BallandBeamDynamics文档格式.docx

BallandBeamDynamics文档格式.docx

《BallandBeamDynamics文档格式.docx》由会员分享，可在线阅读，更多相关《BallandBeamDynamics文档格式.docx（7页珍藏版）》请在冰豆网上搜索。

a.ModeloftheangleprocesswithrespecttothemotorvoltageHφ（s）

b.ModeloftheballpositionwithrespecttothebeamangleHx（s）

Thetotaltransferfunctionfromtheinputvoltagetothevoltagethatindicatestheballpositionisthen

3.MathematicalModelDerivation

a.Modelofbeamanglevs.inputvoltage

TherelationshipbetweentheinputvoltageandtheangleofthebeamisdefinedbytheDCmotortransferfunction.TheDCmotor,usedforanglecontrolapplicationmaybethoughtofasthe‘dirtyintegrator’ortheintegratorwithafilteractionasshowninfigure1onthenextpage.

Figure1:

GeneralDCmotorblockdiagramforanglecontrolapplication

TheKisthemotorconstantandthetauisthemotortimeconstant.Theactualmodelofthemotorusedfortheprojectisshowninfigure2.

Figure2:

ActualblockdiagramoftheDCmotorused

Themeasuredconstantsaresummarizedbelow:

ThereforetheDCmotortransferfunctionbecomes

Modelofballpositionvs.beamangle

Considerthefollowingsketch

Figure3:

Rollingballfree-bodydiagram

Theinclinationisconsideredthex-coordinate.

Letaccelerationoftheballbedenotedas

Theforceduetotranslationalmotionisthen

Thetorquedevelopedthroughballrotationisdeterminedbytheforceattheedgeoftheballmultipliedbytheradiuswhichcanbefurtherexpressedas:

where

J=momentofinertia（forsolidballdefinedbyJ=2/5*mR2）

Wb=angularvelocityoftheball

Vb=speedoftheballalongxaxis

Theequationisarrangedsuchthatthefinalresultisexpressedsolelyintermsofpositionoritsderivativesaswellasvariablesassociatedwiththeball.

Wenowobtaintherotationalforcebydividingtorqueoftheballbyitsradius

substitutingthemomentofinertiaintotheequationweget

Inordertomakethesystemindependentofthemassoftheballwefurtherexpresstheaboveequationsas

rearrangingforx’’gives

weutilizeapproximation

sincetheangleofthebeamwillnotexceed20-30degreeinclination.Thismeansthatinradians,sineoftheangleisapproximatelytheangleitself,sotheequationisfurtherapproximatedas

takingLaplacetransformofpositionwithrespecttoangle（detailsomitted）gives

Theconstantinthenumeratorisatheoreticalconstantthatneglectssurfaceimperfectionsandfriction.Themeasuredconstantisapproximately0.91therefore

Nowtheoveralltransferfunctionofthesystembecomes

where,10.5isanapproximatedconstant.

Theblockdiagramoftheoverallsystemis

Figure4:

Entiresystemblockdiagram

TheMATLABprovideseasyconversionofthesystemintostatespace.

>

num=[000010.5];

den=[0.41000];

[A,B,C,D]=tf2ss（num,den）

TheLQRcontrolcanbeimplementedbychoosingQandRvalues.Thecontrollabilityisverifiedfirstasfollows:

rank（ctrb（A,B））

ans=

4

Thisindicatesthattheballandbeamsystemiscompletelystatecontrollable.NextweselectQandRandcalculateforcontrollergainsinMATLAB.Weget

Q=[10000;

01000;

00100;

00010];

R=1;

[K,S,E]=lqr（A,B,Q,R）

K=

3.604810.50918.74443.1623

S=

10.509155.411150.220519.3050

8.744450.220572.591333.2327

3.162319.305033.232727.6524

E=

-3.9559

-0.9059

-0.6215+0.7044i

-0.6215-0.7044i

Wecanseethatthegainvaluesarereasonableandthereforetheactualsystemmayperformwell.

Thesimulationinsimulinkisdoneusingthefollowingblockdiagram:

Figure5:

Simulinkstatespacemodel

TheScopemeasurestheoutputwhiletheScope1monitorsthecontroleffort.Thesnapshotsareasshowninfigures6and7.

Figure6:

OutputconvergingfromIC

Figure7:

Controleffort

Theinitialconditionwassetto10cmfromthecentreatanangleof5.7degrees.Itiscanbeseenthatthesystemconvergesveryslowly.

Thestatevariablesare:

-angularacceleration

-angleofthebeam

-ballacceleration

-positionoftheball

Thepositionandtheanglecanbemeasureddirectlywithsensorswhiletheangularaccelerationandtheballaccelerationwillhavetobemathematicallyestimated.ThecontrolvoltageVisthen

（注：

可编辑下载，若有不当之处，请指正，谢谢!

）