ps10sol

ps10sol - MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department...

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Unformatted text preview: MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Aeronautics and Astronautics 16.060: Principles of Automatic Control Fall 2003 PROBLEM SET 10 Solutions Problem 1 1. • The Bode gain is K Bode = 1, and there are no integrators, so the low-frequency asymptote is M dB = 0. Again because there are no integrators, the phase at low frequencies is . • The break frequency of the pole is = 1. At this frequency, the slope of the magnitude plot changes by-20 dB/dec. The phase changes by − 5 one decade before the break frequency (at = . 1), by − 45 at the break frequency (at = 1), and by − 85 one decade after the break frequency (at = 10). At high frequencies, the phase approaches − 90 . G 1 (s) Phase (deg) Magnitude (dB) −25 −20 −15 −10 −5 0 −90 −45 0 −1 0 1 10 10 10 Frequency (rad/sec) 1 2. • K Bode = 1 and there is one integrator, so the low-frequency asymptote has a slope of-20 dB/dec and a magnitude of 1 (= dB) at = 1. Because there is one integrator, the phase at low frequencies is − 90 . • The pole at s = − 1 breaks at = 1, so the slope of the magnitude plot changes by-20 dB/dec at that point. (It goes from-20 dB/dec to-40 dB/dec.) The phase is approximately − 95 at = 0 . 1, − 135 at = 1, and − 175 at = 10. The phase at high frequencies approaches − 180 . G 2 (s) Phase (deg) Magnitude (dB) −60 −50 −40 −30 −20 −10 0 10 20 −180 −135 −90 −1 0 1 10 10 10 Frequency (rad/sec) 2 3. • K Bode = 1 and there are no integrators, so the low-frequency asymptote is M dB = and the phase at low frequencies is zero. There is a complex conjugate set of poles given by s 2 + 2 n s + 2 = s 2 + s + 10. So n = 10 n • and = 1 . 16. 2 10 The break frequency for the complex conjugate set of poles is at = n = 10 (half-way • between = 1 and = 10 on a log scale). At this frequency, the slope of the magnitude plot changes by-40 dB/dec. The phase changes by − 180 between...
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This note was uploaded on 11/06/2011 for the course AERO 100 taught by Professor Willcox during the Fall '03 term at MIT.

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ps10sol - MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department...

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