Control Tutorials for MATLAB and Simulink - Introduction_ Simulink Control

20/01/2016 ControlTutorialsforMATLABandSimulinkIntroduction:SimulinkControl

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INTRODUCTION CRUISECONTROL MOTORSPEED MOTORPOSITION SUSPENSION

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Introduction:SimulinkControl

RelatedTutorialLinks

Temperature

ControlActivity

Motor Speed

ControlActivity

Simulink

Interaction with

MATLAB

RelatedExternalLinks

Modeling

ChallengesVideo

Contents

Theopenloopplantmodel

ImplementingaPIDcontrollerinSimulink

Runningtheclosedloopmodel

ExtractingamodelintoMATLAB

ControllerdesignwithinSimulink

SYSTEM

MODELING

ANALYSIS

CONTROL

PID

ROOTLOCUS

FREQUENCY

STATESPACE

DIGITAL

SIMULINK

MODELING

CONTROL



Theopenloopplantmodel

In theIntroduction:SimulinkModelingpagewedemonstratedhowSimulink canbeemployed to simulatea

physicalsystem.Moregenerally,Simulinkcanalsosimulatethecompletecontrolsystem,includingthecontrol

algorithminadditiontothephysicalplant.Asmentionedpreviously,Simulinkisespeciallyusefulforgenerating

the approximate solutions ofmathematicalmodels thatmay be prohibitively difficult to solve "by hand." For

example,considerthatyouhaveanonlinearplant.Acommonapproachistogeneratealinearapproximationof

theplantand thenuse the linearizedmodel todesignacontrollerusinganalytical techniques.Simulinkcan

thenbeemployed to simulate the performance of your controller when applied to the full nonlinearmodel.

SimulinkcanbeemployedforgeneratingthelinearizedmodelandMATLABcanbeemployedfordesigningthe

controllerasdescribedintheotherIntroductionpages.VariouscontroldesignfacilitiesofMATLABcanalsobe

accesseddirectlyfromwithinSimulink.Wewilldemonstratebothapproachesinthispage.

Recall the Simulinkmodel of the toy train system derived in the Introduction: SimulinkModeling page and

picturedbelow.



Youcangeneratethismodelyourself,oryoucandownloadthecompletedmodelhere.Assumingthatthetrain

onlytravelsinonedimension(alongthetrack),wewanttoapplycontroltothetrainenginesothatitstartsand

comestorestsmoothly,andsothatitcantrackaconstantspeedcommandwithminimalerrorinsteadystate.

ImplementingaPIDcontrollerinSimulink

LetusfirstcreatethestructureforsimulatingthetrainsysteminunityfeedbackwithaPIDcontroller.Inorderto

makeourSimulinkmodelmoreunderstandable,wewillfirstsavethetrainmodelintoitsownsubsystemblock.

Toaccomplishthis,deletethethreescopeblocksandreplaceeachonebyanOut1blockfromtheSinkslibrary.

LabeleachOut1blockwiththecorrespondingvariablename,"x1_dot","x1",and"x2".ThendeletetheSignal

Generator block and replace itwith an In1 block from theSources library. Label this input "F" for the force

generatedbetweenthetrainengineandtherailroadtrack.Yourmodelshouldnowappearasfollows.



Nextselectalloftheblocksinyourmodel(CtrlA)andselectCreateSubsystemfromtheEditmenuatthetop

ofthemodelwindow.Withalittlerearrangingandrelabeling,yourmodelwillappearasshownbelow.



Nowwecanaddacontrollertooursystem.WewillemployaPIDcontrollerwhichcanbeimplementedusinga

PIDController block from theContinuous library.Placing this block in serieswith the train subsystem, your

modelwillappearasfollows.Inthefollowing,wemodelthecontrollerasgeneratingtheforce"F"directly.This

neglects the dynamics with which the train engine generates the torque applied to the wheels, and

subsequentlyneglectsthedynamicsofhowtheforceisgeneratedatthewheel/trackinterface.Thissimplified

approachistakenatthispointsinceweonlywishtointroducethebasicfunctionalityofSimulinkforcontroller

designandanalysis.



DoubleclickingonthePIDControllerblock,wewillinitiallysettheIntegral(I)gainfieldequalto0andwillleave

theProportional(P)andDerivative(D)gainsastheirdefaultsof1and0,respectively.NextaddaSumblock

fromtheMathOperationslibrary.DoubleclickonthisblockandmodifytheListofsignsfieldto"|+".Sincewe

wishtocontrolthevelocityofthetoytrainengine,wewillfeedbacktheengine'svelocity.Thisisaccomplished

by4ringalineoffofthe"x1_dot"signalandconnectingittothenegativesignoftheSumblock.Theoutputofthe

Sum block will be the velocity error for the train engine and should be connected to the input of the PID

Controllerblock.Connectingtheblocksasdescribedandaddinglabels,yourmodelshouldappearasfollows.



NextaddaSignalBuilderblockfromtheSourceslibrarytorepresentthevelocitycommandedtothetrain.Since

wewish todesignacontroller tobring the trainsmoothlyup tospeedandsmoothly to rest,wewill test the

systemwithavelocitycommandthatstepsupto1m/sfollowedbyastepbackdownto0m/s(recallthatour

systemisatoytrain).Togeneratethistypeofcommandsignal,doubleclickontheSignalBuilderblock.Then

chooseChangetimerangefromtheAxesmenuatthetopoftheblock'sdialogwindow.SettheMaxtimefield

to"300"seconds.Next,setthestepuptooccurat10secondsandthestepdowntooccurat150seconds.This

isaccomplishedbyclickingonthecorrespondingportionsofthesignalgraph(leftandrightverticallines)and

eitherdraggingthelinetothedesiredposition,orenteringthedesiredtimeintheTfieldatthebottomofthe

window.Whendone,yoursignalshouldappearasfollows.



Also add a Scope block from the Sinks library and use it to replace theOut1 block for the train's velocity.

Relabelingtheblocks,yourmodelwillappearasfollows.



Wearenowreadytoruntheclosedloopsimulation.Ifyouwishtoskiptheabovesteps,youmaydownloadthe

completedmodelwithcontrolhere.

Runningtheclosedloopmodel

Beforerunningthemodel,weneedtoassignnumericalvaluestoeachofthevariablesusedinthemodel.For

thetrainsystem,wewillemploythefollowingvalues.

=1kg

=0.5kg

=1N/sec

=1N

=0.02sec/m

=9.8m/s^2

Createanewmfileandenterthefollowingcommands.

M1=1;

M2=0.5;

k=1;

F=1;

mu=0.02;

g=9.8;

Execute yourmfile in theMATLABcommandwindow todefine thesevalues.Simulinkwill recognize these

MATLABvariablesforuseinthemodel.Nextweneedtosetthetimeforwhichoursimulationwillruntomatch

thetimerangeofthecommandfromtheSignalBuilderblock.ThisisaccomplishedbyselectingParameters

fromtheSimulationmenuatthetopofthemodelwindowandchangingtheStopTimefieldto"300".Now,run

thesimulationandopenthe"x1_dot"scopetoexaminethevelocityoutput(hitautoscale).Theresultasshown



belowdemonstratesthattheclosedloopsystemisunstableforthiscontroller.

Since the performance achieved above is unsatisfactory, we need to redesign our controller. We will first

demonstrate how to extract a model from Simulink into MATLAB for analysis and design. Then we will

demonstratehowtodesignthecontrolfromdirectlywithinSimulink.

ExtractingamodelintoMATLAB

TheSimulinkControlDesigntoolboxoffersthefunctionalitytoextractamodelfromSimulinkintotheMATLAB

workspace.This is especially useful for complicated, or nonlinear simulationmodels. This is also useful for

generatingdiscretetime(sampled)models.Forthisexample,letusextractacontinoustimemodelofourtrain

subsystem.Firstweneedtoidentifytheinputsandoutputsofthemodelwewishtoextract.Theinputtothetrain

systemistheforce .Wecandesignatethisfactbyrightclickingonthesignalrepresenting"F"(outputofthe

PID block) and choosing Linearization Points > Input Point from the resulting menu. Likewise, we can

designate theoutput of the train systemby rightclickingon the "x1_dot" signal and choosingLinearization

Points>OutputPointfromtheresultingmenu.Theseinputsandoutputswillnowbeindicatedbysmallarrow



Points>OutputPointfromtheresultingmenu.Theseinputsandoutputswillnowbeindicatedbysmallarrow

symbolsasshowninthefollowingfigure.Sincewewishtoextractamodelofthetrainbyitself,withoutcontrol,

weneedto furtherdeletethefeedbacksignal,otherwisewewillextract theclosedloopmodel from to .

Yourmodelshouldnowappearasfollows.

WecannowextractthemodelbyopeningtheLinearAnalysisTool.ThisisaccomplishedbyselectingControl

Design>LinearAnalysisfromundertheToolsmenuatthetopofthemodelwindow.Followingthesestepswill

openthewindowshownbelow.



ThistoolgeneratesanLTIobjectfroma(possiblynonlinear)Simulinkmodelandallowsyoutospecifythepoint

aboutwhichthelinearizationisperformed.SinceourSimulinkmodelisalreadylinear,ourchoiceofoperating

pointwillhavenoeffectandwecanleaveitasthedefaultModelInitialCondition.Inordertogeneratethe

linearizedmodel,selecttheLinearizebuttonintheabovefigure,whichisindicatedbythegreentriangle.The

LinearAnalysisToolwindowshouldnowappearasshownbelow.



Inspectingtheabove,thestepresponseofthelinearizedmodelwasautomaticallygenerated.Comparingthis

stepresponsetotheonegeneratedbythesimulationoftheopenlooptrainsystemintheIntroduction:Simulink

Modelingpage,youcanseethattheresponsesareidentical.Thismakessensesincethesimulationmodelwas

already linear. Additionally, the linearization process generated the object linsys1 shown in the Linear

AnalysisWorkspaceabove.ThisLTIobjectcanbeexportedforusewithinMATLABbysimplydraggingthe

objectintotheMATLABWorkspacewindow.

Havingextractedthismodel,wecannowemployallofthefacilitiesthatMATLABoffersforcontrollerdesign.For

example,letusemploythefollowingcommandstogenerateandanalyzetheclosedloopsystemreflectingthe

Simulinkmodelcreatedabove.



sys_cl=feedback(linsys1,1);

pole(sys_cl)

ans=

1.5261

0.0000

0.0670+1.1977i

0.06701.1977i

Examinationoftheabovedemonstratesthattheclosedloopsysteminitscurrentstatehaspoleswithpositive

realpartand,therefore,isunstable.Thisagreeswiththeresultofourclosedloopsimulationfromabove.We

can thenemployMATLAB todesign a new controller. Instead,wewill demonstrate how to access someof

MATLAB'sfunctionalityfromdirectlywithinSimulink.

ControllerdesignwithinSimulink

Rather than performing the controller design in MATLAB, we can also launch interactive tools to tune our

controllerfromwithinSimulink.OnemannerinwhichthiscanbedoneistodoubleclickonthePIDControllerin

themodelandselecttheTunebuttontolaunchthePIDTunerGUI.Ratherthandothis,will launchthemore

generalSimulinkControlDesignGUIby selectingLinearAnalysis>CompensatorDesign fromunder the

Toolsmenulocatedatthetopofthemodelwindow.FollowingthesestepswillopentheControlandEstimation

ToolsManagerwindowshownbelow.



Thefirstthingthatneedstobedoneistoidentifythecontrollerblockthatistobetuned.Thisisaccomplishedby

firstclickingontheSelectBlocksbutton,andthenselectingthePIDControllerblockfromtheresultingwindow

asshownbelow.NextclicktheOKbutton.Notethatcontrollersrepresentedinothertypesofblocks(Transfer

Function,StateSpace,etc.)canalsobetuned.



Beforeweproceedtotuneourcontroller,wemustfirstidentifytheinputsandoutputsoftheclosedloopsystem

we wish to analyze. This is done in the same manner we did when extracting a model into MATLAB.

Specifically, rightclick on the velocity command signal (output of the Signal Builder block) and choose

LinearizationPoints > Input Point from the resultingmenu to identify the input of our closedloop system.

Similarly,rightclickonthetrainenginevelocitysignal("x1_dot")andselectLinearizationPoints>OutputPoint

fromthemenutochoosetheoutputofoursystem.Yourmodelshouldnowappearasfollowswherethesmall

arrowsymbolsidentifytheinputandoutputofthemodel.



Nowthatwehaveidentifiedtheblocktotuneandour inputandoutputsignals,wecannowcommencewith

tuning thecontroller.Select theTuneBlocksbutton in theControlandEstimationToolsManagerwindow.

ThiswillopentheDesignConfigurationWindowshownbelowwhichprovidessome introductiononhow to

employtheinteractivedesigntool.Inessence,thisGUIistheSISODesignTool that isavailable fromwithin

MATLABaswell.

ClickingtheNextbutton,wewillchoosethedesignplotswewishtoemployfordesigningourcontroller.Inthis



example,wewillemployarootlocusdesignapproachandhencewillchooseaPlotTypeofRootLocus for

Plot1asshownbelow.Sincetherootlocusapproachtodesignemploysaplotfromtheopenloopsystemfor

placingtheclosedlooppoles,wewillleavethechoiceofOpen/ClosedLoopsasOpenLoop1(thisisouronly

choice!).

ClickingtheNextbuttonagainwill allowus tochooseouranalysisplots.Weuse thestep responseplot to

assesshowwellweareabletomeetourgoalofbringingthetrainuptospeedsmoothlywithminimalsteady

stateerrortoaconstantspeedcommand.Therefore,wewillchooseaPlotTypeofStepfromthedropdown

menuunderPlot1asshownbelow.WewillalsoselectPlot1underthePlotContentsportionofthewindowfor

theonlysystemthatisdefined.Thereisonlyonesystemavailablebecausewehavedefinedonlyasingleinput

andsingleoutputforoursystem.



SelectingtheNextButtonwillthenopentheSISODesignToolwitharootlocusplotandastepresponseplot.

Therootlocusplotshownbelowdisplaystheclosedlooppolelocationsofthetrainsystemplantundersimple

proportionalcontrol.Examiningtheplot,onecanseethatmanyvaluesofloopgainwillplaceclosedlooppoles

intherighthalfplaneleadingtoanunstableresponse.



Ifwedecreasetheloopgainsufficiently,wecanmovetheclosedlooppolesintothelefthalfplaneandwecan

stabilizeoursystem.Thiscanbeaccomplishedgraphicallyby"grabbing"thepinkboxesmarkingtheclosed

looppole locationsanddragging them toward theopenlooppole locations (markedbyx's).A loopgainof

approximately 0.1will stabilize the system. Examining the corresponding step response,whichwill change

automaticallyinresponsetothegainchangeiftheRealTimeUpdateboxischeckedintheLTIViewerwindow,

youcanseethatwhiletheresponseisstableitssteadystateerrorisquitelarge.



(1)

Recallthataddingintegralcontrolisonewaytoreducethesteadystateerrorforasystem.Inthiscase,adding

anintegratorviathecontrollerwillmakethesystemtype1,wheretype1systemscantrackstepreferenceswith

zerosteadystateerror.RecallthefollowingformofaPIcontroller.

Therefore, aPI controllerwill add an integrator anda zero to our openloop system.The integrator can be

addedtothesystembyrightclickinginthefieldoftherootlocusplotandselectingAddPole/Zero>Integrator

fromtheresultingmenu.Similarly,thezerocanbeaddedbyrightclickingontherootlocusplotandselecting

AddPole/Zero>Realzerofromtheresultingmenu.Thenclickwherealongtherealaxisyouwishtoplacethe

zero.Wewillplacethezerojusttotherightoftheplantpoleontherealaxis.Youcanmovethezerobyclicking

onitanddraggingittoanewlocation.Onceyouhaveplacedthezero,thengrabthepinkboxesrepresenting

theclosedlooppolesandattempt to line the threedominantpolesupso that theyhave thesamerealpart.

Therealsowillbearealclosedlooppoletotheleftthatis"faster"thantherest,andaclosedlooppoleatthe

originthatiscancelledbyaclosedloopzeroattheorigin.Theresultingrootlocusplotisshownbelow.



Thecompensatorcanalsobeeditedbydirectlytypinginpoleandzerolocations.Thiscanbedonebychoosing

EditCompensatorfromtheDesignmenulocatedatthetopoftheSISODesignTaskwindow.Thewindowthat

opensisshownbelow.Wewillmorepreciselyplacethezeroat0.15andwillchoosealoopgainequalto0.01.



Theresultingclosedloopstepresponseplotisshownbelowdemonstratingthatthetrainengineisbroughtto

speedsmoothlyandwithzerosteadystateerrorforaconstantspeedcommand.



Thecontrolgains thathavebeenchosencan thenbeapplied to theSimulinkmodelbyclicking theUpdate

SimulinkBlockParametersbuttonundertheCompensatorEditor tabof theControlandEstimationTools

Managerwindow(seeabove).Thesimulationcanthenberunwiththisnewlytunedcontroller.Clickingonthe

Scopeblockforthetrainengine'svelocityandselectingautoscalewillproduceaplotliketheoneshownbelow.



Overall this responseseems tomeetourgoalsofbringing the trainup tospeedand to restsmoothly,while

maintainingminimalsteadystateerror.ThisresponsematchestheresultgeneratedwiththeSISODesignTool

abovebecausethatanalysisandtheSimulinkmodelusedtheexactsamelinearmodel.

PublishedwithMATLAB7.14

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