Solution Manual for Modern Control Systems, 14th Global Edition By Richard C. Dorf, Exams of Control Systems

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Chapters 1 - 13, Complete

14th Global Edition By Richard C. Dorf

Solution Manual for Modern Control Systems,

T A B L E - O F - C O N T E N T S

• 1. Introduction to Control Systems
• 2. Mathematical Models of Systems
• 3. State Variable Models
• 4. Feedback Control System Characteristics
• 5. The Performance of Feedback Control Systems
• 6. The Stability of Linear Feedback Systems
• 7. The Root Locus Method
• 8. Frequency Response Methods
• 9. Stability in the Frequency Domain.........................................................
• 10. The Design of Feedback Control Systems.............................................
• 11. The Design of State Variable Feedback Systems.................................
• 12. Robust Control Systems
• 13. Digital Control Systems

2 CHAPTER 1 Introduction to Control Systems

Emitter/

current zworld-record zis z 236 zft). zThe zfly zlure zis zattached zto za zshort zinvisible

z leader z about z 15-ft z long, z which z is z in z turn z attached z to z a z longer z and z thicker

z Dacron z line. z The z objective z is z cast z the z fly z lure z to z a z distant z spot z with z dead-

z eye z accuracy z so z that z the z thicker z part z of z the z line z touches z the z water

z first z andzthen zthe zfly zgently zsettles zonzthe zwater zjust zas zan zinsect zmight.

Actual

z positio

n z of

zthe zfly

E1.4 An zautofocus zcamera zcontrol zsystem:

One-way ztrip ztime zforzthe zbeam

Conversionzfactor

1 (speed zof zlight

zor z sound)

Distance zto zsubject

Lens

Controller

Mind and body of the fly-fisher

cast

Measurement

of

Wind

of (^) -

fly

Visual indication

of the position of

the fly

Exercises 3

E1.5 Tacking z a z sailboat z as z the z wind z shifts:

Desired

z sailboa

t

z directio

n

Actual

z sailbo

at

z directio

n

E1.6 An zautomated zhighway zcontrol zsystem zmerging ztwo zlanes zof ztraffic:

Desired

z gap

Actual

z gap

E1.7 z Using z the z speedometer, z the z driver z calculates z the z difference z between z the

z measured z speed z and z the z desired z speed. z The z driver zthrootle z knob z or

z the z brakes z as znecessary zto zadjust z the z speed. z If zthe zcurrent z speed z zis z znot z ztoo

z much z over z the z desired z speed, z the z driver z may z let z friction z and z gravity

z slow z the zmotorcycle zdown.

Desired

z spee

d

Actual

z motorcycl

e z speed

Sailor and (^) Sailboat

compass

Wind

Error

computer steering

Active

Radar

Error

Measured gap

Driver (^) Throttle or brakes

Motorcycle

Speedometer

Error

Exercises 5

E1.11 Anzinvertedzpendulumzcontrolzsystemzusingzanzopticalzencoderztozmeasure

z the zangle zofzthe zpendulum zand za zmotor zproducing za zcontrol ztorque:

Desired

z angle

Angle

E1.12 z In zthe z video z game, zthe z player zcan z serve zas z both zthe zcontroller z and z the z sen-

z sor. zThe zobjective z of zthe z game z might z be z to z drive z a z car z along za z prescribed

z path. z The z player z controls z the z car z trajectory z using z the z joystick z using

z the z visual zqueues zfrom zthe zgame zdisplayed zonzthe zcomputer zmonitor.

Desired

z game

z objectiv

e

Game

z objectiv

e

Controller

Error Voltage^ Torque

Measured

angle

(eyesight, tactile, etc.)

Error

6 CHAPTER 1 Introduction to Control Systems

Problems

P1.1 An zautomobile zinterior zcabin ztemperature zcontrol zsystem zblock zdiagram:

Desired

z temperatur

e z set zby

zthe

z driver

Error

Controller

Thermostat z and z air zconditioning z unit

Automobile z cabin

Automobile

cabin ztemperature

Measured ztemperature

Temperature

P1.2 A zhuman zoperator zcontrolled zvalve zsystem:

Desired

z fluid

z output

z*

* z= zoperator zfunctions

Fluid

z outpu

t

P1.3 A zchemical z composition zcontrol z block z diagram:

Desired

z chemical

z compositio

n

Error

Controller

Valve

Chemical

zcompositio

n

Measured zchemical

z composition analyzer

Controller

Valve

Tank

Meter

Error *

Visual indication

of fluid output *

8 CHAPTER 1 Introduction to Control Systems

P1.7 zAssume z that z the z cannon z fires z initially z at z exactly z 5:00 z p.m.. z We z have

z a z positive z feedback z system. z Denote z by z ∆ t z the z time z lost z per z day, z and

z the z net^ ztime^ zerror^ zby^ z ET z.^ zThen^ zthe^ zfollwoing^ zrelationships^ zhold:

∆ tz = z 4 / 3 z min .z + z 3 z min_. z_ = z 13 / 3 z min_._

and

ET z = z 12 zdaysz×z 13 / 3 zmin ./ day z_._

Therefore, zthe znet ztime zerror zafter z 15 zdays zis

ET z z = z 52 z min_._

P1.8 The zstudent-teacher zlearning zprocess:

Desired

z knowledg

e

Error

Controller

Teacher

Lectures

Student

Knowledge

Measurement

Measured zknowledge

P1.9 Azhumanzarmzcontrolzsystem:

Desired

z arm

z locatio

n

Arm

z locatio

n

Arm &

Measurement

Eyes and pressure

u e

• Nerve^ signals

d

y

z

arm location

Problems 9

Fantail

Error Torque

P1.10 AnzaircraftzflightzpathzcontrolzsystemzusingzGPS:

Desired

flight zpath

z from zair

ztraffic

z controllers

Error

Controller

Computer

z Auto-

pilot

rudder,zand z engine zpower

Aircraft Flight

z pat

h

Measured zflight zpath

System

P1.11 z The z accuracy z of z the z clock z is z dependent z upon z a z constant z flow z from

z the z orifice; z the z flow z is z dependent z upon z the z height z of z the z water z in

z the z float z tank. zThe zheight zof zthe zwater zis zcontrolled zby zthe zfloat. zThe zcontrol

zsystem z controls z only z the z height z of z the z water. z Any z errors z due z to

z enlargement z of z the zorifice zor zevaporation zof zthe zwater zin zthe zlower ztank

zis znot zaccounted z for. zThe zcontrol zsystem zcanzbe zseen zas:

Desired

z height zof

z the zwater

z in zfloat

ztank

Controller

Float zlevel Flowzfrom upperztank

Actual

height

P1.12 zAssume z that z the z turret z and z fantail z are z at z 90 ◦^ , z if z θw z /= z θF z -90◦^. z The z fantail

zoperates zon zthe zerror zsignal z θw z - z θT z , z and zas zthe z fantail zturns, zit z drives z the

z (^) turret zto zturn.

y

Fantail

F

Wind

W

T

W z =zWind zangle F z =^ zFantail

zangle

T z =^ zTurret

zangle

Turret

x

W T

Problems 11

P1.16 z With z the z onset z of z a z fever, z the z body z thermostat z is z turned z up. z The

z body z adjusts z by z shivering z and z less z blood z flows z to z the z skin z surface.

z Aspirin z acts z to zlowers zthe zthermal zset-point zin zthe zbrain.

Desiredztemperature

orzset-point zfrom zbody (^) -

thermostat zin zthe zbraizn

withinzthe z body

Body

Body

z temperatur

e

Measured zbody ztemperature

sensor

P1.17 z Hitting z a z baseball z is z arguably z one z of zthe z most zdifficult z feats z in z all z of z sports.

z Given z that z pitchers z may z throw z the z ball z at z speeds z of z 90 z mph z (or z higher!),

z batters z have z only z about z 0.1 z second z to z make z the z decision z to z swing—with

z bat z speeds z aproaching z 90 z mph. z The z key z to z hitting z a z baseball z a z long

z dis- z tance z is z to z make z contact z with z the z ball z with z a z high z bat z velocity.

z This z is z more zimportant zthan zthe zbat’s zweight, zwhich zis zusually zaround z 33

zounces z (compared zto zTy zCobb’s zbat zwhich zwas z 41 zounces!). zSince zthe

zpitcher zcan z throw z a z variety z of z pitches z (fast zball, z curve zball, z slider, zetc.), z a

zbatter zmust z decide z if z the z ball z is z going z to z enter z the z strike z zone z and z if

z possible, z decide z the ztype zof zpitch. zThe zbatter zuses zhis/her zvision zas zthe

zsensor zin zthe zfeed- z back zloop. zA zhigh zdegree zof zeye-hand zcoordination zis

zkey zto zsuccess—that z is, zanzaccurate zfeedback zcontrol zsystem.

P1.18 z z Define z the z following z variables: z p z = z output z pressure, z fs z z = z spring z force

= z Kx , z fd z = zdiaphragm z force z = z z Ap , z and z fv z z = z zvalve z force z = z z fs z z - z fd.

z The zmotion zof zthe zvalve zis zdescribed zby z y ¨ z= z fv/m z where z m z is zthe zvalve

z (^) mass. zThe zoutput zpressure zis zproportional zto zthe zvalve zdisplacement, zthus

z pz =z cyz,z wherez cz iszthezconstantzofzproportionality.

Screw

z displacemen

t

x ( t )

Spring

f s f v

K

position

y

Constantzof

c

Output z pressur e p ( t )

area

f d

A

12 CHAPTER 1 Introduction to Control Systems

P1.19 A zcontrol z system zto z keep z a zcar zat z a zgiven z relative z position z offset zfrom za

z lead zcar:

Throttle

car

Position zof

z follower

Fuel

z (fue

l)

car

Position -

lead

Actuator u^ z z Controller

• z Relative^ &zprocessing algorithms Reference

z photo

Desired zrelative zposition

P1.20 A zcontrol z system z for z a z high-performance z car z with z an z adjustable z wing:

Desired

z road

z adhesio

n

Road

z adhesio

n

P1.21 Azcontrolzsystemzforzaztwin-liftzhelicopterzsystem:

Measured zseparation

z distance Radar

Desiredzseparation

z distance

Pilot

Helicopter

Separation z distance

Desired z altitude Altitude

Measured zaltitude

Altimeter

Computer (^) wing

Tire internal

Road

Measured road adhesion

14 CHAPTER 1 Introduction to Control Systems

P1.25 A zfeedback zcontrol zsystem zfor zthe zspace ztraffic zcontrol:

Controller

Desired z

z orbit

zposition

Error

Control z law

Jet

z command

s

Reaction

Applied

z forces Satellite

Actual

orbit zposition

Measured zorbit zposition

P1.26 Earth-based zcontrol zof za z microrover z to zpoint zthe zcamera:

Camera z Positio n

P1.27 Control z of za z methanol z fuel zcell:

Desired

z Charg

e

z Level

Charge

z Level

G(s) Gc(s)

Camera position

Rover

Fuel Cell

Gc(s) G(s)

H(s)

Advanced z z Problems 15

Advanced Problems

AP1.1 Control z of z a z robotic z microsurgical z device:

End-effector

z Position

AP1.2 An zadvanced zwind zenergy zsystem zviewed zas za zmechatronic zsystem:

AP1.3 z The z automatic z parallel z parking z system z might z use z multiple z ultrasound

z sensors z to z measure z distances z to z the z parked z automobiles z and z the

z curb. z The zsensor zmeasurements zwould zbe zprocessed zby zan zon-board

zcomputer z to z determine z the z steering z wheel, z accelerator, z and z brake

z inputs z to z avoid z collision zand zto zproperly zalign zthe zvehicle zin zthe zdesired

zspace.

SENSORS

Wind speed and direction sensor

ACTUATORS

COMPUTER EQUIPMENT FOR CONTROLLING THE SYSTEM

SAFETY MONITORING SYSTEMS

CONTROLLER ALGORITHMS

DATA ACQUISTION:WIND SPEED AND DIRECTION

ROTOR ANGULAR SPEED

AERODYNAMIC DESIGN

STRUCTURAL DESIGN OF THE TOWER

ELECTRICAL AND POWER SYSTEMS

CONTROL SYSTEM DESIGN AND ANALYSIS

ELECTRICAL SYSTEM DESIGN AND ANALYSIS

Physical System Modeling POWER^ GENERATION^ AND^ STORAGE

Sensors and Actuators

WIND ENERGY (^) Signals and Systems SYSTEM

Software and Data Acquisition

Computers and Logic Systems

Gc(s) G(s)

H(s)

End-effector -

Design Problems 17

Design Problems

CDP1.1 The zmachine z tool z with z the z movable z table z in z a z feedback z control z configu-

z ration:

Desired

z positio

n

x

Actual

z positio

n

x

DP1.1 Use zthe zstereo zsystem zandzamplifiers zto zcancel zout zthe znoise zby zemitting

z signals z 180 ◦^ z z^ out zof zphase zwith zthe znoise.

Desired

z noise z=

z 0

Noise

• byz^180 zdeg^ motor

Machine z tool zwith z table

Noisezin

z cabin

Microphone

DP1.2 An zautomobile zcruise zcontrol zsystem:

1/ K motor Valve^

Automobile engine

K

sensor

Desired

shaft

speed

Desired

speed

of auto

set by

driver

Actual

speed

of auto

Measured shaft speed Drive^ shaf^ t^ speed

Amplifier motor

Machine tool with table

sensor

Error

Measured position

18 CHAPTER 1 Introduction to Control Systems

DP1.3 An z automoted z cow z milking z system:

Milk

DP1.4 A zfeedback zcontrol zsystem z for za zrobot zwelder:

Desired

z positio

n

Error

• amplifier

and

Voltage

Motorzand z arm

Weld

z top

z positio

n

Measured z zposition camera

DP1.5 A zcontrol zsystem zfor zone zwheel zof za ztraction zcontrol zsystem:

slip

Measured

Actuator

Motor and gears

Robot arm and cup gripper

Cow location

of cup

Desired cup

location

Cow and milker

Measured cup location

Wheel

Antislip

1/ R w

Vehicle

Antiskid

torque

Wheel

speed

slip

speed

torque