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Unformatted text preview: 394 Chapter 6 The Frequency—Response 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) KG(s) = 53%?
b KG = K
( ) (S) (s+1om
(C) KG(s) = W (s +100)(s + 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 root—locus plot. (a) KG(s) = %S+t51_)) (b) KG(s) = 5%(5Jrt—110L) (e) KG(s) = m
(d) KG(s) = fig—:3?) Problemsfor Section 6.3: The Nyquist Stability Criteriorz 6.18 (a) Sketch the Nyquist plot for an open100p system with transfer function 1/52; that
is, sketch 1 '
S2 S=C1 ,
where C1 is a contour enclosing the entire RHP, as shown in Fig. 6.17. (Hint:
Assume C1 takes a small detour around the poles at s = 0, as shown in Fig. 6.27.)
(b) Repeat part (a) for an openloop system whose transfer function is G(s) =
1/(s2 + mg).
6.19 Sketch the Nyquist plot based on the Bode plots for each of the following systems, and
then compare your result with that obtained by using the MATLAB command nyq uist: (a) KG(s) = 5%? . b G = __K.__
()K (S) (s+10)(s+2)2
(c) Kan) = W11 (5 + lOO)(s + 2)3 ((1) Using your plots7 estimate the range of K for which each system is stable, and
qualitatively verify your result by using a rough sketch of a root—locus plot. 6.20 Draw a Nyquist plot for ' ’ K (s + l) ————, (6.77)
s(s + 3) KG(S) = Figure 6.89 Control syste
Problem 6.2'. :Fig. 6.18) are stable for
magining the magnitude
nswers by using a very stable by making a Bode )r down until instability
:ootlocus plot. ! 1sfer function 1 /s2; that wn in Fig. 6.17. (Him:
as shown in Fig. 6.27.) er function is G(s) = : following systems, and
AAB command nyquist: :h system is stable, and
a rootlocus plot. (6.77) Figure 6.89 Control system for
Problem 6.21 Problems 3 choosing the contour to be to [the right of the singularity on the jwaxis. Next, using
Nyquist criterion, determine the range of K for which the system is stable. Then
the N yquist plot, this time choosing the contour to be to the left of the singularity 01
imaginary axis. Again, using the Nyquist criterion, check the range of K for whicl
system is stable. Are the answers the same? Should they be? 6.21 ‘Draw the N yquist plot for the system in Fig. 6.89. Using the Nyquist stability criter determine the range of K for which the system is stable. Consider both positive
negative values of K. and the nonminimumphase system 571 G =—
(S) s+1O 7
s=jw noting that 4(ja) — 1) decreases with w rather than increasing. (b) Does an RHP zero affect the relationship between the 71 encirclements on a p(
plot and the number of unstable closed—loop roots in Eq. (6.28)? (c) Sketch the phase of the following unstable system for a) = 0.1 to 100 rad/sec: s+1
5—10 s=jw G(s) : (d) Check the stability of the systems in (a) and (c) using the Nyquist criterion on KGt
Determine the range of K for which the closed—loop system is stable, and check yr
results qualitatively by using a rough rootlocus sketch. 6.23 Nyquist plots and the classical plane curves: Determine the Nyquist plot, using M2
LAB, for the systems given below, with K = 1, and verify that the beginning point 2 end point for the ja) > 0 portion have the correct magnitude and phase: (a) The classical curve called Cayley’s Sextic, discovered by Maclaurin in 1718: KG(s) = K (Si—D3. (b) The classical curve called the Cissoid, meaning ivy—shaped: KG(s) = K s(s + 1) '
s (c) The classical curve called the Folium of Kepler, studied by Kepler in 1609: 1 W = Km 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): l W = Km (e) The classical curve called the Neﬁhroid, meaning kidneyshaped: 2(s +1)(s2 — 45 +1) KG(s) = K (s _1)3 (f) The classical curve called Nephroid of Freeth, named after the English mathemati
cian 'l‘. J. Freeth: 2
l
KG(s) = Kw
4(s , 1)3
(g) A shifted Nephroid of Freeth: (52 +1) KG(S) = K(5 _1)3. 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 or = 0.4,
,8 = 1.3, and <l> = 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. ‘ Im[G(s)] 6.25 The Bode plot for
100[(.v/10)+ 1] G“) : sr<s/1>—1][<s/100)+ 1] is shown in Fig. 6.91. Figure 6.91 Bode plot for
Problem 6.25 Figure 6.92 Control systr
Problem 6.2 Problems 401 6.38 The Nyquist diagrams for two stablejopen—loop systems are sketched in Fig. 6.100. The
proposed operating gain is indicated» as K0, and arrows indicate increasing frequency. In each case give a rough estimate of the following quantities for the closed—loop (unity
feedback) system: (a) Phase margin (1)) Damping ratio (c) Range of gain for stability (if any)
(d) System type (0, l, or 2) Figure 6.100 Nyquist plots for .38 A
Re[G(S)l ‘ K for which the
s sketch. )ply the Nyquist 'hich the system (a) (b) Y for WhiCh the 6.39 The steering dynamics of a ship are represented by the transfer function
sketch.
V(s) _ G“) _ K[—(s/0.142) +1]
8r(s) _ _ S(s/O.325 + l)(s/0.0362 + 1)’
where V is the ship’s lateral velocity in meters per second, and 8, is the rudder angle in
radians.
(3) Use the MATLAB command bode to plot the log magnitude and phase of G(ja))
for K : 0.2. (b) On your plot, indicate the crossover frequency, PM, and GM.
(c) Is the ship steering system stable with K = 0.2? (d) What value of K would yield a PM of 30°, and what would the crossover
frequency be? ly the Nyquist ich the s t
ys em 6.40 For the open—loop system K (s + 1)
s2 (S + 10)?
determine the value for K at the stability boundary and the values of K at the points
where PM = 30°. for which the KGCY) =
.ketch. Problems for Section 6.5: Bode ’S Gain—Phase Relationship 6.41 The frequency response of a plant in a unity feedback conﬁguration is sketched in
Fig. 6.l01. Assume that the plant is open—loop stable and minimumphase. (a) What is the velocity constant K, for the system as drawn? (b) What is the damping ratio of the complex poles at w = 100? ...
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 Fall '09

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