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Unformatted text preview: a. 502 N increased by at least a factor of 5. A lag compensator
of the form _ (s + 0.5)
GclS) — is to be used. [Section: 9.4] a. Find the gain required for both the compensated
and the uncompensated systems. b. Find the value of Kit) for both the compensated and
the uncompensated systems. c. Estimate the percent overshoot and settling time
for both the compensated and the uncompensated
systems. d. Discuss the validity of the secondorder approx—
imation used for your results in Part c. e. Use MATLAB or any other computer
program to simulate the step
response for the uncompensated
and compensated systems. What do you notice
about the compensated system's response? f. Design a lead compensator that will correct the
objection you notice in Part e. MATLAE ;€'E§./\For the unify feedback system in Figure P91, with K _ a ._ .le
G“) = (Wm/9W design a PIID controller that will yield a”peak time of
1.047 seconds and a damping ratio of 0.8, with zero
error for a step input. [section: 9.4] 26. For the unity feedback system in Figure P9.l, with 27. Redo Problem 26 using MATLAB in the K \IllleYPLUS
=p+4xy+mo+1m do the following: 6(5)
Enntrol Solutions a. Design a controller that will yield no more than
25% overshoot and no more than a 2 second
settling time for a step input and zero steadyState
error for step and ramp inputs. “ATLAB b. Use MATLAB and verifyyour design. following way: ﬂ.
r 1' Chapter 9 Design via Root Locus 29. 30. a. MATLAB will ask for the desired percent over_
shoot, settling time, and PI compensator zero._.i b. MATLAB will design the PD controller’s zero. i; c. MATLAB will display the root locus of the PID;
compensated system with the desired percent
overshoot line. d. The user will interactively select the intersection
of the root locus and the desired percent over
shoot line. 9. MATLAB will display the gain and transient
response characteristics of the PIDcompen
sated system. ‘ f. MATLAB will display the step response of the
PIDcompensated system. 9. MATLAB will display the ramp response of the;
PIDcompensated system. ‘ . If the system of Figure P93 operates with {damping A ratio of 0.517 for the dominant Secondorder poles, ﬁnd the location of all closedloop poles and zeros. m (s + 2)
Figure P93
For the unity feedback system in Figure,P9.1, with
K
G = —._.——.
m so+mo+nw+m do the following: [Section: 9.5] a. Design rate feedback to yield a step response with
no more than 15% overshoot and,no more than3 seconds settling time. Use Approach 1.
' HimAs b. Use MATLAB and simulate your
compensated system. / Given the system of Figure P9.4: [Section: 9.5] a. Design the value of K 1, as well as a in the feedback
path of the minor loop, to yield a settling time of 1
second with 5% overshoot for the step response. b. Design the value of K to yield a majorloop
response with 10% overshoot for a step input. ' c. Use MATLAB or any other computer
program to simulate the step
response to’tiié'entire closed—loop
system. d. Adda PI compensator to reduce the
major—loop steady—state error to
zero‘and simulate the step response
using MATLAB or any other com
‘puter program. smug MATlAs Figure P9.4 31. Identify and' realize the following controllers with
operational ampliﬁers. [Section: 9.6] s+0.01
s
b. 5 +2 a. a. 32. Identify and realize the following com
pensators with passive networks. [Section: 9.6]
s + 0.1
s + 0.01 “‘uIo 5+2
[3' 5+5 c s+0.l 5+1
' s+0.01 Si—lO 33. Repeat Problem 32 using operational ampliﬁers.
[Section: 9.6] a. Design Problems 34. The room temperature of an 11 m2 room is to be
controlled by varying the power of an indoor radiator.
For this speciﬁc room the openloop transfer function
from radiator power, (2(5), to temperature, T(s), is
(Thomas, 2005) _ T_(.Q _ {1 x10’6)52 +0314 x 1W9): +9.66 x1043} Gm Q(s) _ 53 + 0.0016352 + (5.272 x [OJUs + (3.538 x 10‘“) .m 3; 503 Probiems .N .r The system is assumed to be in the closedloop
conﬁguration shown in Figure P9.l. 'n" a. For a unit step input, calculate the steady—state
error of the system. *b. Try using the procedure of Section 9.2 to design a
PI controllehto obtain zero steadystate error for
step inputs without appreciably changing the tran
sient response. Then explain why it is not possible
to do so. c. Design a PI controller of the form 06(5) =
K(s + z)
s
to zero while not changing signiﬁcantly the tran
sient response. (Hint: Place the zero of the com
pensator in a position where the closed—loop poles
of the uncompensated root locus will not be affected signiﬁcantly.) d. UseSimulinktosimulatethesystems
of Parts b and c and to verify the
correctness of your design in Part c. that will reduce the stepresponse error samulink 35. Figure P9.5 shows a twotank system. The liquid
inﬂow to the upper tank can be controlled using a
valve and is represented by F0. The upper tank’s
outﬂow equals the lower tank‘s inflow and is repre
sented by F1. The outﬂow of the lower tank is F 2. The
objective of the design is to control the liquid level,
y(t), in the lower tank. The openloop transmission Y(S) _ 021613 F0(s) w 52 +(a1—i— a4)s + (11614 (Romagnoli, 2006). The system will be controlled in a loop analogous to that of Figure P9.l, where the for this system is F0 Figure P9.5 Chapter 9 Design via Root Locus lower liquid level will be measured and compared to a corresponding controller, TC 22, that compares the,
setpoint. The resultingen'orwill be fed to acontroller, actual temperature with a desired temperature set
which in turn will open or close the valve feeding the 5! point, SP. The controller autornatically opens or
upper tank. closes a valve to allow or prevent the ﬂow of steam
to change the temperature in the tank. The cone
3. Assuming a; = 0.04, a; = 0.0187, 4513 = l, and a4 spending blockdiagram for this system is shown in
2 0.227, design a lag compensator to obtain a Figure,P9.6(b) (SmithagOOZ). Assume the following
stepresponse steadystate error of 10% without transfer functions:
affecting the system’s transient response appreci~
ably Gv(s) = 0'02 ; 01(5) = 70 ; yo) = _1——.
. Verify your design through MATLAB “Am” 45 + 1 505 + 1 125 +1
simulatiOns. ‘
a. Assuming GAS) = K, ﬁnd the value of K that will
. Figure P9.6(a) shows a =treatexchanger process whose “33”” i" a (lominam PO]e With E = 07 Obtain the
purpose is to maintain the temperature of a liquid at a conespondmg Tr
preseribed temperature. b. Design a PD controller to obtain the same damping
The temperature is measured usingasensor anda factor as Part a but with a settling time 20%
transmitter, TT 22, that sends the measurement to a smaller. Condensate
return Figure P9.6 a. Heatexchanger process; b. block diagram at .2;
K Problems 505 c. Verifyyourresults through MATLAB “ATM
simulation: 37. Repeat Problem 36, Parts b and c, using a lead
compensator. 38. a. Find the transfer function of a motor whose torque
speed curve and load are given in Figure P9.7. Torque
(b)
0.5 N—m Figure P9.8
5V
you cannot use a different motor, an ampliﬁer and
I tachometer are inserted into the loop as shown in
0 .. , RPM Figure P9.8(b). Find the values of K1 and Kf to
=0 60.000 . . 2” yield a percent overshoot of 25% and a settling time of 0.2 second.
ﬁlo—r] 1 c. Evaluate the steadystate error speciﬁcations for
both the uncompensated and the compensated systems.
4 4 “ 40. A position control is to be designed with a 20% 10 a overshoot andxa settling time of 2 seconds. You . r have on hand an ampliﬁer and a power ampliﬁer
1 Nms/rad _ whose caScaded transfer function is K1 /(5 + 20)
F'gure P9'7 with which to drive the motor. Two lOturn ‘pots
h are available to convert shaft position into voltage.
b. Design a tachometer compensator to yield a damp A V0hage 0f i5” VORS is PIaCEd across the POW A dc
ing ratio of 0.5 for a position control employing a 1110‘“ Whose transfer funCtiOH i5 0f the form
power ampliﬁer of gain 1 and a preampliﬁer of
gain 5000. 90(5) K_
c... Compare the transient and steadystate character Egg) 2 5(3 + a) istics of the uncompensated system and the com ensated s stem. . . . .
p y 1s also available. The transfer function of the motor 15 l 39_ You are given the motor whose transfer Weypws found. experimentally as follows. The motor and function is Shown in Figure P9_8(a)_ geared load are driven openloop by applying a large.
short, rectangular pulse to the armature. An oscillo gram ofthe response shows that the motor reached 63% of its ﬁnal output value at 1/2 second after the
_ application of the pulse. Further, with a constant 10 Of a "Pity feedback SXStemﬁCalcmate the patent volts dc applied to the armature, the constant output overshoot and settling time that #could be Speed was 100 rad/S_ expected. b. You want to improve the closedloop response. a. Draw a complete block diagram of the system,
Since the motor constants cannot be changed and specifying the transfer function of each compo ,_ I 5
Qunlrol Sulutl"f1
a.4 If this motor were the forward transfer function ,. 506 Ampliﬁer Chapter 9 Design via Root Locus Figure P9.9 nent when the system is Operating with 20%
overshoot. b. What will the steadystate error be for a unit ramp
input? c. Determine the transient response characteristics. d. If tachometer feedback is‘u sed around the motor, as
shown in Figure P9.9, ﬁnd the tachometer and the
ampliﬁer gain to meet the original speciﬁcations.
Summarize the transient and steadystate charac
teristics. 41. A position control is to be designed with a 10% overshoot, a settling time of 1 second, and K, =
1000. You have on hand an ampliﬁer and a power
ampliﬁer whose cascaded transfer function is K1 / (s +
40) with which to drive the motor. Two 10turn pots
are available to convert shaft position into voltage. A
voltage of :i:203r volts is placed across the pots. A dc motor whose transfer function is of the form 60 (s) K Ea(s) _ 5(3 + a) is also available. The following data are observed
from a dynamometer test at 50 V. At 25 N—m of
torque, the motor turns at 1433 rpm. At 75 Nm of
torque, the motor turns at 478 rpm. The speed
measured at the load is 0.1 that of the motor. The
equivalent inertia, includingtheload, at the motor
armature is 100:.kgm2, and the equivalent viscous
damping, including the load, at the motor armature
is 50 N—ms/rad. a. Draw a complete block diagram of the system,
specifying the transfer function of each compo
nent ‘56 b. Design a passive compensator to meet the require—
ments in the problem statement. c. Draw the schematic of the compensator showing
all component values. Use an operational ampliﬁer
for isolation where necessary. d. Use MATLAB or any other computer
program to simulate your system
and Show that all requirements
have been met. “intAs 42. Given the system shown in Figure P910, ﬁnd the values of K and Kf so that‘rthe closedloop dominant
poles will have a damping ratio of 0.5 and the under‘
damped poles of the minor loop will have a damping
ratio of 0.8. ~ Power ampliﬁer Tachometer Figure P9.10 43. Given the system in Figure P9.1 1, ﬁnd the values of K and Kf so that the closedloop system will have a
4.32% overshoot and the minor loop will‘have a. :
damping ratio of 0.8. Compare the expected perfor
mance of the system without tachometer compensa
tion to the expected performance with tachometer
compensation. Power ampliﬁer Amplifier Plat“ Figure P9.11 44. In Problem 55 of Chapter 8, a headposition control 46. _. Clonsi‘derthe temperature control system \tt‘tleyPtt/s system for a ﬂoppy disk drive was designed to yield a settling time of 0.1 second through gain adjustment
alone. Design a lead compensator to decrease the
settling time to 0.05 second without changing the
percent overshoot. Also, ﬁnd the required loop gain. for a chemical process shown in Figure
P9. 1B2. The unc0mpensated system is  '_
operating with a rise time approxi Qqniml Salutiof'il
mately‘ the same as a second~order system with a
peak time of 16 seconds and 5% overshoot. There is
also considerable steadystate error. Design a PID
controller so that the compensated system will have a
rise time approximately equivalent to a secondorder
system with a peak time of 8 seconds and 5% over
shoot, and zero'steadystate error for a step input. s Steamdriven power generators rotate at a constant
speed via a governor that maintains constant steam
pressure in the turbine. In addition, automatic gen
eration control (AGC) or load frequency control
(LFC) is added to ensure reliability and consistency
despite load variations or other disturbances that
can affect the distribution line frequency output. A _ PID
DESll'ed Controller Amplifier
temperature
set point + 507 Problems ._;
speciﬁc turbinegovernor system can be described
only using the block diagram of Figure P9.l in
which 6(5) = Gc(s)Gg(s)G,(s)Gm(s), where (Kho dabakksh ian, 2005) 63(3) 2 is the govemor’s transfer function
1 . . . G,(s) = 6m 1s the turbine transfer function
1 . Gm(s) = represents the machme and load transfer functions GAS) is the LFC compensation to be designed a. Assuming Gc(s) = K, ﬁnd the value of K that will
result in a dodiinant pole with l; = 0.7. Obtain the
corresponding Ts. b. Design a PID controller to obtain the same damp
ing factor as in Part a, but with a settling time of
2 seconds and zero steadystate error to step input
commands. c. Verify your results using a MATLAB
simuation. “ATLAa 47. Repeat Problem 46 using a lag—lead compensator instead of a PID controller. Design for a steadystate
error of 1% for a step "input command. 48. Digital versatile disk (DVD) players incorporate several control systems for their operations. The con—
trol tasks include (I) keeping the laser beam focused
on the disc surface, (2) fast track selection, (3) disc
rOt'ation speed control, and (4) following a track
accurately. In order to follow a track, tithe pickup
=head radial position is controlled via a voltage that
operates a voice coil embedded in a magnet conﬁg Actuator Chemical
and heat
Valve process
Actual
temperature 0.7 1
52 + 1.7s + 0.25 Temperature SCHSOI‘
Figure P9.12 Chemical process temperature control system 508 10(3) = uration. For a speciﬁc DVD player, the transfer func
.tion is given by E Vts)
0.63 v. = 0.36 32 0.04 52
l + s —l— l + s + 305.4 305.42 248.2 248.22)
where x(t) = radial pickup position and v(r) = the coil input voltage (Birz‘anri, 2002). Assume that the system will be controlled in a
closedloop conﬁguration, such as the one shown
in Figure P91. Assuming that the plant, P(5), is
cascaded with a proportional compensatoL Gc(s)
=' K, plot the root locus of the system. b. Repeat Bart ausjng MATLAB if your root locus plot was created by any
other tool. c. Find the range of ‘K for closed—loop stability, the
resulting damping factor range, and the smallest
settling time. a. “mm; d. Design a notch ﬁlter compensator so that the sys
tem’s dominant poles have a damping factor of E = 0.7 with a closedloop settling time of 0.1 second.
“Mum e. Simulate the system‘s step reSponse
for Part c using MATLAB. f. Add a PI compensator to the system to achieve zero
steadystate error for a step input without appre
ciably affecting the transient response achieved in
Part D. g. Simulate the system’s step response
for Part e using MATLAB. Compensator Chapter 9 Design via Root Locus 49 ""‘i i l
l
l
i . A coordinate measuring machine (CMM) measures
coordinates on threedimensional objects. The accuracy
of CMMs is affected by temperature changes as well as
by mechanical resonances due to joint elasticity. These
resonances are more pronounced when the machine has
to go over abrupt changes of dimension, such as sharp
comers at'high speed. Each of the machine links can be
controlled in a closedloop conﬁguration, such as the
one shown in Figure P9.13 for a speciﬁc machine with
prismatic (sliding) links. In the ﬁgure, ch(s) is the
commanded position and X(s) is the actual position.
The minor loop uses a tachometer generator to obtain .. the joint speed, while the main loop controls the joint's 50. Cl.046§s(s2 + 1.155 + 0.33) position (02:81, 2003). a. Find the value of K that will result in a minor loop
with i; z 0.5. b. Use a notch ﬁlter compensator, 643), for the
external loop sothat it results in a closedloop damping factor of = 0.7 with T, #4 seconds;
“Mus c. Use MATLAB to simulate the compen
sated system’s closedloop step
response. Magnetic levitation systems are now used to elevate
and propel trains along tracks. A diagram of a demon
stration magnetic levitation system is shown in Figure
P9. [4(a). Action between a permanent magnet
attachedto the PingPong ball, the object to be levi—_ tated, and an electromagnet provides the lift. The amount of elevation can be controlled through ‘V, applied to the electromagriet as shown in Figure, . P9.14(a). The elevation is controlled by using a photo
detector pair to detect the elevation of the PingPong
ball. Assume that the elevation. control system is
represented by FigureiP9.14(b) and do the following
(Cho, 1993): Plant 574.98
5(52 +14245 + 3447.91) Tachometer generator Figure P9.13 Problems 509 Ampliﬁer
circuit Electromagnet
Control computer Permanent magnet PingPong ball Datum line .
(a) Compensator I : Plant Photocell
(b)
Figi] re P9.14 3. Magnetic levitation system (© 1993 lEEE); I). block diagram .3. Design a compensator, G45), to yield a settling Commanded Actual
tlme of 0.1 second or less 1f the step response 15 to angle of angle of
have no more than 1% overshoot. Specify the ﬁlm" Controller Aircraft “Wk compensator’s poles, zeros, andigain. b. Cascade another compensator to minimize the
“steadystate error and have the total settling time
not exceed 0.5 second. This compensator should
.pot appreciably affect the transient response
designed in Part a. Specify the poles and zeros
of this compensator. c. Use MATLAB or any other computer program to simulate the system to
check your design. 51. The transfer function for an AFTI/Fl6 aircraft relat
ing angle of attack, or(t), to elevator deﬂection, 69 (I), is Figure P9.15 Simpliﬁed block diagram for angle of attack
control MATME a. Find the range of K for stability.
b. ‘Plot or sketch a root locus. c. Design a cascade compensator to yield 'zero steady
state error, a settling time of about 0.05 second, and a percent overshoot not greater than 20%. given by “mus
(1(5) d. Use MATLAB or any other computer
Gm = 5 (S) 2 program to simulate the system
9 to check your design.
_ 0 072 (s + 23 )(s2 + 0.053 + 0.04)
' (S n 07)(S + 1.?)(S2 + 9.083 + 0.04) 52. Figure P9.16 is a simpliﬁed block diagram of a self—
Assume the blockdiagram shown inﬁgure P9. 15 for guiding vehicle's bearing angle control. Design a lead
controlling the angle of attack, or, and do the following compensator to yield a closed100p step response with (Monahemr', 1992): 10% overshoot and a settling time of 1.5 seconds. wawmdﬂwméggg :._.:: ' ‘  . .. 1 510 . ll .
Deslred Contra er bearing angle Figure P9.16 Simpliﬁed block diagram of a selfguiding vehicle‘s bearihg angle control Progressive Analysis and Design Problems 53. Highspeed rail pantograph. Problem 20 in Chapter 1 discusses the active control of a pantograph
mechanism for highspeed rail systems. In Problem
74(b). Chapter 5 , you found the block diagram for the
active pantograph control system. In Chapter 8, Pro
blem 64, you designed the gain to yield a closed—loop
step response with 38% overshoot. A plot of the step
response should have shown a settling time greater
than 0.5 second as well as a highfrequency oscilla
tion superimposed over the step response (O’Conner;
1997). We want to reduce the settling time to about 0.3
second, reduce the step response steadystate error to
zero, and eliminate the highfrequency oscillation. A
way of eliminating the highfrequency oscillation is
to cascade a notch ﬁlter with the plant. Using the notch
ﬁlter, 52 + 165 + 9200 GAS) = (s + 60)2 do the following: a. Design a PD centroller to yield asettling time of
approximately 0.3 second with no more then 60%
overshoot. b. Add a PI controller to yield zero steadystate error
for step inputs. c. Use MATLAB to plot the
PlD/notchcompensated closed—
.Ioop step response. MATLAE — ACYBER, :EXyFllORATION “LABORATORY Experiment 9.1 Objectives To perfOrr'n a tradeoff study for lead..compensatiori. To design a controller and see its effect upon steadystate error. 1 Minimum required software packages MATLAB, and the Control System Toolbar . J. 3‘. nus Chapter 9 Design via Root Locus s1+10s +50 54. Control of HIV/AIDS. It was shown in Chapter ti, Vehicle
Stealing dynamics Actual
bearing angle. Problem 66 that when the virus levels in an HIV/AIDS patient are controlled using RTIs the linearized plant
model is Y(s) _ —5205— 10.3844
U1(s) _ .93 + 2.68m2 + 0.11s + 0.0126 13(5) = Assume that the system is embedded in a conﬁgura 
tion, such as the one shown in Figure P9.l, where, '
6(5) = GE (5) P0). Here, 06(5) is a cascade compen— ..
sator. For simplicity in this problem, choose the dc ‘
gain of Gc(s) less then zero to obtain a negative
feedback system (the negative signs of (Ms) and Pp)
cancel out) (Craig, 2004). a. Consider the uncompensated system with GAS) : r
—K. Find the value of K that Will place all closed
loop poles on the real axis. b. Use MATLAB to simulate the unit
step response of the gaincompen
sated system. Note the %OS and
the T; from the simulation. c. Designta PI compensator so that the steadystate '
error for step inputs is zero. Choose a gain value to
make all poles real. C]. Use MATLABto simulate the design
in Part c for a unit step input.
Compare the simulation to Part b. “ATLAS 590 Chapter 10 Frequency Response Techniques 31PR 0 B L E M S
1. Find analytical expressions for the mag Wequ nitude and phase response for each 6(3)
below. [Section: 10.1] i 5" G“) =m
_ (3+5) '3' G“) ' (s+ 2)(s+4) c. 6(5) _ (s+3)(s+5) _ 5(5 + 2)(s +4) 2. For each function in Problem 1, make a plot the log
magnitude and the phase, using logfrequency in rad/s
as the ordinate. Do not use asymptotic approxima
tions. [Section: 10.1] 3. For each function in Problem I, make a polar plot of
the frequency response. [Section: 10.1] 4. For each function in Problem 1, sketch the Bode
asymmwlmd asymptotic phase plots.
Compare your resu ts w1t "your answers to Problem .1 .
[Section: 10.2] 5. Sketch the Nyquist diagram for each of the systems in
Figure P10.1. [Section: 10.4] 6. Draw the Figure P1031
polar plot 0 ,  _= .. Il—IIII J llllllIIIIIII iiiiii ""ll IIIIIIII i!
magnitude and phase curves shown in Figure P10.2. [Section: 10.1] _30 40 IIIIIIII 20 log M
['0
C Iiiiiiﬂlllllll IIlIlIi'!!!!!!_!!!!!!!
ElllIlllllllIIIIIIII Phase (degreesj

5
G IlllIlllllllIIIIIIII
7200 g
0.1 1 10 100 Frequency (rad/s)
4" Figure P10.2  Problems 591 7. Draw the separate magnitude and phase curves
from the polar plot shown in Figure P103.
[Section: 10.1] x 10‘3 ‘ System 1 Imaginary axis Figure No.4 Real axis x 10’3
Figure P103 11. For each system of Problem 10, ﬁnd the Warm”:
gain margin and phase margin if the L
value of K in each part of Problem 10 ._
is [Section: 10.6] 00mmls°mtiaﬂs 8. Write a program in MATLAB that will MATLAB do the following: a. K: 1000
a. Plot the Nyquist diagram of a system b' K2100
C. K:0.1' b. Display the realaxis crossing value and TLA
frequency 12. Write a program in MATLAB that will “A a do the following: Apply your program to a unity feedback system with MS + 5) a. Allow a value of gain, K, to be entered from the
G = ___——._.—
(S) (s2 +65 +100)(s2 +4s+25) keyboard
btiDIspIay the Bode plots of a system for the
9. Using the Nyquist criterion, ﬁnd out whether each entered Value Of K
system of Problem 5 is stable. [Section: 10.3] c. Calculate and display the gain and phase margin for the entered value of K
10. Using the N yqui st criterion, ﬁnd the range wieyPLUs
of K for stability for each of the systems Test your program on a unity feedback system in Figure PlO.4. [Section: 10.3] ' With 6(5) = K / [5(5 + 3)(5 + 12)]}' 592 13. 14. 15. 16. 17. 18. 19. Use MATLAB’S LTi Viewer to find the
gain margin, phase margin; zero dB
frequency, and 180° frequency for a
unity feedback system with q 10,000
(5 + 5)(s+18)(s + 30) Use the following methods: GUI Too! 6(5) = a. The Nyquist diagram
b. Bode plots Derive Eq. (10.54), the CIOSedloop bandwidth in
terms of C and a)" of a twopole system. [Section: 10.8] For each closed~loop System with the
following performance characteristics, ﬁnd
the closedloop bandwidth: [Section: 10.8] \ﬂl‘iBYPLUS e W .
“"trur Solut'on a. I: = 0.2, T5 = 3 seconds
b. i: = 0.2, Tp .2 3 seconds
c. T, = 4 seconds, TP 2 2 seconds
d. l: 2 0.3, T, = 4 seconds Consider the unity feedback system of stilevPLUS
Figure 10.10. For each 6(3) that follows, ..
use the M and N circles to make a plot of 1 ‘
the closed—loop frequency response: Clint,“ Samuel‘s
[Section: 10.9] 10
a 60 = 1000
b. 00) = W
i 50(5 + 3)
6 0(3)  Repeat Problem 16, using the Nichols chart in place of
the M and N circles. [Section: 109] Using the results of Problem 16, estimate the percent
overshoot that can be expected in the step response for
each system shown. [Section: 10.10] Use the results of Problem 17 to estimate the percent
overshoot if the gain term in the numerator of the
forward path of each part of the problem is respec
tively changed as follows: [Section: 10.10] I .
3. . . c as Chapter 10 Frequency Response Techniques 20. 21. 22. .a. Make a Nichols plot of an open a. From 10 to 30
b. From 1000 to 2500
c. From 50 to 75 Write a program in MATLAB that will
do the following: a. Allow a value of gain, K, to be entered from the
keyboard b. Display the closedloop magnitude and phase
frequency response plots of a unity feedback
system with an openloop transfer function, K 6(5) c. Calculate and display the peak magnitude, fre
quency of the peak magnitude, and bandwidth
for the closedloop frequency response and the ;
entered value of K H Figure P10.5 ,rv Test your program on the system of Figure P105
for K 2 40. Use MATLAB‘s LTI Viewer with the
Nichols plot to find the gain margin,
phase margin, zero dB frequency,
and 180° frequency for a unity feed—
back system with the forward—path
transfer function GUI Too, _ _ 7(5—i—S)
6(5) _s(sl +4s+ 10) Write a «program in MATLAB that will
do the following: loop transfer function b. Allow the user to read the Nichols plot display .
and enter the value of Mp ‘ f c. Make closedloop magnitude and phase plots 1 d. Display the expected values of percent over
shoot, settling time, and peak time e. Plot the closedloop step response Test your program on a unity feedback system with
the fonNardpath transfer function 6(5) = 23. Using Bode plots, estimate the transient response of
the systems in Figure P106. [Sectionz l0.10] 24. For the system of Figure Pl0.5, do the following:
[Section: 10.10] Problems 593
ii" “i if c. Use MATLAB orany other program to wA‘l'LAB
check your assumptions by simulat— ing the step response of the system.
thiewas 7(5 + 5): 5(52 + 45 + 10) 25. The Bode plots for a plant, C(s), used
in a unity feedback system are shown in Figure 1310.7. Do the following: nlrul Solutions 3. Find the gain margin, phase margin, zero dB
frequency, 180° frequency, and the closedloop
bandwidth. b. Use your results in Pan a to estimate the damping
ratio, percent overshoot, settling time, and peak
time. 50(s + 3)(s + 5) 3(3 + 2)(s + 4)(s + 6) 26. Write a program in MATLAB that will minus
use an openloop transfer function, 6(5), to do the following: System 2
Figu re P10.6l a. Make a Bode plot b. Use frequency response methods to estimatethe
percent overshoot, settling time, and peak time c. Plot the closedloop step response a. Plot the Bode magnitude and phase plots. b. Assuming a secondorder approximation, estimate
the transient response of the system if K: 40. Phase (degrees) Test your program by comparing the results to
those obtained for the systems of Problem 23. ESI!!!!!!IIIIIIIIIIIIII
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40 .IIIIIIIIIIIIIIIEEJIIIIII
40 .IIIIlllIlllllhslﬂlll
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400 IIIIIIIIIIIIIIIIIIIIIIII 0.1 l 10 100 Frequency (rad/s) IIIIIIIIIIIIIIIlIIIIIIII
“00 iilliiiIllllllIIIIIIII
—150 "  “' a.
so, IIIIIIIlIIIl..lIIIIIII IIIIIIIlllllllllh'lllllll —250 4 'he n“.. IIIIIIIlIIIIIIlIIIIIIII
0.] 1 IO 100 Frequency (rad/s)
Fig ure P1 0.7 594 27. The open—loop frequency response shown in Fig
ure P108 was experimentally obtained from a unity
feedback system. Estimate the percent overshoot and steady—state error of the closed—loop system. [Sec
tions: 10.10, 10.11] 28. Consider the system in Figure P109. [Section: 10.12] 100
(3+ 5)(s +10) Figure P10.9 a. Find the phase margin if the system is stable for
time delays of 0, 0.1, 0.2, 0.5, and 1 second. b. Find the gain margin if the system is stable for each
of the time delays given in Part a. C. For what time delays mentioned inPart a is the
system stable? (I. For each time delay that makes the system
unstable, how much reduction in gain is required
for the system to be stable? 60
40 EEIIIIIIIIIIEII l i
l
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l
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0.1 29. 30. 31. 

a




 Chapter 10 Frequency Response Techniques Given a unity feedback system with the
forwardpath transfer function K
G s = ——————
U (s +1)(s+ 3)(s+,6)
and a delay of 0.5 second, find the range of gain, K, to
yield stability. Use Bode plots and frequency response
techniques. [Section: 10.12]— Given a unity feedback system with the forwardpath
transfer function W) = K 3(3 + 3)(s+ 12) and a delay of 0.5 second, make a secondorder
approximation and estimate the percent overshoot
if K = 40. Use Bode plots and frequency response
techniques. [Section: 10.12] ﬂ Use the MATIAB function pade(T,n) to modei
the delay in Problem 30. Obtain the
unit step response and evaluate your secondorder approximation in
Problem 30. “Mus IIIIIIIIIIIIIII
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10 Frequency (rad/s) 450 _IIIIII 400 .—.—....mgqunIIllllllllIIIIIIII
4,0 IIIIIIIIIn.E!l!lIlllllllIIIIIIIII IIIIllllIIIIIIIESIIIIIIIIIIIIIIII
IIIIIIllIIIIIIIIﬂIIIIIIIlIIIIIIII —l40 Phase (degrees)
.1; a
8 93
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I


E
7‘
I

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E
I
I



= IIIIIIIIIIIIIIIIIlilllllIIIIIIII
"l
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IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII
0.1 1 10 100 Frequency (rad/s) ‘25 Figure P1D.8 Problems 595 
80
60 g!!!“ III l I IIIIII
40 IIIl! II I IIIIII
i 20 IIIIII I IIIIII
g 0 IIIII  IIII IIIIII
,20 IIIIII II IIIL! !III
40 IIIIIIIIIIIIIIIIII IIIIII I!!!“
_6o __ IIIIIIIIIIIIIIIIII IIIIII IIIIII
0.0l 0.1 1 IO EGG
Frequency (rad/s)
—80 """" lIIIIIIIIII
400 IIIIIII—!!_ _. IIIIIIIIIIIIIIIIIILIIIIllllIIIIIIII IIIIIIIIIIIlllllIi"lIIIII
3
E —140 i... _‘
A .IIIIIIIlIIIIIIllIIIIIIIINIIIIIIII 0.01 O. IIIIIIIIIIIIIIIIIIn!!!!l.
1 1 10 100 Frequency (radjs)
Figure P10.10 32. For the Bode plots shown in Figure 1510.10, determine 34. An overhead crane consists of a horizontally moving
the transfer function by hand or via MATLAB. trolley of mass mrdragging a load of mass mL, which
[Section: 10.13] dangles from its bottom surface at the end of a rope of ﬁxed length, L. The position of the trolley is 33. Repeat Problem 32 for the Bode plots shown in Figure
P10.ll. [Section: 10.13] m ' IIIII
IIIII ll
0 II l llllllll
v —100 f i—ﬁ
% IIIIIIIIIIII IIIIII
E II 100 1 l0 Frequency (rad/s)
Figure P10.11 596 35. controlled in the feedback conﬁguration shown in
Figure 10.20. Here, 0(5) = KP(s), H "= 1, and Xﬂs) __F "1— 32 + (05 HS) = FMS) — m1" s2 (52 + awﬁ) The input is fT (t), the input force applied to the trolley.
The output is x70), the trolley displacement. Also, we = ﬁanda = (mL + mﬂ/m:n (Marrtt'nen, 1990). Make a qualitative Bode plot of the system assuming» a>1. A room’s temperature can be controlled by varying  the radiator power. Inia speciﬁc room, the transfer 36. 37. function from indoor radiator power, Q, to room
temperature, Tin °C is (Thomas, 2005) __ T(s)
Qts)
(1 x 106);2 + (1.314 x 10‘9)s+ (2.66 x 1013) : s3 +0.00163s2 + (5.272‘Y107)s + (3.538 x 10“) The system is controlled in the closedloop conﬁg
uration shown in Figure 10.20 with C(s) = KP(s),
H = l. a. Draw the corresponding Nyquist diagram for
K = l. b. Obtain the gain and phase margins. c. Find the range of K for the closedloop stability.
Compare your result with that of Problem 61,
Chapter 6. The openloop dynamics from dc voltage armature to
angular position of a robotic manipulator joint is 48500 given by P(s) z m (Low, 2005). a. Draw by hand a Bode plot using asymptotic approximations for magnitude and phase.
“mus b. Use MATLAB to plot the exact Bode
plot and compare with your sketch
from Part a. Problem 49, Chapter 8 discusses a magnetic levitation
system with a plant transfer function P(s) = 1300
m (Galvfto, 2003). Assume that the plant is in cascade with an M(s) and that the system will be controlled by the loop shown in Figure 10.20, where
G(s) = M (§)P(s) and H: 1. For each M(s) shown Chapter 10 Frequency Response Techniques 38. 39. P0) = 40. below, draw the Nyquist diagram when K = 1, and
ﬁnd the range of closedloop stability for K > 0. a: M5) = .K
K(s + 200)
s + 1000 c. Compare your results with those obtained in
Problem 49, Chapter 8. b. Mo) = — The simpliﬁed and linearized model for the transfer
function of a certain bicycle from steer angle (8) to
roll angle (cp) is given by (Astrom, 2005) _ 99(3) _ 10(5 + 25) PmTW— 52125 Assume the rider can be represented by a gain K, and
that the closed100p system is shown in Figure 10.20
with G(s) = KP(s) and H = 1. Use the Nyquist sta
bility criterion to ﬁnd the range of K for closedloop
stability. The control of the radial pickup position of a digital
versatile disk (DVD) was discussed in Problem 48,
Chapter 9. There, the openloop transfer function
from coil input voltage ~to radial pickup position
was given as (Bittann‘, 2002) " 0.63 t
e1+0.36 + s2 1+0.04 + 5‘2
305.45 305.42 248525 243.32 Assume the plant is in cascade with a controller,
_ 0.5(s + 1.63) Mm s(s + 0.27) and in the closedloop conﬁguration shown in Figure
10.20, where 0(5) = M(s)P(s) and H = 1. Do the
following: a. Draw the openloop frequency response in a
Nichols chart. b. Predict the system’s response to a unit step input.
Calculate the %0S, cﬁnal, and Ts. c. Verify the results of Part b using
MATLAB simulations. “ATLAE The soft Arm, used to feed people with disabilities,
was discussed in Problem 57 in Chapter 6. Assuming
the system block diagram shown in Figure P1012, use frequency response techniques to determine the fol
lowing (Kara, I992): 41. 42. Desired
spoon
position Controller 1000§s + 0.01 its +6!
3(5' + 20)(s + 100) Desired
position Controller
X0014
." 1001 000 I (s + SOON: + 800) I s(s + 100) I I Problems 597 Actual
spoon
position
C(r) Robot arm Figure P10.13 Floppy disk drive block diagram a. Gain margin, phase margin, zero dB frequency,
and 180° frequency ' b. Is the system stable? Why? A ﬂoppy disk dIiVe was discussed in
Problem 55 in Chapter 8. Assuming
the system block diagram shown in
Figure P10. 13, use frequency response
techniques to determine the following: a. Gain margin, phase margin, zero dB frequency,
180° frequency, and closed—loop bandwidth b. Percent overshoot, settling time, and peak time c. Use MATLAB to simulate the closed tamAs
mlgop step response and compare the
results to those obtained in Part b. Industrial robots, such as that shown in Figure P10. 14,
require accurate models for design of high perfor
mance. Many transfer function models for industrial
robots assume interconnected rigid bodies with the
drivetorque source modeled as a pure gain, or ﬁrst
order system. Since themotith associated with the
robot are connected to the drives through ﬂexible
linkages rather than rigid linkages, past modeling
does not explain the resonances observed. An accu
rate, smallmotion, linearized model has been devel
oped that takes into consideration the ﬂexible drive.
The transfer function (s2 + 8.945 + 44.72) 6(5) .= 99913Wm relates the angular velocity of the robot base to
electrical current commands (Good, 1985). Make a Bode plot of the frequency response and identify the
resonant frequencies. . The chargecoupled device (CCD) that is used in video movie cameras to convert images into electrical
signals can be used as part of an automatic focusing
system in cameras. Automatic focusing can be imple
mented by focusing the center of the image on a
chargecoupled device array through two lenses.
The separation of the two images on the CCD is
related to the focus. The camera senses the separation,
and a computer drives the lens and focuses the image. FIGURE P10.14 AdeptOne, a four or fiveaxis industrial robot,
is used for assembly, packaging, and other manufacturing tasks. r'iril‘ .i "‘M
n l l ‘
! ﬁl "I ll n: .9—.——‘ .. _. 598 Desired
lens 1mm“ + Computer and CCD _ system .a....;.:‘.:s,.c FIGURE P10.15 a. A cutaway view of a Nikon 35min camera showing parts of the CCD automatic focusing system; 1}. functional block diagram; c. block diagram the three references to Popular Photography in this
chapter’s Bibliography.) The automatic focus system
is a position control, where the desired position of the
lens is an input selected by pointing the camera at the
subject. The output is the actual position of the lens.
The Nikon camera in Figure P10.15(a) uses a CCD
automatic focusing system. Figure P10.15(b) shows
the automatic focusing feature represented as a posi
tion control system. Assuming the simpliﬁed model Desired Roll
roll angle angleﬂd“) + Transducer FigUre P10.‘6 Block diagram of a ship’s rollstabilizing system Chapter 10 Frequency Response Techniques Actual
lens
position shown in Figure P10.15(c), draw the Bode plots and
estimate thepercent overshoot for a step input. 44. A ship’s roll can be stabilized with a control system. A
voltage applied to the ﬁns’ actuators creates a roll torque
that is applied to the ship. The ship, in response to the
roll torque, yields a roll angle. Assuming the block dial
gram for the roll control system shown in Figure P1016,
determine the gain and phase margins for the system. Ship
F11] roll ' Actual
actuatOr dynamIcs mur
angle, 60(5) 4
52 + O.9s+9 45. The linearized model of a particular netWork link working under TCP/IP and controlled using a random
early detection (RED) algorithm can be described by
Figure 1020 where G(s) = M (s)P(s), H = l, and
(Holler, 2001) 00051 14062534115
Mm _ s + 0.005 ’ P“) ” (s + 2.67)(s +10) a. Plot the Nichols chart for L = 1. Is the system
closed100p stable?
b. Find the range‘ of L for closedloop stability. c. Use the Nichols chart to predict %OS and T5 for L
= 0.95. Make a hand sketch of the expected unit
step response. d. Verify Part c with a Simulink unit
step response simulation. gunu link 46. In the TCP/IP network link of Problem 45, let L = 0.8,
but assume that the amount of delay is an unknown
variable. a. Plot the Nyquist diagram of the system for z‘ero
delay, and obtain‘ the phase margin. b. Find the maximum delay allowed for closedloop
stability. 47. Thermal ﬂutter of the Hubble Space Telescope (HST)
produces errors for the pointing control system. Ther
mal flutter of the solar arrays occurs when the space
em: passes from sunlight to darkness and when the
spacecraft is in daylight. In passing from daylight to
darkness, an endtoend bending'oscillation of fre—
quency f1 rad/s is experienced. Such oscillations
interfere with the pointing control’ system of the
HST. A ﬁlter with the transfer function G (S) _ 196(52 +5 + 0.25)(s2 +1.26s+ 9.87)
f _ (s2 + 0.0155 + 0.57)(s% + 0.0835 +172) is proposed to be placed in cascade with the controller to reduce the bending (Wie, 1992). a. Obtain the frequency response of the ﬁlter and
estimate the bending frequencies that will be
reduced. b. Explain why this ﬁlter will reduce the bending
oscillations if these oscillations are thought to Problems 599 'ts be disturbances at the output of the Control
system. Progressive Analysis and Design Problems 48. Highspeed rail pantograph. Problem 20 in Chapter 1 discusses active control of a pantographu
mechanism for highspeed rail systems. In Problem
74(a), Chapter 5, you found the block diagram for the
active pantograph control system. In Chapter 8, Pro—
blem 64, you designed the gain to yield a closedloop
step response with 30% overshoot. A plot of the step
response should have shown a settling time greater
than 0.5 second as well as a highfrequency oscillation
superimposed over the step response. In Chapter 9,
Problem 53, we reduced the settling time to about 0.3
second, reduced the step response steady—state error to
icro, and eliminated the highfrequency oscillations
by using a notch ﬁlter (O’Connor, 1997). Using the
equivalent forward transfer function found in Chapter
5 cascaded with the notch ﬁlter speciﬁed in Chapter 9,
do the following using frequency response techniques: a. Plot the Bode plots for a total equivalent gain of l
and ﬁnd the gain margin, phase margin, and 180°
frequency. b. Find the range of K for stability. c. Compare your answer to Part b with your
answer to Problem 65, Chapter 6. Explain any
differences. 9. Control of HIVIAIDS. The linearized model for an HIV/AIDS patient treated with RTIs was obtained in
Chapter 6 as (Craig, 2004); Yo) _ —520s — 10.3844
U1(s) _ s3 4; 2.681752 + 0.11s + 0.0126 P0) 3 a. Consider this plant in the feedback conﬁguration in
Figure 10.20 with G(s) = P(s) and H(s) = 1.
Obtain the Nyquist diagram. Evaluate the system
for closedloop stability. Vb. Consider this plant in the feedback conﬁguration in Figure 10.20 with 0(5) = —P(s) and H(s) : 1.
Obtain the;Nyquist diagram. Evaluate the system
for closedloop stability. Obtain the gain and phase
margins. ...
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