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chapter1. Introduction

来源：乌哈旅游

The1stSemesteroftheAcademicYear2009/2010

AutomaticControlTheoryA:LinearControlSystems

2009.09.28-2010.01.13

QingheWu

SchoolofAutomationBeijingInstituteofTechnology

2009.08

Contents

1Introduction

1.1Introduction......................1.2HistoryofAutomaticControl............1.3TwoExamplesoftheuseoffeedback........1.4ControlEngineeringPractice............1.5ExamplesofMordernControlSystems.......1.6AutomaticAssemblyandRobots..........1.7TheFutureEvolutionofControlSystems.....1.8EngineeringDesign..................1.9ControlSystemsDesign...............1.10DesignExample:TurntableSpeedControl.....1.11DesignExample:InsulinDeliveryControlSystem1.12SummaryandProblems...............

1.12.1Summary...................1.12.2ExercisesandProblems...........

112333333344444555567771011121519192025252729303132323535364141424343

2MathematicalModelsofDynamicSystems(Chapter3)

2.1Introduction.........................................

2.1.1WhatisaMathematicalModelofaSystemandItsPurpose:.........2.1.2DeﬁnitionsandNomenclatures..........................2.1.3OutlineofthisChapter:..............................2.2DiﬀerentialEquationsofPhysicalSystemsandtheStateSpaceModels(Section3-4)

2.2.1TheTwoVariablesDescribingtheBehaviorofaPhysicalComponent....2.2.2ElectricalComponentsandCircuits(Section3-8)................2.2.3MechanicalTranslationalSystems(Section3-7).................2.2.4MechanicalRotationalSystems..........................2.3LinearApproximationsofPhysicalSystems(Section3-10)...............2.4TheLaplaceTransform..................................2.5TheTransferFunctionofLinearSystems(Section3-2).................

2.5.1TheTransferFunctionfromadiﬀerentialequation...............2.5.2TheTransferFunctionfromaState-SpaceModel................2.5.3TheState-SpaceModelfromaTransferFunction(Sections3-5,11-2,11-3).a.ControllableCanonicalForm..............................b.ObservableCanonicalForm..............................c.TheDiagonalCanonicalForm.............................d.TheJordanCanonicalForm..............................2.6BlockDiagramModels(Section3-3)...........................

2.6.1TheElementsofaBlockDiagram........................2.6.2BlockDiagramofaPositionControlSystem..................2.6.3BlockDiagramsofState-SpaceModelsinCanonicalForms..........2.6.4BlockDiagramofMultivariableSystem.....................2.6.5Rulesofblockdiagramalgebra..........................2.7SignalFlowGraph(Section3-9).............................

2.7.1Flow-GraphDeﬁnitions..............................2.7.2Mason’sSignal-FlowGainFormula........................2.8Summaryandhomework..................................

2.8.1Summary......................................

CONTENTS

2.8.2

iii

Homework............................................43

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3SolutionofDiﬀerentialEquations(Sections11-4,11-5)

3.1Introduction...............................3.2SolutionofLinearTime-invariantDiﬀerentialEquations......3.3SolutionoftheStateEquations....................

3.3.1SolutiontotheHomogeneousEquation...........3.3.2CompleteSolutionoftheStateEquation..........3.4EvaluatingtheStateTransitionMatrix................

3.4.1TransformingAintoDiagonalorJordanCanonicalForm.a.Aissimple..............................b.ACanBeTransformedintoJordanForm.............3.4.2UsingtheMinimalPolynomialTheory............a.AnnihilatingandtheMinimalPolynomialofaSquareMatrixAb.EvaluatingeAt............................3.4.3ByLaplaceTransform(A-11-15)...............3.5Problems................................4Control-SystemCharacteristics(Chapter5)

4.1Introduction...............................4.2TransientResponseAnalysis......................

4.2.1FirstOrderSystems......................a.FirstOrderSystemasaUnityFeedbackSystem.........b.Unit-StepResponse.........................c.Unit-RampResponse........................d.Unit-ImpulseResponse.......................4.2.2SecondOrderSystems.....................a.TheStandardSecondOrderSystem................b.Unit-StepResponse.........................c.TimeResponseSpeciﬁcations....................d.TransientResponseSpeciﬁcationsforSecond-OrderSystems..e.ServoSystemWithVelocityFeedback...............4.2.3TransientRespondofHigh-OrderSystems..........4.2.4StabilityAnalysisIntheComplexPlane...........4.2.5DominantClosed-LoopPoles.................4.3Routh-HurwitzStabilityCriterion..................

4.3.1SimpleStabilityTestforLow-DegreePolynomials.....4.3.2Routh’sCriterion........................a.TheBasicCriterion.........................b.SomeSpecialCases.........................4.3.3HurwitzStabilityCriterion..................4.3.4DeterminingtheStabilityRegionoftheSystemParametersa.StabilityRegionfortheOpen-LoopGain.............b.StabilityRegionforOtherSystemParameters..........4.4Steady-StateResponseAnalysis....................

4.4.1FeedbackSystemTypes....................4.4.2TheEnd-ValueTheoremofLaplaceTransform.......4.4.3StaticPositionError......................4.4.4StaticVelocityError......................4.4.5StaticAccelerationError...................4.4.6Steady-stateErrorwithRespecttoSinusoidalInputs....4.4.7Steady-StateErrorsWithRespecttoDisturbances.....4.5SummaryandProblems........................

4.5.1Summary............................4.5.2Problems............................

iv5The

5.15.25.35.4

RootLocusApproach(Chapter6)

Introduction.............................RootLocusofATypicalSecond-OrderSystem.........ConstructionRulesforRootLocus................SomeSpecialCases.........................5.4.1RootLocusforOtherSystemParametersthantheGain5.4.2RootLocusforPositive-FeedbackSystem........5.4.3ConditionallyStableSystem...............5.4.4Non-minimum-PhaseSystems...............SystemswithTransportLag....................SummaryandProblems......................5.6.1Summary..........................5.6.2TheProblems........................

CONTENTS

100.100.101.102.114.114.115.115.115.116.121.121.121122.122.122.122.122.122.122.122.122.122.123.128.128.128.129.132.132.132.132.132.132.132.132133.133.133.134.135.135.135.137.138.141.141.142.142.142.143.143.143.144.144.144.145

5.55.6

................Factor..............................................................................................

..................................

......................................................

6Root-LocusCompensation:Design(Chapter7)

6.1Introduction..............................

6.1.1PerformanceSpeciﬁcation..................6.1.2SystemCompensation....................6.1.3SeriesCompensationandFeedbackCompensation....6.1.4TheCompensators......................6.1.5DesignProcedure.......................6.2PreliminaryDesignConsiderations.................6.3LeadCompensation.........................

6.3.1LeadCompensatorRealizationbyPassiveNetwork....6.3.2LeadCompensationBasedonRootLocusApproach...6.3.3PD-Controller.........................6.4LagCompensation..........................

6.4.1LagCompensatorRealizationbyPassiveNetwork....6.4.2LagCompensationBasedonRootLocusApproach....6.4.3PI-Controller.........................6.5Lead-LagCompensation.......................

6.5.1Lead-LagCompensationRealizationbyPassiveNetwork.6.5.2Lead-LagCompensationBasedonRootLocusApproach6.5.3PID-Controller........................6.6Summary,ExercisesandProblems.................

6.6.1Summary...........................6.6.2ExercisesandProblems...................

7FrequencyResponseAnalysis(Chapter8)

7.1Introduction........................................

7.1.1Steady-StateResponseofanLTISystemtoSinusoidalInput........7.1.2DetermineG(jω).................................7.1.3GraphicalRepresentationsofFrequency-Response..............7.2NyquistStabilityCriterion...............................

7.2.1TheMappingTheorem.............................7.2.2NyquistStabilityCriterion...........................7.2.3TheRelationshipbetweenNyquistCriterionandtheRoot-LocusApproach7.3DrawingtheNyquistPlots................................

7.3.1TheFirst-OrderSystems............................7.3.2TheTypicalSecond-OrderSystems.......................7.3.3TheSystemswithTransportLag(DeadTime)................7.3.4NyquistPlotsforSystemsofType0,1,2...................a.Type0ControlSystem................................b.Type1ControlSystem................................c.Type2ControlSystem................................7.3.5DrawingNyquistPlotsUsingMATLAB....................7.4StabilityAnalysisUsingNyquistCriterion.......................

7.4.1SomeExamples..................................7.4.2ConditionallyStableSystems..........................

CONTENTS

7.4.3Multi-LoopSystems..............................7.4.4ChecktheD-StabilityUsingNyquistCriterion...............RelativeStabilityAnalysis...............................7.5.1RelativeStabilityAnalysisbyConformalMapping.............7.5.2AnInterpretationoftheRelativeStabilityMeasurefromtheViewpointof7.5.3GainMarginandPhaseMargin........................BodeDiagram.....................................7.6.1Introduction..................................7.6.2BodeDiagramfortheElementaryFactors..................a.BodeDiagramforConstants............................b.BodeDiagramforIntegratororDiﬀerentiatorss±1...............c.BodeDiagramfortheFactors(1+sT)±1.....................

󰀁󰀃󰀂±1󰀄

d.BodeDiagramforQuadraticFactors........ωn+1

.......

.....................Robustness.................................................

............

v147148149149150154155155157157157157

7.5

7.6

........159

.........................162162163164165165166166166167169169170171172172173174174175176176177177177

7.7

7.8

7.9

7.6.3BodeDiagramsforSystemsofType0,1,2........................a.BodeDiagramsforSystemsofType0andtheStaticPositionErrorConstant.......b.BodeDiagramsforSystemsofType1andtheStaticVelocityErrorConstant.......c.BodeDiagramsforSystemsofType2andtheStaticAccelerationErrorConstant.....7.6.4Minimum-PhaseandNonminimum-PhaseSystems....................7.6.5DeterminetheGainMarginandPhaseMarginFromtheBodeDiagram........7.6.6PlottingtheBodeDiagramwithMATLAB........................Closed-LoopFrequencyResponse..................................7.7.1Closed-LoopFrequencyResponseofUnity-FeedbackSystems..............7.7.2CorrelationbetweenStepTransientResponseandClosed-LoopFrequencyResponse.7.7.3ConstantMagnitudeLocus(MCircles)andConstantPhase-AngleLocus(NCircles).a.ConstantMagnitudeLocus(MCircles).............................b.ConstantPhase-AngleLocus(NCircles)............................c.DeterminetheClosed-LoopFrequencyResponseUsingtheMandNCircles........Log-MagnitudeVersusPhasePlot.................................7.8.1Introduction.........................................7.8.2DrawtheNicholsPlot....................................7.8.3Log-MagnitudeVersusPhasePlotsofMandNCircles(NicholsChart)........a.TranscribetheMCircles.....................................b.TranscribetheNCircles.....................................c.PeriodicityoftheNicholsChart.................................7.8.4DeterminetheClosed-LoopLog-MagnitudeandPhaseversusFrequencyCharacteristics

UsingNicholsChart.....................................

Summary,ProblemsandExercises.................................7.9.1Summary...........................................7.9.2ProblemsandExercises...................................

8Frequency-ResponseCompensationDesign(Chapters9and10)

8.1Introduction................................8.2LeadCompensation...........................

8.2.1CharacteristicsofaLeadCompensator.............8.2.2LeadCompensationTechniquesBasedonBodeDiagrams..8.3LagCompensation............................

8.3.1CharacteristicsofaLagCompensator.............8.3.2LagCompensationTechniquesBasedonBodeDiagrams...8.4Lead-LagCompensation.........................8.5PIDControl(Chapter10)........................

8.5.1PIDControlofPlants......................8.5.2RealizationoftheD-component.................8.5.3Ziegler-NicholsTuningRulesforControllers..........8.6Two-Degrees-Of-FreedomControl....................8.7SummaryandProblems.........................

8.7.1Summary.............................8.7.2Problems.............................181.181.181.181.182.185.185.186.189.189.189.189.190.191.192.192.192

9AnalysisofControlSystemsinState-Space(Sections11-6and11-7

9.1Introduction..................................9.2ControllabilityAnalysis...........................

9.2.1Controllability............................9.2.2TestsforControllability.......................9.2.3ControllabilityoftheCanonicalSystems..............a.ControllabilityNormalFormII.....................b.TheDiagonalForm............................c.TheJordanForm.............................9.3ObservabilityAnalysis............................

9.3.1Observability.............................9.3.2TestsforObservability........................9.3.3Popov-Belevitch-HautusTest....................9.3.4ObservabilityoftheCanonicalSystems..............9.3.5DualityPrincipleandDualSystems................9.4TransformaState-SpaceRepresentationintoCanonicalForms.....

9.4.1Introduction.............................9.4.2ControllableNormalFormsforSISOSystems...........a.ControllableNormalFormI.......................b.ControllableNormalFormII.......................9.4.3ObservableNormalFormsforSISOSystems............a.ObservableNormalFormI........................b.ObservableNormalFormII.......................9.5LinearTransforminState-Space(Continued)...............9.6DecompositionoftheState-Space......................

9.6.1ControllableDecomposition.....................9.6.2ObservableDecomposition......................9.6.3KalmanDecomposition.......................9.7ControllabilityandObservabilityinFrequencyDomain..........

9.7.1Controllability,ObservabilityandPole-ZeroCancellation....9.7.2ControllabilityandObservabilityofCascadedSystems......9.7.3ControllabilityandObservabilityofParallelSystems.......9.7.4ControllabilityandObservabilityofFeedbackSystems......9.8Summary,ProblemsandExercises.....................

9.8.1Summary...............................9.8.2ProblemsandExercises.......................

)...................................

CONTENTS

193.193.193.193.194.197.197.198.199.206.206.206.208.208.209.210.210.210.211.212.214.214.214.214.215.216.217.218.219.219.220.221.222.224.224.224225.225.225.225.226.228.228.228.228.228.228.230.231.231.231.231.231.231.233.234.235

..........................................................................................

.......................................................

10DesignofControlSystemsinState-Space(Chapter12)

10.1Introduction....................................10.2PolePlacement..................................

10.2.1StateFeedback..............................10.2.2NecessaryandSuﬃcientConditionforPolePlacement........10.2.3TheStabilizationProblem........................10.2.4OutputFeedback.............................10.2.5SelectRtofulﬁlltheSteady-StateErrorSpeciﬁcations........10.2.6SolvingthePolePlacementProblemwithMATLAB.........10.3Observer-BasedStateFeedback.........................

10.3.1TheConceptofStateObserver.....................10.3.2OutputInjectionControl........................10.3.3ProcedurefortheDesignofObserver-BasedState-Feedback.....10.3.4Reduced-OrderObserver.........................10.3.5SolvingtheObserver-BasedPolePlacementProblemwithMATLAB10.4YoulaParameterizationofAllStabilizingControllers.............10.5TheRegulatorProblemandtheInternalModePrinciple...........

10.5.1TheRegulatorProblem.........................10.5.2TheInternalStability..........................10.5.3CriterionforSteady-State-ErrorFree..................10.5.4SolvingtheRegulatorProblembyAugmentedState-Feedback...

CONTENTS

10.5.5SolvingtheRegulatorProblembyObserver-BasedState-Feedback10.6Summary,ProblemsandExercises......................

10.6.1Summary................................10.6.2ExercisesandProblems........................ASomeBackgroundResultsinLinearAlgebra

A.1MatrixPolynomialsandElementaryOperations......

A.1.1TheNotionofaMatrixpolynomial.........A.1.2ElementaryOperationsandEquivalence......A.2TheNormalForms......................

A.2.1TheSmithCanonicalFormforMatrixPolynomialsA.2.2TheNaturalNormalForm..............A.2.3TheJordanCanonicalForm.............

............................................................................................................................................................................................

vii236237237237

239.239.239.240.241.241.241.244

Chapter1

Introduction

1.1

Introduction

•Controlsystemengineersareconcernedwithunderstandingandcontrollingsystemstoprovideeconomicproductsforsociety.

•Thetwingoalsofunderstandingandcontrolarecomplementary:eﬀectivesystemcontrolrequiresthatthesystemsbeunderstoodandmodelled,controlengineeringmustconsiderthecontrolofpoorlyunderstoodsystems.

•Controlengineeringisbasedonthefoundationsoffeedbacktheoryandlinearsystemanalysis,andintegratestheconceptsofnetworktheoryandinformationtheory.Itisnotlimitedtoanyengineeringdiscipline.

•Acontrolsystemisaninterconnectionofcomponentsformingasystemconﬁgurationthatwillprovidedesiredsystemresponse.Acomponentoraprocesstobecontrolledcanbepresentedbyablock,asillustratedinFig.6.2

output-Figure1.1:Processtobecontrolled

•Open-loopsystemsarethoseinwhichtheoutputhasnoeﬀectupontheinput,asillustratedinFig.1.2

-Process

desired

-Compar.response

(referenceinput)

-actuatorSensor󰀛

Figure1.3:closed-loopsystem

-output

•Multivariablecontrolsystem(showninFig.1.4)

2CHAPTER1.INTRODUCTION

1.2HistoryofAutomaticControl

•Hero’sdeviceforopeningthedoorsofatemple

1.3.TWOEXAMPLESOFTHEUSEOFFEEDBACK

–time-domainformulationinSovietUnion;

•AlargeimpetustothetheoryandpracticeofautomaticcontroloccuredduringWorldWarII.

•Duringthe1950’s,theemphasisincontrolengineeringtheorywasonthedevelopmentanduseofthes-planemethods,particularlytherootlocusapproach.

•Duringthe1980’s,theutilizatinofdigitalcomputersforcontrolcomponentsbecomesroutine.•Thespaceage.

1.3TwoExamplesoftheuseoffeedback

•TheconceptofnegativefeedbackbyBlack;•Theconceptofservo-mechanismbyParkinson.

1.41.5

ControlEngineeringPractice

ExamplesofMordernControlSystems

•Basicidea:usingthediﬀerencebetweenthedesiredoutputresponseandtheactualoutputtocontroltheprocess,asillustratedinFig.1.9.

•Amanuallycontrolledclosed-loopsystemforregulatingthelevelofﬂuidinatank,asshowninFig.1.10.•Asimpliﬁedmodelforaboiler-generatorsystemshowninFig.1.12.Thisisagoodexampleformul-tivariablecontrolsystemssinceinapowersystemtwovariablesareimportantandmustbecontrolled:Thevoltageandthefrequency.

1.61.7

AutomaticAssemblyandRobots

TheFutureEvolutionofControlSystems

Toprovideextensiveﬂexibilityandahighlevelautonomy.

1.81.9

EngineeringDesignControlSystemsDesign

1.Establishthecontrolgoals;2.Identifythevariablestocontrol;3.Writethespeciﬁcationsforthevariables;

4.Establishthesystemconﬁgurationandtheactuator;5.Obtainamodelfortheprocess,theactuatorandthesensor;6.Describeacontrollerandselectkeyparameterstobeadjusted;7.Optimizetheparametersandanalysetheperformance;8.Judge

4CHAPTER1.INTRODUCTION

1.10DesignExample:TurntableSpeedControl

1.Goal:Theactualspeedofrotationiswithinaspeciﬁedpercentageofthedesiredspeed;2.Thevariabletocontrol:3.speciﬁcationforthevariable:

4.Actuator:Adcmotor.Systemconﬁguration:

•Usingabatteryandapotentiometertegeneratethedesiredspeed•open-looporclosed-loop?

Inthecaseofopen-loopcontrol,thevoltageforspeedsettingwillbeampliﬁedbyaDCampliﬁerandappliedtotheactuator(theDCmotor);=⇒Fig.1.20;Inthecaseofclosed-loopcontrol,

•Usingatachometertomeasurethespeedoftheshaft;•...,=⇒Fig.1.21.

1.11DesignExample:InsulinDeliveryControlSystem

•TheGoal:toregulatethebloodsugarconcentrationofadiabetic;•Thevariabletocontrol:

•Speciﬁcationforthevariable:toprovideabloodglocuseconcentrationofadiabeticthatcloselyap-proximatetheglocuseconcentrationofahealthypersonshowninFig.1.22;

•Theactuator:aninsulinpumpimplantedinthebody;Thesystemconﬁguration:open-loopcontrolbecase...

1.12

1.12.1

SummaryandProblems

Summary

•Inthissectionwehaveintroducedtheconceptsofsystems,controlsystems,automaticcontrolsystems,theopen-loopcontrolsystems,theclosed-loopcontrolsystems.Variousexamplesareusedtoexplainthesebasicconcepts;

•shownthecontrolsystemsdesignprocess.Usingtwodesignexamples,aturntablespeedcontrolsystemandaninsulindeliverycontrolsystem,wehaveexplainedthisprocess.Afurtherexampleisprovidedinthereferencebook.

1.12.2ExercisesandProblems

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