线性系统理论期中大作业.docx

线性系统理论期中大作业.docx

《线性系统理论期中大作业.docx》由会员分享，可在线阅读，更多相关《线性系统理论期中大作业.docx（14页珍藏版）》请在冰点文库上搜索。

线性系统理论期中大作业

线性系统理论期中大作业

在下列问题中选择一个问题，完成如下工作：

1）建立系统的状态空间模型；

2）分析开环系统的运动规律，并仿真其状态轨迹；

3）分析系统的能控能观性；

4）分析开环系统的稳定性

问题一：

DCMotorSpeedControl

AcommonactuatorincontrolsystemsistheDCmotor.Itdirectlyprovidesrotarymotionand,coupledwithwheelsordrumsandcables,canprovidetransitionalmotion.Theelectriccircuitofthearmatureandthefreebodydiagramoftherotorareshowninthefollowingfigure:

Forthisexample,wewillassumethefollowingvaluesforthephysicalparameters.ThesevalueswerederivedbyexperimentfromanactualmotorinCarnegieMellon'sundergraduatecontrolslab.

*momentofinertiaoftherotor（J）=0.01kg.m^2/s^2

*dampingratioofthemechanicalsystem（b）=0.1Nms

*electromotiveforceconstant（K=Ke=Kt）=0.01Nm/Amp

*electricresistance（R）=1ohm

*electricinductance（L）=0.5H

*input（V）:

SourceVoltage

*output（theta）:

positionofshaft

*Therotorandshaftareassumedtoberigid

Themotortorque,T,isrelatedtothearmaturecurrent,i,byaconstantfactorKt.Thebackemf,e,isrelatedtotherotationalvelocitybythefollowingequations:

InSIunits（whichwewilluse）,Kt（armatureconstant）isequaltoKe（motorconstant）.

FromthefigureabovewecanwritethefollowingequationsbasedonNewton'slawcombinedwithKirchhoff'slaw:

问题二：

DCMotorPositionControl

AcommonactuatorincontrolsystemsistheDCmotor.Itdirectlyprovidesrotarymotionand,coupledwithwheelsordrumsandcables,canprovidetransitionalmotion.Theelectriccircuitofthearmatureandthefreebodydiagramoftherotorareshowninthefollowingfigure:

Forthisexample,wewillassumethefollowingvaluesforthephysicalparameters.ThesevalueswerederivedbyexperimentfromanactualmotorinCarnegieMellon'sundergraduatecontrolslab.

*momentofinertiaoftherotor（J）=3.2284E-6kg.m^2/s^2

*dampingratioofthemechanicalsystem（b）=3.5077E-6Nms

*electromotiveforceconstant（K=Ke=Kt）=0.0274Nm/Amp

*electricresistance（R）=4ohm

*electricinductance（L）=2.75E-6H

*input（V）:

SourceVoltage

*output（theta）:

positionofshaft

*Therotorandshaftareassumedtoberigid

Themotortorque,T,isrelatedtothearmaturecurrent,i,byaconstantfactorKt.Thebackemf,e,isrelatedtotherotationalvelocitybythefollowingequations:

InSIunits（whichwewilluse）,Kt（armatureconstant）isequaltoKe（motorconstant）.

FromthefigureabovewecanwritethefollowingequationsbasedonNewton'slawcombinedwithKirchhoff'slaw:

问题三：

InvertedPendulum

Thecartwithaninvertedpendulum,shownbelow,is"bumped"withanimpulseforce,F.Determinethedynamicequationsofmotionforthesystem,andlinearizeaboutthependulum'sangle,theta=Pi（inotherwords,assumethatpendulumdoesnotmovemorethanafewdegreesawayfromthevertical,chosentobeatanangleofPi）.Findacontrollertosatisfyallofthedesignrequirementsgivenbelow.

Forthisexample,let'sassumethat

M

massofthecart

0.5kg

m

massofthependulum

0.5kg

b

frictionofthecart

0.1N/m/sec

l

lengthtopendulumcenterofmass

0.3m

I

inertiaofthependulum

0.006kg*m^2

F

forceappliedtothecart

x

cartpositioncoordinate

theta

pendulumanglefromvertical

ForthePID,rootlocus,andfrequencyresponsesectionsofthisproblemwewillbeonlyinterestedinthecontrolofthependulumsposition.Thisisbecausethetechniquesusedinthesetutorialscanonlybeappliedforasingle-input-single-output（SISO）system.Therefore,noneofthedesigncriteriadealwiththecart'sposition.Forthesesectionswewillassumethatthesystemstartsatequilibrium,andexperiencesanimpulseforceof1N.Thependulumshouldreturntoitsuprightpositionwithin5seconds,andnevermovemorethan0.05radiansawayfromthevertical.

Thedesignrequirementsforthissystemare:

∙Settlingtimeoflessthan5seconds.

∙Pendulumanglenevermorethan0.05radiansfromthevertical.

However,withthestate-spacemethodwearemorereadilyabletodealwithamulti-outputsystem.Therefore,forthissectionoftheInvertedPendulumexamplewewillattempttocontrolboththependulum'sangleandthecart'sposition.Tomakethedesignmorechallengingwewillbeapplyingastepinputtothecart.Thecartshouldachieveit'sdesiredpositionwithin5secondsandhavearisetimeunder0.5seconds.Wewillalsolimitthependulum'sovershootto20degrees（0.35radians）,anditshouldalsosettleinunder5seconds.

ThedesignrequirementsfortheInvertedPendulumstate-spaceexampleare:

∙Settlingtimeforxandthetaoflessthan5seconds.

∙Risetimeforxoflessthan0.5seconds.

∙Overshootofthetalessthan20degrees（0.35radians）.

Forceanalysisandsystemequations

BelowarethetwoFreeBodyDiagramsofthesystem.

SummingtheforcesintheFreeBodyDiagramofthecartinthehorizontaldirection,yougetthefollowingequationofmotion:

Notethatyoucouldalsosumtheforcesintheverticaldirection,butnousefulinformationwouldbegained.

SummingtheforcesintheFreeBodyDiagramofthependuluminthehorizontaldirection,youcangetanequationforN:

Ifyousubstitutethisequationintothefirstequation,yougetthefirstequationofmotionforthissystem:

（1）

Togetthesecondequationofmotion,sumtheforcesperpendiculartothependulum.Solvingthesystemalongthisaxisendsupsavingyoualotofalgebra.Youshouldgetthefollowingequation:

TogetridofthePandNtermsintheequationabove,sumthemomentsaroundthecentroidofthependulumtogetthefollowingequation:

Combiningtheselasttwoequations,yougettheseconddynamicequation:

（2）

SinceMatlabcanonlyworkwithlinearfunctions,thissetofequationsshouldbelinearizedabouttheta=Pi.Assumethattheta=Pi+ø（ørepresentsasmallanglefromtheverticalupwarddirection）.Therefore,cos（theta）=-1,sin（theta）=-ø,and（d（theta）/dt）^2=0.Afterlinearizationthetwoequationsofmotionbecome（whereurepresentstheinput）:

问题四：

BusSuspensionSystem

Designinganautomaticsuspensionsystemforabusturnsouttobeaninterestingcontrolproblem.Whenthesuspensionsystemisdesigned,a1/4busmodel（oneofthefourwheels）isusedtosimplifytheproblemtoaonedimensionalspring-dampersystem.Adiagramofthissystemisshownbelow:

Where:

*bodymass（m1）=2500kg,

*suspensionmass（m2）=320kg,

*springconstantofsuspensionsystem（k1）=80,000N/m,

*springconstantofwheelandtire（k2）=500,000N/m,

*dampingconstantofsuspensionsystem（b1）=350Ns/m.

*dampingconstantofwheelandtire（b2）=15,020Ns/m.

*controlforce（u）=forcefromthecontrollerwearegoingtodesign.

Designrequirements:

Agoodbussuspensionsystemshouldhavesatisfactoryroadholdingability,whilestillprovidingcomfortwhenridingoverbumpsandholesintheroad.Whenthebusisexperiencinganyroaddisturbance（i.e.potholes,cracks,andunevenpavement）,thebusbodyshouldnothavelargeoscillations,andtheoscillationsshoulddissipatequickly.SincethedistanceX1-Wisverydifficulttomeasure,andthedeformationofthetire（X2-W）isnegligible,wewillusethedistanceX1-X2insteadofX1-Wastheoutputinourproblem.Keepinmindthatthisisanestimation.

Theroaddisturbance（W）inthisproblemwillbesimulatedbyastepinput.Thisstepcouldrepresentthebuscomingoutofapothole.Wewanttodesignafeedbackcontrollersothattheoutput（X1-X2）hasanovershootlessthan5%andasettlingtimeshorterthan5seconds.Forexample,whenthebusrunsontoa10cmhighstep,thebusbodywilloscillatewithinarangeof5mmandreturntoasmoothridewithin5seconds.

Equationsofmotion:

FromthepictureaboveandNewton'slaw,wecanobtainthedynamicequationsasthefollowing:

问题五：

CruiseControlSystem

Themodelofthecruisecontrolsystemisrelativelysimple.Iftheinertiaofthewheelsisneglected,anditisassumedthatfriction（whichisproportionaltothecar'sspeed）iswhatisopposingthemotionofthecar,thentheproblemisreducedtothesimplemassanddampersystemshownbelow.

UsingNewton'slaw,modelingequationsforthissystembecomes:

（1）

whereuistheforcefromtheengine.Forthisexample,let'sassumethat

m=1000kg

b=50Nsec/m

u=500N

Designrequirements

Thenextstepinmodelingthissystemistocomeupwithsomedesigncriteria.Whentheenginegivesa500Newtonforce,thecarwillreachamaximumvelocityof10m/s（22mph）.Anautomobileshouldbeabletoaccelerateuptothatspeedinlessthan5seconds.Sincethisisonlyacruisecontrolsystem,a10%overshootonthevelocitywillnotdomuchdamage.A2%steady-stateerrorisalsoacceptableforthesamereason.

Keepingtheaboveinmind,wehaveproposedthefollowingdesigncriteriaforthisproblem:

Risetime<5sec

Overshoot<10%

Steadystateerror<2%

问题六：

APitchController

Physicalsetupandsystemequations

Theequationsgoverningthemotionofanaircraftareaverycomplicatedsetofsixnon-linearcoupleddifferentialequations.However,undercertainassumptions,theycanbedecoupledandlinearizedintothelongitudinalandlateralequations.Pitchcontrolisalongitudinalproblem,andinthisexample,wewilldesignanautopilotthatcontrolsthepitchofanaircraft.

Thebasiccoordinateaxesandforcesactingonanaircraftareshowninthefigurebelow:

Assumethattheaircraftisinsteady-cruiseatconstantaltitudeandvelocity;thus,thethrustanddragcanceloutandtheliftandweightbalanceouteachother.Also,assumethatch