lecture_29

# lecture_29 - ME 421 Mechanical Dynamics and Control Fall...

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ME 421 Mechanical Dynamics and Control Fall 2006, Lecture 29 Today’s topics (Secs. 11-6) • Lead-Lag Compensator Design Using Root- Locus;

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How to determine the location of a pole?

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Design Example:

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Design Example:
A few comments about lead-lag Compensator design • It may not always lead to the desired performance; • Iterations might be required: – Desired pole location; – Choice of the low-frequency gain improvement; – Choice of the lead compensator’s zero; – Choice of the lag compensator’s zero; • Tuning can be applied to further improve the performance;

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I:\CourseWork\ME421_Fall06\Lecture\Lecture_28.m Page 1 December 5, 2006 9:29:11 AM %=================================== % ME 421 Lecture 29 Fall 2006 % Lead-lag compensator design example % Qingze Zou %=================================== % Define the plant dynamics; Num_0 = 0.8*[1 3]; Den_0_a = [1 2]; Den_0_b = [1 0.2 5]; G_o = tf(Num_0, conv(Den_0_a, Den_0_b)); % Define the desired damping ratio and desired natural frequency % (i.e., the pair of desired dominant close poles); xi = 0.9; omega_n = 8; % Desired dominant pole: p_des = -xi*omega_n + sqrt(1-xi^2)*omega_n*j; % Plot the step response of the open-loop system and the root locus; figure(1), step(G_o); figure(2), rlocus(G_o), % Mark the location of the desired damping ratio line and natural % frequency; figure(2), sgrid(xi, omega_n); % Design the lead-compensator first. % First set the zero of the lead compensator to cancel one of the % plant's pole (NOTE must be a real pole); num_l = Den_0_a; % Second, compute the needed phase lead from the lead compensator; G_l = tf(num_l, 1); G_l_p = evalfr(G_l, p_des); Phase_z = angle(p_des + 3); Phase_p = angle(p_des^2 + 0.2*p_des + 5); % This line applied the angle condition; angle_p = Phase_z - Phase_p - pi; % Determine the pole of the lead compensator; p = imag(p_des)/tan(angle_p) - real(p_des); % Determine the lead compensator with a unit gain; G_lead = tf(num_l, [1 p]); % Next, find the gain of the lead compensator by the gain condition; G_oc = G_lead*G_o; G_oc = tf(Num_0, conv([1 p], Den_0_b)); figure(3), rlocus(G_oc); sgrid(0.6, 2);
I:\CourseWork\ME421_Fall06\Lecture\Lecture_28.m Page 2 December 5, 2006 9:29:11 AM % Find the gain of the lead compensator; K_l = 1/abs(evalfr(G_oc, p_des)); % Finally, we get our lead compensator; G_lead = K_l*G_lead; % Form the close-loop system with the lead compensator; G_oc = G_lead*G_o; G_cl_1 = feedback(G_oc, 1); % Design the lag compensator; % We want to increase the DC-Gain of the feedforward path by 10 times; K_lag = 10; % Set the pole of the lag compensator to close to the zero; p_lag = 0.1; % Set the zero according to the desired gain increase; z_lag = K_lag*p_lag; % Form the lag compensator and the close-loop system with both lead % and lag compensator;

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lecture_29 - ME 421 Mechanical Dynamics and Control Fall...

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