A process whose dynamics is described by a double integrator is controlled by an ideal PD...
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A process whose dynamics is described by a double integrator is controlled by an ideal PD controller with the transfer function C(s) = kas + kp, where the gains are ka = 25wo and kp = wf, and 0 < < 1. 1 (a) What is the loop transfer function (L(s) = P(s)C(s)) for this system? (b) The gain margin is 9m = where |L(iwpc)|' Wpc is the phase crossover frequency (the frequency at which the Nyquist curve crosses the nega- tive real axis: ImL(iwpc) = 0 and ReL(iwpc) <0). When the Nyquist curve never intersects the (strictly) negative real axis, we define the gain margin to be infinite. Show that the gain margin is infinite for any value of $. (c) The phase margin is m = arctan(Rege)), where wgc is the gain crossover frequency (the frequency at which the Nyquist curve crosses through the unit circle: L(iwgc)| = 1). Calculate the phase margin m, and plot m vs. C. Hint: if you do the algebra correctly, you will not need to know the value of wo for your plot. (d) The stability margin is the shortest distance from the Nyquist curve to the critical point: Sm = minw |L(iw) + 1. you Calculate the stability margin sm, and plot sm vs. C. Hint 1: if do the algebra correctly, you will not need to know the value of wo for your plot. Hint 2: the minimizer wmin may depend on the value of C. A process whose dynamics is described by a double integrator is controlled by an ideal PD controller with the transfer function C(s) = kas + kp, where the gains are ka = 25wo and kp = wf, and 0 < < 1. 1 (a) What is the loop transfer function (L(s) = P(s)C(s)) for this system? (b) The gain margin is 9m = where |L(iwpc)|' Wpc is the phase crossover frequency (the frequency at which the Nyquist curve crosses the nega- tive real axis: ImL(iwpc) = 0 and ReL(iwpc) <0). When the Nyquist curve never intersects the (strictly) negative real axis, we define the gain margin to be infinite. Show that the gain margin is infinite for any value of $. (c) The phase margin is m = arctan(Rege)), where wgc is the gain crossover frequency (the frequency at which the Nyquist curve crosses through the unit circle: L(iwgc)| = 1). Calculate the phase margin m, and plot m vs. C. Hint: if you do the algebra correctly, you will not need to know the value of wo for your plot. (d) The stability margin is the shortest distance from the Nyquist curve to the critical point: Sm = minw |L(iw) + 1. you Calculate the stability margin sm, and plot sm vs. C. Hint 1: if do the algebra correctly, you will not need to know the value of wo for your plot. Hint 2: the minimizer wmin may depend on the value of C.
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Related Book For
Thomas Calculus Early Transcendentals
ISBN: 9780321884077
13th Edition
Authors: Joel R Hass, Christopher E Heil, Maurice D Weir
Posted Date:
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