二轴陀螺仪动态平台控制及应用于自动测试系统中华科技大学.docx

二轴陀螺仪动态平台控制及应用于自动测试系统中华科技大学.docx

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二轴陀螺仪动态平台控制及应用于自动测试系统中华科技大学

TheImplementationofPC-BasedReal-TimeControlSystemsUsingPole-ZeroCancellationMethod

PC-Based即時控制系統之極點零點消去法實現

中華科技大學

11581台北市南港區研究院路三段245號

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ABSTRACT

Thegoalofthispaperistoimplementthepole-zerocancellationapproachinPC-basedrealtimecontrolsystemsbyusingmathematicmodelblocksinVisSimsoftwarepackageenvironment.WeusePCasacontrollertoapplypole-zerocancellationapproachinPC-basedrealtimecontrolsystems.Adesignexampleusingtherealcontrolsystem,FB-33servocontrolsystem,basedonthepole-zerocancellationapproachisgiven.Thepole-zerocancellationcontrollercanbeeasilyobtainedaccordingtothedesireddynamicperformancespecificationsoftheclosed-loopsystem.Finally,thePCcontrollerusesthepole-zerocancellationcontrolleranddataacquisitionsystemtocontroltheFB-33servocontrolsystem.Thesatisfiedresultsareshowninthispaper.

Keywords：

PC-BasedReal-TimeControlSystem,Pole-ZeroCancellationMethod,DataAcquisition

摘要

本文以VisSim為系統發展環境，以數學方塊圖模式設計開發極點零點消去法，以PC為主控制器，實現極點零點消去法於PC-Based即時控制系統。

首先以實際FB-33伺服控制系統為例，應用極點零點消去法，以滿足性能規格需求，得到控制器適當之極點與零點；另外以VisSim為系統發展環境，模擬加入控制器後系統動態響應，並與性能規格需求作比對，驗證其正確性。

然後再配合資料擷取模組與動態連結函數庫整合FB-33控制系統，完成PC-Based即時控制系統，實現極點安置演算法於伺服控制系統。

關鍵詞：

PC-Based即時控制系統、極點零點消去法、資料擷取

I.Introduction

Theroot-locusapproachtodesignisverypowerfulwhentheperformancespecificationsaregivenintermsoftime-domainquantities.Indesigningacontrolsystem,ifotherthanagainadjustmentisneed,wehavetoreshapetheoriginalrootlocibyaddingasuitablecompensator[1].Thetransferfunctionsofmanysystemscontainoneormorepolesthatareveryclosetotheimagineraxisofthes-plane.Thesepolesmaycausethesystemtobeslowlyresponseorlightlydamped.Wemayinsertapole-zerocancellationcompensatorthathasatransferfunctionwithzeroselected,whichwouldcanceltheundesiredpoleoftheuncompensatedopen-looptransferfunction,andtoplacethepoleofthecompensatoratmoredesirablelocationinthes-planetomeetthedesiredperformancespecifications[2][3].

ThispaperusesVisSimasasystemdevelopingenvironmenttodesignanddevelopadvancemoderncontrolalgorithmsbymathematicmodelblocks.VisSimisaWindows-basedprogramforthemodeling,designandsimulationofcomplexcontrolsystemswithoutwritingalineofcode[4][5].Itcombinesanintuitivedrag-and-dropblockdiagraminterfacewithapowerfulmathematicalengine.Thevisualblockdiagraminterfaceoffersasimplemethodforconstructing,modifyingandmaintainingcomplexcontrolsystemmodels.Furthermore,VisSimoffersunprecedentedease-of-useandconsequentlyashorterlearningcurvethancompetitivesystems.

SettingupasimulationinVisSimissimple.Connectthecontrollerwithpole-zerocancellationtothemathematicalmodelandtoaplotblock.Inpractical,thepoleofthetransferfunctionofcontrolsystemsmayvaryduetoexternaldisturbanceornoiseduringtheoperationofthesystem[6]-[10].However,wechoosethevariedrangeofthepoleoftheplanttobelessthan10%.TheresultingresponsescanbeeasilyanalyzedinVisSim.Onceasatisfactorymathematicalmodelhasbeenobtained,thepole-zerocancellationcompensatorusingoperationalamplifiersareappliedintheFB-33controlsystem.Noticethattheresultingresponsesshowthatexactcancellationisnotnecessarytopreciselynegativetheinfluenceoftheundesirablepoles[2]

Oncethepole-zerocancellationdesigniscomplete,thePCusingVisSim/Real-Timecanbeusedason-lineservocontrollerthroughahigh-speeddataacquisitioncard.PC-basedrealtimecontrolsystemscanbeconfiguredandexecutedbyinterfacingVisSimcontrollermodewiththeFB-33controlsystem.

II.MathematicalModelingofDCMotorControlSystem

ToestablishamathematicalmodelofPMdcmotor[2][6],wehavetheequivalentcircuitdiagraminFigure1.

Figure1Modelofaseparatedexciteddcmotor.

where

istheangulardisplacementofthemotorshaft,

istheangularvelocityofthemotorshaft,

istherotorinertial,

istheviscous-frictioncoefficient,

isthetorqueofthemotor,

istheappliedvoltage,

isthearmatureresistance,

isthearmaturecurrent,

isthearmatureinductance,

isthebackemf,and

isthemagneticfluxintheairgap.

Whenthearmatureisrotating,theback-emfvotageisproportionaltotheangularvelocity

weobtain

（1）

where

istheback-emfconstantofthemotor.

ApplyingKirchhoff’svoltagelawtothesystem,thedifferentialequationforthearmaturecircuitis

（2）

Thetorqueequationsofthesystemis

（3）

TakingtheLaplacetransformofbothsidesofEquation

（1）,

（2）and（3）,weobtain

（4）

（5）

（6）

SubstitutingEquation（5）’s

and（6）’s

toEquation（4），wehave

（7）

Ifthearmatureinductanceisreallyverysmall（

）,itcanbeneglected,andthetransferfunctionrelating

and

isgivenby

（8）

where

isDCmotorgainand

isDCmotortimeconstant.

ConsideraDCmotorpositionsystemwithproportionalcontrollershowninFigure2,theopen-looptransferis

　　　（9）

where

=

，

istheamplifiergain

P1istheattenuatorgain,

=

，

isthemotorgain,and

isthepotentiometergain.

IftheDCmotorpositioncontrolsystemwithPcontrollershowninFigure2isaunit-feedback,theclosed-looptransferfunctionis

（10）

where

istheopen-loopgainand

.

Figure2BlockdiagramoftheDCmotorpositioncontrolsystemwithPcontroller.

III.ControllerDesignbyPole-ZeroCancellation

Theopen-looptransferfunctionoftheDCmotorcontrolsystemshowninFigure2containsonepolethatisclosetotheimagineraxisofthes-planeintherootlocusplot.Thispolemaycausetheclosed-loopsystemtobeslowlyresponseorlightlydamped.Wecaninsertapole-zerocancellationcompensatorthathasatransferfunctionwithzeroselected,whichwouldcanceltheundesiredpoleoftheopen-looptransferfunction,andtoplacethepoleofthecompensatoratmoredesirablelocationinthes-planetomeetthedesiredperformancespecifications[1][2].

Becausethetransferfunctionofelementsincascadeistheproductoftheirindividualtransferfunctions[1],someundesirablepolesandzeroscanbecancelledbyinsertingacompensatingelementincascade.Forexample,thelargetimeconstantinEquation（9）maybecancelledbyuseofthecompensator

asfollows:

If

ismuchsmallerthan

wecaneffectivelyobtainthesmallmotortimeconstant

bycancelingthelargetimeconstant

Inpractical,thetransferfunctionoftheplantisusuallyobtainedthroughtestingandphysicalmodeling;linearizationofanonlinearprocessandapproximationofacomplexprocessareneeded[2].Thus,thetruepolesandzerosofthetransferfunctionoftheplantmaynotbeaccuratelymodeled.Infact,thetrueorderofthesystemmayevenbehigherthanthatpresentedbythetransferfunctionusedformodelingpurpose.Anotherdifficultyisthedynamicpropertiesoftheplantmayvaryduetoexternaldisturbanceornoise,sothepolesandzerosofthetransferfunctionmayvaryduringtheoperationofthesystem.Itisobviousthatexactcancellationisphysicallyimpossiblebecauseofinaccuraciesinvolvedinthelocationofthepolesandzerosofcontrolsystems.Thisproblemcanbesolvedinthefollowingbyshowingthatexactcancellationisnotnecessarytopreciselynegativetheinfluenceoftheundesirablepoles.

Letusassumethattheplantofthesystemisrepresentedby

（11）

where

isthepolethatistobecancelledand

isaverysmallvalueduetoexternaldisturbancesoftheplant..Letthetransferfunctionofthecompensatorbe

（12）

where

isthezeroandcisthepoleofthecontroller.Theopen-looptransferfunctionofthecompensatedsystemis

（13）

Becauseofinexactcancellation,itisobviousthattheterms

cannotbecancelledinthedenominatorofEquation（13）.Theclosed-looptransferfunctionis

（14）

Noticethatoneclosed-looppoleduetoaresultofinexactcancellationisveryclosetotheopen-looppoleat

.Thus,Equation（14）canbeapproximatedas

（15）

Thepartial-fractionexpansionofEquation（10）is

termsduetotheremainingpoles（16）

Wecanshowthat

isproportionalto

whichisaverysmallvalue.Itcanbeseenthatalthoughthepoleat

cannotbecancelledaccurately,theresultingtransient-responsetermduetoinexactcancellationwillhaveaverysmallmagnitude.Theeffectcausingbyinexactcancellationcanbeneglectedforallpracticalpurpose.

IV.VerifyingPole-ZeroCancellationMethodwithVisSim

ToverifythePole-ZeroCancellationmethodforFB-33controlsystem[6]describedabove,itcanbeeasilydoneusingVisSim.ItisconvenienttouseVisSimforthemodeling,designandsimulationofFB-33controlsystemwithoutwritingalineofcode[4]-[5].Itcombinesanintuitivedrag-and-dropblockdiagraminterfacewithapowerfulmathematicalengine.Thevisualblockdiagraminterfaceoffersasimplemethodforconstructing,modifyingandmaintainingcomplexcontrolsystemmodels.

TheFB-33FeedbackcontrolsystemshowninFigure3,theapparatusofautomaticcontrollaboratoryisatypicalsecond-ordersystem[7].Theobjectiveofthissystemistocontrolthepositionofthemechanicalloadinaccordancewiththereferenceposition.Theopen-looptransferfunctionoftheFB-33controlsystemobtainedbytheauthor[7]canbewrittenas

（17）

where

and

.

Thedesiredperformancespecificationsare

Maximumovershootislessthan32%（i.e.

）

Naturalfrequency

Tomeettheperformancespecifications,letusmodifytheclosedlooppolessothatan