计科全套课件

AutomaticControlTheory

SchoolofInformationandEngineering,ZZU1

TextBook:吴麟，自动控制原理（上、下），北京：清华大学出版社

OtherReferenceTeachingBooks:

JohnJ.D’Azzo.LinearControlSystemAnalysisandDesign(线性控制系统分析与设计，第4版).北京：清华大学出版社

RichardC.Dorf,RobertH.Bishop.ModernControlSystems（NinthEdition)（现代控制系统

第九版）.北京：科学出版社胡寿松.自动控制原理（第四版）.北京：科学出版社

2

Period:

Teachinghours:48hours

Exams:

Closed

exam.

Grading:

Homework20%,Finalexam80%3

Bytheendofthiscourse,thestudentshouldbeabletoFormulateamathematicalmodelofagivenphysicalsystemintimeandLaplacedomain.Identifythesystemorderandtype.Determinethesystem’stimeresponseduetoastep,rampandharmonicinput.4Evaluatethesystemstabilityusing:Routh-Hurwitzcriterion,rootlocusandNyquistdiagrams.ApplyclassicalcontrolmethodssuchasBodeplots,todesignclosedloopcontrolofthesystem.Applystatespacerepresentationofamultipleinputmultipleoutput(MIMO)system.DesignacontrollerandobserverforaMIMOsystem.5Chapter1Introduction

1.1AutomaticControl

1.2Open-loopControlSystemandClosed-loopControlSystem

1.3ConstituteofFeedbackControlSystem

1.4ClassifyofControlSystem

61.1AutomaticControl

1、AutomaticControlSystemspermeatelifeinalladvancedsocietiestoday.Technologicaldevelopmentshavemadeitpossibletotraveltothemoon;exploreouterspace.Andthesuccessfuloperationofspacevehicle;thespaceshuttle;spacestation;robot;industrycontrol,suchasthecontroloftemperature,pressure,fluid,lever,andsoon.

7

2、SomeTerminologies

A

controlsystem--------

A

controlsystem

isaninterconnectionofcomponentsformingasystemconfigurationthatwillprovideadesiredsystemresponse.

Referenceinput(Desiredoutput)------Excitationappliedtoacontrolsystemfromanexternalsource.Thereferencesignalproducedbythereferenceselector.Itistheactualsignalinputtocontrolsystem.8

Disturbanceinput-------Adisturbanceinputsignaltothesystemthathasanunwantedeffectonthesystemoutput.

Output(controlledvariable)--------Thequantitythatmustbemaintainedataprescribedvalue,i.e.,itmustfollowthecommandinputwithoutrespondingtodisturbanceinputs.

Feedback-----Theoutputofasystemthatisreturnedtomodifytheinput.9

Error-----Thedifferencebetweentheinputandtheoutput.

Open-loopcontrolsystem–Asysteminwhichtheoutputhasnoeffectupontheinputsignal.

Feedbackelement–Theunitprovidesthemeasurementvalueforfeedingbacktheoutputquantity,orafunctionoftheoutput,inordertocompareitwiththereference.10

Actuatingsignal(errorsignal)–Thesignalthatisthedifferencebetweenthereferenceinputandthefeedbacksignal.Itistheinputtothecontrolunitthatcausestheoutputtohavethedesiredvalue.

Negativefeedback–Theoutputsignalisfeedbacksothatitsubtractsfromtheinputsignal.

11

Closed-loopcontrolsystem–Asysteminwhichtheoutputhasaneffectupontheinputquantityinsuchamannerastomaintainthedesiredoutputvalue.Thatis,asystemthatusesameasurementoftheoutputandcomparesitwiththedesiredoutput.12

3、Controlsystemsareusedtoachieve:(1)increasedproductivity;(2)improvedperformanceofadeviceorsystem.Thecontrolofanindustrialprocess(manufacturing,production,andsoon)byautomaticratherthanmanualmeansisoftencalledautomation.Automationisusedtoimproveproductivityandobtainhigh-qualityproducts.13

4、Historyofautomaticcontrol

(1)Thefirstautomaticfeedbackcontro-llerusedinanindustrialprocessisgenerallyagreedtobeJamesWalt’sflyballgovernor,developedin1769forcontrollingthespeedofasteamengine.ShowninFig.1.1.

14Fig.1.1JamesWatt’sFlyballGovernor(1769)

Theall-mechanicaldevice,showninFig.1.1,measuredthespeedoftheoutputshaftandutilizedthemovementoftheflyballwithspeedtocontrolthevalveandthereforetheamountofsteamenteringtheengine.Asthespeedincreases,theballweightsriseandmoveawayfromtheshaftaxis,thusclosingthevalve.Theflyweightsrequirepowerfromtheenginetoturnandthereforecausethespeedmeasurementtobelessaccurate.

16

(2)J.C.Maxwellformulatedamathema-ticaltheoryrelatedtocontroltheoryusingadifferentialequationmodelofagovernor.(1868)

(3)Conventionalcontroltheoryiseffectivelyappliedtomanycontroldesignproblems,especiallytoSISOsystems.ItsmathematicalfoundationistheLaplacetransform.17

Routh1884;Hurwitz1895,algebrastabilitycriterion;

1932,Nyquist,steady-statefrequency-responsetechniques;

1927,BodeandNichols,frequency-responseanalysis;

1948,Evans,root-locustheory;

A.M.Lyapunov,stabilitytheory.18

(4)Moderncontroltheory(1960)isbasedonstatevariablemethods,forthedesignofmultiple-inputmultiple-output(MIMO)systems.Wiener(1949),Optimumdesign.Bellman(1957),Dynamicprogramming.Pontryagin(1962),Maxmumprinciple.Kalman(1960),ControllabilityandobservabilityKalmanandBuey(1961),Combinationofoptimalfilterandoptimalcontroller,LinearquadraticGaussian(LQG)control.19ClassicalcontroltheoryModerncontroltheory

SISO,linear,time-unvaryingsystem

MIMO,nonlinear,time-varyingsystem

Laplacetransform

matrix,vector,linearalgebra

Time,complexnumber,frequencydomain

Timedomain

output

state(5)Advancedcontroltheory(1980)Robusttheory.(1980s)Intelligentcontroltheory:ArtificialNeuralNetworks(ANNS);FuzzyControl(FC);ExpertSystem(ES).GA,GP,EC,Chaosetc.RETURN21

1.2Open-loopControlSystem

andClosed-loopControlSystem

1、Acontrolsystemisaninterconn-ectionofcomponentsformingasystemconfigurationthatwillprovideadesiredsystemresponse.

Acomponentofprocesstobecontrolledcanberepresentedbyablock,asshowninFig.1.2.

22(Stimulus)(Response)Cause(Desiredresponse)Effect(Actualresponse)23

2、Anopen-loopcontrolsystemutilizesacontrollerorcontrolactuatortoobtainthedesiredresponse,asshowninFig.1.3.

Thecontrolactioniscalculatedattheinitialtimet0andthenappliedtothephysicalsystemovertheentirecontrolhorizon[t0,tf]withoutmodification.Theoutputisnotobserved.

Anopen-loopcontrolsystemutilizesanactuatingdevicetocontroltheprocessdirectlywithoutusingfeedback.

Anotheropen-loopcontrolsystem:253、Themeasureoftheoutputiscalledthefeedbacksignal.Asimpleclosed-loopfeedbackcontrolsystemisshowninFig.1.4orFig.1.5.

Theadvantagesanddisadvantagesofopen-loopcontrolsystemandclosed-loopcontrolsystemTheexamplesofclosed-loopsystem:Example1:Example2:4.Complexcontrolsystem(idealcontrolmethod)

Combineopen-loopcontrolwithclosed-loopcontrolRETURN271.3ConstituteofFeedbackControlSystem

1、ConstituteoffeedbackcontrolsystemPlant+ControllerPlant(process):Thedevice,plantorsystemundercontrol.Measurementcomponent(Sensor);Comparisoncomponent;Amplifier;Actuatingdevice;Compensator;Controller28ComparedeviceCompensatoramplifieractuatorplantDisturbsignalInputdeviceReferenceinputControlsignalsensorMeasuredevicecontrollerstructureofgeneralcontrolsystem292、AnalysisofcontrolsystemStability:themostimportant;Performance:thetransientandsteady-stateresponseperformance;Robustness.30StabilityTransientperformancesteady-stateperformance3、“Control”

“Control”isthedesignandanalysisofsensors,actuators,andcomputationalsystems(analogordigital)tomodifythebehaviorofphysicalsystems.34Typicalsteps:SelectionofactuatorsandsensorsDevelopmentofadynamicmodelofthesystemtobecontrolledDesignofcontrolsystemsbasedonthesoundfundamentalprinciplesImplementationofthecontrollerusinganalogordigitalelectronics354、Designofcontrolsystem(1)Establishthesystemgoals.(2)Identifythevariablestocontrol;(3)Writethespecificationsforthevariables;(4)Establishthesystemconfigurationandidentifytheactuator;(5)Obtainamodeloftheprocess,theactuatorandthesensor;(6)Describeacontrollerandselectkeyparameterstobeadjusted;(7)Optimizetheparametersandanalyzetheperformance.36RETURN1.4ClassifyofControlSystem1.Theformofsystem’smathematicsmodelLinearsystem/Nonlinearsystem:thedynamicequationofsystemislineardifferentialequation/nonlineardifferentialequation.

Alinearsystemsatisfiestheprincipleofsuperposition.38Time-varyingsystemandtime-invariantsystem

Time-varyingsystemisasystemforwhichoneormoreoftheparametersofthesystemmayvaryasafunctionoftime.2.Referenceinputr(t)=constantr(t)=f(t)canbeknownthevaryrulef(t)cann’tbeknown“servo”393.Thetransfersignalsequence-timesystem

discrete-timesystem40Summary

Inordertodesignandimplementacontrolsystem,thefollowingessentialgenericelementsarerequired:1)KnowledgeofthedesiredvalueItisnecessarytoknowwhatitisyouaretryingtocontrol.412)KnowledgeoftheoutputoractualvalueThismustbemeasuredbyafeedbacksensor,againinaformsuitableforthecontrollertounderstand.Inadditional,thesensormusthavethenecessaryresolutionanddynamicresponse,sothatthemeasuredvaluehasaccuracyrequiredfromtheperformancespecification.3)KnowledgeofthecontrollingdeviceThecontrollermustbeabletoacceptmeasure-mentsofdesiredandactualvaluesandcomputeacontrolsignalinasuitableformtodriveanactuatingelement.424)KnowledgeoftheactuatingdeviceThisunitamplifierthecontrolsignalandprovidesthe“effort”tomovetheoutputoftheplanttowardsitsdesiredvalue.5)KnowledgeoftheplantMostcontrolstrategiesrequiressomeknowledgeofthestaticanddynamiccharacteristicoftheplant.Thesecanbeobtainedform

measurementsorformtheapplicationofFundamentalphysicallaws.orcombinationofboth.RETURN43

Aclosed-loopcontrolsystemusesameasurementoftheoutputandfeedbackofthesignaltocompareitwiththedesiredoutput(referenceofcommand).

Ageneralcontrolsystemcanberepresentedas:FeedbackelementActuatingsignal(errorsignal)

Anotherblockdiagramofageneralizedfeedbackcontrolsystem

1.5RETURN

open-loopcontrolsystemThemainadvantages:1)Muchsimpleandlessexpansivetoconstruct;2)Easytohavegoodstability;

Themaindisadvantages:Requiredetailedknowledgeofeachcomponentinordertodeterminetheinputvalueforarequiredoutput.

Theadvantagesanddisadvantagesofopen-loopcontrolsystemandclosed-loopcontrolsystem

closed-loopcontrolsystemThemainadvantages:

1)Havehigheraccuracy;

2)Notrequiredetailedknowledgeofeachcompo-nentoraccuratemodeloftheindividualcomponent;

3)Theabilitytorecoverfromexternal,unwanteddisturbances;

4)Reducedsensitivitytodisturbance;

Themaindisadvantages:

1)Complexandexpensivetoconstruct;

2)thelossofgain;

3)Quiteeasytobecomeunstable.

47Formostcases,theadvantagesfaroutweighthedisadvantages,andafeedbacksystemisutilized.Thereforeitisnecessarytoconsidertheadditionalcomplexityandtheproblemofstabilitywhendesigningacontrolsystem.RETURN

Example1:Awatertemperaturecontrolsystem

1.61.7Observe(throughsenses)Computeacontrolaction(throughbrain)andapplythecontrolAction(throughhands,feet,etc!)RETURNExample2:watertemperaturecontrolsystemwithaautomaticmethod1.8

Theblockdiagramofthewatertemperatureautomaticcontrolsystem

1.9

Comparemanualcontrolmethodtoautomaticcontrolmethod

Brain----Thermostat(Referenceinputordesiredoutput)Nervoussystem----DifferencingjunctionHand----Amplifier,MotorandWheeldrivers

EyesandSkinsensor---Temperaturesensor

MixerValve----MixerValveChapter2SystemModeling2.1

Mathematicalmodels2.2

Dynamicsequation2.3

Statespaceandstateequation2.4

Lineardifferentialequation2.5

Stateequation2.6

Transferfunction(matrix)2.7

Transferfunctionofclose-loopsystem2.8

Fundamentalparts2.9

Signalflowgraphs2.10

Impulseresponseandstepresponse2.11

Summary

Excises552.1

MathematicalModelsThefundamentalconceptofmathematicalmodelThefundamentalformsofmathematicalmodelsThemethodofmodelingThestepsofanalyzingandstudyingadynamicsystemInstructionalobjectivesAppendix:Propertyoflinearsystem56Ifthedynamicbehaviorofaphysicalsystemcanberepresentedbyanequation,orasetofequations,thisisreferredtoasthemathematicalmodelofthesystem.

Modelofsystem–therelationshipbetweenvariablesinsystem.

Whydowemuststudythemodelofcontrolsystem?

Quantitativemathematicalmodelsmustbeobtainedtounderstandandcontrolcomplexdynamicsystems.(analysisanddesign)1.Mathematicalmodel57Mathematicalexpression

DifferentialEquation:

(timedomainmodel)

areusedbecausethesystemsaredynamicinnature.

Impulsetransferfunction;state-spaceequation.Assumptionsareneededbecauseofthecomplexityofsystemsandtheignoranceofalltherelevantfactors..2.Thefundamentalformsofmathematicalmodel58Transferfunction(sdomainmodel)Laplacetransformisutilizedtosimplifythemethodofsolutionforlinearequations.

BlockdiagramandSignalflowdiagramResponseCurve(nonparametermodel)

frequencyresponsecurve;Bodediagram593.ThemethodofmodelingAnalyticalmethod:Itcanbeconstructedfromknowledgeofthephysicalcharacteristics

ofthesystemExperimentalmethodOthers.(NN)60Definethesystemanditscomponents.Formulatethemathematicmodelandlistthenecessaryassumptions.Writethedifferentialequationsdescribingthemodel.Solvetheequationsforthedesiredoutputvariables.Examinethesolutionsandtheassumptions.Ifnecessary,reanalyzeorredesignthesystem.4.Thestepsofanalyzingandstudyingadynamicsystem61

5.InstructionalobjectivesDevelopdynamicmodelsofphysicalcomponents.Derivationoftransferfunctions.Blockdiagramrepresentation.Blockdiagramrulesandsimplifytheblockdiagramtodeterminetheclosed-looptransferfunction.62Themathematicalmodelofasystemislinear,ifitobeystheprincipleofsuperposition(orprincipleofhomogeneity).Thisprincipleimpliesthatifasystemmodelhasresponsesy1(t)andy2(t)toanytwoinputsx1(t)andx2(t)respectively,thenthesystemresponsetothelinearcombinationoftheseinputs:ax1(t)+bx2(t)isgivenbythelinearcombinationoftheindividualoutputs,i.e.ay1(t)+by2(t).6.Appendix:Propertyoflinearsystem63Figure2.1MeaningofalinearsystemReturn642.2

Dynamicsequation

2.2.1

Derivationofthedifferentialequations2.2.2Linearapproximationsofnonlinearequation2.2.3Thedifferentialequationsofcomplexplants2.2.4Singlevariabledifferentialequation’sderivationoforiginalequations2.2.5Dynamicsequationofdiscrete-timesystemReturn652.2.1Derivationofthedifferentialequations

Inordertoanalyzethebehaviorofphysicalsystemsintimedomain,wewritethedifferentialequationsrepresentingthosesystems:

Electricalsystems(KVLand/orKCL,aelectricnetworkcanbemodeledasasetofnodalequationsusingKirchhoff’scurrentlaworKirchhoff’svoltagelaw).

66

Mechanicalsystems(Newton’slawsofmotion)Hydraulicsystems(Thermodynamic&Conservationofmatter)Thermalsystems(Heattransferlaws,Conservationofenergy)

Thestandarddifferentialequationis:outputinput67Thenumberofindependentenergystoragecomponentsinasystemdeterminetheorderofthatsystem.Annthorderdifferentialequationimpliesnindependentenergystoringcomponentsandyouneedninitialconditions.where

n≥m

foraphysicallyrealizablesystem.6869707172737475Example.2.1Displacementsystemofspring-mass-damper76

Example2.2：DifferentialequationofaR-L-Cnetwork.

uc(t)

istheoutputvoltageandu(t)

isthevoltagesource.LRCu(t)uc(t)++i(t)Figure2.2R-L-Cnetwork77TheequationoftheRLCnetworkshowninFig.2.2isobtainedbywritingtheKirchhoffvoltageequation,yielding

Therefore,solvingEq.(2)foriandsubstitutinginEq.(1),wehave78

arethetimeconstantsofthenetwork.

79AnalogousvariablesandsystemsRewritingthefollowingequationsintermsofdisplacementandvoltagerespectivelyItisobviousthattheyareequivalent.Displacementx(t)andvoltageuc(t)areequivalentvariables,usuallycalledanalogousvariables,andthesystemsareanalogous

systems.80Theconceptofanalogoussystemsisaveryusefulandpowerfultechniqueforsystemmodeling.Analogoussystemswithsimilarsolutionsexistforelectrical,mechanical,thermal,andfluidsystems.Theanalystcanextendthesolutionandtheunderstandingofonesystemtoallanalogoussystemswiththesamedescribingdifferentialequations.Analogoussystemshavethesameformsolution.81

Conclusion:

Differentphysicalsystemcangainsimilardifferentialequations.Thedifferentialequationsreflecttheessencecharacteristicsofsystem.82Example2.3Differentialequationofdcmotor.IaU(t)MΩML

EaJLoadΦdRa,Ladcmotorwiringdiagram83Thearmaturecurrentisrelatedtotheinputvoltageappliedtothearmatureas：

whereEa

isthebackelectromotive-forcevoltageproportionaltothemotorspeed,Uisinputvoltage,Iaisthearmaturecurrent,，La

isthemotorinductance,Raismotorresistance.84

Misthemotortorque,MListheloadtorque,Jistherotorinertia,Ωisthe(angular)velocityofthemotorbearings.

where

kdisdefinedasthemotorconstant.85Therearetwofirst-orderdifferentialequationsandtwoalgebraicequations,andsixvariablesU,Ea,,

Ia,M,ML,J,Ω.U

andML

aretheinputvariables,

whichleadtothemotor’smovement.IfweregardasΩbetheoutputvariable,othersbethemiddlevariables.Thedifferentialequationofthemotor-loadcombinationis86isthefieldtimeconstantofthearmature;isthetimeconstantofthemotorarmature;Generally,IfTa

isneglected,theequationisReturn872.2.2LinearApproximations

ofnonlinearequationAgreatmajorityofphysicalsystemsarelinearwithinsomerangeofthevariables.However,allsystemsultimatelybecomenonlinearasthevariablesareincreasedwithoutlimit.Asystemisdefinedaslinearintermsofthesystemexcitationandresponse.Keywords:

operatingpoint;small-signalconditions;Taylorseries;continuous;linearapproximation88Therelationshipoftwovariableswrittenas,wheref(x)indicatesyisafunctionofx.Thenormaloperatingpointisdesignatedbyx0.Becausethecurve(function)iscontinuousovertherangeofinterest,aTaylorseriesexpansionabouttheoperatingpointmaybeutilized.Thenwehave89Theslopeattheoperatingpoint,isagoodapproximationtothecurveoverasmallrangeof(x-x0)(smallperturbation),thedeviationfromtheoperatingpoint.Then,asareasonableapproximation,theequationcanberewrittenasthelinearequationwheremistheslopeattheoperatingpoint.9091Example2.4NonlineardifferentialequationOperatingpoint:（T0，u0）Smallchange:92Linearapproximationofdifferentialequation93Ifthedependentvariableydependsuponseveralexcitationvariables,thenthefunctionalrelationshipiswrittenas

TheTaylorseriesexpansionabouttheoperatingpointx0(x10,x20,…,xn0)isusefulforalinearapproximationtothenonlinearfunction.Whenthehigher-ordertermsareneglected,thelinearapproximationiswrittenasReturn942.2.3Thedifferentialequations

ofcomplexplantsGenerallysteps:

(1)ConfirmtheI/Ovariablesofsystemandeverycomponents;(2)Writethedynamicsequationsofeverycomponentsusingphysicallaws;(3)Checkupthenumberofequationsanddecideifitisequaltothenumberofunknownvariablesornot.(4)Expurgatetheinsidevariablesfromthesetofthedifferentialequations,andchangetothestandardform.95

Example2.5Servo-system

Workprinciple:Twosamechangeableresistoraresuppliedbythesamedcelectricalsource;Thearmofresistor1canrotatebyhandle3.Supposingψ、φrepresentthepositionofthearmofthetworesistorsrespectively.Ifψisnotequaltoφ,thentheerrorsignalup

isformed,whichbeamplifiedby4,finallythefieldcurrentIfofthedcdynamotor’sfieldwindingisformed,whichleadtothevoltage’schangeofdynamotor5.Sothedcmotor8rotates,whichleadtotheload10rotatestoobythegear9.Sothearmofresistor2startstomove,asfarasψ=φ.96up97Writeoutdynamicsequationsofeverycomponents:(1)thesetofchangeableresistances：Inputs:ψ,φ;output:up

（1）Wherekdisthecoefficientoftheresistance.(2)amplifier：Input:up,output:If

;

or（2）

wherekaisthemagnifiedmultipleofvoltage;Rfisthesumresistanceofoutputcircuit;Lfisinductanceoffieldwinding6;Tf=Lf/Rf

isthetimeconstantoffieldcircuit.98(3)dynamotor-motor:Input:If

;output:Ω.Fromexample2.2:

(3)(4)driveinstitution:Input:Ω;output:φ.

(4)

wherektisthedriveratioofthedriveinstitution.99Equation（1）~（4）arethemathematicsmodelsoftheservo-system.Itconsistsofonesecond-orderdifferentialequation,twofirst-orderdifferentialequationsandonealgebraicequation.Therearesixvariablessuchasψ、φ、

up、

If

、

ML、Ω.Tothewholesystem,theinputsareψand

ML,othersaretheinsidedependentvariables,andthenumberisequaltothenumberoftheequations.Return1002.2.4

Scalardifferentialequation’s

derivationoforiginalequations

Return(1)ConfirmtheI/Ovariablesofsystemandeverycomponents;(2)Writethedynamicsequationsofeverycomponentsusingphysicallaws;(3)Checkupthenumberofequationsanddecideifitisequaltothenumberofunknownvariablesornot.(4)Expurgatetheinsidevariablesfromthesetofthedifferentialequations,andchangetothestandardform.1012.2.5

Dynamicsequation

ofdiscrete-timesystem

Thediscrete-timeapproximationisbasedonthedivisionofthetimeaxisintosufficientlysmalltimeincrements.Thevaluesoftheinput/outputareevaluatedatthesuccessivetimeintervals,thatis,t=0,T,2T,…,whereTistheincrementoftime:=T.IfthetimeincrementTissufficientlysmallcomparedwiththetimeconstantsofthesystem,thediscrete-timeserieswillbereasonablyaccurate.102Differenceequation:reflecttherelationshipoftheinputandoutputseriesattheeverysamplepointsofthediscretetimesystem.UsingTdenotesthespacebetweentwosamplingtime.

Differenceequationdescribesthemovementoftheseplants.Detailsinchapternine.103Generalbackwarddifferenceequationoflineartime-invariantsystem:ortheforwarddifferenceequation:Return1042.3Statespaceandstateequation

2.3.1Statevectorandstatespace

2.3.2Stateequationandoutputequation

2.3.3Stateequation’sderivationoforiginalequationsReturn1052.3.1StatevectorandstatespaceDynamicsystem

—asystemthatcanstoreinputinformation.Example:xF(t)mv(t)xm:

themass;a(t):

theaccelerationofmattimet;F(t):

theforceappliedtomfortime

[t0,t];v(t):thevelocityof

m

attime

t;x(t):thedisplacement

of

m.106AccordingtoNewton‘slaw,wehave:Ifwechoosex(t)andv(t)astwostatevariables,thenWherex(t0)andv(t0)aretheinitialstatesvalue.x(t0-),v(t0-)

F(t)(t>t0)_determinex(t)andv(t)candescribethebehaviorofsystemcompletely.107

State—theminimumsetofvariables(calledthestatevariables)whichatsomeinitialtimet0,togetherwiththeinputssignalu(t)fortime,sufficestodeterminethefuturebehaviorofthesystemfortime.Foradynamicsystem,thestateofasystemisdescribedintermsofasetofstatevariables[x1(t),x2(t),…,xn(t)].Example:ExpressthestateofsystemattimetExpresstheinitialstateofsystem108Statevariables—arethosevariablesthatdeterminethefuturebehaviorofasystemwhenthepresentstateofthesystemandtheexcitationsignalsareknown.

Example:

displacementx(t)andvelocity

v(t)aretwostatevariables,anyoneofthemcannotdescribethemasssystemcompletely.109

Statevector—Vector

whichconsistsofnstatevariablesthatdescribedentirelythedynamicsactionofaknownsystem.thatisdescribedas

or

Example:arestatevectorofmasssystem110Statespaceisndimensionsspacewhichtakesx1,x2…xn

as

coordinate.Themannerinwhichthestatevariableschangeasafunctionoftimemaybethoughtofasatrajectoryinndimensionalspace,calledStatespace.0xvExample:twodimensionstatespace

111

Example:

Laplace

then：1)ift≥0，u(t)isknown，andtheinitialcondition