Position Control System - Analysis and Design

8/6/2019 Position Control System - Analysis and Design

1/58


2/58

A Project Report

Positional Control System-Analysis &Design

frojtareyert"tufitnitte{inyartialfttffi ttnentof tfrereyuirementfertde ora"{"frf,eg"ee"fBachelor of Technology in Electrical Engineering

2011By

Dehojit tvlaitraSai"fr.at fiatt a.ciarrya

tuati.fr,Dey' Sen"dniSioSf*Defiottarn Cfrnfrrafiorty

SryiiN Sfi"umc

Under the Guidance of"rof. ejry Cfrmda

Department of Electrical EngineeringJIS College of Engineering

Block-A, Phase-III, Kalyani, Nadia747295


3/58

CERTIflGJTTEThis s to certiS thatDebojitMaitra (UniversityRoll No. : 071230116015 RegistrationNo.z 071230116101015),aikat Bhattacharya (Universify Roll No. z 071230116016&RegistrationNo. : 07123011610101q,ratih Dey (lJniversityRoll No. t 071230116017RegistrationNo. : 071230116101014,andhi Singha(UniversityRoll No. : 071230116018& RegistrationNo. z 071230116101018),ebottam Chakraborty(University Roll No. :071230116019RegistrationNo. : 071230116101019),anjivShome UniversityRoll No. :0712301L6020 RegistrationNo. z 071230116101020)re he students f 4ft yearElectricalEngineeringf JISCollegeof Engineeringn the session f 2007-2011 nd heyhavesubmittedtheirmajor thesiswork in partial fulfilment of the requirementsor the Degreeof BachelorofTechnologyn ElectricalEngineering.t is the result of the bonafide work carriedout on thedissertation itled *POSITIONCONIryOLSYSTEM ANALYSISAND DESIGI\|' during theacademicession20l}-2}ll under heguidanceand supervision f mine. I hereby ecommendthat his thesiss acceptedn partial fulfifunentof the requirementsor the Degreeof BachelorofTechnologyn ElectricalEngineering, rom JIS Collegeof Engineering.During this thesisworkthey werefoundto be sincere, egularand hard working andhavesuccessfully ompletedhethesiswork assignedo them.

Prof.A. Chanda(ProjectGuide)AssistantProfessor

Dept.Of ElectricalEngineeringJISCollege f Engineering,alyani.


4/58

J15 ollegef Engineering8lo* 'A', PhaseJll.Kalyani,H:dia,741335Fhone:+91 33 258?2137,Teh{sx: +91 33 ?582:138Websir*:www-jiscalt+je-ae.in, mail: n@;iisnollege-m. n

CENTIflC^ETE,F*.PPROV*.LThis sto certify hat the projectentitled"POSITIONAIONTROLYSTEM-NALYSISDESfGN"ubmitted y DebojitMaitra,SaikatBhattachotya, ratikDey,SandhiSingha,DebottamChakraborty, aniivShome f Electrical ngineeringepartment,lSCollegeof Engineering,alyani,n the partial ulfillmentof the requirementor the awardofthe degree of Bachelorof ElectricalEngineeringn West BengalUniversityofTechnologyWBUT)uring he academicear2AtA-20Lt.

Directo(Prof. U. Banerjee)

PrincipalJIS College of EngineeringKalyani - 741235

JISCollegeof EnKalvani - 74

HeadElectrical Engine ering Departuent

JIS College of EngineeringKalyani - 741235

(Prof. A Chanda)Guide, Lecturer

Electrical Engineering DepartmentJIS College af Engineering

Kalvani - 741235

(Prof. Asit,5*6x Guha)

(Prof. Subrata K

Corpotere offirel 7, larat 8sr Eoad. Iclletn-IlHl SlU, Plroneq+91 33 ll89 394#5323, Tdefexl +91 l$ 22*9 3945


5/58

"*.glilOIT/I.uDqEITUNTWe would like to extend our sincere gratitude to our respected eacher Mr" A Chanda for hisassistance ndguidance owards the progressof this project.

We also appreciate with deep senseof thankfulness, the co- operation extended to us by the labassistantand some of our frierrds and other teachers.

Lastly we acknowledge our indebtedness o the principal of institution and our parents whosesupport, dedication havegiven us an immense help in commencingthis project.

We have made a humble attempt n this direction to make this project perfect as far aspossible.In spite of efforts some errors might.crept in. we shall be grateful to the teacher if some arebroughtby notice. Comments and criticism is welcome in this regard.

Date:eloslutPlace:",Iry *;

Debojit Maitra(07r230n 01s)SaikatBhattacharya(07r230n 016)Pratik Dey(07r230n 017)SandhiSingha(07r230n 018)

)"u,b,#t l.t^ c" .3 d,raf B\^GU64-Lorla ,

!rSorn;tv ShPcrno

DebottamChakraborty(07r230n 019)Sanjiv Shome(07r230n 020)


6/58

GONTENTTitle pageCertificateCertificateof approvalAcknowledgementContent1. fntroduction

1"1 IntroductionL.2 Objectiveof this project1.3 Basicconceptof positionalcontrol1.8.1 Detectionofpositioningerrorsduring movementI.4 MATLABr"4.L category:Generalpu{poseapplicationsand utilitiesavailableon-all systemsBriefdescriptionf iVIATLABprod.uctsSIMULINK.What is a SIMULINK function

Proportional integral-derivative control& lead -lag compensator2.1 Introduction2.2 Different typesof controller2.2.L Proportional ontrol Gain)2.2.2 Integral control(Automaticreset)2.2.3 Derivativecontrol(pre Act(TM) or rate)2.2.4 Putting it together:pID ControllerLead-LagCompensator

2.3.L Application2.3.2 Theory2.3.3 Implementation2.3.4 IntuitiveExplanationTime response pecifications2.4.t OvershootSignal)

2.4.L.I Definition2.4.1.2 Control heory

IIIIIIryV123444

45o6

6777810T2131313L41515151616

L.4.21.4.3I.4.42.

2.4

f r Vl_


7/58

16181_92A2t2L

25262727282930

3.13.23.3

353637383940404T42

4.

2.5

2.4.2 Rise ime2.4.2.1 Application2.4.3 Settlingtime2.4.3.L Application2"4.4 Steadystateerror2.4.4.1 ApplicationImportanceof PID and lead-lag compensator vertime response pecifications2.5.L Introduction2.5.2 Three-term controller2.5.3 The characteristicsof p, i, and d controllers

22222324

3. I\{ATLAB Prograning & SIMIILINK model of

3.4

amature controlled dc motorIntroductionParametersused n IVIATLABoperationPrograming or stepresponseof open oop system3.3.L ResultPrograming or stepresponseof closedoop system3.4.1 ResultPrograming or stepresponseof a closed oop systemusing PID controller3.5.1 ResultPrograming or stepresponseof closedoop system3.6.1 ResultPrograming or stepresponseof a closedoop systemusing PID controller3.7.1 Result

3.8 SIMULINK model3.8.1 Result

Future aspects of the Proiect4.L DC positioncontrolsystemResult4.1.L Introduction4.1.2 Experiment4.1.3 'Features& sPecifications4.2 Robotics

4.2.1 Introduction4.2.2 Experiment

3.5

3.63.7

3L323334

434344

t tlv lt ,


8/58

4.3 Radarcontrol4.3.1 Introduction4"3.2 The systemapproach

ConclusionReferences

454545

4648

f r vl lt ,


9/58

INTRODUCTION


10/58

IntroductionI.1 INTRODUCTION

Chapter1

This project confines tself to a study of dynamicand steadystatemodeling of seryosystem,henceprogrammingof thedcmotor andanalysisof its different dynamicparameterslike settling img risetime, peakovershoot, teadystateeror by usingMATLAB tools.

The dynamic simulationprocedures derivedconsideringa stateof dataof DC drivefrom the subsystemdifferential and functional relationships"Attempt has been made toundertakehe problan of designing n MATLAB Programingand5IMULINK e,nvironmentfor a closed oop control system. n order to put the idea on a familiar footing we haveconsiderhe designof PID,PI & PD controllerby writing MATLAB script thensimulate hedynamicsusing SIMULINK library functionsblocks.A position conhol systememployinganarmature onfrolledDC servomotoranda DC drive related o designcriteria towards otalnanativedesign procedureof the systernhasbeenundertaken.As in many casesof systemdesign, here s no unique solution o the designproblemof a pID controller.Howeverattempt asbeenmade owards heobjectiven the contoller design s to select he suitablevalues or theparametersp, ki & kd ts satisff thedesired equirements"

zt ,


11/58

IntroductionL.2 OBJECTI1M

Chapter

Positional control stands or confrolling the whole systemparameterand analpingthe dynamic response of that system and makes a comparisonbetween the dynamicresponses,btainedby theMATLAB and SIMULINK analysis.

If we get a certain hansfer function of a system,by the analysisof hansferfunction using MATLAB, we can conclude over the dynamiccharactersuch as rise time,settling ime, overshoot,steadystateerror etc. of that particularsystem"By this process,wecan improve the responseof the systemcomparingwith the standardvalue, such as in aRADAR SYSTEM, steadystateerror must be zero and he settlingtime must be lesser han0.2sec.

We can also obtain the responseof the same ransferfunction by SIMULINKanalysisand can comparehe response ith the response btainedby thepreviousanalysis"

t3 t


12/58

Introduction1.3 BASIC CONCEPT OF POSITIONAL CONTROL

Chapter1

A positional control is a type of automaticconhol in which the input commandsarethedesired ositionof a body. If a bodyor a systemmoves rom its desiredposition, t makesanerror.Throughthis control system,we caneliminatethis positionalerror. By the useof pIDconholler and leadJag compensatorwe can determine he exact value of this error andoptimizehesystem.

1.3.1.DETECTION OF POSITIONING ERRORS DT]RINGMO\MMENTS

Maximum olerableerror mayvary (canbe as smallas25pm)Drivespeedimit = maxerror systemesponseimeE.g.25 pm / 100ms= 0.25mm/s

In case his is in the resonancgangeof the assernbly,he speedshouldbe furtherreduced.Continuous inglestepmode).

1.4 MATI"ABMATLAB is an integrated echnicalcomputing environment hat combinesnumericcomputation,dvanced raphics ndvisualization, ndahigh-levelprograrnminganguage.

1.4.LCATEGORY:GENERAL PIIRPOSE APPLICATIONSAIYDUTILITTES AVAII,ABI,E ON-.ALL SYSTEMS

Short for Matrix Laboratory MATLAB is a technicalcomputing environmentforhigh-performance umeric computationand visualization.It integratesnumerical analysis,mahix computation,signalprocessingand graphics n an easy-to-useenvironmentwhereproblemsand solutions are expressedust as they are written mathematically, withouthaditional programming. Tlpical uses include general-purposenumeric computation,algorithm prototyping, and special-purposeroblem solving with matrix formulations that

t4 t____ _ _'.__--'a t


13/58

Introduction Chapter1arise n disciplinessuch as linear algebra,structuralanalysis,statistics,and digital signalprocessing.MATLAB is included in the default user path. The current version can be evoked bytping'MATLAB" on the commandine.1.4.2BRIEF DESCRIPTION OF IT{ATI"ABPRODUCTS. Thefollowingtools are n MATLAB, whichmakeshesoftwaremore easyanduseful oranykindof Mathematical ndProgramming orks.TheMATLB toolsare----1.CDMA Reference lockset2. Communicationsoolbox3. ControlSystemToolbox4. DataAcquisitionToolbox5. Databaseoolbox6. Developer's it for Texas nstruments SP7. Instrument onholToolbox8. LMI ConholToolbox9. MappingToolbox10.MATLAB CompilerEtc.andmanymore.1..4.3 SIMITLINK

SIMULINK, developedby The Math Works, is a commercial tool for modeling,simulatingand analyzingmultidomaindynamicsystems.ts primary interface s a graphicalblock diagramming ool and a customizableset of block libraries. It offers tight integrationwith the rest of the MATLAB environmentand can either drive MATLAB or be scriptedfrom it. SIMULINK is widely used n control theory and digital signal processing ormultidomain imulation ndModel-Based esign.


14/58

Introduction Chapter

I.4.4 WTIAT IS A SIMI.TLINK FUNCTION?In a State flow chart, a SMULINK function is a graphical object that you fill with

SIMULINK blocks andcall in the actionsof statesand ransitions.This functionprovidesanefficient model designand improvesreadabilityby minimizing graphicaland nongraphicalobjects. lpical applicationsnclude:

Defining a functionthatrequiresSIMULINK blocks, such as ookup tables.Schedulingexecutionof multiple confrollers.

3o

t\-_6 tltt


15/58

ER2A

PROPORTIONAL-INTEGRAL.

DERIVATIYECONTROL&

LEAD-LAGCOMPEI{SATOR


16/58

proportional-lntegral-Derivativeontrol& Lead-Lagcompensator Chapter2

2"! INTRODUCTIONThe pID algorithm is the mostpopular feedbackcontroller usedwithin the process

industries. t hasbeensuccessfullyusedfor over 50 years.It is arobusteasily understoodalgorithm that can provide excellent control performancedespite the varied dynamiccharacteristicsf process lant.Theseecturenotes:. introduceheProportional- ntegral-Derivative(PID) controlalgorithm." discussherole of thethreemodesof thealgorithm'. highlightdifferentalgorithmstructures'nDiscussmethodshathaveevolvedover he ast50yearsasaids n control oop tuning.After completionof this sectionof the coursea studentshouldbe capable f approachingloop tuning problem n a competentandeffrcientmannerandhave sufficient knowledgetoeffectively unea PID control algorithm'2.2 DIFFERENET TWES OF CONTROLLER2.2,I PROPORTIONAL CONTROL (GAIN)

Thefirst elementof pID control o be developeds Proportionalcontrol.The equationis simple:error: measurementsetpoint (directaction)tllorerror: setpoint- measurementreverse ction)tuNotethe actionmaybe eitherdirector reverse.n a direct actingcontrol loop an increasentheprocessmeasurementauses n ncreasen theoutputto thefinal controlelement.TheproportionalonlYequations:OrSput gain x effor+ biasThebias is sometimes nown asthemanual eset.Somecontol systems suchas Foxboroproducts,use proportionalband ratherthan gain. The proportionalband and the gain arerelated y:

lliI


17/58

Proportional-lntegral-Derivativeontrol& Lead-LagCompensator Chapter2TED?OGain =

ProportionalBard =ProportionalBandLEDTzEffi- 2.1111

Gains the atioof thechangen theoutputo thechangen the nput.r{-:* Output clnqgeumn=hpfffi4ge 2.2t2r

Proportionalband s the amount he input would have o changen orderto cause heoutput o move rom 0 to 100% orvice versa)

With proportional only control the confrollerwill not bring the processmeasurernentto the set point without a manual adjusfrnento the bias (or manual reset) term of theequation. n the early daysof contrpl the operator,upon observingan offset in the controlloop would correct heoffsetby manuallyoresetting"thecontroller(adjusting he bias).

2.2.2 INTEGRAL CONTROL (AIITOI\{ATIC RESET)Ratherthan to requirethat the operator"manuallyreset" the control loop whenever

there was a load changecontrol functions were developed o "automatically reset" theconhollerby adjusting hebias term whenever here was an eror. This "automatic eset" isalsoknownsimply as"reset"or as"integral".The mostconrmonway to implementntegralmode n analogcontrollers s to use a positivefeedbacknto theoutput.

t ^ I- , .. .- t 6t ,


18/58

Proportional-lntegral-Derivativeontrol& Lead-LagCompensator Chapter

Figure .1:Theequation or PI control s:Out:gx KrxJ e dtout: gainx (error+ integral(error)dt)

23trJ

The amount of reset used s measuredn termsof "reset ime" in minutesor itsinverse,"resetrate" in repeatsper minute. The following test can be performedon acontrollerwhich s not connectedo theprocess:1.An adjustable ignal s connectedo the nput.2.The output s indicated r recorded.

3. With the controller manual he setpoint and he input areset o the samevalue.4. The controller s switched o automatic.Becausehe error is zero, the outputdoesnotchange"5. The nput to the controller s changed y a smallamount.The outputwill move suddenlydue o the gain"The output will continue o changeat a constant ate. The time is measuredfrom the time of the initial changeuntil the time that the instant change s repeatedby theconstantmovement.The repeat ime, or reset ime, is the time it takes or the reseteffect to

I 1II

IIIIII

M+;;;r;AVariabl+

IIIIIII

t9t


19/58

Proportional-lntegral-Derivativeontrol& Lead-LagCompensator Chapter2repeat(or move the output the same amount as) the gain effect. Its inverse is reset rate,measuredn repeats erminute.

Reset effectGain effect

% +10Eror 0-1 0(-Tt-*

1 "Repeat- timeTime

Figure2.2:2.2.3 DERMTI\m CONTROL(PRE-ACT(TM) OR RATE)

The third term of PID control is derivative, also known as Pre-Act (trademark ofTaylor nstrumentCompanies, ow ABB), and ate.

The derivative term looks at the rate of changeof the input and adjusts he outputbased n the rate of change"The derivative unction can eitherusethe time derivativeof theerror,which would include changesn the setpoint, or of the measurement nly, excludingsetpointchanges.Theequation or the derivativecontribution assuming erivativeon error) is:Out:gxKdxde/dtTheamountof derivativeused s measuredn minutesof derivative.To illustratethemeaningof minutes f derivative, onsiderhe ollowing opNloop test:1. Connect a signal generatorwith a ramp capability to the input of a controller. Thecontroller output is connected o a recorder. Configure the controller with some gain, noreset, ndno derivative.

t t#10 f,


20/58

Proportional-lntegral-Derivative ontrol& Lead-LagCompensator Chapter22. With a constantoutput from the signalgeneratorand the controller in manual,adjust thesetpoint to be equal o the nput from thesignalgenerator.3. Place hecontrollernto automaticmode.4. Start heramp.5. Laterstop heramp.6. Repeathe abovestepswith somederivative.Compare he hend recordsof the controller'sinputand output.Onthe following trend ecord

Figure2.3:Note that when he ramp s started,with no derivative(dashedine) the output ftImps

up due o the changen input and hegain.Using derivative(solid line) the output umps up,rises n a ramp,and hen urnps down.The differencen time between he solid line andthedashedine representsheamountof derivative,n units of time (usuallyminutes).

D e rivat iv e eff e ct%Ouput Ga.in effect

% +1 0Eror O-1 04--ur----*Der iva. t ive t i rne

/ l 11t,


21/58

-Proportional-lntegral-Derivativeontrol& Lead-LagCompensator2.2.4 PUTTING IT TOGETHER: PID CONTROLCombining hethreeelements, ain, ntegral,andderivative,wehave heequation:

out=c(u+RJedt*o#)WhereG: GainR. Reset repeats erminute)D = Derivative minutes)Showngraphically:

Chapter

2.4tt1

ManualH+set

Ouput

Figare2"4:Note that n the equation hegain s multiplied by all three erms.This is important or

thePID equationo be able obe tuned y anyof the standarduningmethods.

12


22/58

Proportional-lntegral-Derivativeontrol& Lead-LagCompensator2.3 LEAD.I,AG COMPENSATOR

Chapter

A lead-lag compensator s a component in a control system hat improves anundesirablerequency esponsen a feedbackand control sptem. It is a fundamentalbuildingblock n classical ontol theory.2.3.1. PPLICATION

Lead-lag compensatorsnfluence disciplines as varied asrobotics, satellitecontrol,automobilediagnostics,aser requencystabilization, and many more. They are an importantbuildingblock in analog ontrolsystems, ndcanalsobe used n digital control.Given a conkol plant, desired specificationscan be achieved using Compensators.,D, PI, PD, and PID, are optimizing controllerswhich are used o improve systemparameterssuchas(steadystateerror, reducing esonant e*, improvesystan response y reducing isetime).All theseoperationsanbedoneby Compensatorsswell.2.3.2THEORY

Both lead compensators nd lag compensatorsnkoduce apole-zero pair into theopen oop hansfer unction.The kansfer unction canbe written in the Laplacedomain asvX s-g.s-p 2.5121whereXis the input to the compe,nsator,is the output,s is the complexLaplacetransformvariable,z is the zero frequencyandp is the pole frequency.Thepole and zero arebottr tlpically negative. n a lead compensator,he pole is left of the zero in thecomplexplang|" l. Ip l,while in a ag compensatorz |> |p l.

A lead-lag compensatorconsists of a lead compensatorcascadedwith a lagcompensator. he overallhansfer unctioncanbe written as(u-"t)( t - t r)(s -pr)(s -pz) '

vX 2.6421Typically pr | > lal> lzzl> l,pz , where l alndplare he zeroandpole of the ead

compensator ndz2andp2are the zero and pole of the lag compensator.The leadcompensatorrovidesphase ead at high frequencies.This shifts the poles o the left, whicht t 13t,


23/58


24/58

Proportional-lntegral-Derivative ontrol& Lead-LagCompensator

2.3.4 INTUITT\IE EXPI.,AI{ATIONChapter2

To begin designinga lead-lagcompensator, n engineermust considerwhether thesystem needing correction can be classified as a lead-network, a lag-network, or acombinationof the two: a lead-lagnetwork (hence he name"lead-lagcompensator").Theelectricalresponse f this networkto an input signal s expressed y the network'sLaplace-domain ransfer unction, acomplexmathematical unction which itself canbe expressedasone of two ways: as the Current-gain atio hansfer function or as the Voltage-gain ratiotransfer unction. Rememberhat a complex unction can be in generalwritten asF(x) : A(x)+ iB(x), where (.r) is the "RealPart" and^B(i) s the "ImaginaryPart" of the single-variablefunctionF(x).

The "phaseande" of the network s the argument fF(x); in the left half planethisis tan- t(f(") / A(x\).If thephaseangle s negative or all signal requenciesn the networkthen he network s classifiedas a "lag network". f thephaseangle s positive or all signalfrequenciesn the network then thp network is classified as a "lead network". If the totalnetwork phaseangle has a combinationof positive and negativephaseas a function offrequency hen it is a "lead-lagnetwork". Depending upon the nominal operation designparametersof a system under an active feedback contol, a lag or lead network cancausenstability andpoor speed nd esponseimes.

2.4 TIME RESPONSE SPECIF'ICATIONS2.4.L OVERSHOOT (SIGNAL)

In signalprocessing, ontrol theory,elechonics,and mathernatics, vershoot s whe,na signal or function exceeds ts target. It arises especially in thestep response fbandlimitedsystemssuch aslow-pass ilters. It is often followed byringing, dd at timesconflatedwith this latter.

t15 ,


25/58

Proportional-lntegral-Derivativeontrol& Lead-LagCompensator

2.4.L.I DEFINITIONChapter2

Ma:rimumOvershoot(signal) s defined in Katsuhiko Ogata'sDiscrete-time controlsystemss "the mildmum peak value of the responsecurve measured rom the desiredresponsef thesystem.utu

2.4.I.2 CONTROL TTIEORYIn control theory,overshootefers to an oubut exceeding its final, steady-state

value.t2l or a step nput, thepercentage vershoot PO) is thema>dmumalue minus thestepvalue divided by the step value. In the caseof the unit step, the overshoot s just themildmum value of the ste,p esponseminus one. Also see the definition of overshootnan electronics ontext.Thepercentage vershoots afunctionof theDamping atio ( and s givenby

l

r r l . r " :(?t%)The damping atio canalsobe foundby

2"gt3t

2.10t312.4.2 RISE TIME

We have seenhow the RC circuit responds o a step and how its behavior can becharacterizedy thetime constant.A relatedmeruiurements that of kise time'.The definition of rise time is "thattimetaken or a linearnetwork'soutputto rise from 10% o90%of its final valuewhen stimulatedby a step nputu. This measurements useful becauseit is easy o measure n an oscilloscope ndcanbe applied o any inearnetwork"

(lnPo)z?rs (lnPO)?

t t, - , - -". - - 1 16(,


26/58

Proportional-lntegral-Derivativeontrol& Lead-LagCompensator Chapter2For our example f 6': I second nd or a 1 volt inputstepwe have he following:

1"11I.6.7.E.5.4.3.2.lo o 1-25 1. 5 'r .75Ti,ne (+ec)

Figure2.5:In caseof the RC network,he 0% level s reached fter0.105 ime constants nd he

90% after2.302 ime constants;hus he risetime is (2.302 0.105) ime constants hich is2.197 ime constantsr 2.197s..Well call t2.2W or simplicity.In the frequency domain, the RC network offers no attenuationor loss at DC (0 Hz);attenuationiseswith frequencyand s 3 dB at a frequencyof l(Zfnc) or llQff[=).This frequencys uzually referred o as he 3 dB bandwidthor the half powerbandwidth"Now multiplying the 3 dB frequency y the rise time:I IQWWx 2.2W= 2.212W:0.35 dimensionless)

OR:Bandwidth RiseTime= 0.35

atgt6cto(J6trL(E,L )

fl L7t,


27/58

Proportional-lntegral-Derivativeontrol& Lead-LagCompensator Chapter2The figure 2.5 often s quotedwhen characterizingheperformance f oscilloscopes.

A 'scopewhose bandwidth s 100MHz and whosestep responses that of a simple RCnetwork will havean internal rise time of 3.5 ns. Therefore,a pulse with rise time shorterthan his cannotbemeasured n sucha'scope and t is fruitless o attempt o measurehe risetime of a pulse if it is shorter han that of the 'scope's.Even signals whose rise timeapproacheshat of the 'scopewill be distorted so that their measuredise time is inqreased.The measuredise ime s givenby:ftmeasured {(ttr"on")' + (rtporr")t}% 2"1{31Rule: f you want o measureherise ime of a pulse;knowwhat he rise time of your'scopeis otherwiseyoumay end up merelymeasuring hat of the scopeand not that of thepulse.2.4.2.I APPLICATION

In control theory, rise time is commonly definedas the time for a waveform to gofrom l0% to 90o/o f its final value. Nise2008)Thequadraticapproximationor normalized ise time for a 2d-order system,step response,no zeross:

t, "{do 2"2sof o.o7s( 1.12 2.t2t3lwhere is the damping atio andcoos the natural requencyof the network.However, he propercalculation or rise time from 0 to 100%of anunder-damped nd-ordersystems:

t, 'c,.16 2.t3t3lwhere is the damping atio andco6s the natural requencyof the network.

1{L-C2

t t 18f ,


28/58

RiseTime D1oop

rT1--.

Pulse Durat ionBasel ine

dershoctJ

Proportional-lntegral-Derivativeontrol& Lead-LagCompensator Chapter2

representative pu lse '*,rravef rmFigure2.6

2.4.3 SETTLING TIMESettling ime is the timerequired or the responseo reachandstay within a specified

tolerance and(usually2% and5%) of its final value. If the damping atio,( is reduced romthe value of unity (corresponding o the critical damping case), the settling time alsodecreases.he settling ime for a tolerance andof 2o/os determinedy--

Ts= alKa): 4TWhere,T {ime constant

(= dampingratioco natural requency

But n caseof a tolerance andwith 5Yo,the pproximate alueof settling ime s 3T.

2J4t41

#19 t


29/58

Proportional-lntegral-Derivative ontrol& Lead-LagCompensator Chapter2

Figore2.7:2.4.3.LAPPLICATIONSettling ime depends n the system esponse nd ime constant.The settlingtime for a 2nd ordr,underdampd system esponding o a stepresponse anbeapproximatedf ttredampingratio( ( l UV

ln (toleran ce fr action)T*: - du*prttg ratio*natural frec1 2.15t51Thus,settling imeto within 2o/e0.02 s:-

lnf0.02) 3.9rs--1;;-=T" (tJ", (U", 2.16rsl

PUT:DIGITAL CHANGEANALOG STEP

UTPUTPONSE

SETTLING TIME..-..-.->I

t l. - . , , l2o t ,


30/58

Proportional-lntegral-Derivative ontrol& Lead-LagCompensator

2.4.4 STEAI}Y STATE ERRORChapter2

In fact,manycontrolsystems redesignedo haveeven ower damping. ustificationof this lies in the fact thatalmostall practicalsystems osses omecoulomb riction andother nonlinearitiese.g. backlash,binding in gears,and linkagesetc. The presence fthesenonlinearities ends o introducesteadystateerror'

It indicates heerror betweenhe actualoutputanddesiredoutputast tends o infinitv i.e.

** : f3 lFc+>-"Cq zJlA)2.4.4.L APPLICATION

Figure.8For the examplesystem, he controlled system often referred o as he plant - is a first ordersystemwith a transfer unction:G(s): Cdc(st+ l)We will consider systemwith the followingpammeters'

. Gd.: l. t :1. K:1.

. Kn canbeset o variousvalues n therangeof 0 to 10,. The nput s always .Here is a simulationyou can run to check how this works. In this simulation, he

systembeing controlled (the plant) and the sensorhave the parametersshwon above.You can adjust he gain up or down by SYa sing he "-rowu buttonsat bottom riglrt. Wecan alsoenterour own gain in the textbon and see he responsor thegain you enter'


31/58

proportional-Integral-Derivative Control& Lead-Lag Compensator

Enter he OCGaln ortheSystemn rkebaxat fte righl.Thenclic* thistxrtlon.- ls

Chapter2

Closedhopffi gain= o.sCls$sdhspTimeSon*t=o.s

gsc.

Figure .9:

2.5 IMPORTA}'ICE OF PID A}ID LEAD-LAG COMPENSATOROVER TIME RESPONSE SPECIFICATIONS

2,5,L INTRODUCTIONThis tutorial will showthe characteristicsof the eachof proportional(P), the integral (I)'

and the derivative (D) controls, and how to usethem to obtain a desiredresponse-n thistutorial.we will considerhefollowing unlty feedback ystem:

Plant:A system o becontrolled


32/58

Proportional-lntegral-Derivative ontrol& Lead-LagCompensator Chapter2Controller: Provides the excitation for the plant; Designedto control the overall syatembehavior.2.5.2 TIIE THREE-TERIVI CONTROI LERThe hansfer unctionof thePID contoller looks ike the following:

ar K, ", , EosI+Kos+K,.8.- +r +l},^iPtr 's2Jg.2l

. Kp: Proportional ain. KI: Integralgain. Kd: Derivativegain

First, lefs take a look at how thePID controller works in a closed-loopsystemusingthe schematicshown above. The variable (e) representshe hacking eror, the differencebetween he desired nput value (R) andthe actualoutput (Y). This error signal(e) will besent o the PID confioller, and the conholler computes oth the derivative and he integralofthis error signal.The signal (u) just past he contoller is now equal o the proportionalgain(Kp) times hemagnitudeof the errorplusthe,integralgain (Ki) times the integralof theemorplusthe derivativegain(Kd) times he derivativeofthe eror.

u: K'B Krfedt- K"# 2.lgt2lThis signal(u) will be sent o the plant,andthe new output (Y) will be obtained.This newoutput Y) will besentbackto thesensor gain o find the new error signal(e).The controllertakes his new error signal and computests derivative and its integral again; This processgoes n andon.

t l 23t ,


33/58

Proportional-lntegral-Derivative ontrol& Lead-LagCompensator Chapter22.5.3 TrrE CTTARACTERTSTICSOF p, I, AIYD DCONTROLLERS

A proportionalcontroller (Kp) will have the effect of reducing the rise time and willreduce,but never eliminate, the steady-state rror. An integral contol (Ki) will have theeffect of eliminatingthe steady-state ror, but it may makethe transient esponseworse.Aderivativeconffol (Kd) will have he effectof increasing he stabilityof the system, educingthe overshoot,and mproving the hansient esponse.Effects of eachof controllers Kp, Kd,andKi on a closedJoopsystemaresummarizedn thetableshownbelow.

. Table .1Note that thesecorrelationsmaj not be exactlyaccurate,becauseKp, Ki, and Kd are

dependeirt f eachother. n fact, changingone of thesevariablescan changehe effect of theother two. For this reason, the table should only be used as a referencewhen we aredetermining he values or Ki, Kp andKd.

CL RESPONSE RISE TIME OYERSIIOOT SETTLING TIME S-SER,RORKp Decrease lncrease SmallChange DecreaseKi Decrease Increase Increase EliminateKd SmallChange Decrease Decrease SmallChange

, r_, _ -- 124 +, ,IItt


34/58

.-.

CHAPTERJ

MATLABPROGRAMING

&SIMULINKMODEL

OF ARVIATURECONTROLLE,DDCMOTOR


35/58

MATLABPrograming& SIMULINKmodelof armaturecontrolledDCmotor Chapter33.1 INTRODUCTION

The servomotor is actually an assenrblyof four things: a normal DC motor, a gearreduction unit, a position-sensing evice(usually a potentiomsfsl-4 volume conhol knob),and a control circuit. Dc servomotors are normally used as prime movers in computers,numerically conholled machinery,or other applications where starts and stops are madequickly and accurately.

In an armature controlled DC motor, the armature inductance s negligibly smallcompared o armature esistance.So, we can assume hat the ftansfer function of the dcservomotors-

G(s)=o* /Ea(s):(Kt/Ra* y[S^2+(B/3f+Kb*Kt/Ra*J)S] 3.1t61Where,

J:Total moment of Inertia of the rotor togetherwith that of the reflected oad on therotor side.

B=Total viscousconstant.KFMotor torqueconstant.Kb:Motor backEMF constant.Ra=Armatureesistance.La:Armature inductance

t t 261'


36/58

MATLABPrograming& SIMULINKmodelof armaturecontrolledDCmotor3.2 PARAMETERS USED IN 1T{ATI,ABOPERATIONLet us assumeheparametersor positioncontrol system-

J:

Chapter3

5.0*10 - 4kgtn2/sec2.6.0*10^-3 N-m/sec.0.085N-m.0.lV/rad/sec.5Crnegligible.

BKtKbRaLa

3.3 PROGRAMINGFOR STEPRESPONSEOF OPEN LOOPSYSTEM:

J:5.0e-4;B:6.0e-3'Ke0.085Kb=0.Ra=5;num:[(Kt)/(Ra*J)]den:[ ((B/J)+(Kb*Kt/(Ra*J)) 0]gnf(num,den)F0:0.01:2step g,t)

Here, num' rqrresents he numeratorof the fransfer irnction.'den' representshe denominator f the transfer unction.'t' gives hetimeperiodover which theresponses required o be computed.'step g,t)will plot the step esponse f theopen oopsystem.

t27 a


37/58

MATLABPrograming SIMULINK odelof armature ontrolledDCmotor Chapter33.3.1 RESI,]LT

The response f the motorpositionwill found to grow continuouslywhen the unitste,p oltage s applied o the armatureerminals.The open oop systems actingas a pureintegratorso there s a need or feedbackhat will stabilize he system or a step nput. Theclosedoop fransfer unction of the systemwith unit feedback anbe obtained-

h:1gtb=feedback(g,h);step(gtb,t)

Figure3.1


38/58

MATLABPrograming SIMULINK odelof armature ontrolledDCmotor Chapter

3.4PROGRAMING FOR STEP RESPONSE OF CLOSED LOOPSYSTEM

i I=5 . 0e-4 ; 8=6 . 0e-3 ;Kt=0 . 085 Kb=0. 1 ;Ra=5,'num= (Xt1 / (Ra*,J) ]d.en= 1 ( (B/, ] ) + (Kb*Kt) / (Ra* 'J) ) 0 l9=tf (num, den)t=0:0.01:2h=1;gfb=f eedback (9, h)step (gfb, t )

Hetre,num' rqlresents henumeratorof thetransfer unction.'den' representshedenominator f thetransfer unction.ot' gives he imeperiodover'whichhe responses required o be computed.'step(gfb,t)will plot the step esponse f theclosed oop systern.

t\ 29t ,


39/58

MATLABPrograming SIMULINKmodelof armature ontrolledDCmotor Chapter3.4.I.RESULT

Figure3.2In this aboveprograming,we usea feedbackwhosegain s I and t is known asunit

feedbackTheopen oop system cts ike an ntegrator.So his feedbackmakes heopen oopsystem nto a close oop systernwhich will stabilize he system.The step response f theabove losedoop systems shownn thenextpage.

Though we use a closed loop systern, he responsewhich is obtained from thepreviousplot is not desirable.Here, we will designa position control systernwith unityfeedback.he design pecificationsor a stable nddesired ystern reas ollows.1.Steady tateerror n angular osition,0ess:0.2. Settling ime, s:0.2 sec.3. Overshoot ustbe equalor less han5%.

From the closed loop response we find that the settling time is1.55(approximately),whichs much more han hedesired alueof 0.2 sec,even hough heovershoots O.Sincehe rising time s muchhigh, soto reduce t we used he PID controtler"It is to be mentionedhat n majority cases PI or PD controllermay be adequate there

t30


40/58

MATLABPrograming& SIMULINKmodelof armaturecontrolledDCmotorMATLABPrograming& SIMULINKmodelof armaturecontrolledDCmotor Chapter3maynot beneed o use hePID controller.However,ull PID controllermaybe designed orbetter performance. n order to improve the settling time, we chooseanotherresponseasfollows.

3.5 PROGRAMING FOR STEP RESPONSE OF A CLOSEDLOOP SYSTEM USING PID CONTROLI;ERi I=5. 0e-4 ;B=5. 0e-3 ;Kt=0. 69$ Kb=0 - L ;Ra=5;num= (Kt) / (Ra*iI) ]den= [1 ( (B/ . ] ) + (Kb*Kt) / (Ra*,J) 0lKP=3ootKd=O.5tKi=O.01nr:m1= tKpl*num1 = [Kd Kp]*num1 [Kd Kp Ki]denl= t1l*den= t1 0lnum2= conv(numl-,num)den2= conv(den1 ,den)g1=tf (num2 den2 )t= 0:0 .005:1-92=feedback (91, 1 )step (92, t )

t t 31t,


41/58

MATLAB rograming SIMULINK odelof armature ontrolledDCmotor Chapter

3.5.1"RESI,]LT

From the previousclosed oop responsewe find that the settling ime is 0.502sec,which s not desirable alue. t is more han he desired alue of 0.2 sec"Full PID controllermay be designed y trial anderrorprocedure. ow changing he valuesof Ki, Kd & Kp wegetanother losedoop responses ollows.

Figure3.3

t l 32TJ


42/58

MATLABPrograming SIMULINKmodelof armature ontrolledDCmotor Chapter33.6 PROGRAMING FOR STEP RESPONSE OF CLOSEDLOOP SYSTEM:

J:5.0e-4;B:6.0e-3'KF0.085Kb:0"1Ra:5;num:[(Kt)/(Ra*J)]den:[l ((B/J)+(Kb*Kt)/(Ra*J)) ]KnlKd:20%Ki=0.01%numl: [Kp]numl : [Kd Kp]%onumlKd Kp Ki]7odenl: ]denl: [l 0]num2: conv(num1,num)den}: conv(denl,den)gl:t(num2derZ)F 0:0.01:2#=feedback(g1,1)step(g2,t)

t33 t


43/58

MATLABPrograming SIMULINKmodelof armature ontrolledDCmotor Chapter

3.6.1RESLILT

Figure3.4Fromthe previousclosed oop response, e find that the settling ime is 0.433 sec,

which s undesirable alue.In a controlsystem esponse,he time constants directly relatedto the settling ime.A decreasen thevalueof the time constantof a control system ndicatesthat he steady tate eaches arlier. Therefore,ower the time constant.e. lower the settlingtime faster s the time response f a controlsystem.So, we choose he value of Kp, Kd andKi againandcreateanother esponse singPID controllerfor the optimaloutput response ndthe settling.The newresponses shownbelow.

, - ----"--{ 34


44/58

MATLAB rograming SIMULINKmodelof armaturecontrolledDCmotor Chapter3.7 PROGRAMING FOR STEP RESPONSE OF A CLOSEDLOOP SYSTEUIUSING PID CONTROLLER:

J=5. 0e-4; B=5 0e-3;Kt=0 " 085;Kb=O 1;Ra=5;num= (Ktr) / (Ra*,J) ]den= [1 ( (B/ \T)+ (I{b*Kt) / (Ra*J) ) 0]Kp=350Kd=50Ki=300tnuml= [Kp]tnuml = [Kd Kp]numl = [Kd Kp Ki]tdenl= tl ldenl= tl - 0Inum2= conv(num1 ,num)den2= conv(den1 ,den)g1=tf (num2,den2 )t= 0 :0 .01-:292=feedback(91, 1 )step (92, t )

tt ?EI erI


45/58

MATLAB rograming SIMULINKmodelof armature ontrolledDCmotor Chapter3.7.1RESLILT

In thisprogram,we increasehevalueof Kp, Kd andKi.We increasehe proportionalcontrolKp to improvethe rise time, addderivativeactionKd and ncreaset to decreaseheovershootand usean integralKi to decreasehe steadystateerror.The stepresponseof theaboveprogram s shown n the nextpart.

In theprevious esponse,t is obtainedhat hesettling ime is0.00985 ec.which liesin the desirableangeof it. The overshoots 0.075%which is alsoacceptable.he iterativedesignprocedure f a PID controller s illustrated n MATLAB environment" s in manycases f systemdesign here s no uniquesolution o the designproblemof a PID controller.Howeverhe final choiceof the PIDparameters aybe dictatedby hardware onskainsandfinallywe obtainasatisfactoryesponsef thesystem.

Figure3.5


46/58

MATLAB rograming SIMULINKmodelof armature ontrolledDCmotor Chapter33.8 SIMT]LINK MODELln the SIMULINK model, we use three tlpes of controller. Theseare-l) Proportionalcontroller2) Derivative controller and 3) Integral controller. trn he following model, thesethreecontrollersare connected n cascadeorm. The gain I is used for the proportionalconholler,where as the gauin is the derivativecontroller and the gain 3 is for the integralcontoller. Actually by using this threegain,we canselect he valueof the Kp, Kd and Kirespectively.

Figure .1:

t3V l - - _.


47/58

MATLABPrograming SIMULINKmodelof armature ontrolledDCmotor Chapter3.8.1 RESI,]LT

Time n seesFigure .6

Herewe use hreecontroller Kp,Kd & Ki) in SIMULINK Model. By theuseof this threecontrollerwe can achievehe above esponse. erewe cansee he responses closelymovetowardshestability"

(l)tAEoa.h(ufr

f t| 38 +_-_ _-1.,


48/58


49/58

Futureaspects fthe project

Thisprojectcould ind its applicationn different ields,as ollows:4.I DC POSITION CONTROL SYSTEM

Chapter4

. Figure .1

4.L"L INTRODUCTIONOneof the most coillmon examplescovered n text books and literature on linearsystemss a DC position control system).This system s easily understoodand hasa secondorder ransfer unction in the standardorm, for which a well developedheoretical reatrnentis available.

Thisunit provides he students nopportunity o study andoperatea practicalelectro-mechanicalangular-position-control ystem. The system is built around a good qualitypermanentmagnet armature-conholledDC motor, speed eduction gear-set,potentiometricerror detector using special 360o revolution servo potentiometers,a tachogenerator orvelocity feedback and associatedelectronic circuits. Unlike simulatedsystems,e.g. ourLINEARSYSTEMSIMULATOR, hepositioncontrolsystem aturallyconsists f non-idealparameters iz. saturationof amplifier and motor current, deadzone and backlash,non-linearity in the motor and gears, mperfections n mechanical abrication and somewhatuncertainorder of the completesystem due to filters, various time constantsand loadparameters. xperimentalwork on this systemwould enable he students o appreciate hedifferencen performance etween dealized ystems tudied n the theoryclassesand the

rr#40


50/58

Futureaspects fthe projectsystems ncounteredn practice.

Chapter4

A difficulty which is facedwhile working with manypracticalcontrol systemss thattheir responses re rather slow (Notethat in a simulatedsystein he common practice is toscale-uphefrequency o ensureaproperviewing on aCRO).StorageCROor anX-Y plotteris therefore equired for studying the waveforms.Both these nstrumentsare too expensiveand/ordelicate,and are thereforenot usuallyavailable o theundergraduatetudentsn mostinstitutions.Thepresentunit has a built-in mP basedwaveformcapture/display ystemwhichstoreshe step response f the control systemn a RAM andthendisplays t on a measuringCRO for furttrerstudies.This arrangements extremelysimple o operateandconforms o theaccuracy eedsof a class oom experiment.

The motor unit is housed n a separateabinetwith transparentanels or easyviewing.Interconnectionwith the main unities througha standard9-pin D-t1pe connector.All powersuppliesand step input signal are intemally provided. In addition a 3% digit DVM isavailableon the panel for the measrirement f various signals.A good quality measuringCRO s the only accessoryhatwouldberequired.

4.T.2 EXPERIMENTOperationof the positioncontrol system or. Studyof the effect of velocity feedback

on thedifferent valuesof the forwardgainto angular ransientandsteadystateperformanceof thepositioncommands ystemaswell as ts stability.Step esponse tudies or variousvaluesof forward gain. Study of the effectof velocity feedbackon the transientand steadystateperformance f the systernaswell as ts liability.

The experiments would involve calibration and operation of the waveformcapture/displayect)onasafirst step, t maybe mentionedhere hatdue o thenon-linearitiesandother mperfectionsand uncertainties xisting n a physicalsystemof thepresent 1pe,aquantitative erification of the resultswith theoreticalanalysis s not recommended. his isbest done on a simulated system. Of course the experimental work does includedeterminationof rise time, overshoot, teadystate errors etc. for various conditionsfor anevaluation f the systemperformance.

, \ ALr_t ' - t


51/58

Futureaspects fthe projectFEATURES & SPECIFICATIONS

Chapter

operation iteratureandpatchcords ncludedEssential ccessoriesa CRO.

Figare4.2

f l 421J


52/58


53/58

Futureaspects fthe project4.2.2 DGERIMENT

Chapter4

In order to implementa force conhol scheme nto the systemseveralexperiment scarried out to achieve he key requirementof the project. The first experiment nvolves theoperationof the robot system n free spacemotion. Several estswere performed n worldcoordinatesn x-axis direction where he force is appliedto the force/torquesensorand thedata measured s then routed to the robot controller. This generatesncrremental ositiondemand or the robot system and the robot arm will move at a velocity proportional to theapplied force. Determinationof specificgain value for a conventional orce controller inprinciple is relatively straightforward.However, he problem ariseswhen the specific valueofKis unknownor variable.Furtherexperimentmustbe done o determinehe dealKvalueto avoid any serious consequencesn practical robotic systemsparticularly in systemstability.A secondexperimentwill be carriedout using conventional ID control for forceconholler to improve stability performanceand manoeuwability of the manipulatorarm infreespacemotion test as well as experimentn contact with the environmentwhich is a keyrequirementf this project.This will be closed oop conhol systern. he forceconkol loopcompensateshe differenceor error between he desiredand actualforce and then send hedesiredposition to the robot conholler. Further experimentwill be done to develop forcecontrolschemeor TMS robotic systerrbasedon hybrid position/forcecontrol scherne.

t l#44 t,


54/58


55/58

CONCLUSION


56/58

Conclusion Chapter56.1 CONCLUSION

The processeswhich are taken in this report for MATLAB programing are notcompletelyoptimum.Some approxirnationaretaken to complete he report and the value ofKp,Kd andKi are chosenby hial and errormethod. Throug[r, he results which areobtainedfrom theresponses re n the desirableange.

In the first MATLAB programingweChoose nopen oop transfer unction andobtain ts step response. rom theresponse,t is clearthat the system s not stable.To makethe systemstable, n the next MATLAB programing.We use a unity feedbackand get aclosed oop transfer unction. In this response,we find that the responsies desirablebut theresponse ararnetersike, settling imq overshootetc.are not in the desirable ange.To takethispararnetersnto desirable ange,we usePID conholler and choosedifferent valuesof Kp,Kd, andKi. It is to bementionedhat n majority cff;esa PI or PD controllermaybe adequateand hreemay not be need o use he PID controller. However, ull PID controller is designedforcomplete rotectionof the response.

Before completing he project,Wehave ust the bookish knowledge on the MATLABprograming.But, it becomesmoreknowledgeableo usafter the completionof this project.

At last, we can conclude hat our project "positional control-design andanalysis" will be very mucheffective vor he vast ield of radarand satellitepositioningand tabilizationf anykind of system

t t# 47t,


57/58

6. REFERENCES1. Equations:

t1 PID controllerWikipedia " lYikipedia,he ree encyclopediat2l PIDTutorial,"TheUniversity fMichigan,l8l7".[3] ControlTheoryby Katsihuko Ogat4"ModerncontrolEngineering"[4] I.J.Nagrath,M.Gopal,"ontrol systemEngineering",5sEdition 2007New ageInternationalFublisher

Nise,N. "Control Systemsngineering," dEd.,2000,JohnWiley & Sons.Shooter,S.andMcNeill, M. JulS 2002,"'InterdisciplanaryollaborativeLeaming in Mechatronicsat BuclotellUniversi4r,"Journalof EngineeringEducation.B.S.Manke,"inear Contro!Systemu,4hdition, *rannapublishers.

tslt6l

2.


58/58

Documents

Position Control System - Analysis and Design