Control loops 1
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Controlloopsandcontroltheory I. Feedbackloopandtransferfunction II. PIcontrollers III. Stabilitycriteria IV. PIcontroller+fastloop V. 1HzClocklaserexample El8 Controlloopexample:driving PeriodlengthL Forthedriver,the(periodic)oscillationsintheroadhavean apparentfrequencyof݂ ൌ ݒȀܮ. 1. Atwhichfrequency݂ (variablespeed)ݒdoyouexpectthe car/drivertolosecontrolofthecar?(perfectgrip) 2. Whichfactorscontributetothislossofcontrol? Capturerange El9 Feedbackloop Disturbance ܷሺݏሻ Systemܣ௦ ሺݏሻ Controlsignal ܥሺݏሻ Complexfrequency ࢙ ࣌ ؠ ࣓ Systemoutput ܺሺݏሻ Errorsignal ܺܺ=ݎሺݏሻ Controllerߚሺݏሻ Referencevalue()ݎ Example:Proportionalcontrollerࢼ ࢙ = Assumption:nodelay,constantgainܣ System(Strecke)isslow,hasdelay߬ ֜ phaseresponsedelayed,by120° at݂ .For thisdelay,effectivenegativefeedback becomesimpossible. ܣ ܣ ڄ௦ Hence,totalgainߚ ݂ ܣ ڄ ሺ݂ ሻ shouldbe smallerthanone,toavoid resonance/introducedamping. ܣ ܣ ڄ௦ Drawbacks: Limitedthegainatlowfrequencies. Exponentiallyslowapproachtoreference=0. Constantoffset ݔሺλሻ െ Ͳ ് ݎifͲ ് ݎ. El11 Tietze+Schenk,Chap19 Openloopgain Disturbance ܷሺݏሻ ௦ andtransferfunction Systemܣ௦ ሺݏሻ Controlsignal ܥሺݏሻ Complexfrequency ࢙ ࣌ ؠ ࣓ Systemoutput ܺሺݏሻ Errorsignal ܺܺ=ݎሺݏሻ Controllerߚሺݏሻ Referencevalueݎ Responseofsystemtoinput?Withoutfeedback: ௧ᇲ ݐ ݔൌ ݐ ݑᇱ ܽ ڄ௦ ݐെ ݐᇱ ݀ ݐᇱ ܽ כ ݑ ؠ௦ ܺ ݏൌ ܷ ܣ ڄ ݏ௦ ݏ ؠכConvolution Convolution՜ Multiplikation.Solveinfrequencydomain,finalIFT Closedloop()Ͳ ؠ ݎ: ܺ ݏൌ ܷ ܣ ڄ ݏ௦ ݏെܺ ܣ ڄ ݏ ߚ ڄ ݏ௦ ݏ ܺ ݏൌ ݏ ܨൌ ୢ୧ୱ୲୳୰ୠୟ୬ୡୣ ୬ୣୟ୲୧୴ୣୣୣୢୠୟୡ୩ ೞ ܷ ڄሺݏሻ ଵାఉڄೞ ೞ iscalledtransferfunctionorimpulseresponse,becausewithߜimpulse ଵାఉڄೞ ݐ ݑൌ ߜ ݐെ Ͳ ֜ ܷ ݏൌ ͳǢ ܺ ݏൌ ݐ ݔ ֜ ͳ ڄ ݏ ܨൌ ݂ሺݐሻ El12 ܷሺݏሻ Systemincl.Feedback:ݏ ܨ ܺሺݏሻ Transferfunction ܷሺݏሻ andstability Systemܣ௦ ሺݏሻ Controllerߚሺݏሻ Systemresponse ௦ ܺሺݏሻ ௦ ܷሺݏሻ Systemincl.Feedback:ݏ ܨ ܺሺݏሻ Suppressionofdisturbancesbyfeedback Thelargerߚሺݏሻ,thebettersuppressionofdisturbancesatthatfrequencyݏ Stability:ࢼ ࢙ ࢙ ڄሺ࢙ሻ shouldneverbecomeൌ െ becausethenpositivefeedback,instable Goals? • Stability,i.e.ߚ ܣ ڄ ݏ௦ ് ݏെͳݏ • Largegainࢼ atlowfreqencies • Why:Mostdisturbancesaretypicallyacousticandhenceatfrequencies<2kHz Twoviewpoints: 1. Study ߚ ܣ ڄ ݏ௦ ݏinBodediagrams,avoid߮ ߚ ܣ ڄ௦ ൌ െͳͺͲι AND|ߚ ܣ ڄȁ ͳ • Moreintuitive.Thislecture. 2. Studypolesandzerosof ݏ ܨincomplex ݏൌ ߪ ݅߱ plane.Mathematicallymore powerful(residuetheorem). • Severalstabilitycriteriae.g.polesmustbeinߪ<0halfplane.RouthHurwitzcriteria See2ndlecture. El13 Laplacetransform Laplacetransform=Fouriertransformgeneralisedtocomplexݏ: ஶ ݏ ܨൌන ݁ ି௦ڄ௧ ͳ ఊାஶ ௦ڄ௧ න ݁ ݏ݀ ݏ ܨ ݂ ݐ ݂ ֞ ݐ݀ ݐൌ ʹߨ݅ ఊିஶ Complexfrequency ࢙ ࣌ ؠ ࣓ Integration&Differentiation֞ Division&Multiplicationbyݏ ௧ ݆ ݐൌ න ݂ ݐᇱ ݀ ݐᇱ ֞ ݏ ܬൌ ݀ ݐൌ ݏ ܨ ߱ ܨ ߱ ܬൌ ݏ ݅߱ ݂݀ ݐ ֞ ݏ ܦൌ ߱ ܦ ݏ ܨ ڄ ݏൌ ݅߱ܨሺ߱ሻ ݀ݐ ஶ ஶ ௧ ି௦ڄ௧ Proof: ݏൌ ݁ถ ܬ ڄ ݐᇱ ݐ݀ ݐൌ ݂ ݐᇱ ݀ ݐᇱ ݀ ݐൌ ݅ ڄ ݑȁஶ ถ െ ݑ ௨ᇲ ܫ ୀ ௧ ܫ Integration షೞ ೞ Differentiation ͳȀ݂ െͻͲι El14 െͳͺͲι ݂ ݂ ߮ሺܫሻ ͳȀ݅ ி ௦ ௦ ߮ሺܫሻ െͻͲι െͳͺͲι ݂ ݅ Integrate byparts Cornerfrequency݂ூ ൌ PIController ଵ ଶగோమ మ Unitygainpoint ܣூ ߱ ൌ ܣ ͳ Proportionalgain athighfrequencies ʹߨ݂ூ ݅߱ Integralgain atfrequencies݂ ൏ ݂ Advantages: • Muchhighergainatlowfrequencies݂: • Limitedbycornerfrequency݂ூ : ು ൌ ು • Zerooffsetݔሺλሻ െ ݎൌ Ͳ fromref.valueͲ ് ݎ. ֜ PIcontrollermuchbetterthanPcontroller! But:Extra90° delay.Noproblem,only120° is. Currentܫଵ ൌ ܷ௫ Ȁܴଵ chargesܥଶ ,changingܷ . Untildeviationfromdesiredreference=xr=ܷ௫ iszero,i.e.perfectlock. R3 R2 ܴଵ ܫଵ ܷ௫ u u – + i3 ܥଶ Math: Gainൌ ܫଵ Control signalܷ మ భ ൌ ோమ ା ఠమ షభ ோభ ൌ ͳ ோమ మ షభ ఠ ൌ ܣ ͳ ଶగ ఠ ܼଶ islargeforlowf,whereܥଶ iseffectiveblock.Atlargef,ܥଶ becomes conducting,andܴଶ dominates ܴଷ limitsgainatverylowfrequencies,i.e.allowsdischargingofܥଶ El15 TietzeSchenk,Chap.19 PIcontrollertips GeneralGuideline: Maximumgainatlow,acousticf‘s! • MoreimportantthanBandwith݂ௐ asfigureofmerit. Def.:݂ௐ ǣ ߮ ߚሺ݂ௐ ሻ ܣ ڄ௦ ሺ݂ௐ ሻ ൌ120ι • Butthehigherthebandwidth,thehigher݂ூ ֜ higher thegainatlowfrequencies. • Checkthedelaysofeachrelevantpart:System,detector, partsofthecontrollerlikepreamp,actuator(e.g.piezo, AOM).Optimizelimitingone. • Increasebandwithbyaddingsecond,fastercontrol element,e.g.AOM.՜2ndlecture. UseonlyIcontrollerwithoutP • • Yes,yourbandwithwillgodownduetoextra90°.Butyou aretakingadvantageofthe1/fbehaviouralways. Andyouwillnothavetochoose݂ூ correctly Phaseandamplitudelinkedbycausality/FourierTrafo • • El16 ோమ ோଵ Candirectlydeduce ֞ ܫ( ߮ ֞ ܣെͻͲι; ֞ ܦͻͲι) providedphaseshiftscausedbylowpassbehaviourand notcabledelays AdvancedPIcontrollertips Dodoubleintegration:PI2 • • • • Moregainatlowfrequency െͳͺͲι delayatlowI2frequenciesisnotaproblem,aslongas ݂ூమ ݂ د ȀͶ (prooflater) Limitdoubleintegrationatlowf,otherwiseinitallockinghard. Integrator is a lowpass ֜ can implement 2nd I passively. However, 1 integrator must be active, to be able to charge capacitor to setpoint. Implement PI2 with normal PIlockbox (corner ݂ூమ ) + extra external lowpass with ݂ ̱ͳͲͲ Hz. Sharppiezoresonances • • • causestrongphasedelays>120° andhencelimitbandwidth Sometimeshelpstosuppressgainat݂௭௦ by(higherorder) lowpasswith݂ ൏ ݂௭௦Ǥ Thentotalgaincanbeincreased. >moregainatlowfrequency Normally:Donotusedifferentialgain(PID) • • Gainincreases‚infinitely‘ forincreasingf.Butthe߮ ൌ െͳʹͲι still ariseduetosystemdelay,despite+90°.HereA>1>instable DpartgoodwhencancellingsomeIpartinthesystem,e.g.for currentfeedbackonlaserdiodes(whichare~capacitors). • Goodforveryslowloopsliketemperaturecontrollers El17 Exerciseincircuitdiagramreading– PIinputampstage Errorsignal+and in amplified errorsignal Circuittoprovideoffset correctionforINA (notimportant) El18 Integrator+outputstage LogicDither(LD):Alogic(5V,0V)signaltellingthecircuit Wetherheshould„dither“ortrytolock. Dither:Asawtoothsignaltosweep outputacrossentire+10Vregion whennotlocked.Toseelockingsignal LD Relock(logic): Whenlockislost, integrator&outputsaturates.> Relockisputtozero(elsewhere), dischargingintegratorC El20 SomePIcontrollermodelstoconsider • PIcontrollerScheich=Hänschgroupelectronician:Anton.Scheich@physik.unimuenchen.de • Good,PIpart,extrafasterPpart.Yougetcircuitdiagramandcanmakechanges. Notveryfast(<1MHz).Cheap~800€. • TopticaLockbox.SimilartoScheich.Prize?Circuitdiagram? • NewportLB1005:10MHzfastanalogPIlockbox.Verygood.Price~1700€? • Vescent: 10MHzfastlockboxwithPI2D,goodcontrols,3500$ • TopticaFALC.Extremelyfastlockbox(45MHz).OneslowPID(fore.g.Laserpiezo)andone fast(Laserdiodecurrent).Idealforhighbandwidthlocks:Phaselockswithfastfeedbackto laserdiode. • TEMNoiseeater:Continouslock(nodither).Goodforlaserpowerstabilisation • TopticaDigilock:FPGAorDSPbaseddigitallockbox.2slowPID(1MHzfore.g.laserpiezo) andonefast(21MHz,forLaserdiodecorrent). • Allvaluesadjustableviacomputer:݂ூ ǡ ݂ ,severalfilters,gains,relock,Controlvaluelike e.g.cavitytransmissiononwhichtoswitchfromdithertolock.Diverseextrafunctions • InternalPoundDreverhallfunction • ~4000€.NoPotis! • TEMLaselockdigital: SimilartoDigilock.~3400€ • NationalInstrumentsCRIO:FPGAbasedlogicwithvariousAnalog/DigitalInandout perhiphery.Cane.g.realiseupto~16PIlockswithbandwidth~10kHz,16bitD/AA/Doutput. • Lotsofchannels:digitalFunctionalitysuchassample+hold,logic,freelyprogrammable • Realivelyslowwhenmanychannels:Notcheapeither:16ch~10‘000€.Digitalnoise. • Goodforexampleforpiezostraingaugelocks,uncriticallocks,temperaturecontrol, El23 interlocks… analog
