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CompareandConclusion19
Afterdesign20
Appendix22
Reference22
1.Preface:
Withthehighpaceofhumancivilizationdevelopment,thecarhasbeenacommontoolsforpeople.However,someproblemsalsoariseinsuchtendency.Amongmanyproblems,thevelocitycontrolseemstoasignificantchallenge.
Inaautomatedhighwaysystem,usingthevelocitycontrolsystemtomaintainthespeedofthecarcaneffectivelyreducethepotentialdangerofdrivingacarandalsowillbringmuchconveniencetodrivers.
Thisarticleaimsatthediscussionaboutvelocitycontrolsystemandthecompensatortoamelioratethepreferenceoftheplant,thusmeetsthecomplicateddemandsfrompeople.ThediscussionisbasedonthesimulationofMATLAB.
Keyword:
PIcontroller,rootlocus
2.TheDesignIntroduction:
Figure2-1automatedhighwaysystem
Thefigureshowsanautomatedhighwaysystem,andaccordingtocomputingandsimulation,avelocitycontrolsystemformaintainingthevelocityifthetwoautomobilesisdevelopedasbelow.
Figure2-2velocitycontrolsystem
Theinput,R（s）,isthedesiredrelativevelocitybetweenthetwovehicles.Ourdesigngoalistodevelopacontrollerthatcanmaintainthevehiclesinseveralspecificationbelow.
DS1Zerosteady-stateerrortoastepinput
DS2Steady-stateerrorduetoarampinputoflessthan25%oftheinputmagnitude.
DS3Percentovershootlessthan5%toastepinput.
DS4Settlingtimelessthan1.5secondstoastepinput（usinga2%criteriontoestablishsettlingtime）
3.RelativeKnowledge:
Controllerhereactuallyservesasacompensator,andwehavesomecompensatorsfordifferentspecificationandsystem.
Table3-1Compensators
compensator
Appilicablecondition
Adjuststeadyerror
Addthetypeofsystem,thusadjustthesteadyerror
PIcontroller:
adjuststeadyerrorandaddanzero
Increasethephasemargin
Increasethephasemargin,keepthesteadyerrorunchanged
PIDcontroller:
balanceallthepreference（steadypreference,aswellasdynamicpreference）
4.DesignandAnalysis:
4.1Specificationanalysis
Accordingtotherelativeknowledgeabove,ImayconsideraPI
controllertocompensate------------
.
Ds1:
zerosteadyerrortostepresponse:
Tointroduceanintegralparttoaddthesystemtypeisenough.
Ds2:
Steady-stateerrorduetoarampinputoflessthan25%ofthe
inputmagnitude.
Ds3:
overshootlessthan5%toastepresponse.
AccordingtoDS3andDS4,wecandrawthedesiredregion
toplace
ourclose-looppoles.（astheshadowindicate）
Figure4-1Desiredregionforlocatingthedominantpoles
Afteraddingthecontroller,thesystemtransferfunctionbecome:
ThecorrespondingRoutharrayis:
1Kp+ab
a+bKi
0
Ki
Forstability,wehave
Foranotherconsideration,weneedtoputthebreakpointof
rootlocustotheshadowareainFigure4-1toensurethedominant
polesplacedontheleftofs=-2.66line.
Inall,thespecificationisequaltoaPIcontrollerwithlimitbelow.
4.2Designprocess:
4.2.1Controllerverification:
Attheverybeginning,wetakethesystemwithG（s）=
andthecontroller（providedbythebook）with
foraninitialdiscussion.
Figure4-2stepresponse（a=2,b=8,Kp=33,Ki=66）
Figure4-3rampresponse（a=2,b=8,Kp=33,Ki=66）
Fromfigure4-2,wecanseethattheovershootis4.75%,andthesettlingtimeis1.04swithzeroerrortothestepinput.
Fromfigure4-3,itisclearthattherampsteady-stateerrorisalittlelessthan25%.
Thus,thecontrollerwith
completelymeetsthespecification.
4.2.2furtheranalysis:
Fornextprocedure,Iwillhavesomemorespecificdiscussionabouttheapplicablerangeofthiscontrollertoseehowmuchcanaandbvaryyetallowthesystemtoremainstable.
Wedon’tchangetheparameterofthecontroller,andinserttheKi=66,Kp=33intotheinequalityandgetthis:
Ifwesupposethesystemtobeaminimumphasesystem,a,b>
0,thusitiseasytoverifythe3rdinequality.Now,wedrawtoseetherangeofaandb.
Figure4-4therangeofa,bforcontroller（Kp=33,Ki=66）
Actually,theshadeareacannotcompletelymeetsthespecification,fortheconstraintconditionsrepresentedinthe3inequalityisnotenough,weneedtodrawtherootlocusforacertainsystem（aandb）tolocatetheactuallimitforcontroller.
However,thistaskisratherdifficult,inaway,the4variables（a,b,Ki,Kp）allvaryintermsofothers’change.Thuswecanapproximatelylocatetherangeofaandbfromthefigureabove.
4.2.3Alternativesdiscussion：
Accordingtoinequality
Therangeofaandbbearsomerelationwiththeinequalitybelow:
Basingourassumptionontherangeinthepreviousdiscussion,wecaneasilyseethatinordertoincreasetherange,wecanincreaseKianddecreasetheratioofKitoKp.
Thus,Iadjusttheparameterto
Figure4-5therangeofa,bforcontroller（Kp=64,Ki=80）
Asthefigureindicate,（therangebetweendottedlinesreferstothepreviouscontroller,whiletherangebetweenredlinesreferstothenewalternatives）,therangeincreaseasweexpect.
Nextstep,wemaykeepthesystemofG（s）=
fixed,anddiscussthedifferentcompensatingeffectofdifferentPIparameter.
Whencarefullycheckingthecontroller,wemayfindthatthecontrolleractuallyaddazero（-Ki/Kp）,anintegralpartandagainpart,sowecanonlychangethezeroanddrawthelocusrootandexaminethestepresponseandrampresponse.
:
Figure4-6therootlocus（
）
Usingrlocfind,wefindthemaximumKp=34.8740
Sowechoose3groupsofparameter（[35,52.5],[30.45],[25,37.5]）toexaminethereponse
Figure4-7thestepresponse（
It’sclearthatthestepresponsepreferenceisnotsatisfyingwithtoolongsettlingtime
Figure4-8therootlocus（
Usingrlocfind,wefindthemaximumKp=34.3673
Sowechoose3groupsofparametertoexaminetheresponseandrampresponse.
Figure4-9thestepresponse（
Figure4-10therampresponse（
Figure4-11therootlocus（
Usingrlocfind,wefindthemaximumKp=31.47
Similarly,wechoose3groupsofparametertoexaminetheresponseandrampresponse.
Figure4-12thestepresponse（
Virtually,theovershoot（Kp=30,Ki=75）doesn’tmeetthespecificationasweexpect.Iguess,thatmaycomefromtheeffectofzero（-2.5）,thus,gobacktothestepresponseof
duetotheeliminationbetweenzero（-2）andpoles,thusthepreferenceiswithinourexpectation.
Figure4-13therampresponse（
5.CompareandConclusion
Mainlyfromthestepresponseandrampresponse,itcanbeconcludedthat,inacertainratioofKitoKp,thelargerKpbringssmallerrampresponseerror,aswellaslargerrangeofapplicablesystem.Nevertheless,thelargerKpmeansworsestepresponsepreference（includingovershootandsettlingtime）.Thiscontradictionisrathercommonincontrolsystem.
Inall,togetthemostsatisfyingpreference,weneedtobalancealltheparametertomakeacompromise,butnotasingleparameter.
Fromwhatwearetalkingabout,wefindthecontrollerprovidedbythebook（Kp=33,Ki=66）maybeoneofthebestcontrollerincomparisontosomedegree,withsatisfyingstepresponseandrampresponsepreference,aswellasawiderrangeforthevariationofaandb,further,ituseazero（s=-2）totransferthe3rdordersystemto2ndordersystem,indoingso,wemayeliminatesomeunexpectedinfluencefromthezero.
Thecontrollerverifiedabove（inFigure4-9andFigure4-10）withKp=34,Kp=68maybealittlebetter,butonlyalittle,anditdoesn’tleavesomemargin.
6.AfterDesign
这是一次艰难，且漫长的大作业，连续一个星期，每天忙到晚上3点，总算完成了这个设计，至少我自己是很满意的。
其实与其说是大作业，不如说就是一次课程设计。
运用所学的自控知识，加上matlab操作知识，去探究了一下用根轨迹法去研究校正的问题。
这次选题很多，有超前滞后校正，有状态反馈校正，但这两种在前不久的自控实验中都已经做过，所以这一次挑战一下自己，选择这个根轨迹法来做。
一开始觉得这个选题并不难，而且书上也要代码等，但真正做起来，发现很有点棘手。
由于题目中要求研究某种控制器对某些系统的校正能力，相当于PI控制器中的Kp，Ki和系统中的两个极点全是变化的。
一段时间琢磨和不断仿真试验后，我决定换个角度去思考，分别控制系统不变和控制器不变，去研究控制器的控制范围，以及各种控制器对一个特定系统的矫正效果，最后在通过比较分析论证。
题目要求找到一个适用范围更广的控制器，但经过我不断摸索，证明出课后题中所给控制器已经是最优解了，再优的也只是提高一点点的极限值，虽然我没有找到更优解了，但这个过程中我充分了解到设计和探究的步骤，不管结果是否正确，我在这个探究过程中收获颇多。
7.
holdoff
clg
n=[1];
d=[11016];
tau=2.5;
nc=[1tau];
dc=[10];
[num,den]=series（n,d,nc,dc）;
rlocus（num,den）
holdon
plot（[-2.66-2.66],[-2020]）;
z=0.69;
plot（[0-20*z],[020*sqrt（1-z^2）],[0-20*z],[0-20*sqrt（1-z^2）]）
grid
rlocfind（num,den）
%drawtherootlocus%
Appendix：
a=[0:
0.01:
20];
b=6.57-a;
c=[0:
0.10:
20]
d=20./c
plot（a,b,c,d）
%plottheinequality%
Kp=30;
Ki=75;
nc=[KpKi];
[numa,dena]=series（n,d,nc,dc）;
g=tf（numa,dena）
f=feedback（g,1）;
h=f*tf（[1],[1,0]）;
step（h）,grid
holdon;
t=[0:
100];
plot（t,t,'
r-'
）;
%rampresponse%
[na,da]=cloop（numa,dena）;
step（na,da）,grid
%stepresponse%
8.Reference：
1）RobertH.Bishop，ModernControlSystemAnalysisAndDesignUsingMATLABAndSimulink，Beijing：
TsinghuaUniversityPress，2003，12
2）胡寿松，《自动控制原理》（第五版），北京：
科学出版社，2007，6
3）胡寿松，《自动控制原理简明教程》（第四版简明版），北京：
科学出版社，2003,4
4）张德丰，《MatlabSimulink建模与仿真》，北京，电子工业出版社，2009,3
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