Solution Manual for Feedback Control of Dynamic Systems, 8th Edition

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100
Solutions Manual: Chapter 1
8th Edition
Feedback Control of Dynamic
Systems.
.
Gene F. Franklin
.
J. David Powell
.
Abbas Emami-Naeini
.
.
.
.
Assisted by:
H. K. Aghajan
H. Al-Rahmani
P. Coulot
P. Dankoski
S. Everett
R. Fuller
T. Iwata
V. Jones
F. Safai
L. Kobayashi
H-T. Lee
E. Thuriyasena
M. Matsuoka

Chapter 1
An Overview and Brief
History of Feedback Control
1.1 Problems and Solutions
1. Draw a component block diagram for each of the following feedback control
systems.
(a) The manual steering system of an automobile
(b) Drebbelís incubator
(c) The water level controlled by a áoat and valve
(d) Wattís steam engine with áy-ball governor
In each case, indicate the location of the elements listed below and
give the units associated with each signal.
the process
the process desired output signal
the sensor
the actuator
the actuator output signal
the controller
the controller output signal
the reference signal
the error signal
Notice that in a number of cases the same physical device may per-
form more than one of these functions.
Solution:
(a) A manual steering system for an automobile:
101

1.1. PROBLEMS AND SOLUTIONS 103
2. Identify the physical principles and describe the operation of the thermo-
stat in your home or o¢ ce.
Solution:
A thermostat is a device for maintaining a temperature constant at a
desired value. It is equipped with a temperature sensor which detects
deviation from the desired value, determines whether the temperature
setting is exceeded or not, and transmits the information to a furnace or
air conditioner so that the temperature in the room is brought back to the
desired setting. Examples: Tubes Ölled with liquid mercury are attached
to a bimetallic strip which tilt the tube and cause the mercury to slide
over electrical contacts. A bimetallic strip consists of two strips of metal
bonded together, each of a di§erent expansion coe¢ cient so that temper-
ature changes bend the metal. In some cases, the bending of bimetallic
strips simply cause electrical contacts to open or close directly. In most
cases today, temperature is sensed electronically using,for example, a ther-
mistor, a resistor whose resistance changes with temperature. Modern
computer-based thermostats are programmable, sense the current from
the thermistor and convert that to a digital signal.
Fig 1.12 A Paper Making Machine

104CHAPTER 1. AN OVERVIEW AND BRIEF HISTORY OF FEEDBACK CONTROL
3. A machine for making paper is diagrammed in Fig. 1.12. There are two
main parameters under feedback control: the density of Öbers as controlled
by the consistency of the thick stock that áows from the headbox onto the
wire, and the moisture content of the Önal product that comes out of the
dryers. Stock from the machine chest is diluted by white water returning
from under the wire as controlled by a control valve (CV). A meter supplies
a reading of the consistency. At the ìdry endî of the machine, there is a
moisture sensor. Draw a signal graph and identify the seven components
listed in Problem 1 for
(a) control of consistency
(b) control of moisture
Solution:
(a) Control of paper machine consistency:
(b) Control of paper machine moisture:
4. Many variables in the human body are under feedback control. For each
of the following controlled variables, draw a graph showing the process
being controlled, the sensor that measures the variable, the actuator that
causes it to increase and/or decrease, the information path that completes
the feedback path, and the disturbances that upset the variable. You may
need to consult an encyclopedia or textbook on human physiology for
information on this problem.

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1.1. PROBLEMS AND SOLUTIONS 105
(a) blood pressure
(b) blood sugar concentration
(c) heart rate
(d) eye-pointing angle
(e) eye-pupil diameter
Solution:
Feedback control in human body:
Variable Sensor Actuator Inform ation path Disturbances
a) Blood pressure -Arterial -Cardiac output -A§erent nerve -Bleeding
baroreceptors -Arteriolar/venous Öb ers -Drugs
dilation -Stress,Pain
b) Blood sugar -Pancreas -Pancreas secreting -Blood áow to -Diet
concentration insulin pancreas -Exercise
(Glucose)
c) Heart rate -Diastolic volum e -Electrical stimulation -M echanical draw -Horm one release
sensors of sino-atrial node of blood from heart -Exercise
-Cardiac sym pathetic and cardiac muscle -Circulating
nerves epinephrine
d) Eye p ointing -Optic nerve -Extraocular muscles -Cranial innervation -Head m ovem ent
angle -Im age detection -M uscle twitch
e) Pupil diam eter -Rods -Pupillary sphincter -Autonom ous -Ambient light
muscles system -Drugs
f ) Blood calcium -Parathyroid gland -Ca from b ones to blood - Parathorm one -Ca need in b ones
level detectors -Gastrointestinal horm one a§ecting -Drugs
absorption e§ector sites
5. Draw a graph of the components for an elevator-position control. Indi-
cate how you would measure the position of the elevator car. Consider a
combined coarse and Öne measurement system. What accuracies do you
suggest for each sensor? Your system should be able to correct for the
fact that in elevators for tall buildings there is signiÖcant cable stretch as
a function of cab load.
Solution:
A coarse measurement can be obtained by an electroswitch located before
the desired áoor level. When touched, the controller reduces the motor
speed. A ìÖneî sensor can then be used to bring the elevator precisely
to the áoor level. With a sensor such as the one depicted in the Ögure,
a linear control loop can be created (as opposed to the on-o§ type of the
coarse control).Accuracy required for the course switch is around 5 cm;
for the Öne áoor alignment, an accuracy of about 2 mm is desirable to
eliminate any noticeable step for those entering or exiting the elevator.

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106CHAPTER 1. AN OVERVIEW AND BRIEF HISTORY OF FEEDBACK CONTROL
6. Feedback control requires being able to sense the variable being controlled.
Because electrical signals can be transmitted, ampliÖed, and processed
easily, often we want to have a sensor whose output is a voltage or current
proportional to the variable being measured. Describe a sensor that would
give an electrical output proportional to:
(a) temperature
(b) pressure
(c) liquid level
(d) áow of liquid along a pipe (or blood along an artery) force
(e) linear position
(f) rotational position
(g) linear velocity
(h) rotational speed
(i) translational acceleration
(j) torque
Solution:
Sensors for feedback control systems with electrical output. Exam-
ples
(a) Temperature: Thermistor- temperature sensitive resistor with resis-
tance change proportional to temperature; Thermocouple; Thyristor.
Modern thermostats are computer controlled and programmable.
(b) Pressure: Strain sensitive resistor mounted on a diaphragm which
bends due to changing pressure

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1.1. PROBLEMS AND SOLUTIONS 107
(c) Liquid level: Float connected to potentiometer. If liquid is conductive
the impedance change of a rod immersed in the liquid may indicate
the liquid level.
(d) Flow of liquid along a pipe: A turbine actuated by the áow with a
magnet to trigger an external counting circuit. Hall e§ect produces
an electronic output in response to magnetic Öeld changes. Another
way: Measure pressure di§erence from venturi into pressure sensor
as in Ögure; Flowmeter. For blood áow, an ultrasound device like a
SONAR can be used.
(e) Position.
When direct mechanical interaction is possible and for ìsmallî dis-
placements, the same ideas may be used. For example a potentiome-
ter may be used to measure position of a mass in an accelerator (h).
However in many cases such as the position of an aircraft, the task is
much more complicated and measurement cannot be made directly.
Calculation must be carried out based on other measurements, for
example optical or electromagnetic direction measurements to several
known references (stars,transmitting antennas ...); LVDT for linear,
RVDT for rotational.
(f) Rotational position. The most common traditional device is a poten-
tiometer. Also common are magnetic machines in which a rotating
magnet produces a variable output based on its angle.
(g) Linear velocity. For a vehicle, a RADAR can measure linear velocity.
In other cases, a rack-and-pinion can be used to translate linear to
rotational motion and an electric motor(tachometer) used to measure
the speed.
(h) Speed: Any toothed wheel or gear on a rotating part may be used to
trigger a magnetic Öeld change which can be used to trigger an elec-
trical counting circuit by use of a Hall e§ect (magnetic to electrical)
sensor. The pulses can then be counted over a set time interval to
produce angular velocity: Rate gyro; Tachometer

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108CHAPTER 1. AN OVERVIEW AND BRIEF HISTORY OF FEEDBACK CONTROL
(i) Acceleration: A mass movement restrained by a spring measured by
a potentiometer. A piezoelectric material may be used instead (a ma-
terial that produces electrical current with intensity proportional to
acceleration). In modern airbags, an integrated circuit chip contains
a tiny lever and íproof massíwhose motion is measured generating a
voltage proportional to acceleration.
(j) Force, torque: A dynamometer based on spring or beam deáections,
which may be measured by a potentiometer or a strain-gauge.
7. Each of the variables listed in Problem 6 can be brought under feedback
control. Describe an actuator that could accept an electrical input and be
used to control the variables listed. Give the units of the actuator output
signal.
Solution:
(a) Resistor with voltage applied to it or mercury arc lamp to generate
heat for small devices. a furnace for a building..
(b) Pump: Pumping air in or out of a chamber to generate pressure.
Else, a ítorque motoríproduces force..
(c) Valve and pump: forcing liquid in or out of the container.
(d) A valve is normally used to control áow.
(e) Electric motor
(f) Electric motor
(g) Electric motor
(h) Electric motor
(i) Translational acceleration is usually controlled by a motor or engine
to provide force on the vehicle or other object.
(j) Torque motor. In this motor the torque is directly proportional to
the input (current).
8. Feedback in Biology
(a) Negative Feedback in Biology: When a person is under long term stress
(say a couple of weeks before an exam!), hypothalamus (in the brain) se-
cretes a hormone called CRF (Corticotrophin Releasing Factor) which
binds to a receptor in the pituitary gland stimulating it to produce ACTH

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1.1. PROBLEMS AND SOLUTIONS 109
(Adrenocorticotropic hormone), which in turn stimulates the adrenal cor-
tex (outer part of the adrenal glands) to release the stress hormone Glu-
cocorticoids (GC). This in turn shuts down (turns o§ the stress response)
for both CRF and ACTH production by negative feedback via the blood-
stream until GC returns to its normal level. Draw a block diagram of this
closed-loop system.
(b) Positive Feedback in Biology: This happens in some unique circum-
stances. Consider the birth process of a baby. Pressure from the head
of the baby going through the birth canal causes contractions via secre-
tion of a hormone called Oxytocin which causes more pressure which in
turn intensiÖes contractions. Once the baby is born, the system goes back
to normal (negative feedback). Draw a block diagram of this closed-loop
system.
Solution:
(a) Negative Feedback in Biology - Stress
Stress induced negative feedback
(b) Positive Feedback in Biology - Child birth
Child birth induced positive feedback

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2000
Solutions Manual: Chapter 2
8th Edition
Feedback Control
of Dynamic Systems
.
.
Gene F. Franklin
.
J. David Powell
.
Abbas Emami-Naeini
.
.
.
.
Assisted by:
H. K. Aghajan
H. Al-Rahmani
P. Coulot
P. Dankoski
S. Everett
R. Fuller
T. Iwata
V. Jones
F. Safai
L. Kobayashi
H-T. Lee
E. Thuriyasena
M. Matsuoka
Copyright (c) 2019 Pearson Education
No part of this publication may be reproduced, stored
in a retrieval system, or transmitted, in any form or by any means, electronic,
mechanical, photocopying, recording, or otherwise, without the prior permission
of the publisher.

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Chapter 2
Dynamic Models
Problems and Solutions for Section 2.1
1. Write the di§erential equations for the mechanical systems shown in Fig. 2.43.
For (a) and (b), state whether you think the system will eventually decay
so that it has no motion at all, given that there are non-zero initial condi-
tions for both masses, and give a reason for your answer. Also, for part
(c), answer the question for F=0.
2001

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2002 CHAPTER 2. DYNAMIC MODELS
Fig. 2.43 Mechanical systems
Solution:
The key is to draw the Free Body Diagram (FBD) in order to keep the
signs right. For (a), to identify the direction of the spring forces on the
object, let x2 = 0 and Öxed and increase x1 from 0. Then the k1 spring
will be stretched producing its spring force to the left and the k2 spring
will be compressed producing its spring force to the left also. You can use
the same technique on the damper forces and the other mass.
(a)m1 m2
x1 x2
k (x -x )2 1 2
k (x -x )2 1 2
k (x3 2 - y)k x1 1
b x1 1
. Free body diagram for Problem 2.1(a)
m1 x1 = k1x1 b1 _x1 k2 (x1 x2)
m2 x2 = k2 (x2 x1) k3 (x2 y)
There is friction a§ecting the motion of mass 1 which will continue
to take energy out of the system as long as there is any movement of
x1:Mass 2 is undamped; therefore it will tend to continue oscillating.
However, its motion will drive mass 1 through the spring; therefore,
the entire system will continue to lose energy and will eventually
decay to zero motion for both masses.

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2003m1 m2
x1 x2
x2k (x -x )2 1 2
k (x -x )2 1 2
k x1 1
b x1 2
.
x2
x2k3
Free body diagram for Problem 2.1(b)
m1 x1 = k1x1 k2(x1 x2)
m2 x2 = k2(x2 x1) b1 _x2 k3x2
Again, there is friction on mass 2 so there will continue to be a loss
of energy as long as there is any motion; hence the motion of both
masses will eventually decay to zero.m1 m2
x1 x2
k (x - x )2 1 2 k (x - x )2 1 2
k x1 1
b (x - x )1 1 2 b (x - x )1 1 2
. .. .
F
Free body diagram for Problem 2.1 (c)
m1 x1 = k1x1 k2(x1 x2) b1( _x1 _x2)
m2 x2 = F k2(x2 x1) b1( _x2 _x1)
The situation here is similar to part (a). It is clear that the relative
motion between mass 1 and 2 would decay eventually, but as long
as mass 1 is oscillating, it will drive some relative motion of the two
masses and that will cause energy loss in the damper. So the entire
system will eventually decay to zero.
2. Write the di§erential equations for the mechanical systems shown in Fig. 2.44.
State whether you think the system will eventually decay so that it has
no motion at all, given that there are non-zero initial conditions for both
masses, and give a reason for your answer.

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2004 CHAPTER 2. DYNAMIC MODELS
Fig. 2.44 Mechanical system for Problem 2.2
Solution:
The key is to draw the Free Body Diagram (FBD) in order to keep the
signs right. To identify the direction of the spring forces on the left side
object, let x2 = 0 and increase x1 from 0. Then the k1 spring on the left
will be stretched producing its spring force to the left and the k2 spring
will be compressed producing its spring force to the left also. You can use
the same technique on the damper forces and the other mass.m1 m2
x1 x2
x2
k (x -x )2 1 2
k x1 1 k1
b (x - x )2 1 2 b (x - x )2 1 2
. . . .
Free body diagram for Probelm 2.2
Then the forces are summed on each mass, resulting in
m1 x1 = k1x1 k2(x1 x2) b1( _x1 _x2)
m2 x2 = k2(x1 x2) b1( _x1 _x2) k1x2
The relative motion between x1 and x2 will decay to zero due to the
damper. However, the two masses will continue oscillating together
without decay since there is no friction opposing that motion and áexure
of the end springs is all that is required to maintain the oscillation of the
two masses. However, note that the two end springs have the same spring
constant and the two masses are equal If this had not been true, the two
masses would oscillate with di§erent frequencies and the damper would
be excited thus taking energy out of the system.
3. Write the equations of motion for the double-pendulum system shown in
Fig. 2.45. Assume the displacement angles of the pendulums are small
enough to ensure that the spring is always horizontal. The pendulum

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