Control Systems Engineering

Norman S. Nise

Chapter 1

Introduction - all with Video Answers

Educators

Chapter Questions

Problem 1

A variable resistor, called a potentiometer, is shown in Figure Pl.1. The resistance is varied by moving a wiper arm along a fixed resistance. The resistance from $A$ to $C$ is fixed, but the resistance from $B$ to $C$ varies with the position of the wiper arm. If it takes 10 tums to move the wiper arm from $A$ to $C$, draw a block diagram of the potentiometer showing the input variable, the output variable, and (inside the block) the gain, which is a constant and is the amount by which the input is multiplied to obtain the output

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 2

A tenperature control system operates by sensing the difference between the thermostat setting and the actual temperature and then opcring a fuel valve an amount proportional to this difference. Draw a functional closed-loop block diagram similar to Figure $1.9(d)$ identifying the input and output transducers, the controller, and the plant. Further, identify the input and output signals of all subsysterns previously clescribed.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 3

An aircraft's attitude varies in roll, pitch, and yaw as defined in Figure $\mathrm{p} 1.2$ Draw a functional block diagram for a closed-loop system that stabilizes the roll as follows: The system measures the actual roll angle with a gyro and compares the actual roll angle with the desired roll angle. The ailerons respond to the roll-angle error by undergoing an angular deflection. The aircraft responds to this angular deflection, producing a roll angle rate. Identify the input and output transducers, the controller, and the plant. Further, identify the nature of each signal

James Kiss

Numerade Educator

Problem 4

Many processes operate on rolled material that moves from a supply reel to a take-up recl. Typically, these systems, called winders, coritrol the material so that it travels ar a constant velocity, Beside velocity, complex winders also control tension, compensate for roll inertia while accelerating or decelerating, and regulate acceleration due to sudden changes, A winder is shown in Figure P1.3. The force transducer measures tension; the winder pulls against the mip rolls, which provide an opposing force; and the bridle provides slip. In order to compensate for changes in speed, the material is looped around a dancer. The loop prevents rapid changes from causing excessive slack or darnaging the material If the dancer position is sensed by a potentiometer or other device, speed variations due to buildup on the take up reel or other causes can be controlled by comparing the potentiometer voltage to the commanded speed, The system then corrects the speed and rescts the dancer to the desired position (Ayers, 1988 ). Draw a functional block diagram for the speed control system, showing each component and signal.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 5

A university wants to establish a cootrol systern model that represents the student population as an outpur, with the desired student population as an input. The administration determines the rate of adrnissions by comparing the current and desired student populations. The admissions office then uses this rate to admit studenss. Draw a functional block diagrarn showing the administration and the admissions office as blocks of the system. Also show the following signals:
the desired student population, the actual student population, the desired student rate as determined by the administration, the actual student rate us generated by the admissions office, the dropout rate, and the net rate of influx.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 6

In a nuclear power generating plant, heat from a reactor is used to generate steam for turbines. The rate of the fission reaction determines the amount of heat generated, and this rare is controlled by rods inserted into the radioactive core. The rods regulate the flow of neutrons. If the rods are lowered into the core. the rate of fission will diminish; if the rods are raised, the fission rate will increase. By automatically controlling the position of the rods, the amount of heat generated by the reactor can be regulated. Draw a functional block diagram for the nuclear reactor control system shown in Figure PI.4. Show all blocks and signals

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 7

We can build a control system that will automatically adjust a motorcycle's radio volume as the noise generated by the motorcycle changes, The noise generated by the motorcycle increases with speed. As the noise increases, the system increases the volume of the radio. Assume that the amount of noise can be represented by a voltage generated by the speedometer cable, and the volume of the radio is controlled by a de voltage (Hogan, 1988 ). If the de voltage repre sents the desired volume disturbed by the motorcycle noise, draw the functional block diagram of the automatic vobime control xystem, showing the input transducer, the volume control carcuit, and the speed transducer as blocks. Also show the following signals: the desired volune as an input, the actual volume as an cutput, and voltages representing speed, desired volume, and actual volume.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 8

Your bathtub at home is a control system that keeps the water level constant. A constant flow from the tap yickds a constant water level, because the flow rate through the drain increases as the water level increases, and decreases as the water level decreases. After equilibrium has been reached, the level can be controlled by controlling the input flow rate. A low input flow rate yields a lower level, while a higher input flow rate yields a higher level.
a. Sketch a control system that uses this principle to precisely control the fluid Icvel in a tank. Show the intake and drain valves. the tank, any sensors and transducers, and the interconnection of all corrponerits.
b. Draw a functional bluck diagram of the system, identifying the input and output signals of esch block.

Lainey Roebuck

Numerade Educator

Problem 9

Your bathtub at home is a control system that keeps the water level constant. A constant flow from the tap yickds a constant water level, because the flow rate through the drain increases as the water level increases, and decreases as the water level decreases. After equilibrium has been reached, the level can be controlled by controlling the input flow rate. A low input flow rate yields a lower level, while a higher input flow rate yields a higher level.
a. Sketch a control system that uses this principle to precisely control the fluid Icvel in a tank. Show the intake and drain valves. the tank, any sensors and transducers, and the interconnection of all corrponerits.
b. Draw a functional bluck diagram of the system, identifying the input and output signals of esch block.

Lainey Roebuck

Numerade Educator

Problem 10

During a medical operation an anesthesiologist controls the depth of unconsciousness by controlling the concentration of isoflurane in a vaporized mixture with oxygen and nitrous oxide. The depth of anexthesia is measured by the patient's blood pressure. The anesthesiologist also regulates ventilation, fluid bal ance, and the administration of other drugs In order to free the anesthesiologist to devote more time to the latter tashs, and in the interest of the patient's safety. we wish to automate the depth of anesthesia by autornating the control of isoflurure concentration. Draw a functional block diagram of the systern showing pertinent signals and subsysterns (Meier, 1992 )

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 11

The vertical position, $x(t)$, of the grinding wheel shown in Figure P1.5 is controlled by a closed-loop system. The input to the systern is the desired depth of grind, and the output is the actual depth of grind. The difference between the desired depth and the actual depth drives the motor, resulting in a force applied to the work. This force results in a feed velocity for the grinding wheel (Jenkins.
1997). Draw a closed-loop functional block diagram for the yrinding process showing the input, output, force, and grinder fced rate.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 12

A high-speed proportional solenoid valve is shown in Figure $\mathrm{P}$ 1.6. A voltage proportional to the desired position of the spool is applied to the coil. The resulting magnetic field produced by the current in the coil causes the armature to move. A push pin connected to the armature moves the spool. A linear voltage differential transformer (LVDT) that outputs a voltage proportional to clisplacement senses the spool's position. This voltage can he used in a feedback path to impiement closed-loop operation (Vaughan, 1996 ). Draw a functional block diagram of the valve, showing input and output positions, coil voltage, coil cur rent, and spool force.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 13

Given the electric network shown in Figure $\mathbf{P}$ I. 7 .
a. Write the differential equation for the network if $v(t)=u(t)$, a unit step.
b. Solve the differential equation for the current. if . if there is no initial energy in the network.
c. Make a plot of your solution if $\boldsymbol{R} \boldsymbol{L}=1$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 14

Repeat Problem 13 using the network shown in Figure $P$ I. 8 . Assume $R=$ $1 \Omega, L-n 5 H$ and $1<C=30$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 15

Solve the following differcntial equations using classical meahods. Assume zero initial conditions. a. $\frac{d x}{d t}+7 x=5 \cos 2 t$
b. $\frac{d^{2} x}{d t^{2}}+6 \frac{d x}{d t}+8 x=5 \sin 3 t$
c. $\frac{d^{2} x}{d t^{2}}+8 \frac{d x}{d t}+25 x=10 u(t)$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 16

Solve the following differential equations using classical methods and the given initial conditions: a. $\frac{d^{2} x}{d t^{2}}+2 \frac{d x}{d t}+2 x=\sin 2 t$
$x(0)=2 ; \quad \frac{d x}{d t}(0)=-3$
b. $\frac{d^{2} x}{d t^{2}}+2 \frac{d x}{d t}+x=5 e^{-2 t}+t$
$x(0)=2 ; \quad \frac{d x}{d t}(0)-1$
c. $\frac{d x}{d t^{2}}+4 x=t^{2}$
$x(0)=1 ; \quad \frac{d x}{d t}(0)=2$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 17

Some high-speed rail systems are powered by electricity supplied to a pantograph on the train's roof from a catenary overhead, as shown in Figure $\mathrm{Pl}$.9. The force applied by the pantograph to the catenary is regulated to avoid loss of contact due to excessive transient motion. A proposed method to regulate the force uses a closed-loop feedback system, whereby a force, $F_{\text {up }},$ is applied to the botom of the pantograph, resulting in an output force applied to the cateriary at the top. The contact between the head of the pantograph and the catenary is represented by a spring. The output force is proportional to the displacement of this spring, which is the difference between the catenary and pan- tograph head vertical positions (O'Connor, 1997 ). Draw a functional block diagram showing the following signals: the desired output force as the input; the force, $F_{u p}$ applied to the bottom of the pantograph: the difference in clisplacement between the catenary and pantograph head: and the output contact force. Also, show blocks representing the input transducer, controller, actuator generating $F_{\mathrm{up}}$ pantograph dynamics, spring described above, and output sensor. All forces and displacements are measured from equilibrium.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator