0 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 % k_c=5; G_c=k_c; % open loop transfer function % G_ol=G_c*G_p; % Nyquist plot % nyquist(G_ol); 3. (20 pts) An unstable exothermic reactor is modeled by the following transfer function: y ( s ) = 1 ( s + 2)( s - 1) m ( s ) 5
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A PI controller is chosen to stabilize the process. Using the Rouths Criterion, find conditions on the controller
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