Table Of ContentCustom Preliminary Edition
MEM255
INTRODUCTION TO CONTROLS
by Harry G. Kwatny and Bor-Chin Chang
Drexel University
Winter 2019
cognella®
CUSTOM
Harry G. Kwatny∗ Bor–Chin Chang†
Introduction to Control Systems
Engineering
December24,2018
∗Department of Mechanical Engineering & Mechanics, Drexel University, Philadelphia, PA 19104;
[email protected]
†Department of Mechanical Engineering & Mechanics, Drexel University, Philadelphia, PA 19104;
[email protected]
ii
(cid:13)c H.G.KwatnyandB.C.Chang
Dedication
Preface
Controlsystemengineeringintodaystechnologyenvironmentrequiresamuchbroadersetofcon-
cepts and tools than even a decade ago. This book is focused on bringing attention to essential
ideasincludingsometypicallyomittedinanundergraduatecourseoncontroldesign.Sinceclass
timeisfixedandrigorisessential,anenlargedscoperequiresreducingthetimedevotedtosome
topics traditionally taught. Consequently, subjects having lesser import today have less emphasis
herein.Ofcourse,suchjudgmentsarepersonalandnotalwayseasytomake.Indeterminingthese
choices,wehavebeenguidedbytwocentralobjectives.First,thecontentmustbealignedwiththe
the needs of present day applications and make use of modern computational tools. Second, the
narrativemustreflectarespectandacknowledgementofthehistoricalevolutionofcontrolanalysis
anddesigntheory.
Thisbookisintendedforatwoorthreequarter/semestercourse.Correspondingly,thebookis
structuredinthreemainparts.Thefirsttwofocusonlinearsystems.PartIintroducesbasiclinear
systemanalysisandmodelassemblyconcepts.Anessentialrequirementofcontrolsystemdesign
is that the engineer has a full grasp of the system physical behavior in order to be successful.
Controldesignintodaysworldbeginswithamathematicalmodelofthesystemtobecontrolled.It
isessentialthatthecontroldesignteamunderstanditsformulationandtheassumptionsonwhich
it is based even if they do not assemble the model. With this in mind, the first course contains
basic model building concepts. Fundamental analysis tools for linear systems are introduced and
discussedincorrelationwithmathematicalmodelsofrealphysicalsystems.
Part II is focused on linear control system design. Both frequency domain and time domain
methodsareincludedandbothsingleinput singleoutput(SISO)systemsandmultipleinput mul-
tipleoutput(MIMO)systemsarediscussed.Again,realphysicalsystemsareexaminedusingthe
methods and tools introduced. Some of these systems require more advanced modeling concepts
thataredeferreduntilPartIII,butthesystemsandmodelsthemselvesarediscussedinPartIIina
ChapterentitledThemeExamples.Thepurposeofthismaterialistoprovideandexplainrealistic
modelsofsystemshavingcontemporaryinterestthatcanbeselectivelyusedinaparticularcourse.
PartIIIaddressesmoreadvancedtopicsincludinganalysis,designandmodelingmethodsfor
nonlinearandhybridsystems.Here,wetrytomaintainabalancedapproachbetweenmathematical
rigor and engineering necessity in applications. However, contemporary applications for control
and automation increasingly involve nonlinearity as well as the mixing of dynamics with logical
constraints.
Thereisnodoubtthatthisbookwillpresentachallengetostudents.Hopefullyitwillalsobe
aninspiration.
Contents
PartI BasicConceptsofLinearSystems
1 Introduction............................................................... 3
1.1 ControlEngineering .................................................... 3
1.2 ALittleHistory ........................................................ 4
1.3 ImpactoftheDigitalComputer........................................... 5
1.4 AFirstExample:CruiseControl.......................................... 6
1.5 ContentOverview ...................................................... 7
2 LinearSystemsAnalysisI................................................... 9
2.1 TypicalFirst-orderDynamicSystems...................................... 9
2.1.1 MathematicalEquivalencyamongFirst-OrderDynamicSystems ........ 10
2.1.2 CharacterizationofTypicalFirst-OrderDynamicSystems .............. 11
2.2 ABriefReviewofComplexNumbers ..................................... 12
√
2.2.1 SignificanceoftheImaginaryNumber −1 ......................... 12
2.2.2 PolarForm,RectangularForm,andEulerFormula .................... 13
2.2.3 GeometricalAspectsofComplexNumbers .......................... 15
2.3 ABriefReviewoftheLaplaceTransform .................................. 17
2.3.1 LaplaceTransformPairs .......................................... 17
2.3.2 LaplaceTransformProperties...................................... 20
2.4 Time-DomainResponseofTypicalFirst-orderDynamicSystems .............. 22
2.4.1 TheResponseoftheTypicalFirst-orderSystemsduetoInitialCondition . 23
2.4.2 TheResponseoftheTypicalFirst-orderSystemsduetoUnitStepInput .. 24
2.5 Frequency-DomainPropertiesofTypicalFirst-orderSystems.................. 26
2.5.1 Transferfunctionsvs.DifferentialEquations ......................... 26
2.5.2 CharacteristicEquationandSystemPoles............................ 27
2.5.3 TheResponsesoftheTypicalFirst-orderSystemsduetoSinusoidalInputs 27
2.5.4 FrequencyResponsesandtheBodePlots ............................ 30
2.6 ExerciseProblems...................................................... 33
3 LinearSystemsAnalysisII .................................................. 37
3.1 TypicalSecond-orderDynamicSystems ................................... 37
3.1.1 MathematicalEquivalencyamongTypicalSecond-OrderDynamicSystems 38
3.1.2 CharacterizationofTypicalSecond-OrderDynamicSystems............ 39
X Contents
3.2 TransferFunction,CharacteristicEquationandSystemPoles ................. 40
3.3 Time-DomainResponseofTypicalSecond-orderDynamicSystems ............ 41
3.3.1 TheResponseoftheTypicalSecond-orderSystemsduetoInitialConditions 42
3.3.2 TheResponseoftheTypicalSecond-orderSystemsduetoUnitStepInput 43
3.4 CharacterizationoftheUnderdampedSecond-OrderSystems.................. 49
3.4.1 GeometryofConjugateSystemPolesontheComplexPlane ............ 49
3.4.2 StepResponseoftheUnderdampedSecond-OrderSystem.............. 50
3.4.3 GraphicalInterpretationoftheUnderdampedSecond-OrderStepResponse 51
3.5 AnalysisandDesignofaMass-Damper-SpringSystem ...................... 54
3.6 AnalysisandDesignofaSimpleDCMotorPositionControlSystem ........... 56
3.7 Steady-StateSinusoidalResponseandBodePlotsofTypicalSecond-Order
Systems .............................................................. 57
3.8 ExerciseProblems...................................................... 59
4 ModelingofMechanicalSystems............................................. 65
4.1 TranslationalMechanicalSystems ........................................ 65
4.1.1 d’Alembert’sPrinciple............................................ 67
4.1.2 ABriefIntroductionoftheLagrangeApproach....................... 69
4.1.3 AQuarter-CarSuspensionSystem.................................. 70
4.2 RotationalMechanicalSystems........................................... 72
4.2.1 d’Alembert’sPrincipleinRotationalSystems......................... 73
4.2.2 TheLagrangeApproachforRotationalSystems ...................... 75
4.2.3 ATwo-WheelwithOneShaftRotationalSystem...................... 76
4.3 ARotationalSystemwithaGearTrain .................................... 78
4.4 Cart-InvertedPendulumSystem .......................................... 80
4.4.1 TheNewtonianApproachModelingoftheCart-InvertedPendulumSystem 80
4.4.2 LagrangeApproachModelingoftheCart-InvertedPendulumSystem .... 82
4.4.3 PreliminaryAnalysisoftheCart-InvertedPendulumSystem ............ 83
4.4.4 LinearizedState-spaceModeloftheCart-invertedPendulumSystem..... 86
4.4.5 StabilizingControllerDesignfortheCart-invertedPendulumSystem .... 87
4.5 ExerciseProblems...................................................... 91
5 ModelingofElectricalSystems .............................................. 95
5.1 BasicElectricalCircuitElementsandCircuitConventions .................... 96
5.2 BasicTime-domainCircuitModelingApproachesandKirchhoff’sLaws ........ 97
5.2.1 CircuitModelingUsingthe2kEquationsApproach ................... 98
5.2.2 TheNode-To-Datum(NTD)VoltagesApproach ...................... 99
5.2.3 TheMeshCurrentsApproach ..................................... 100
5.3 BasicImpedanceCircuitModelingApproaches ............................. 100
5.3.1 TheImpedanceNTDVoltagesApproach ............................ 102
5.3.2 TheImpedanceMeshCurrentsApproach ........................... 102
5.4 TheLagrangeApproachforCircuitModeling............................... 103
5.5 CircuitModelingUsingTheState-spaceApproach .......................... 107
5.6 OperationalAmplifierCircuits............................................ 111
5.7 DCMotor............................................................. 117
5.7.1 Ampere’sForceLawandFaraday’sLawofInduction.................. 118
5.7.2 AssemblingEquationsfortheDCMotorSystem...................... 119
5.7.3 Torque-speedRelationship ........................................ 120
Contents XI
5.7.4 ADCMicromotorExample ....................................... 121
5.8 ExerciseProblems...................................................... 126
6 SystemsRepresentationsandInterconnectedSystems .......................... 129
6.1 BlockDiagrams........................................................ 130
6.2 SignalFlowGraphsandtheMason’sGainFormula.......................... 134
6.2.1 SignalFlowGraphs .............................................. 134
6.2.2 TheMason’sGainFormula........................................ 135
6.3 State-spaceModel...................................................... 138
6.4 StateTransitionMatrix.................................................. 142
6.4.1 TheoremsandPropertiesoftheStateTransitionMatrix ................ 143
6.4.2 ComputingtheStateTransitionMatrix .............................. 144
6.5 SolutionoftheStateEquation ............................................ 148
6.6 State-spaceModelsandTransferFunctions................................. 151
6.6.1 FindTransferFunctionfromState-spaceModel....................... 152
6.6.2 ConstructaState-spaceModelfromanInterconnectedSystemswith
TransferFunctions ............................................... 153
6.7 ExerciseProblems...................................................... 160
References..................................................................... 163
Part I
Basic Concepts of Linear Systems
1
Introduction
Feedback control is a centuries old tool applied to quite primitive machinery in its earliest days.
The last century saw an explosion of applications as mankind conceived of new machines and
devicesformanufacturing,communication,landandairtravel,spaceexplorationandmore.These
applicationsprovokednewfunctionalitiesofcontrolthatrequiredadeeperunderstandingofhow
control systems worked. Hence, control systems emerged as a true engineering science. In this
chapter the basic structure of control systems will be discussed along with a brief history of the
evolution of the control discipline to where it is today. The chapter ends with a summary of the
remainderofthebook.
1.1 ControlEngineering
Controlengineeringisadisciplinedealingwiththedesignofdevices,calledcontrollers,thatman-
age the performance of a system through manipulation of control inputs. The control input com-
mandscanbebasedsolelyontheanticipatedresponseofthesystemortheycanadjustinaccor-
dancewithobservationsofsystembehavior.Theformeriscalledopen-loopcontrolandthelatter
isfeedbackcontrol.Feedbackcontrolisimportantwhenthereisimpreciseknowledgeofhowthe
systemwillrespond,orthereareenvironmentaluncertaintiesthatcanaffectbehavior.Mechanisms
ofthissorthavebeenemployedforcenturiesbuttodaytheyaretrulyubiquitous.Controlsystems
are an essential part of chemical and manufacturing processes, communication systems, electric
powerplantsandsystems,groundvehicles,ships,aircraftandspacecraft,robotsandmanipulators,
computersandsoon.
Duringthelastcenturyengineeringwastransformedfromacraftintoascience.Thoseinter-
estedinprofitingfromsociety’sthirstfornewtechnologyhavefounditimpossibletorelyontime
consumingtrialand errortodevelopnew productsorresolveproblems inexistingones.Modern
technologies like automobiles, aircraft, telecommunications, and computers are too complex to
thrivesolelyonvastcompilationsofempiricaldataanddecadesofexperience.Someintellectual
constructs that organize and explain essential facts and principles are required. So engineering,
in general and control engineering in particular, has come to adopt the style and methods of the
naturalsciences.
Atthecoreofthispointofviewisthedistinctionbetweentwothoughtprocesses:thephysical,
andthemathematical.Whileengineersconceiveofproblemsinthephysicalworldandconstruct
solutions intended for application in the physical world, the solution is almost always developed
inthemathematicalworld.Today’sengineersmustbecomfortablewithtranslatingbetweenthem.
