INSTRUCTOR’S SOLUTIONS MANUAL DITION M odern C ontrol S ySteMS Fourteenth E Richard C. Dorf University of California, Davis
Robert H. Bishop
University of South Florida
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ISBN-13: 978-0-13-730729ISBN-10: 0-13- 27307296
PREFACE Ineachchapter,therearefiveproblemtypes: Exercises Problems
AdvancedProblems
DesignProblems/ContinuousDesignProblem ComputerProblems
Intotal,thereareover980problems.Theabundanceofproblemsofincreasingcomplexitygivesstudentsconfidenceintheirproblem-solving abilityastheyworktheirwayfromtheexercisestothedesignand computer-basedproblems.
Itisassumedthatinstructors(andstudents)haveaccessto MATLAB and theControlSystemToolboxortoLabVIEWandtheMathScriptRTModule.Allofthecomputersolutionsinthis SolutionManual weredeveloped andtestedonanAppleMacBookProplatformusingMATLAB R2020a andtheControlSystemToolboxVersion9.6andLabVIEW2020. Itisnot possibletoverifyeachsolutiononalltheavailablecomputerplatforms thatarecompatiblewithMATLAB andLabVIEWMathScriptRTModule.Pleaseforwardanyincompatibilitiesyouencounterwiththescripts toProf.Bishopattheemailaddressgivenbelow.
TheauthorsandthestaffatPearsonEducationwouldliketoestablish anopenlineofcommunicationwiththeinstructorsusing ModernControlSystems.WeencourageyoutocontactPearsonwithcommentsand suggestionsforthisandfutureeditions.
RobertH.Bishoprobertbishop@usf.edu
1.IntroductiontoControlSystems.....................
2.MathematicalModelsofSystems...................... ..........23
3.StateVariableModels..............................
4.FeedbackControlSystemCharacteristics............. ..........136
5.ThePerformanceofFeedbackControlSystems...........
6.TheStabilityofLinearFeedbackSystems.............. ........237
7.TheRootLocusMethod...............................
8.FrequencyResponseMethods.........................
9.StabilityintheFrequencyDomain....................
10.TheDesignofFeedbackControlSystems...............
11.TheDesignofStateVariableFeedbackSystems......... .......604
12.RobustControlSystems............................
13.DigitalControlSystems...........................
IntroductiontoControlSystems Thereare,ingeneral,nouniquesolutionstothefollowingexercisesand problems.Otherequallyvalidblockdiagramsmaybesubmittedbythe student.
Exercises E1.1 Describetypicalsensorsthatcanmeasureeachofthefollowing:
a.Linearposition → ultrasonictransducer
b.Velocity(orspeed) → Dopplerradar
c.Non-gravitationalacceleration → inertialmeasurementunit
d.Rotationalposition(orangle) → rotaryencoder
e.Rotationalvelocity → gyroscope
f.Temperature → thermocouple
g.Pressure → barometer
h.Liquid(orgas)flowrate → velocimeter
i.Torque → torquemeter
j.Force → loadcell
k.Earth’smagneticfield → magnetometer
l.Heartrate → electrocardiograph
E1.2 Describetypicalactuatorsthatcanconvertthefollowing:
a.Fluidicenergytomechanicalenergy → hydrauliccylinder
b.Electricalenergytomechanicalenergy → electricmotor
c.Mechanicaldeformationtoelectricalenergy → piezoelectricactuator
d.Chemicalenergytokineticenergy → automobileengine
e.Heattoelectricalenergy → thermoelectricgenerator
E1.3 Amicroprocessorcontrolledlasersystem:
E1.4 Adrivercontrolledcruisecontrolsystem:
E1.5 Althoughtheprincipleofconservationofmomentumexplainsmuchof theprocessoffly-casting,theredoesnotexistacomprehensivescientific explanationofhowafly-fisherusesthesmallbackwardandforwardmotionoftheflyrodtocastanalmostweightlessflylurelongdistances(the currentworld-recordis236ft).Theflylureisattachedtoashortinvisible leaderabout15-ftlong,whichisinturnattachedtoalonger andthicker Dacronline.Theobjectiveiscasttheflyluretoadistantspotwithdeadeyeaccuracysothatthethickerpartofthelinetouchesthewaterfirst andthentheflygentlysettlesonthewaterjustasaninsectmight.
Anautofocuscameracontrolsystem:
Tackingasailboatasthewindshifts:
E1.8 Anautomatedhighwaycontrolsystemmergingtwolanesoftraffic:
E1.9 Askateboardridermaintainingverticalorientationanddesiredspeed:
E1.10
Humanbiofeedbackcontrolsystem:E1.11 Measurement
E1.11 E-enabledaircraftwithground-basedflightpathcontrol:
Desired Flight Path Flight Path
Exercises 5
E1.12 Unmannedaerialvehicleusedforcropmonitoringinanautonomous mode:
E1.13 Aninvertedpendulumcontrolsystemusinganopticalencodertomeasure theangleofthependulumandamotorproducingacontroltorque:
E1.14 Inthevideogame,theplayercanserveasboththecontroller andthesensor.Theobjectiveofthegamemightbetodriveacaralongaprescribed path.Theplayercontrolsthecartrajectoryusingthejoystickusingthe visualqueuesfromthegamedisplayedonthecomputermonitor.
E1.15 Aclosed-loopbloodglucosesystemwithacontinuousglucosemeasurementinformingthedecisiontoinjectinsulinornot:
P1.1
Anautomobileinteriorcabintemperaturecontrolsystemblockdiagram:
P1.2 Ahumanoperatorcontrolledvalvesystem:
P1.3 Achemicalcompositioncontrolblockdiagram:
P1.4 Anuclearreactorcontrolblockdiagram:
P1.5 Alightseekingcontrolsystemtotrackthesun:
P1.6 Ifyouassumethatincreasingworker’swagesresultsinincreasedprices, thenbydelayingorfalsifyingcost-of-livingdatayoucouldreduceoreliminatethepressuretoincreaseworker’swages,thusstabilizingprices.This wouldworkonlyiftherewerenootherfactorsforcingthecost-of-living up.Governmentpriceandwageeconomicguidelineswouldtaketheplace ofadditional“controllers”intheblockdiagram,asshownintheblock diagram.
P1.7 Assumethatthecannonfiresinitiallyatexactly5:00p.m..Wehavea positivefeedbacksystem.Denoteby∆t thetimelostperday,andthe nettimeerrorby ET .Thenthefollwoingrelationshipshold:
3min
P1.8
and
Therefore,thenettimeerrorafter15daysis
Thestudent-teacherlearningprocess:
P1.9
Ahumanarmcontrolsystem:
P1.10
CHAPTER1IntroductiontoControlSystems AnaircraftflightpathcontrolsystemusingGPS:
P1.11
P1.12
Theaccuracyoftheclockisdependentuponaconstantflowfromthe orifice;theflowisdependentupontheheightofthewaterinthefloat tank.Theheightofthewateriscontrolledbythefloat.Thecontrolsystem controlsonlytheheightofthewater.Anyerrorsduetoenlargementof theorificeorevaporationofthewaterinthelowertankisnot accounted for.Thecontrolsystemcanbeseenas:
Assumethattheturretandfantailareat90◦,if θw = θF -90◦.Thefantail operatesontheerrorsignal θw - θT ,andasthefantailturns,itdrivesthe turrettoturn.
P1.13 Thisschemeassumesthepersonadjuststhehotwaterfortemperature control,andthenadjuststhecoldwaterforflowratecontrol.
P1.14 Iftherewardsinaspecifictradeisgreaterthantheaveragereward,there isapositiveinfluxofworkers,since
Ifaninfluxofworkersoccurs,thenrewardinspecifictradedecreases, since
P1.15 Acomputercontrolledfuelinjectionsystem:
CHAPTER1IntroductiontoControlSystems P1.16 Withtheonsetofafever,thebodythermostatisturnedup.Thebody adjustsbyshiveringandlessbloodflowstotheskinsurface. Aspirinacts tolowersthethermalset-pointinthebrain.
P1.17 Hittingabaseballisarguablyoneofthemostdifficultfeatsinallofsports. Giventhatpitchersmaythrowtheballatspeedsof90mph(orhigher!), battershaveonlyabout0.1secondtomakethedecisiontoswing—with batspeedsaproaching90mph.Thekeytohittingabaseballalongdistanceistomakecontactwiththeballwithahighbatvelocity.Thisis moreimportantthanthebat’sweight,whichisusuallyaround33ounces. Sincethepitchercanthrowavarietyofpitches(fastball,curveball,slider, etc.),abattermustdecideiftheballisgoingtoenterthestrikezoneand ifpossible,decidethetypeofpitch.Thebatteruseshis/hervisionasthe sensorinthefeedbackloop.Ahighdegreeofeye-handcoordinationiskey tosuccess—thatis,anaccuratefeedbackcontrolsystem.
P1.18 Definethefollowingvariables: p =outputpressure, fs =springforce = Kx, fd =diaphragmforce= Ap,and fv =valveforce= fs - fd Themotionofthevalveisdescribedby¨ y = fv/m where m isthevalve mass.Theoutputpressureisproportionaltothevalvedisplacement,thus p = cy, where c istheconstantofproportionality.
P1.19 Acontrolsystemtokeepacaratagivenrelativepositionoffsetfroma leadcar:
P1.20 Acontrolsystemforahigh-performancecarwithanadjustablewing:
P1.21 Acontrolsystemforatwin-lifthelicoptersystem:
CHAPTER1IntroductiontoControlSystems P1.22 Thedesiredbuildingdeflectionwouldnotnecessarilybezero.Ratherit wouldbeprescribedsothatthebuildingisallowedmoderate movement uptoapoint,andthenactivecontrolisappliedifthemovementislarger thansomepredeterminedamount.
P1.23 Thehuman-likefaceoftherobotmighthavemicro-actuators placedat strategicpointsontheinteriorofthemalleablefacialstructure.Cooperativecontrolofthemicro-actuatorswouldthenenabletherobottoachieve variousfacialexpressions.
P1.24 Wemightenvisionasensorembeddedina“gutter”atthebaseofthe windshieldwhichmeasureswaterlevels—higherwaterlevelscorresponds tohigherintensityrain.Thisinformationwouldbeusedtomodulatethe wiperbladespeed.
P1.25 Afeedbackcontrolsystemforthespacetrafficcontrol:
P1.26 Earth-basedcontrolofamicrorovertopointthecamera:
P1.27 Controlofamethanolfuelcell:
AdvancedProblems AP1.1 Controlofaroboticmicrosurgicaldevice:
AP1.2 Anadvancedwindenergysystemviewedasamechatronicsystem:
AP1.3 Theautomaticparallelparkingsystemmightusemultipleultrasound sensorstomeasuredistancestotheparkedautomobilesandthecurb. Thesensormeasurementswouldbeprocessedbyanon-boardcomputer todeterminethesteeringwheel,accelerator,andbrakeinputstoavoid collisionandtoproperlyalignthevehicleinthedesiredspace.
AP1.4
Eventhoughthesensorsmayaccuratelymeasurethedistance between thetwoparkedvehicles,therewillbeaproblemiftheavailablespaceis notbigenoughtoaccommodatetheparkingcar.
AP1.5
Therearevariouscontrolmethodsthatcanbeconsidered,includingplacingthecontrollerinthefeedforwardloop(asinFigure1.3).Theadaptive opticsblockdiagrambelowshowsthecontrollerinthefeedbackloop,as analternativecontrolsystemarchitecture.
Thecontrolsystemmighthaveaninnerloopforcontrollingtheaccelerationandanouterlooptoreachthedesiredfloorlevelprecisely.
AP1.6 Anobstacleavoidancecontrolsystemwouldkeeptherobotic vacuum cleanerfromcollidingwithfurniturebutitwouldnotnecessarilyputthe vacuumcleaneronanoptimalpathtoreachtheentirefloor.Thiswould requireanothersensortomeasurepositionintheroom,adigitalmapof theroomlayout,andacontrolsystemintheouterloop.
AP1.7 Theattitudecontrolofthedroneshiprequiresmeasuringtheyawand rollusingagyro.Oftenthegyromeasuresattituderate,therefore,itmay benecessarytointegratethegyrooutputtocomputethemeasuredroll andyaw.
DesignProblems 19
DesignProblems
CDP1.1 ration:
Themachinetoolwiththemovabletableinafeedbackcontrol configu-
DP1.1 Usethestereosystemandamplifierstocanceloutthenoiseby emitting signals180◦ outofphasewiththenoise.
DP1.2 Anautomobilecruisecontrolsystem:
DP1.3 Utilizingasmartphonetoremotelymonitorandcontrolawashingmachine:
DP1.4 Anautomatedcowmilkingsystem:
DP1.5 Afeedbackcontrolsystemforarobotwelder:
DesignProblems
DP1.6 Acontrolsystemforonewheelofatractioncontrolsystem:
DP1.7 AvibrationdampingsystemfortheHubbleSpaceTelescope:
DP1.8 Acontrolsystemforananorobot:
Manyconceptsfromunderwaterroboticscanbeappliedtonanorobotics withinthebloodstream.Forexample,planesurfacesandpropellerscan
DP1.9
providetherequiredactuationwithscrewdrivesproviding thepropulsion.Thenanorobotscanusesignalsfrombeaconslocatedoutsidethe skinassensorstodeterminetheirposition.Thenanorobots useenergy fromthechemicalreactionofoxygenandglucoseavailableinthehuman body.Thecontrolsystemrequiresabio-computer–aninnovationthatis notyetavailable.
Thefeedbackcontrolsystemmightusegyrosand/oraccelerometersto measureanglechangeandassumingtheHTVwasoriginallyinthevertical position,thefeedbackwouldretaintheverticalpositionusingcommands tomotorsandotheractuatorsthatproducedtorquesandcouldmovethe HTVforwardandbackward.
DP1.10
Therearetwoloopsinthiscontrolsystem,onetocontrolthe automobilevelocityandonetocontroltherelativepositionofthe twovehicles. Sincewehavenowaytomeasurethevelocityoftheforwardvehicle,we relyontheradartoproviderelativepositioning.Thecontrollerwillneed toaccountforboththevelocityerrorandtherelativepositionerrorin computingthedesiredacceleration.
CHAPTER2 MathematicalModelsofSystems Exercises E2.1 Wehavefortheopen-loop y = r 2 andfortheclosed-loop e = r y and y = e 2
So, e = r e2 and e2 + e r =0 .
FIGUREE2.1 Plotofopen-loopversusclosed-loop.
Forexample,if r =1,then e2 + e 1=0impliesthat e =0 618.Thus, y =0.382.Aplot y versus r isshowninFigureE2.1.
E2.2 Define
and
Then,
when R0 =10, 000Ω.Thus,thelinearapproximationiscomputedby consideringonlythefirst-ordertermsintheTaylorseriesexpansion,and isgivenby
R = 135∆T.
E2.3 Thespringconstantfortheequilibriumpointisfoundgraphicallyby estimatingtheslopeofalinetangenttotheforceversusdisplacement curveatthepoint y =0.5cm,seeFigureE2.3.Theslopeofthelineis K ≈ 1.
y=Displacement (cm) For ce (n) Spring compresses Spring breaks
FIGUREE2.3 Springforceasafunctionofdisplacement.
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Exercises 25
E2.4 Since R(s)= 1 s wehave Y (s)= 4(s +50) s(s +20)(s +10)
Thepartialfractionexpansionof Y (s)isgivenby Y (s)= A1 s + A2 s +20 + A3 s +10 where
UsingtheLaplacetransformtable,wefindthat
Thefinalvalueiscomputedusingthefinalvaluetheorem: lim t→∞ y(t)=lim s→0 s 4(s +50) s(s2 +30s +200) =1
E2.5 ThecircuitdiagramisshowninFigureE2.5.
FIGUREE2.5 Noninvertingop-ampcircuit.
Withanidealop-amp,wehave
CHAPTER2MathematicalModelsofSystems
where A isverylarge.Wehavetherelationship v = R1 R1 + R2 vo
Therefore,
vo = A(vin R1 R1 + R2 vo), andsolvingfor vo yields vo = A 1+ AR1 R1 +R2 vin
Since A ≫ 1,itfollowsthat1+ AR1 R1+R2 ≈ AR1 R1 +R2 .Thentheexpressionfor vo simplifiesto
E2.6 Given
vo = R1 + R2 R1 vin
y = f (x)= ex
andtheoperatingpoint xo =1,wehavethelinearapproximation y = f (x)= f (xo)+ ∂f
x=xo (x xo)+ ···
where f (xo)= e, df dx x=xo=1 = e, and x xo = x 1
Therefore,weobtainthelinearapproximation y = ex
E2.7 TheblockdiagramisshowninFigureE2.7.
FIGUREE2.7 Blockdiagrammodel. Copyright ©2022 Pearson Education, Inc.
Exercises 27
Startingattheoutputweobtain I(s)= G1(s)G2(s)E(s).
But E(s)= R(s) H(s)I(s),so I(s)= G1(s)G2(s)[R(s) H(s)I(s)] .
Solvingfor I(s)yieldstheclosed-looptransferfunction I(s) R(s) = G1(s)G2(s) 1+ G1(s)G2(s)H(s)
E2.8 TheblockdiagramisshowninFigureE2.8.
FIGUREE2.8 Blockdiagrammodel.
Startingattheoutputweobtain
), so
Substituting W (s)= KE(s) H1(s)Z(s)intotheaboveequationyields
)Z(s)]
CHAPTER2MathematicalModelsofSystems
andwith E(s)= R(s) Y (s)and Z(s)= sY (s)thisreducesto
Y (s)=[ G1(s)G2(s)(H2(s)+ H1(s)) G1(s)H3(s) 1 s G1(s)G2 (s)K]Y (s)+ 1 s G1(s)G2(s)KR(s)
Solvingfor Y (s)yieldsthetransferfunction
Y (s)= T (s)R(s), where T (s)= KG1(s)G2(s)/s 1+ G1(s)G2(s)[(H2(s)+ H1(s)]+ G1(s)H3(s)+ KG1(s)G2(s)/s
E2.9 FromFigureE2.9,weobservethat
Ff (s)= G2(s)U (s) and FR(s)= G3(s)U (s)
Then,solvingfor U (s)yields U (s)= 1 G2(s) Ff (s)
anditfollowsthat
Again,consideringtheblockdiagraminFigureE2.9wedetermine
But,fromthepreviousresult,wesubstitutefor FR(s)resultingin
Solvingfor Ff (s)yields
FIGUREE2.9 Blockdiagrammodel.
E2.10 TheshockabsorberblockdiagramisshowninFigureE2.10.Theclosedlooptransferfunctionmodelis
Piston travel measurement
FIGUREE2.10
Shockabsorberblockdiagram.
E2.11 Let f denotethespringforce(n)and x denotethedeflection(m).Then K = ∆f ∆x . Computingtheslopefromthegraphyields:
(a) xo = 0.14m → K =∆f/∆x =10n/0.04m=250n/m
(b) xo =0m → K =∆f/∆x =10n/0.05m=200n/m
(c) xo =0.35m → K =∆f/∆x =3n/0.05m=60n/m
CHAPTER2MathematicalModelsofSystems
E2.12 ThesignalflowgraphisshowninFig.E2.12.Find Y (s)when R(s)=0. Y(s) -1 K 2 G(s) -K 1 1 Td(s)
FIGUREE2.12 Signalflowgraph.
Thetransferfunctionfrom Td(s)to Y (s)is Y (s)= G(s)Td(s) K1K2G(s)Td(s) 1 ( K2G(s)) = G(s)(1 K1K2)Td(s) 1+ K2G(s)
Ifweset K1K2 =1 , then Y (s)=0forany Td(s).
E2.13 Thetransferfunctionfrom R(s), Td(s),and N (s)to Y (s)is
)=
Therefore,wefindthat Y (s)/Td(s)= 1 s2 +25s + K and Y (s)/N (s)= K s2 +25s + K
E2.14 Sincewewanttocomputethetransferfunctionfrom R2(s)to Y1(s),we canassumethat R1 =0(applicationoftheprincipleofsuperposition). Then,startingattheoutput Y1(s)weobtain Y1(s)= G3(s)[ H1(s)Y1(s)+ G2(s)G8(s)W (s)+ G9(s)W (s)] , or
Consideringthesignal W (s)(seeFigureE2.14),wedeterminethat W (s)= G5(s)[G4(s)R2(s) H2(s)W (s)] ,
FIGUREE2.14
Blockdiagrammodel. or
Substitutingtheexpressionfor W (s)intotheaboveequationfor Y1(s) yields
E2.15 Forloop1,wehave
Andforloop2,wehave
E2.16 Thetransferfunctionfrom R(
TheblockdiagramisshowninFigureE2.16a.Thecorrespondingsignal flowgraphisshowninFigureE2.16bfor P (s)/R(
FIGUREE2.16
(a)Blockdiagram,(b)Signalflowgraph.
E2.17 Alinearapproximationfor f isgivenby
where xo =1/2,∆f = f (x) f (xo),and∆x = x xo
E2.18 Thelinearapproximationisgivenby
(a)When xo =1,wefindthat yo =2 4,and yo =13 2when xo =2.
(b)Theslope m iscomputedasfollows: m = ∂y ∂x x=xo =1+4 2x 2 o
Therefore, m =5 2at xo =1,and m =18 8at xo =2.
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Exercises 33
E2.19 Theoutput(withastepinput)is
Y (s)= 30(s +1) s(s +5)(s +6)
Thepartialfractionexpansionis
Y (s)= 5 s 20 s +3 + 15 s +2 .
TakingtheinverseLaplacetransformyields
y(t)=5 20e 3t +15e 2t
E2.20 Theinput-outputrelationshipis Vo V = A(K 1) 1+ AK where K = Z1 Z1 + Z2
Assume A ≫ 1.Then, Vo V = K 1 K = Z2 Z1 where Z1 = R1 R1C1s +1 and Z2 = R2 R2C2s +1 .
Therefore, Vo(s) V (s) = R2(R1C1s +1) R1(R2C2s +1) = 2(s +1) s +2
E2.21 Theequationofmotionofthemass mc is
TakingtheLaplacetransformwithzeroinitialconditionsyields
So,thetransferfunctionis
(
)
CHAPTER2MathematicalModelsofSystems
E2.22 Therotationalvelocityis ω(s)= 2(s +4) (s +5)(s +1)2 1 s .
Expandinginapartialfractionexpansionyields
TakingtheinverseLaplacetransformyields
E2.23 Theclosed-looptransferfunctionis
(s) R(s) = T (s)=
E2.24 Let x =0.6and y =0.8.Then,with y = ax3,wehave 0.8= a(0.6)3 .
Solvingfor a yields a =3 704.Alinearapproximationis y yo =3ax 2 o(x xo) or y =4x 1 6,where yo =0 8and xo =0 6.
E2.25 Theclosed-looptransferfunctionis Y (s) R(s) = T (s)= 10 s2 +21s +10
E2.26 Theequationsofmotionare
TakingtheLaplacetransform(withzeroinitialconditions)andsolving for X2(s)yields X2(s)= k (
Then,with m1 = m2 = k =1,wehave
X2(s)/F (s)= 1 s2(s2 +2)
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E2.27 Thetransferfunctionfrom Td(s)to Y (s)is Y (s)/Td(s)= G2(s) 1+ G1G2H(s) .
E2.28 Thetransferfunctionis
E2.29 (a)If G(s)= 1 s2 +15s +50 and H(s)=2s +15 , thentheclosed-looptransferfunctionofFigureE2.28(a)and(b)(in Dorf&Bishop)areequivalent.
(b)Theclosed-looptransferfunctionis
T (s)= 1 s2 +17s +65
E2.30 (a)Theclosed-looptransferfunctionis
FIGUREE2.30 Stepresponse. Copyright ©2022 Pearson Education, Inc.
CHAPTER2MathematicalModelsofSystems
(b)Theoutput Y (s)(when R(s)=1/s)is
or
(c)Theplotof y(t)isshowninFigureE2.30.Theoutputisgivenby y(t)=0 5(1 1 1239e 2 5t sin(4 8734t +1 0968));
E2.31 Thepartialfractionexpansionis
V (s)= a s + p1 + b s + p2 where p1 =5 8 66j and p2 =5+8 66j.Then,theresiduesare a = 5 77jb =5 77j.
TheinverseLaplacetransformis
