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Dynamic Systems: Modeling,
Simulation, and Control
Craig A. Kluever
UniversityofMissouri-Columbia
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LibraryofCongressCataloging-in-PublicationData
Kluever,CraigA.(CraigAllan)
Dynamicsystems:modeling,simulation,andcontrol/CraigKluever.
pagescm
Includesbibliographicalreferencesandindex.
ISBN978-1-118-28945-7(hardback)
1.Systemanalysis.2.Dynamics–Mathematicalmodels.3.Mechanicalengineering–Mathematicalmodels.I.Title.
T57.6.K482014
620.001’185–dc23
2014023118
PrintedintheUnitedStatesofAmerica
10987654321
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Contents
Preface vi
1 IntroductiontoDynamicSystemsandControl 1
1.1 Introduction 1
1.2 ClassificationofDynamicSystems 2
1.3 ModelingDynamicSystems 4
1.4 ObjectivesandTextbookOutline 5
References 6
2 ModelingMechanicalSystems 7
2.1 Introduction 7
2.2 MechanicalElementLaws 7
2.3 TranslationalMechanicalSystems 13
2.4 RotationalMechanicalSystems 24
Summary 30
References 30
Problems 30
3 ModelingElectricalandElectromechanicalSystems 43
3.1 Introduction 43
3.2 ElectricalElementLaws 43
3.3 ElectricalSystems 46
3.4 Operational-AmplifierCircuits 54
3.5 ElectromechanicalSystems 57
Summary 68
References 69
Problems 69
4 ModelingFluidandThermalSystems 80
4.1 Introduction 80
4.2 HydraulicSystems 80
4.3 PneumaticSystems 93
4.4 ThermalSystems 99
Summary 104
References 105
Problems 105
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iv Contents
5 StandardModelsforDynamicSystems 115
5.1 Introduction 115
5.2 State-VariableEquations 115
5.3 State-SpaceRepresentation 119
5.4 Linearization 128
5.5 Input-OutputEquations 133
5.6 TransferFunctions 136
5.7 BlockDiagrams 139
5.8 StandardInputFunctions 143
Summary 145
Problems 146
6 NumericalSimulationofDynamicSystems 156
6.1 Introduction 156
6.2 SystemResponseUsingMATLABCommands 157
6.3 BuildingSimulationsUsingSimulink 162
6.4 SimulatingLinearSystemsUsingSimulink 165
6.5 SimulatingNonlinearSystems 170
6.6 BuildingIntegratedSystems 178
Summary 183
References 184
Problems 184
7 AnalyticalSolutionofLinearDynamicSystems 197
7.1 Introduction 197
7.2 AnalyticalSolutionstoLinearDifferentialEquations 197
7.3 First-OrderSystemResponse 205
7.4 Second-OrderSystemResponse 213
7.5 Higher-OrderSystems 230
7.6 State-SpaceRepresentationandEigenvalues 232
7.7 ApproximateModels 235
Summary 237
Reference 238
Problems 238
8 SystemAnalysisUsingLaplaceTransforms 247
8.1 Introduction 247
8.2 LaplaceTransformation 247
8.3 InverseLaplaceTransformation 254
8.4 AnalysisofDynamicSystemsUsingLaplaceTransforms 259
Summary 271
References 271
Problems 271
9 Frequency-ResponseAnalysis 277
9.1 Introduction 277
9.2 FrequencyResponse 277
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Contents v
9.3 BodeDiagrams 290
9.4 Vibrations 306
Summary 313
References 313
Problems 313
10 IntroductiontoControlSystems 322
10.1 Introduction 322
10.2 FeedbackControlSystems 322
10.3 FeedbackControllers 329
10.4 Steady-StateAccuracy 344
10.5 Closed-LoopStability 350
10.6 Root-LocusMethod 352
10.7 StabilityMargins 373
10.8 ImplementingControlSystems 381
Summary 385
References 385
Problems 386
11 CaseStudiesinDynamicSystemsandControl 397
11.1 Introduction 397
11.2 VibrationIsolationSystemforaCommercialVehicle 397
11.3 SolenoidActuator–ValveSystem 409
11.4 PneumaticAir-BrakeSystem 417
11.5 HydraulicServomechanismControl 427
11.6 FeedbackControlofaMagneticLevitationSystem 444
Summary 455
References 455
AppendixA Units 457
AppendixB MATLABPrimerforAnalyzingDynamicSystems 459
B.1 Introduction 459
B.2 BasicMATLABComputations 459
B.3 PlottingwithMATLAB 462
B.4 ConstructingBasicM-files 463
B.5 CommandsforLinearSystemAnalysis 464
B.6 CommandsforLaplaceTransformAnalysis 465
B.7 CommandsforControlSystemAnalysis 467
AppendixC SimulinkPrimer 469
C.1 Introduction 469
C.2 BuildingSimulinkModelsofLinearSystems 469
C.3 BuildingSimulinkModelsofNonlinearSystems 478
C.4 SummaryofUsefulSimulinkBlocks 482
Index 485
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Preface
This textbook is intended for an introductory course in dynamic systems and control, typically required in
undergraduate mechanical engineering andsomeaerospace engineering curricula.Suchacourseisusually
takeninthejuniororsenioryear,afterthestudenthascompletedcoursesinmechanics,differentialequations,
andelectricalcircuits.Themajortopicsofadynamicsystemsandcontrolcourseinclude(1)mathematical
modeling, (2) system-response analysis, and (3) an introduction to feedback control systems. The primary
objectiveofthistextbookisacomprehensiveyetconcisetreatmentofthesemajortopicswithanemphasis
ondemonstratingphysicalengineeringapplications.Ithasbeenmyexperiencethatundergraduatestudents
remainengagedinasystemdynamicscoursewhentheconceptsarepresentedintermsofrealengineering
systems (such as a hydraulic actuator) instead of academic examples. This textbook is a distillation of
20 years of course notes and strategies for teaching system dynamics in the Mechanical and Aerospace
EngineeringDepartmentattheUniversityofMissouri-Columbia.Itisthusbasedonmyextensiveclassroom
experienceandstudentfeedback,andtheendresultisatextwhosekeyfeaturesdifferfromcurrentsystem
dynamicstextbooks.
Chapter1introducesdynamicsystemsandcontrol,includingdefinitionsoftherelevanttermsandcat-
egoriesofdynamicsystems.Thefollowingthreechaptersalldealwithdevelopingmathematicalmodelsof
physicalengineeringsystems.Chapter2introducesthefundamentaltechniquesusedtoderivethemodeling
equationsformechanicalsystems.Mechanicalsystems,perhapsmoreintuitivelyunderstoodbyundergradu-
atestudentsastheyinvolveNewton’slawsofmotion,aretreatedfirst.Chapter3introducesthefundamental
methodsofdevelopingthemathematicalmodelsforelectricalandelectromechanicalsystems.Modelsinthis
chapterarederivedbyapplyingKirchhoff’sandFaraday’slawsandtheelectricalelementlawsthatgovernthe
interactionsamongelectricalcharge,current,magneticflux,andvoltage.Chapter4presentsthefundamental
techniquesinvolvedindevelopingmodelsforfluidandthermalsystems.Itisthelastchapterdedicatedtothe
derivationofmathematicalmodels.Fluid(hydraulicandpneumatic)modelsarebasedontheconservationof
mass,whilethermalmodelsarederivedusingtheconservationofenergy.
Chapter5presentsstandardformsforrepresentingthemathematicalmodelsofdynamicsystems:state-
variableequations,state-spacerepresentation,input-outputequations,transferfunctions,andblockdiagrams.
Thelinearizationprocessisalsodescribedinthischapter.ThekeyconceptemphasizedinChapter5isthat
eachstandardformissimplyaconvenientrepresentationofthesystemmodel(i.e.,thedifferentialequations)
that lends itself to analysis of the system’s dynamic response. Therefore Chapter 5 serves as a transition
betweendevelopingthemathematicalmodel(Chapters2–4)andobtainingthesystem’sresponseusingeither
numericalsimulationoranalyticaltechniques(Chapters6–10).
Chapter6presentsnumericalsimulationmethodsforobtainingtheresponseofdynamicsystems.Here,
theMATLABsimulationsoftwareisusedexclusivelyasithasbecomethestandardcomputationalplatform
foracademiaandindustry.SimulatingtheresponseoflinearsystemsusingMATLABcommandsispresented
first.ThegraphicalsoftwareSimulinkispresentednext,anditisthefocusofthischapterandtheprimarysim-
ulationtoolusedthroughouttheremainderofthetextbook.Simulinkisusedtosimulatelinearandnonlinear
systemsusingthestandardformspresentedinChapter5.
The next three chapters involve the analytical solution of linear systems. Chapter 7 covers analytical
methodsforobtainingthesystem’sresponseinthetimedomainwithanemphasisonfirst-andsecond-order
system response. Here, the two key concepts are (1) the correlation between the roots of the characteristic
equationandtheformofthefree(ortransient)response;and(2)theequivalenceofthecharacteristicroots,
polesofthetransferfunction,andeigenvaluesofthesystemmatrix.Chapter8presentsabriefoverviewof
Laplacetransformtheoryanditsuseinobtainingtheresponseoflineardynamicsystems.Chapter9involves
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Preface vii
frequencyresponse,orthesystemresponsetoperiodicinputfunctions.Inthischapter,theemphasisisonthe
Bodediagramasagraphicaldepictionoftheinformationrequiredforcompletefrequency-responseanalysis.
Chapter10introducesfeedbackcontrolsystemswherethePIDcontroller(anditsvariants)isempha-
sized.Twographicaltechniques,theroot-locusmethodandtheBodediagram,areusedtoanalyzetheclosed-
loop response, design controllers, and assess stability. The chapter closes with a brief discussion of how
controllersareimplementedasdiscrete-timealgorithmsinadigitalcomputer.
Servingasacapstoneforthetextbook,Chapter11presentscasestudiesindynamicsystemsandcontrol.
Mixed-discipline,integratedengineeringsystems,inspiredbytheresearchliterature,serveasthecasestudies.
The five case studies illustrate the major topics of the textbook: (1) developing mathematical models, (2)
predicting the system’s behavior using analytical and numerical methods, and (3) selecting the important
system parameters in order to improve performance. Simulink is used extensively to obtain the dynamic
responseofthesesystemsthatofteninvolvenonlinearitiesandothercomplexities.
NumerousexamplesareprovidedatkeylocationsthroughoutChapters2to10inordertoillustratethe
topic discussed by the particular section. Chapters 2–10 contain end-of-chapter problems that are grouped
intothreecategories:(1)conceptualproblems,(2)MATLABproblems,and(3)engineeringapplications.In
manycases,physicalengineeringsystems(suchasasuspensionsystem,solenoidactuator,orfiltercircuit)are
revisited throughout the textbook in the chapter examples and problems. As with the capstone Chapter 11,
many of the example and end-of-chapter problems illustrate concepts in dynamic systems and control by
presenting“real-world,”physicalengineeringsystems.
Appendix A presents the basic and derived units used in this textbook. Appendix B provides a brief
introduction to MATLAB, M-files, and the commands that pertain to solving problems involving dynamic
systemsandcontrol.AppendixCisaprimeronSimulinkandexpandsonthebriefSimulinktutorialcovered
inChapter6.
As previously mentioned, this textbook is an outgrowth of 20 years of teaching a system dynamics
course. In the long time that I have taught this course, I have often employed a trial-and-error approach to
determininganoptimumsetofstrategiesformaximizingstudentunderstandingofkeytopics.Someofthese
strategies are unique to my teaching, and I outline them here so that the reader can see how this textbook
marksadeparturefromothersinthefield.
1. Because system analysis and control system design begin with developing the appropriate math-
ematical models, the foundations for modeling all physical systems are presented in sequence by
Chapters 2–4.Thegoal ofthesechapters istopresentmodeling explanations thatareconcise and
clear, and this objective is achieved by focusing on essential, representative systems. Chapter 5
succinctly presents all standard forms for representing system models so that the fundamentals of
systemmodelingarecoveredearlyinthetextbookandnotspreadthroughoutthemanuscript.
2. NumericalsimulationoflinearandnonlinearsystemsusingMATLABandSimulink(Chapter6)are
coveredbeforeanalyticalmethods(Chapters7–9).Myexperienceisthatpresentingmodelingand
simulation of real engineering applications early in the semester engages the students in the sub-
jectmaterial.IndustryhaslongusedSimulinktomodelandanalyzerealengineeringsystems.My
classroomexperiencehasshownthatundergraduatestudentsarequitecapableof – andenthusias-
tic about! – using Simulink to simulate complex, integrated systems that involve real effects such
as nonlinear friction,turbulent flow, and modeling discontinuities.Instructors whowishtopresent
analyticalmethodsbeforenumericalsimulationmaysimplycoverChapter6afterChapters7and8.
3. ThemethodsforobtaininganalyticalsolutionsdonotrelyonLaplacetransformtheory.InChapter7
(time-domainresponse)thecharacteristicequationandtheassociatedcharacteristicroots(ortransfer
functionpolesoreigenvalues)areusedtodeterminethetransientresponse.Thefrequencyresponse
isderivedinChapter9bydeterminingtheforcedresponsetotheinputestwheres=𝜎+j𝜔isacom-
plexvariable.Transferfunctionsareusedextensivelyinthistextbookandarederivedwithoutusing
Laplace transforms. I believe that using Laplace transforms (and the associated inversion process)
toobtaintheresponseisundulytedious;furthermore,myexperienceisthatmoststudentsfindtheir
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viii Preface
use nonintuitive. Therefore, my approach is to focus on the characteristic roots r and the form of
i
the system response in terms of erit. Chapter 8 presents the Laplace transformation and its use to
obtainthedynamicresponse.Therefore,instructorswhowishtoutilizeLaplacetransformmethods
maydoso.However,Chapter8maybeomittediftheinstructordoesnotcaretoincludetheLaplace
transformmethod.
4. Chapter 11 presents five engineering case studies in dynamic systems and control. The five case
studies, inspired by research articles, are (1) vibration isolation in a vehicle suspension system,
(2) an electromechanical (solenoid) actuator, (3) a pneumatic air-brake system, (4) hydraulic ser-
vomechanismcontrol,and(5)feedbackcontrolofamagneticlevitationsystem.Thesecasesillustrate
topicsfromthepreviouschapters:modeling,simulation,linearization,analyticalmethods,andcon-
trol.InstructorsmaywishtointroducethevariousapplicationscontainedinChapter11earlyinthe
semester as certain topics are covered and then revisit the cases studies as the course progresses.
Again, I cannot overstate the importance of presenting analysis of real, complex engineering sys-
tems to undergraduate students in order to motivate them, engage their interest, and illustrate key
conceptsindynamicsystemsandcontrol.
Several people have contributed to the production of this textbook. I would like to thank Roger C.
Fales,mycolleagueattheUniversityofMissouri,forhisexpertknowledgeandcommentsonfluidsystems.
TwostudentsattheUniversityofMissouriprovidedtremendousassistance:AnnemarieHoyersearchedthe
engineering literature and helped with the solution manual, and James Smith created illustrations for the
draft manuscript. Many reviewers provided valuable suggestions for improving this textbook and they are
listedhere:
BradleyT.Burchett,Rose-HulmanInstituteofTechnology
MarkB.Colton,BrighamYoungUniversity
EricConstans,RowanUniversity
R.ReesFullmer,UtahStateUniversity
RajeshGanesan,GeorgeMasonUniversity
SeonHan,TexasTechUniversity
S.GrahamKelly,TheUniversityofAkron
KathleenLamkin-Kennard,RochesterInstituteofTechnology
JavadMohammadpour,UniversityofGeorgia
NejatOlgac,UniversityofConnecticut
AndrewR.Plummer,UniversityofBath
ErnestW.Tollner,TheUniversityofGeorgia
Hwan-SikYoon,TheUniversityofAlabama
It was a sincere pleasure to work with the editorial and production staff at Wiley and I particularly
thank Linda Ratts and Hope Ellis. I also thank my wife Nancy M. West for her skilled editorial work and
heartfeltencouragementthroughoutthisproject.Finally,Iacknowledgeandthankmymentorswhohavehad
animmenseimpactonmylife:myadvisorBionL.PiersonofIowaStateUniversity,JohnP.RiehlofNASA
GlennResearchCenter,andmyparentsAllanandJanetKluever.
Andfinally,Ithankmyson,SilasWestKluever,towhom,alongwithmyparents,Idedicatethisbook.
At 12, he has all the makings of a mechanical engineer, although at present he is much more interested in
becomingamajorleaguebaseballplayer.
CraigA.Kluever
Mechanical&AerospaceEngineering
UniversityofMissouri-Columbia
January2015
