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Unformatted text preview: 6.3 6.4 6.5 Sketch the asymptotes of the Bode plot magnitude and phase for each or the Iollowmg openloop transfer functions. After completing the hand sketches, verify your result
using MATLAB. Turn in your hand sketches and the MATLAB results on the same scales.
(a) L(s) = 57% '(b) L(s) = W
(c) L(s) = magma—73
@Lm=gﬁmﬁg;$
@Lm=$:%§£;3
©L®=Eﬁ%%$%ﬁ%
@L®=R%ﬁ%?ﬂﬁ
(h) L(s) = mLm= S Real poles and zeros. Sketch thc asymptotes of the Bode plot magnitude and phase for each of the listed ope verify your result using MATLAB. Turn in your hand sketches and the MATLAB results on the same scales. @Lm=mﬁﬁmgm:m
(b) Ms) Z S<S— 1)g;§))(s+10) me=miﬁg%gﬁm \
(d) Lo) = (S + 2)(s + 4) Complex poles and zeros. Sketch the asymptotes of the Bode plot magnitude and phase
for each of the listed open—loop transfer functions, and approximate the transition at the
secondorder break point, based on the value of the damping ratio. After completing
the hand sketches, verify your result using MATLAB. Turn in your hand sketches and the MATLAB results on (s + l)(s +10)(52 + 2s + 2500) s(s +1)(s + 5)(s +10) nloop transfer functions. After completing the hand sketches, the same scales, _ 1
(a) LO) ‘ 52 + 35 +10
1
b :
( ) MS) 5(s2——3s~ 10)
(32 — 2s + 8) L : (c) (5) 5(57‘ + 25 —— 10) Problems 391
(d) Ms) = (s2 +25+122 s62 + 2s + 10) 2
L 2 3 +1)
(e) (S) S(S2 S _ (9+4)
a) L<)_s(52+1) 6.6 Multiple poles at the origin. Sketch the asymptotes of the Bode plot magnitude and phase
for each of the listed openloop transfer functions. After completing the hand sketches,
verify your result with MATLAB. Turn in your hand sketches and the MATLAB results
on the same scales. L = _.1__. (a) (5) S2 (s + 8)
__ 1 (b) [19” ‘ ﬁts + 8) L = ——1 (C) (S) S4 (S __ 8) _ (s + 3)
((1) MS) — ————SZ(S __ 8) (e) MS) = M Qo+® (s+1)2 L ' :————— (D (A) s (s+4)
__ (s+1)2
(g) MS) _ s3(s+10)2 6.7 Mixed real and complex poles. Sketch the asymptotes of the Bode plot magnitude and
phase for each of the listed open—loop transfer functions. Embellish the asymptote plots
with a rough estimate of the transitions for each break point. After completing the
hand sketches, verify your result with MATLAB. Turn in your hand sketches and the MATLAB results on the same scales. = (s + 2)
(a) Ms) s(s + 10) (52 + 2s + 2)
(s + 2)
b L = T._.____.——
( ) (S) s (s +10)(s2 + 63 —— 25)
(c) Lo) = (‘+ 32 52(s +10)(s2 + 65 —— 25) . _ M
(d) LO) ‘ 52(s+10)(s2 +4.9“ 85) _ [(M1)2 +11
(6) MS) — ———Sz(s __ MS + 3) 392 Chapter 6 The FrequencyResponse Design Method Figure 6.87 Magnitude portion of
Bode plot for
Problem 6.9 6.8 ' 6.9 6.10 6.11 6.12 Right half—plane poles and zeros. Sketch the asymptotes of the Bode plot magnitude
and phase for each of the listed openloop transfer functions. Make sure that the phase asymptotes properly take the RHP singularity into account by sketching the complex
plane to see how the 1L(s) changes as s goes from 0 to +joo. After completing the hand sketches, verify your result with MATLAB. Turn in your hand sketches and the
MATLAB results on the same scales. (a) L(s) = Eéﬁzosz 1— 1 (the mode] for a case of magnetic levitation with lead
compensation)
(b) L(s) = ii—J— (The magnetic levitation system with integral control S(S 4 10) (52 __ 1)
and lead compensation) _ S—1
(0 L(s)— s2 2
d‘ =: S dZSd—l
( ) MS) s(s+20)2(32 —2s+2)
(s—l—Z)
s(s — l)(s ‘— 6)2 _ l
(D L“) _ (s 1)[(s —— 2)2 + 3] A certain system is represented by the asymptotic Bode diagram shown in Fig. 6.87.
Find and sketch the response of this system to a unit—step input (assuming zero initial
conditions). (6) Lts) = 1 10 100 1000 Prove that a magnitude slope of ——l in a Bode plot corresponds to —20 db per decade
or —6 db per octave. A normalized secondorder system with a damping ratio r = 0.5 and an additional Zero isgivenby
s/a+l
Gs =——~.
U s2+s+1 Use MATLAB to compare the M17 from the step response of the system for a = 0.01,
0.1, l, 10, and 100 with the M, from the frequency response of each case. Is there a correlation between M r and Mp? A normalized second—order system with r = 0.5 and an additional pole is given by
l [rs/p) + no2 +s +1)' Draw Bode plots with p = 0.01, 0.1, l, 10, and 100. What conclusions can you draw
about the effect of an extra pole on the bandwidth compared with the bandwidth for the secondorder system with no extra pole? G(s) : Problems 393 6.13 For the closed—loop transfer function { (02
NS) = . s + 25mm: + can derive the following expression for the bandwidth wBW of T(s) in terms of a)" and z: wBW : cum/l — 2:2 +1/2+4z4 —4;2. Assuming that a)" = l, plot cow for 0 s 4“ 5 l.
6.14 Consider the system whose transfer function is
A a) 3
am 2 #7.
Q3 + tags + wOQ
This is a model of a tuned circuit with quality factor Q. (a) Compute the magnitude and phase of the transfer function analytically, and plot
them for Q = 0.5, l, 2, and 5 as a function of the normalized frequency w/wo. (b) Deﬁne the bandwidth as the distance between the frequencies on either side of
(00 where the magnitude drops to 3 db below its value at £00, and show that the bandwidth is given by
BW = i g .
271' Q (c) What is the relation between Q and g? 6.15 A DC voltmeter schematic is shown in Fig. 6.88. The pointer is damped so that its
maximum overshoot to a step input is 10%. (a) What is the undamped natural frequency of the system?
(b) What is the damped 'natural frequency of the system? (0) Plot the frequency response using MATLAB to determine what input frequency will
produce the largest magnitude output? (d) Suppose this meter is now used to measure a lV AC input with a frequency of
2 rad/sec. What amplitude will the meter indicate after initial transients have died
out? What is the phase lag of the output with respect to the input? Use a Bode plot
analysis to answer these questions. Use the lsi m command in MAT LAB to verify your answer in part (d). Ire 6.88
meter schematic I z 40 x io—ékg '1112 k = 4 X 10—6 kg  m2/sec2
T = input torque = Kmv v = input voltage Km=1Nm/V 394 Chapter 6 The FrequencyResponse Design Method Problems for Section 6.2: Neutral Stability 6.16 Determine the range of K for which the closedloop systems (see Fig. 6.18) are stable for
each of the cases below by making a Bode plot for K = 1 and imagining the magnitude plot sliding up or down until instability results. Verify your answers by using a very rough sketch of a rootlocus plot. a m = KG =
(b) (S) (s + 10)(s + 1)?
(c) KG(S) = K(S +10)(S +1) (s + 100m + 5)3
6.17 Determine the range of K for which each of the listed systems is stable by making a Bode
plot for K = 1 and imagining the magnitude plot sliding up or down until instability results. Verify your answers by using a very rough sketch of a rootlocus plot.
(a) Keg) : Ligﬁ : K(s +1)
s2(s+10) K
(s + 2) (s2 + 9) (b) KG(s) (c) mm) = _ K(s+1)2 (d) KG“) — s3(s +10) Problems for Section 6.3 .' The Nyquist Stability Criterion 6.18 (:1) Sketch the Nyquist plot for an Openloop system with transfer function 1/32; that is, sketch
1 32 S=C1 7
re RHP, as shown in Fig. 6.17. (Hint: where C1 is a contour enclosing the enti
s = 0, as shown in Fig. 6.27.) Assume C1 takes a small detour around the poles at
(b) Repeat part (a) for an openloop system whose transfer function is G(s) =
2 2 ‘
l/(s +w0). 6.19 Sketch the Nyquist plot based on the Bode
then compare your result with that obtained plots for each of the following systems, and
by using the MATLAB command nyquist: (a) KG(s) = [$551622
K
b G =
( ) K (s) (5+10)(S+2)2
(c) KG(s) : W (s +100)<s + 2)3 ((1) Using your plots, estimate the range of K for which each system is stable, and qualitatively verify your result by using a rough sketch of a rootlocus plot. 6.20 Draw a Nyquist plot for K(S+l) (677)
s(s+3), ' KG(s) = 396 Chapter 6 The Frequency—Response Design Method Figure 6.90 Nyquist plot for
Problem 6.24 (d) The classical curve called the Folium (not Kepler’s): 1 (e) The classical curve called the Nephroid, meaning kidney—shaped: 2(s+ l)(52 —4s+ 1) KG(s) = K (s _1)3 (f) The classical curve called Nephroid of Freeth, named after the English mathemati cian T. J. Freeth:
(s + no2 + 3) KG 2 K
(S) 4(s ~ 1)3
(g) A shifted Nephroid of Freeth:
2
KG(s) = K6 +1).
(5 — D3 Problems for Section 6.4: Stability Margins 6.24 The Nyquist plot for some actual control systems resembles the one shown in Fig. 6.90. What are the gain and phase margin(s) for the system of Fig. 6.90, given that at = 0.4,
,8 = 1.3, and 4) = 40°. Describe what happens to the stability of the system as the gain
goes from zero to a very large value. Sketch what the corresponding root locus must
look like for such a system. Also, sketch what the corresponding Bode plots would look like for the system. T Im[G(s)]
B + a + 1 Re[G(s)i V 6.25 The Bode plot for
100[(s/10) + l] S[(S/1) — 1l[(S/100) + 1] G(s) = is shown in Fig. 6.91. Problems 397 (3) Why does the phase start at —270° at the low frequencies?
(b) Sketch the Nyquist plot for G(s).
(c) Is the closedloop system shown in Fig. 6.91 stable? ((1) Will the system be stable if the gain is lowered by a factor of 100? Make a rough
sketch of a root locus for the system, and qualitatively conﬁrm your answer. e 6.91
plot for 1000
am 6.25
100
10
lGl 1 db
0.1
0.01
0.001
0.01 0.1 1 10 100 1000
w (rad/sec)
A
—9o° —r~— w"  _ —~v L—
40 180° T ‘_—*
270° F"
0.01 0.1 1 10 100 1000
w (rad/sec)
6.26 Suppose that in Fig. 6.92,
25 l
s(s + 2)(s2 + 25 +16)
Use MATLAB’s margin to calculate the PM and GM for G(s) and, on the basis of
the Bode plots, conclude which margin would provide more useful information to the
control designer for this system.
6.92 [system for
n 6.26 398 Chapter 6 The Frequency—Response Design Method 6.27 Consider the system given in Fig. 6.93. (3) Use MATLAB to obtain Bode plots for K : 1, and use the plots to estimate the
range of K for which the system will be stable. (b) Verify the stable range of K by using margin to determine PM for selected values
of K.
(c) Use rlocus to determine the values of K at the stability boundaries. ((1) Sketch the Nyquist plot of the system, and use it to verify the number of unstable
roots for the unstable ranges of K. i (e) Using Routh’s criterion, determine the ranges of K for closed—loop stability of this
i system. i‘; 1 Figure 6.93
‘ . l Controlsystem for 1 Problem 6.27 ‘ l . . R
i v; 6.28 Suppose that in Fig. 6.92, (
.2 1
0(3) = #iisil—d.
s(s + 2)(s2 + 0.2s +16)
Use MATLAB ’s margin to calculate the PM and GM for G(s), and comment on whether
you think this system will have welldamped closed—loop roots. F.
11
6.29 For a given system, show that the ultimate period PM and the corresponding ultimate '
gain Ku for the Zeigler~Nichols method can be found by using the following: Bl'
(a) Nyquist diagram
(b) Bode plot
(c) Root locus
6.30 If a system has the openloop transfer function
i ‘ ‘ G(s) — _——.w% H! — 5(S+2§wn) C0
P
l‘ ‘ with unity feedback, then the closedloop transfer function is given by n
(0% T5 = .
“ 0 52+2§wns+aﬁ i1 3 Verify the values of the PM shown in Fig. 6.37 for g“ = 0.1, 0.4, and 0.7.
6.31 Consider the unity—feedback system with the open—loop transfer function K = M
G“) s(s +1)[(s2/25)+ 0.4(5/5) + 11' l ‘ 5 (3) Use MATLAB to draw the Bode plots for G(jw), assuming that K : 1.
i (b) What gain K is required for a PM of 45°? What is the GM for this value of K? (c) What is Kv when the gain K is set for PM = 45°? )4 (a) 399 Problems (d) Create a root locus with respect to K, and indicate the roots for a PM of 45°.
6.32 For the system depicted in Fig. 6.94(a), the transfer—function blocks are deﬁned by G(s) and H (s) = _(s+2)2(s+4) s+i‘
(a) Using rlocus and rlocﬁnd, determine the value of K at the stability boundary. (b) Using rlocus and rtocﬁnd, determine the value of K that will produce roots with
damping corresponding to g = 0.707. (c) What is the gain margin of the system if the gain is set to the value determined in
part (b)? Answer this question without using any frequency—response methods. (d) Create the Bode plots for the system, and determine the gain margin that results for
PM = 65°. What damping ratio would you expect for this PM? (6) Sketch a root locus for the system shown in Fig. 6.94(b). How does it differ from
the one in part (a)? (t) For the systems in Figs. 6.94(a) and (b), how does the transfer function Y2(s)/R(s)
differ from Y1 (s) / R (3)? Would you expect the step response to r(z) to be different
for the two cases? (b) [ram for Problem 6.32: (a) unity feedback; (b) H(5) in feedback )5 stem for
.33 6.33 For the system shown in Fig. 6.95, use Bode and rootlocus plots to determine the gain
and frequency at which instability occurs. What gain (or gains) gives a PM of 20°?
What is the gain margin when PM = 20°? (3+ l)(s + 2)
K 2 2
5 (5+ 3)(s + 25+ 25) 6.34 A magnetic tapedrive speed—control system is shown in Fig. 6.96. The speed sensor is
slow enough that its dynamics must be included. The speedmeasurement time constant
is rm 2 0.5 see; the reel time constant is rr : J / b : 4 sec, where b = the output shaft
damping constant = 1 N msec; and the motor time constant is T1 = 1 sec. (a) Determine the gain K required to keep the steady~state speed error to less than 7%
of the reference—speed setting. (b) Determine the gain and phase margins of the system. Is this a good system design?
6.35 For the system in Fig. 6.97, determine the Nyquist plot and apply the Nyquist criterion (a) to determine the range of values of K (positive and negative) for which the system
will be stable, and ...
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 Fall '09

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