PID Control

Analog PID control using OP-AMPS

1. 1
EXPERIMENT6
ANALOGPIDCONTROL
USINGOP-AMPS
GROUPMEMBERS
Name UIN EmailId Date
ChenLin(GroupLeader) [email protected] 12/7/2018
WeLi [email protected]/7/2018
AngieKampert [email protected]/7/2018
ShriramVasudevan [email protected]/7/2018
VijayAnandVelmurugan [email protected]/7/2018

2. 2
Contents
1.SUMMARY................................................................................................................................3
2.DESCRIPTIONOFTHEEXPERIMENT.........................................................................................4
2.1BasicConcepts..............................................................................................................4
2.2AnalysisoftheCircuit...................................................................................................7
3.LISTOFCOMPONENTS...........................................................................................................12
4.Porcedures..............................................................................................................................13
5.Results......................................................................................................................................14
6.Conclusion................................................................................................................................20

3. 3
SUMMARY
Thebelowobjectiveswereaccomplishedwithresultsdescribingthedifferentmodesof
operationusingvoltagecharacteristics.
Understandingthetheoryofsumming,inverting,differential,derivative,andintegrator
op-amps.
BuildacompleteanalogPIDcontrolcircuit.
Testtheinput–outputsignalrelationofaPIDcircuit(i.e.,P-only,Donly,Ionly,PID
versionsofthecircuit).
Comparetheresultsandevaluatethereasonsfordifferencebetweenrealand
simulatedoutputs.
Figuringoutconclusions.

4. 4
DESCRIPTIONOFTHEEXPERIMENT
1.BasicConcept
The“pure”derivativehasalargegainathighfrequencyandwillamplifythenoiseintheclosed
loop,andhenceleadtostabilityproblems.Toreducethegainofthepurederivativeathigh
frequency,apracticalderivativeop-ampcircuitismodifiedsothatithasafirst-orderpolein
additiontothederivative,hencereducingthehighfrequencygainofthetransferfunction
therebyreducingtheproblemofnoiseamplification.
Figure1.PIDcontrol
addingaresistorR1inserieswiththecapacitorC.Letusderivethetransferfunctionforthis
practicalderivativecircuit.Noticethatv+=GNDandv+=v−attheinputterminalsoftheop-
amp.Sincetherecannotbecurrentdrawnintotheop-amp,theni1(t)=i2(t)wherei1(t)isthe
currentontheinputsideoftheop-ampthroughR1andC,andi2(t)isthecurrentonthe
feedbackloopoftheop-ampthroughR2.
1.1Derivative
Aderivativetermdoesnotconsidertheerror(meaningitcannotbringittozero:apureD
controllercannotbringthesystemtoitssetpoint),buttherateofchangeoferror,tryingto
bringthisratetozero.Itaimsatflatteningtheerrortrajectoryintoahorizontalline,damping
theforceapplied,andsoreducesovershoot(errorontheothersidebecausetoogreatapplied
force).Applyingtoomuchimpetuswhentheerrorissmallandisreducingwillleadto
overshoot.Afterovershooting,ifthecontrollerweretoapplyalargecorrectionintheopposite
directionandrepeatedlyovershootthedesiredposition,theoutputwouldoscillatearoundthe
setpointineitheraconstant,growing,ordecayingsinusoid.Iftheamplitudeoftheoscillations

5. 5
increaseswithtime,thesystemisunstable.Iftheydecrease,thesystemisstable.Ifthe
oscillationsremainataconstantmagnitude,thesystemismarginallystable.
1.2Controldamping
Intheinterestofachievingacontrolledarrivalatthedesiredposition(SP)inatimelyand
accurateway,thecontrolledsystemneedstobecriticallydamped.Awell-tunedposition
controlsystemwillalsoapplythenecessarycurrentstothecontrolledmotorsothatthearm
pushesandpullsasnecessarytoresistexternalforcestryingtomoveitawayfromtherequired
position.Thesetpointitselfmaybegeneratedbyanexternalsystem,suchasaPLCorother
computersystem,sothatitcontinuouslyvariesdependingontheworkthattheroboticarmis
expectedtodo.Awell-tunedPIDcontrolsystemwillenablethearmtomeetthesechanging
requirementstothebestofitscapabilities.
1.3Proportionalterm
Figure2.ResponseofPVtostepchangeofSPvstime,forthreevaluesofKp(KiandKdheldconstant)
Theproportionaltermproducesanoutputvaluethatisproportionaltothecurrenterrorvalue.
TheproportionalresponsecanbeadjustedbymultiplyingtheerrorbyaconstantKp,calledthe
proportionalgainconstant.
Theproportionaltermisgivenby
Pout=Kp*e(t)
Ahighproportionalgainresultsinalargechangeintheoutputforagivenchangeintheerror.If
theproportionalgainistoohigh,thesystemcanbecomeunstable.Incontrast,asmallgain

6. 6
resultsinasmalloutputresponsetoalargeinputerror,andalessresponsiveorlesssensitive
controller.Iftheproportionalgainistoolow,thecontrolactionmaybetoosmallwhen
respondingtosystemdisturbances.Tuningtheoryandindustrialpracticeindicatethatthe
proportionaltermshouldcontributethebulkoftheoutputchange.
1.4Integralterm
Figure3.ResponseofPVtostepchangeofSPvstime,forthreevaluesofKi(KpandKdheldconstant)
Thecontributionfromtheintegraltermisproportionaltoboththemagnitudeoftheerrorand
thedurationoftheerror.TheintegralinaPIDcontrolleristhesumoftheinstantaneouserror
overtimeandgivestheaccumulatedoffsetthatshouldhavebeencorrectedpreviously.The
accumulatederroristhenmultipliedbytheintegralgain(Ki)andaddedtothecontrolleroutput.
Theintegraltermisgivenby
Iout=Ki∫0te(τ)dτ.
1.5PIDcontroller
Aproportional–integral–derivativecontroller(PIDcontrollerorthreetermcontroller)is
acontrolloopfeedbackmechanismwidelyusedinindustrialcontrolsystemsandavarietyof
otherapplicationsrequiringcontinuouslymodulatedcontrol.APIDcontrollercontinuously
calculatesanerrorvalueasthedifferencebetweenadesiredsetpoint(SP)anda
measuredprocessvariable(PV)andappliesacorrectionbasedonproportional,integral,
andderivativeterms(denotedP,I,andDrespectively)whichgivethecontrolleritsname.
Inpracticaltermsitautomaticallyappliesaccurateandresponsivecorrectiontoacontrol
function.Aneverydayexampleisthecruisecontrolonaroadvehicle;whereexternal

7. 7
influencessuchasgradientswouldcausespeedchanges,andthedriverhastheabilitytoalter
thedesiredsetspeed.ThePIDalgorithmrestorestheactualspeedtothedesiredspeedinthe
optimumway,withoutdelayorovershoot,bycontrollingthepoweroutputofthevehicle's
engine.
Thefirsttheoreticalanalysisandpracticalapplicationwasinthefieldofautomaticsteering
systemsforships,developedfromtheearly1920sonwards.Itwasthenusedforautomatic
processcontrolinmanufacturingindustry,whereitwaswidelyimplementedinpneumatic,and
thenelectronic,controllers.TodaythereisuniversaluseofthePIDconceptinapplications
requiringaccurateandoptimizedautomaticcontrol.
2AnalysisoftheCircuit
Figure4.ThecircuitoftheanalogPIDcontroller

8. 8
2.1DifferentialInputOp-Amp
Figure5.Differentialinputamplifier
Thedesiredfunctionistodeterminethedifferencebetweentwosignalsandpossiblymultiplythe
differencewithagain,
whichisusedinclosedloopcontrolcircuitsasthesummingjunctionthatisfindthedifference
betweenacommandsignalandsensorsignal.(Figure5)showsadifferentialinputop-amp
circuit.Initsgeneralform,theinput–outputrelationshipcanbeobtainedusingthe
superpositionprinciple.Theoutputisthesumoftheoutputsduetotheinvertinginputandthe
non-invertinginput.Thesuperpositionprinciplecanbeusedinthederivation:(i)connectV2to
groundandsolveforv’o=K1⋅V1,and(ii)connectV1togroundandsolveforv”o=K2⋅
V2.Then,addthemtogethertogetVo=v’o+v”o.Theoutputduetoinputatitsnon-
invertingterminalis
Andtheoutputduetoinputatitsinvertingterminalis

9. 9
Thetotaloutputis
NotethatwhenR1=R2=R3=R4,theinput–outputrelationshipis
Similarly,whenR1=R3=RandR2=R4=K⋅R,
Oneofthemainusagesofdifferentialop-ampsisinamplifyingnoisesensitivesignals.As
discussedinFigure5,single-endedsignalsarereferencedwithrespecttoground.Anynoise
inducedonthesignalwirecomingintotheop-ampwouldbeamplified.Thisisparticularly
problematicwhenthenoisesignaliscomparabletotheactualsignalmagnitude.Insuchcases,
itisbesttotransmitthesignalvoltageindifferential-endedformat.Thatisusingtwowiresand
thesignalinformationisthevoltagedifferencebetweenthetwowires.Ifanynoiseisinduced
duringthetransmission,itwouldbeinducedonbothlinesandthedifferencebetweenthem
wouldstillbeunaffectedbynoise.
2.2DifferentiatorOp-Amp
Figure6.Differentiatoramplifier
Thedesiredfunctionofaderivativeopamp,showninFigure6,wastoamplifythe
derivativeoftheinputvoltagetooutputvoltagewithagain.

10. 10
Figure6showsanop-ampcircuitfordifferentiation.Usingtheidealop-ampassumptions,
theinput–outputrelationshipisderivedasfollows,
Hence,
Inourexperimentsetting,wemodifiedthederivativeop-amptoreducethenoisefrom
highfrequencygain.WeaddedaresistorR1inseriestothecapacitorasshownincircuit
diagram.Hence,
2.3IntegratorOp-Amp
Figure7.Integratoramplifier

11. 11
TheIntegratorop-ampwasdesignedtoamplifytheintegraloftheinputvoltageoveraperiod
oftimewithagain.Ifwechangethelocationsoftheresistorandcapacitorinthederivativeop-
amp,weobtainanintegratingop-ampcircuit(Figure7).Thedesiredfunctionis,
whereVo(0)istheinitialvoltage.ThederivationoftheI/Orelationshipisstraightforward,
wheretheinitialvoltagevaluesintheintegrationshavebeenneglected.Inpractice,aresistor
wasaddedinparallelwiththecapacitor,shownincircuitdiagram,toreducethephaselagat
lowfrequencycontentandimprovethestabilityofintegratorop-amp.Thus,theinput-output
ofthismodifiedintegratorop-ampwasobtainedasfollowing,
2.4TheRelationshipbetweenInputandOutput
FortheProportionalController,thetransferfunctionwas
ForthemodifiedDifferentiatorController,thetransferfunctionwas
ForthemodifiedIntegratorController,thetransferfunctionwas

12. 12
Therefore,thecompletetransferfunctionforthePIDcontrollerinpracticewas,
LISTOFCOMPONENTS
Component Quantity
LM358Op-AmpIC 5
Resistor220Ω 1
Resistor1kΩ 8
Resistor4.7kΩ 4
Resistor100kΩ 4
Resistor330kΩ 1
Capacitor0.22μF 2
PowerSupply 1
DMM 1
Breadboard 2
Connectorwires 1set
Oscilloscope 1
FunctionGenerator 1

13. 13
Procedures
1.Assemblethecircuitonthebreadboardasshowninpart4CircuitDiagram.Sincethecircuit
isinvolvedwith5op-amps,over15componentsandnumerouswires,webuildthecircuit
on2breadboards.ThePIDcontrollerisbuiltinonebreadboard.Theinputcomparatorand
outputamplifierarebuiltinanotherbreadboard.Inthisway,wecanavoidthetouching
betweencomponentsanditiseasierforustocheckthecircuit.
2.Setupthefunctiongeneratortoproducethesquarewavesignalwithanamplitudeof6V.
Thefrequencyofthewavefunctionscanbeeasilyadjustedonthefunctiongenerator.
ConnectthefunctiongeneratorsignaltoVi(feedbacksignal).ConnecttheVreftotheground.
3.Connecttheoscilloscopetotheinputsignaloftheproportionalop-amp,andtheoutputof
theproportionalop-amp.Then,takeapictureoftheoscilloscopescreen.
4.Connecttheoscilloscopetotheinputsignalofthederivativeop-amp,andtheoutputofthe
derivativeop-amp.Then,takeapictureoftheoscilloscopescreen.
5.Connecttheoscilloscopetotheinputsignaloftheintegratingop-amp,andtheoutputof
theintegratorop-amp.Then,takeapictureoftheoscilloscopescreen.
6.Connecttheoscilloscopechannel1totheinputsignalofthewholePIDcontrollercircuit,
andoscilloscopechannel2totheoutputofthesummingop-amp(whichistheoutputof
thePIDcontrollercircuit).Then,takeapictureoftheoscilloscopescreen.
7.DerivethecompletetransferfunctionofthePIDcontroller.Calculatetheproportional,
derivative,andintegratorgains:KP,KD,KI,andtheadditionalpolelocationofthemodified
derivativeterm,andpolelocationandgainofthemodifiedintegralterm.

14. 14
Results
DifferentialInputOp-Amp
Figure8.Non-InvertedAmplifierAssembly
Gainfordifferential:

L

–

E
–

E




F





–
Simplifying,

L
–


F



–,
–L
꿘ቩ
L


L

–



F
–

=
–

=
tΩ
tΩ
=1
UsingresistorsofonlytΩ,thegainoftheproportionalcontrolleris1.
Figure9showsthesquaregraphsobtainedfromoscilloscopeandcorrespondingpure
differentialoutputs.

15. 15
a. b.
Figure9.Squaregraphsobtainedfromoscilloscopeandcorrespondingpuredifferentialoutputswith1kΩ
resistors.
Fig.9demonstratestheamplificationoftheinputerrorbymultiplicationofthegainofthe
proportionalcontroller,1.
DerivativeOp-Amp
Figure10.InvertingAmplifierAssembly
Gainforpurederivative:

LF⩗

Simplifying,

ቩ⩗L


LF?––F??oΩ⩗L-1.034e-3s

ቩLF?ǤF

16. 16
Gainformodifiedderivative:




LF

–⩗

⩗E
Simplifying,


LF
?––F??oΩ⩗
?––F??oΩ⩗E
LFE-1.034e-3s

ቩLFE-1.034e-3s
Poleformodifiedderivative:
⩗LF



LF

?o??––F
LF?ǤeF


LFE-1.034e-3*F?ǤeFL?ǤǤǤǤǤǤm

ቩN
Thepureandmodifiedgainsandpoleforthederivativeop-ampcanbeseeninTable1.
Table1.DerivativeGainsandPoleResults
PureDerivativeGain,
ቩ F?ǤF
ModifiedDerivativeGain,
ቩ
Modifiedpole,s F?ǤeF
Figure11showsthesquaregraphsoftheinput-outputsignalrelationshipsforthederivative
op-amp.
a.
Figure11.Squaregraphfromthepurederivativeoutputwitha4.7kΩresistorand.22?i.
Thefiguredemonstrateshighinputandoutputvoltagesatthespikeoutputs.

17. 17
IntegratingOp-Amp
Figure12.InvertingAmplifierAssembly
Gainforpureintegrator:

LF

⩗


Simplifying,

⩗L?––F?––ǤΩ⩗LF?Fm⩗

LF?Fm
Gainformodifiedintegrator:




LF


⩗E
Simplifying,




LF

?––F?––ǤΩ⩗E
LF

?Fm⩗E

LF

?Fm⩗E
Poleformodifiedintegrator:
⩗LF



LF

?Fm
LF–––?

18. 18

LF

?Fm–––?E

LǤ?m
TheresultingpureandmodifiedgainsandpolecanbeseeninTable2
Table2.IntegratorGainsandPoleResults
PureIntegratorGain,
F?Fm
ModifiedIntegratorGain,
Ǥ?m
Modifiedpole,s F–––?
Figure13showsthesquaregraphsoftheinput-outputsignalrelationshipsfortheintegrating
op-amp.
Figure13.Squaregraphfromthepurederivativeoutputwitha220Ωresistorand.22?i.
Themodifiedintegratorcreatesasawteethwavewithsomenoisecontainedinthecircuit.
PIDController
Figure14.ThecircuitofanalogPIDcontroller

19. 19
Theproportional–integral–derivativecontrollerutilizesthemodifiedcontrollers.Thevaluesusedforthe
PIDfrompreviousthepreviousmodifiedcircuitscanbeseeninTable3.
Table3.PIDGainValues
ProportionalGain,
1
ModifiedDerivativeGain,
ቩ
ቩN
Modifiedpole,s F?ǤeF
ModifiedIntegratorGain,
Ǥ?m
Modifiedpole,s F–––?
Figure15showsthesquareandsinewavesforthePIDcontroller.
a.
b. c.
Figure15.Thesquaregraphsareshownina.andb.forthePIDcontroller.Thesinewavecanbeseeninc.

20. 20
Conclusion
Inthislab,webuiltananalogPIDcontrollerwithop-ampsLM358andseveralresistorsand
capacitorsbyputtingthedifferential,proportional,derivativeandintegralop-ampstogether.
Thecontrolsignalisthusasumofthreeterms:aproportionaltermthatisproportionaltothe
error,anintegraltermthatisproportionaltotheintegraloftheerror,andaderivativeterm
thatisproportionaltothederivativeoftheerror.AgoodPIDcontrollershouldhavequickand
stableresponse.AfterbuildingandtestingthePIDcontroller,wefoundthatthere’ssomenoise
beingamplified.Wetriedtochangethecapacitorsandtheop-ampsseveraltimesandfinally
gotprettygoodoutputgraph.