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hw8soln_F2009

# hw8soln_F2009 - MAE170 Homework 8 Solution Fall 2009 P9.1...

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MAE170 Homework 8 Solution Fall 2009 P9.1 The Nyquist plots shown in the following ﬁgures are obtained by the use of Matlab’s nyquist command. For sketching them by hand, ﬁrst sketch the polar plot L ( ) , ω = 0 + + guided by the Bode plots of L ( ). Then, the plot of L ( ) , ω = 0 - → -∞ is the symmetric graph with respect to the real axis. Finally, if there N poles at s = 0, (as in case (d)), complete the plot (to a closed curve) with an arc of N × 180 0 , running clockwise from L ( j 0 - ) to L ( j 0 + ). (a) The ﬁrst open-loop transfer function is L ( s ) = 1 (1 + 0 . 5 s )(1 + 2 s ) In this case, the number of open-loop unstable poles is P = 0, the number -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Nyquist Diagram Real Axis Imaginary Axis Figure 1: Nyquist Plot for problem P9.1a of clockwise encirclements of the (-1,0) point by the Nyquist plot is N = 0 (see Figure 1), therefore the number of unstable closed-loop poles is Z = N + P = 0. Hence, the closed-loop system is stable. (b) The second open-loop transfer function is L ( s ) = 10( s 2 + 1 . 4 + 1) ( s - 1) 2 1

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-8 -6 -4 -2 0 2 4 6 8 10 -10 -8 -6 -4 -2 0 2 4 6 8 10 Nyquist Diagram Real Axis Imaginary Axis Figure 2: Nyquist Plot for problem P9.1b In this case, P = 2 because there are 2 open-loop unstable poles at s = - 1, N = - 2 because there are 2 counterclockwise encirclements of the (-1,0) point by the Nyquist plot (note that in Figure 2, the Nyquist plot consists of going twice around the indicated plot—check the Bode plot!), therefore the number of unstable closed-loop poles is Z = N + P = 0. So, the closed-loop system is stable. You can verify that this system has a Gain Margin GM = - 16 . 9 dB , which indicates that multiplying the open-loop
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hw8soln_F2009 - MAE170 Homework 8 Solution Fall 2009 P9.1...

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