r + - FINAL June 12, 2012 MAE 143B Linear Control Prof. M. Krstic

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1 MAE 43B Linear Control Prof. M. Krstic FINAL June, One sheet of hand-written notes (two pages). Present your reasoning and calculations clearly. Inconsistent etchings will not be graded. Write answers only in the blue book. Total points:. Time: hour minutes. Problem : Nyquist Plots ( points) For each of the given transfer functions, sketch the Nyquist plot and find the range of K > that makes the following closed-loop system stable. r + - K G(s) y (a) (3 points) G(s) =.(s )(s + ) (s + ) Solution. G(s) = (s+) ( s + ) ( s + ). We first draw the Bode plot The phase plot indicates that the Nyquist plot () starts at the negative real axis, () moves in a clockwise way, (3) crosses the positive real axis, (4) turns around and moves back in a counter-clockwise way, and () converges to the positive real axis. The magnitude plot indicates that the Nyquist plot () starts at due to db and 8, () gets closer to the origin as ω increases, and (3) ends at. due to 4dB and.

2 Nyquist Diagram Since G(s) is stable, the closed-loop system is stable if, and only if, there is no encirclement around ( K, j). Therefore, < K <. s(s + ) (b) (3 points) G(s) = s +.s + ( Solution. G(s) = s s + ). We first draw the Bode plot. s +.s The phase plot indicates that the Nyquist plot () starts at the positive imaginary axis, () moves in a clockwise way, (3) crosses the positive real axis, (4) turns around and moves back in a counter-clockwise way, and () converges to the positive real axis.

3 The magnitude plot indicates that the Nyquist plot () starts at the origin, () moves, first, away from and, then, towards the origin, and (3) ends at due to db and. 6 4 Nyquist Diagram Since G(s) is stable, the closed-loop system is stable if, and only if, there is no encirclement around ( K, j). Therefore, K > (c) (3 points) G(s) = (s ) s (s + s + ) Solution. G(s) =. (s ) s ( ) s + s +). We first draw the Bode plot

4 The phase plot indicates that the Nyquist plot () starts with phase 8, () moves in a clockwise way, and (3) converges to the negative real axis. The magnitude plot indicates that the Nyquist plot () comes from, () moves towards to the origin, and (3) converges to the origin. 3 Nyquist Diagram () The number of poles of G(s) on the right half plane is. () Around ( K, j), we have #CCW = and #CW = for any K >. Therefore, the Nyquist stability criterion is not satisfied for any K >, which means that the closed-loop system is unstable for any K >. 4

5 Problem : Root Locus, Bode Plot, and Nyquist Plot (3 points) We can learn a lot about the behavior of a system from its Bode plot. Suppose we perform an experiment on a dynamic system, and we obtain the following Bode plot: (a) (4 points) Sketch the Nyquist plot. The phase plot indicates that the Nyquist plot () starts at the positive real axis, () moves in a clockwise way until it reaches and crosses the positive real axis, and (3) converges to the negative imaginary axis. The magnitude plot indicates that the Nyquist plot () starts at due to db and, () moves towards the origin except for the peak, and (3) converges to the origin.. Nyquist Diagram

6 (b) ( points) Determine the gain and phase margin. When the phase plot crosses 8, the magnitude plot crosses about 7dB, which means that the gain margin is about 7dB. When the magnitude plot crosses db, the phase plot crosses about 6, which means that the phase margin is about 6 ( 8 ) = 4. (c) (4 points) Determine the transfer function of the open-loop system. (Use powers of for your breakpoints) (Hint: Use the magnitude and frequency corresponding to the phase of 8 in order to find ζ for the resonant pair of poles.) From the Bode plot, the transfer function is determined as G(s) = s + ζs + ( s ) + s + = (s ) (s + ζs + )(s + ) And from the phase plot, the frequency corresponding to the phase 8 is about.8 rad/sec. At.8 rad/sec, we have ( ( j.8).8 G(j.8) = = arctan (.4 + j3.6ζ)( + j.8) arctan 3.6ζ ) arctan.8, which should be 8. Thus, we have ζ =.4 ( 3.6 tan arctan.8 ).8 arctan =.443 Therefore, the transfer function is G(s) = (s ) (s s + )(s + ) (d) (3 points) Sketch the Root Locus to confirm that high gain leads to instability. () The open-loop zeros are at s =, () The open-loop poles are at s =,.443± j.443 (3) The relative degree is. Thus, there is one asymptote. Root Locus Root Locus magnified around the origin

1 (s + 3)(s + 2)(s + a) G(s) = C(s) = K P + K I

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