Homework 10 Solution - AME30315, Spring 2013
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Problem 9.37 [12 pts](both sketch by hand and verify with Matlab function nyquist).
Note that there is one unstable open-loop pole, P = 1. Z = N + P . Need Z = 0 for stability. Since P = 1,
need 1 counterclockwise encirclement of -1 to make N = −1.
Z=number of unstable closed-loop poles.
P=number of unstable open-loop poles.
N=number of clockwise encirclements of the point -1.
When K=1 there are no encirclement of -1 ⇒ Z = 0 + 1 = 1 unstable closed-loop pole.
When K=15 there is one counterclockwise encirclement of -1 ⇒ Z = −1 + 1 = 0 unstable closed-loop poles.
When K=70 there is one clockwise encirclement of -1 ⇒ Z = 1 + 1 = 2 unstable closed-loop poles. [2 pts]
Figure 1: Nyquist plot for k=1 [1 pt] .
Figure 2: Nyquist plot for k=15 [1 pt].
Hand sketches are shown at the end of the document Fig.8 and Fig.9.[2 pts for each]
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Figure 3: Nyquist plot for k=70 [1 pt].
Figure 4: rootlocus (Pole at -4 is always stable,unstable OL pole moves in LHP as gain increases,poles move
in RHP as gain increases more) [1 pt]
Problem 2 [8 pts] Determine the range of k for which each of the following plants, under unity feedback,
is stable by hand-sketching a Bode plot for k = 1 and imagining the magnitude plot sliding up or down in
until instability results. Validate your results using the Matlab function margin.
kG(s) =
kG(s) =
k(s + 1)
1
=k· ·
s(s + 5)
5
s
1
+11
·
1 s
k
2
=k· ·
(s + 0.5)(s2 + 1.2s + 9)
9
s
0.5
s
5
1
+1
9
1
·
+ 1 s2 + 1.2s + 9
a) [2 pts]From the Bode plot, the frequency never hits −180o . Therefore, K can be increased as much as you
want and you will still have stability. K > 0 leads to stability.
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Figure 5: Bode plot for equation a [1 pt] .
b) [2 pts]From the Bode plot, the gain is below 0 dB for all frequencies. So increasing gain cause the entire
mag plot to shift up. Instability occurs when K > 11.82.
Figure 6: Bode plot for equation b [1 pt].
Hand sketches are shown at the end of the document Fig.10 [1 pt for each].
Problem 3: [5 pts] The Nyquist diagrams for two stable, open-loop systems are sketched in Fig. 7. The
proposed operating gain is 1 and arrows indicate increasing frequency. In each case give a rough estimate of
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the following quantities for the closed-loop (unity feedback) system:
a)
b)
-1
-1
(Franklin, Powell, 2002)
Figure 7: Nyquist plots for Problem 3
(a) [2 pts]Phase margin
system a: 15o ; system b: 45o
(b) [2 pts]Range of gain for stability
system a: k > 0.5; system b:k < 2;
(c) [1 pt]system type
system a:low frequency phase is ±180o . Given OL system is stable, this corresponds to a type 2 system;
system b:low frequency phase is ±90o . Given OL system is stable, this corresponds to a type 1 system;;
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Hand Sketch
Figure 8: Hand sketch for Problem 1
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Figure 9: Hand sketch for Problem 1
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Figure 10: Hand sketch for Problem 2