Conditions and Methods for Offset-Free Performance in Discrete Control Systems

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Conditions and Methods for Offset-Free Performance in Discrete Control Systems
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Qian, Yuzhou
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Master's ( M.S.)
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Chemical Engineering
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Crisalle, Oscar D
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Latchman, Haniph A

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disturbance-model -- integral-action -- mpc -- pi-controller
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Abstract:
The offset-free controller drives controlled outputs to their desired target, and this thesis addresses the problem of offset-free controller when tracking the constant or inconstant references. Conditions that guarantee detectability of the augmented system model are discussed both in proportional–integral–derivative control (PID) framework and model predictive control (MPC) framework. Also two examples are presented to introduce a new MPC method to guarantee the best closed-loop offset performance for unstable reference.
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by Yuzhou Qian.
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Thesis (M.S.)--University of Florida, 2012.
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Adviser: Crisalle, Oscar D.
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CONDITIONSANDMETHODSFOROFFSET-FREEPERFORMANCEINDISCRETE CONTROLSYSTEMS By YUZHOUQIAN ATHESISPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF MASTEROFSCIENCE UNIVERSITYOFFLORIDA 2012

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2012YuzhouQian 2

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Toeveryoneunderthesamesky 3

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ACKNOWLEDGMENTS TheauthorissincerelythankfultoProf.OscarD.Crisalle, DistinguishedTeaching ScholarandProfessor intheChemicalEngineeringDepartmentofUniversityofFlorida, forhelpfulsuggestionsfromtheearlystageofthisworkanddoctoralcandidatesM. RafeBiswasandShyamP.Mudirajfortheirconstructivefeedbackontherevisionsofthis thesis. 4

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TABLEOFCONTENTS page ACKNOWLEDGEMENTS.................................4 LISTOFTABLES......................................7 LISTOFFIGURES.....................................8 ABSTRACT.........................................9 CHAPTER 1INTRODUCTION...................................10 1.1Background...................................10 1.1.1StateFeedbackPIDController....................10 1.1.2Stateobserverandestimator.....................11 1.1.3ModelPredictiveControllerMethod..................11 1.2DescriptionofContents............................12 2LITERATUREREVIEW...............................14 2.1IntegralActioninPIControllers........................14 2.2Challenges...................................15 2.3OffsetPerformanceinMPCControlSystems................17 2.4Offset-freePerformanceConsideringInconstantSignals..........18 3ANALYSISANDDESIGNOFPICONTROLLERS................20 3.1SystemwithoutDisturbance..........................20 3.1.1DescriptionofSystemwithoutDisturbance..............20 3.1.2Controllability..............................21 3.1.3OffsetPerformanceofSystem.....................22 3.1.4RankoftheIntegralGain K 1 ......................24 3.2SystemwithaConstantDisturbanceModel.................26 3.2.1IntroductionoftheSystemwithaConstantDisturbanceModel...26 3.2.2ProblemDescription..........................26 3.2.3TheOffsetPerformanceoftheSystem................27 3.3Example.....................................28 3.3.1Example1................................28 3.3.2Example2:CSTRPlantSystem....................31 3.3.2.1Modelingequations.....................31 3.3.2.2Linearizationofthesystem.................32 3.3.2.3Offsetperformance.....................34 5

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4ANALYSISANDDESIGNOFTHEOBSERVER..................37 4.1StabilityofRegulationProblem........................38 4.2Example.....................................40 5ANALYSISANDDESIGNOFTHEMPCCONTROLLER.............43 5.1Offest-freeMPCDesignConditionsforConstantReference........43 5.2TheOffsetintheMPCWhenTrackingNonconstantReference......46 5.2.1Zero-offsetinOutputwithConstantReference............46 5.2.2CauseoftheOffsetwithNonconstantReference..........50 5.2.3NewControlStructureandMethodSummary............52 5.2.3.1Method1...........................53 5.2.3.2Method2...........................53 5.3ExampleandComparison...........................53 5.3.1RampReference............................53 5.3.2OscillatingReference..........................56 6FUTUREWORK...................................58 6.1TrackingNonconstantReferencebyModifyingtheDisturbanceModel..58 6.2FutureWork...................................59 APPENDIX ARANKPRESERVINGTHEOREM.........................61 BTHECAYLEY-HAMILTONTHEOREM.......................62 REFERENCES.......................................63 BIOGRAPHICALSKETCH................................65 6

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LISTOFTABLES Table page 3-1TheparametersoftheCSTR.............................33 7

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LISTOFFIGURES Figure page 1-1Atypicalblockdiagramofafeedbackcontroller..................11 1-2Atypicalblockdiagramofasystemwiththeobserver...............11 1-3AblockdiagramofMPCcontroller..........................12 2-1AblockdiagramofMPCcontroller..........................17 3-1ThediagramofCSTRtanksystem.........................32 5-1Comparisonofcloseloopresponsetoanrampsignal...............56 5-2Comparisonofcloseloopresponsetoanunstablesignal.............57 8

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AbstractofaThesisPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofMasterofScience CONDITIONSANDMETHODSFOROFFSET-FREEPERFORMANCEINDISCRETE CONTROLSYSTEMS By YuzhouQian August2012 Chair:Oscar.D.Crisalle Major:ChemicalEngineering Theoffset-freecontrollerdrivescontrolledoutputstotheirdesiredtarget,andthis thesisaddressestheproblemofoffset-freecontrollerwhentrackingtheconstantor inconstantreferences.Conditionsthatguaranteedetectabilityoftheaugmentedsystem modelarediscussedbothinproportionalintegralderivativecontrolPIDframework andmodelpredictivecontrolMPCframework.Alsotwoexamplesarepresentedto introduceanewMPCmethodtoguaranteethebestclosed-loopoffsetperformancefor unstablereference. 9

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CHAPTER1 INTRODUCTION Thischapterisintendedtobeaguidetothetopicscoveredbythisthesis.The objectivesofthisworkincludetheanalysisofthestabilityanddetectabilityofa proportional-integralcontroller,vericationofthepossibleunstableperformancein certainMPCcontrollers,andnallyintroduceanMPCmethodtoobtaintheimproved offsetperformanceintrackinginconstantreference. Severalgeneralstate-feedbackcontrollersandthechallengesassociatedwith theirdesignaredescribedinsubsection1.1.Anexamplefortheplantanditsdetailed informationisprovidedinsubsection1.2.andsubsection1.3,andtheorganizationof thisthesisisexplainedinthedescriptionofcontentssection. 1.1Background 1.1.1StateFeedbackPIDController ThePIDcontrol,whichistheabbreviationofproportionalintegralderivative control,isageneralcontrolmechanismusedintheindustry.UsuallyPIDcontroller calculatesthedifferencebetweenameasuredprocessvariableandadesiredsetpoint astheerror.Thedesignofacontrollerattemptstominimizetheerrorbyadjustingthe controllerparameters.ThePIDcontrollerinvolvesthreeterms:theproportionalterm, theintegraltermandthederivativeterm,andtheseareusuallydenotedasP,I,andD respectively.Theintegraltermisincorporatedtotrackerrorstatestoreachitstarget value,andobtainthetheoffset-freeperformance.. Intheabsenceofknowledgeoftheprocessdetails,awelldesignedPIDcontroller isthebestcontroller.APIDcontrolleriscalledaPIcontrollerwhenthederivativecontrol isswitchedoff.PIcontrollersarefairlycommoninthechemicalrelatedindustriessince derivativeactionissensitivetounpredictableandmeaninglessnoise. Thedisadvantageofthisapproachisthatananti-windupalgorithmisrequiredfor theintegraltermtopreventanunnecessarypenalty. 10

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Figure1-1.Atypicalblockdiagramofafeedbackcontroller. 1.1.2Stateobserverandestimator Astateobserverprovidesanestimateofthestateoftheplantbymeasuringthe inputandoutputwithnecessaryknowledgeofthesystem.Theobserverstructureis typicallyinvolvedwithcomputer-aidmathematicalproblems.Ifasystemisobservable, itispossibletofullyreconstructthesystemstatefromitsoutputmeasurementsusing thestateobserver.However,inmostpracticalcases,thephysicalstateofthesystem cannotbedeterminedbydirectobservation.Instead,theindirectmethodsareused. Figure1-2.Atypicalblockdiagramofasystemwiththeobserver. 1.1.3ModelPredictiveControllerMethod ModelPredictiveControlMPCisanoptimalcontrolmethodbasedononline numericaloptimization.Thecontrolinputsandplantoutputsarepredictedbyusinga estimatedsystemmodel,whichisoptimizedatregularintervalsaccordingtothecost 11

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functionandconstraints.TheMPCmethodoriginallyisacomputationaltechnique usedtoimprovecontrolperformanceinapplicationsinprocessindustries.Since then,predictivecontrolbecamethemostwidespreadadvancedcontrolmethodinthe industry.MPCcontrollercanworkinthelargescalemultivariablesystems,andprovide asystematicmethodofdealingwithconstraintsoninputsandstateswithsimpledesign andtuning. OnedisadvantageofMPCisthattheaugmentationalsofacesthewind-upproblem whensuddenchangeoccursinthereference.AlsotheexistingMPCmethodsonly considerthesteporconstantreferencesignalsforthezerooffsetperformance,whilein practiceitisoftendesirabletohaveainconstantreference. Figure1-3.AblockdiagramofMPCcontroller. 1.2DescriptionofContents ThethesismaterialhasevolvedatUniversityofFloridaoverthelastoneanda halfyears.Thethesisisdividedintofourpartstoexamineandeliminatetheoffset performanceforPIandMPCcontrolsystemwithconstantorinconstantreference signals. PartIprovidesanintroductiontothisthesis,includingthebriefdescriptionabout PIDcontrolmethod,stateobserverandestimatorandtheMPCcontrolmethod. 12

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PartIIChapter2isconcernedwiththeliteraturereview.TheconditionsofoffsetfreeperformanceinthesystemwithPIcontrollerandMPCcontrollerwiththeintegral actionsobtainedbypreviousresearchersareprovidedanddiscussed.Severaldisturbancemodelsusedbythepioneerstoeliminatetheinuenceofpersistentdisturbances andplantmodelmismatcharepresentedanddiscussed. PartIIIChapter3through5addressestheanalysisanddesignofPIandMPC controllersunderrequirementofzerooffsetperformanceforbothconstantandinconstantreferencesignal.Thefundamentalmathematicsmethodisusedtodiscussthe rankoftheintegralgain K 1 andtheexistenceofintegralterm z 1 ,andthedisturbance models.Alsosimulationsoftheplantsystemwithdifferentparametersareusedto checkthetheorydiscussedinthispart. PartIVChapter6isconcernedwithconclusionsandrecommendationsforfuture work. PartVaddressestheappendix,thereferenceandthethebiographicalsketchof thethesis.Theappendixiscomposedofthemathematicalmethodsnecessaryforthe thesisandtheunitconversiontablefortheplantsystemexample. 13

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CHAPTER2 LITERATUREREVIEW 2.1IntegralActioninPIControllers Integralcontrolshavelongbeenusedinchemicalengineeringindustriessince 1970stoachievethelongtermoffset-freeperformancewhentheprocessissubjected totheunmeasuredconstantdisturbance.Ogata consideredadiscretesquare systemwithstateequationandoutputequation, x k +1 = Ax k + Bu k y k = Cx k inwhich x k 2 R n istheplantstate, y k 2 R m isoutputormeasuredvariable, u k 2 R m isthecontrolormanipulatedvariable.Thematrices A 2 R n n B 2 R n m and C 2 R m n .Thepair A B isassumedtobecontrollableandthepair C A is assumedtobeobservable.Theintegratorstateequationisdenedas z k = z k )]TJ/F26 11.9552 Tf 11.955 0 Td [(1 + r k )]TJ/F44 11.9552 Tf 11.955 0 Td [(y k whereristhereferenceandthecontrollerdesignofthecontrolvector u k givenby u k = )]TJ/F44 11.9552 Tf 9.298 0 Td [(K 2 x k + K 1 z k Theaugmentedstatevectorischosenas 2 6 4 x k u k 3 7 5 Ifthereferenceisconstantorastepfunction, u 1 and x 1 existunderthefollowing conditions: 1.Thepair A B iscontrollableand 14

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2.Thematrix A )]TJ/F44 11.9552 Tf 11.955 0 Td [(IB C 0 isfullrank. Since u k isdesignedtoholdtherelationshipwith x k z k as u k = )]TJ/F44 11.9552 Tf 9.298 0 Td [(K 2 x k + K 1 z k Ifthe z 1 exists,forastepinput,theaugmentedstatevectorwouldapproachthe constantvectorwithvalues u 1 x 1 and z 1 respectively.Thefollowingequation atthesteadystateisobtained z 1 = z 1 + r )]TJ/F44 11.9552 Tf 11.956 0 Td [(y 1 Therefore,thereisnooffseterrorintheoutputwhentheinputisasteporconstant. 2.2Challenges Whenthecontroliswelldesignedas u k = )]TJ/F44 11.9552 Tf 9.298 0 Td [(K 2 x k + K 1 z k ,theterms u 1 and x 1 areconstants. u 1 = )]TJ/F44 11.9552 Tf 9.298 0 Td [(K 2 x 1 + lim k !1 K 1 z k whichmeanslim k !1 K 1 z k isaconstantvector.However,thisdoesn'tassurethe z 1 existsasaconstantvector. Thekeyproblemiswhetherthesquarematrix K 1 isfullrank.FromAckermann's formula, K = 00 0 I m HGH G n )]TJ/F27 7.9701 Tf 6.586 0 Td [(1 H )]TJ/F27 7.9701 Tf 6.587 0 Td [(1 G 15

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where K = K 2 K 1 and G )]TJ/F44 11.9552 Tf 11.955 0 Td [(HK isthecharacteristicequationof G )]TJ/F44 11.9552 Tf 11.955 0 Td [(HK ,and G = 2 6 4 A 0 )]TJ/F44 11.9552 Tf 9.298 0 Td [(CAI m 3 7 5 H = 2 6 4 I n )]TJ/F44 11.9552 Tf 9.298 0 Td [(C 3 7 5 B Since G )]TJ/F44 11.9552 Tf 12.841 0 Td [(HK satisesitsowncharacteristicequation, G )]TJ/F44 11.9552 Tf 11.955 0 Td [(HK =0 ,fromthe Cayley-Hamiltontheorem.Thus,theexpressionof K isobtainedas K = 00 0 I m HGH G n )]TJ/F27 7.9701 Tf 6.586 0 Td [(1 H )]TJ/F27 7.9701 Tf 6.587 0 Td [(1 G K 2 K 1 = K + 0 I m 2 6 4 A )]TJ/F44 11.9552 Tf 11.955 0 Td [(I n B CACB 3 7 5 )]TJ/F27 7.9701 Tf 6.587 0 Td [(1 hencefromthetwoequationsabove, K 1 = 00 0 I m HGH G n )]TJ/F27 7.9701 Tf 6.586 0 Td [(1 H )]TJ/F27 7.9701 Tf 6.587 0 Td [(1 G + 0 I m # 2 6 4 A )]TJ/F44 11.9552 Tf 11.955 0 Td [(I n B CACB 3 7 5 )]TJ/F27 7.9701 Tf 6.587 0 Td [(1 2 6 6 6 6 6 6 6 4 0 0 I m 3 7 7 7 7 7 7 7 5 Objectiveistoprovethat K 1 isfullrank.Asimulationwithrandominputsisperformed usingMatlabusing A asa 3 3 randommatrix, B asa 3 2 randommatrix, C asa 2 3 randommatrix.ThesethreematricessatisfytherelationshipsfromEquation2to Equation2.Thecodeoperatesfortenmillioncycles,nallyobtaining 130 examples amongwhichatleastoneoftheabsolutevalueofpole K 1 < 0.001 .Thisresultdoesn't showstrongconclusionwhetherthematrix K 1 isfullrankornot.Thecomplexityof G 16

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andthefactthat K 1 isasubmatrixofalargermatrixclearlyincreasesthedifcultyto provedirectly.AnotherindirectmethodisappliedinthisthesisintheSection3. 2.3OffsetPerformanceinMPCControlSystems Mostmodelpredictivecontrolmethodsobtainoffset-freecontrolperformance byaddingtheintegratingdisturbancetotheprocessmodel.Thepurposeofthese additionaldisturbancesistoeliminatethemismatchbetweenrealsystemandthemodel orunmeasureddisturbancesbythecorrespondingintegralaction.Ithasbeenprovedto beeffectiveforparticularconditions.Thisstrategyissimilartotheintegrationoftheerror inPIDcontorllers,whichalsobringssimilardisadvantages.Sincetheerrorintegration isindependentofthecontroller,thismethodmayleadtowindupinconstrainedsystem. Thus,anti-windupmechanismsarerequiredgenerally. Themethodsaimtoavoidthisproblembyemployingadisturbanceestimator approach.Thereby,thestateupdateequationsusedforthepredictionareaugmented bythereferenceanddisturbance.Anobserverisusedtoestimatethedisturbance states,andtheMPCisdesignedtorejecttheestimateddisturbanceandtrackthe reference,andsolvewindupproblem. Figure2-1.AblockdiagramofMPCcontroller. 17

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2.4Offset-freePerformanceConsideringInconstantSignals Theexistingmethodsessentiallyconsiderconstantdisturbanceandreference andhenceremoveoffsetatsteady-state.Formoregeneralsignals,suchasramps andsines,thesemethodswillfailtoremoveoffset.U.Maeder providesa generalizationofthedisturbanceestimationapproachtoarbitraryunstabledynamics. Thereferencesignalisgeneratedbyaautonomousdynamicsystem x r k +1 = A r x r k +1 r k = C r x r k where A r 2 R n r n r and C r 2 R n y n r ,andthematrix A r maybeunstable.Thesignalis generatedbymode withorder p ,ifthereexistsalinearsystemsuchthat s k = C s x s k x s k +1 = J p x s k k =0,1,... where J p isaJordanblockmatrixfor withorder p J = 2 6 6 6 6 6 6 6 6 6 6 4 10 0 0 1 0 00 . . 1 0000 3 7 7 7 7 7 7 7 7 7 7 5 Offset-freeperformanceisachievedundertheassumptionthattheobserverisstable andthefollowingdecompositionsexist, y = m X i =1 y i p i u = m X i =1 u i p i 18

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where i isthe i theigenvalueof A withthelongestJordanchainoflength p i .This crucialpointsofthemethodarethechoiceofdisturbancemodelsatisfyingtheinternal modelcondition,andtheadditionoftargettrajectoryconditionstotheMPCproblems. Thedisadvantagesofthemethodincludeincreaseofcomputationalcostofthe trajectoryoptimizationandverylimitedimprovementoftheoffsetperformancefor certainreferenceduetothefactthatthedecompositionsabovedon'tholdgenerally, whichisdiscussedinChapter6. 19

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CHAPTER3 ANALYSISANDDESIGNOFPICONTROLLERS 3.1SystemwithoutDisturbance 3.1.1DescriptionofSystemwithoutDisturbance Adiscrete-time,linear,timeinvariantmodelispresentedandassumedtobe completelystatecontrollable. x k +1 = Ax k + Bu k y k = Cx k inwhich x k 2 R n istheplantstate, y k 2 R m isoutputvector, u k 2 R m isthe controlvectorwith A 2 R n n B 2 R n m and C 2 R m n .Thepair A B isassumedto becontrollableandthepair C A isassumedtobeobservable.Theintegratorstate equationisdenedas z k = z k )]TJ/F26 11.9552 Tf 11.955 0 Td [(1 + r k )]TJ/F44 11.9552 Tf 11.955 0 Td [(y k wherethereference r k 2 R m andthecontrolisdesignedas u k = )]TJ/F44 11.9552 Tf 9.298 0 Td [(K 2 x k + K 1 z k Thusthestateandtheintegratorstateatthetime k +1 are z k +1 = z k + r k +1 )]TJ/F44 11.9552 Tf 11.956 0 Td [(y k +1 = z k + r k +1 )]TJ/F44 11.9552 Tf 11.956 0 Td [(C [ Ax k + Bu k ] = )]TJ/F44 11.9552 Tf 9.299 0 Td [(CAx k + z k )]TJ/F44 11.9552 Tf 11.955 0 Td [(CBu k + r k +1 x k +1 = Ax k + Bu k = Ax k + B [ )]TJ/F44 11.9552 Tf 9.298 0 Td [(K 2 x k + K 1 z k ] 20

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Thestatesequationsabovecanalsobewrittenas 2 6 4 x k +1 z k +1 3 7 5 = 2 6 4 A )]TJ/F44 11.9552 Tf 11.955 0 Td [(BK 2 BK 1 )]TJ/F44 11.9552 Tf 9.298 0 Td [(CA + CBK 2 I m )]TJ/F44 11.9552 Tf 11.955 0 Td [(CBK 1 3 7 5 2 6 4 x k z k 3 7 5 + 2 6 4 0 I m 3 7 5 r k +1 orinthefollowingformat 2 6 4 x k +1 z k +1 3 7 5 = G 2 6 4 x k z k 3 7 5 + Hw k + 2 6 4 0 I m 3 7 5 r where w k = )]TJ/F44 11.9552 Tf 9.298 0 Td [(K 2 6 4 x k z k 3 7 5 G = 2 6 4 A 0 )]TJ/F44 11.9552 Tf 9.299 0 Td [(CAI m 3 7 5 H = 2 6 4 I n )]TJ/F44 11.9552 Tf 9.299 0 Td [(C 3 7 5 B K = )]TJ/F35 11.9552 Tf 11.291 16.856 Td [( )]TJ/F44 11.9552 Tf 9.298 0 Td [(K 2 K 1 Thedynamicsofthesystemaredeterminedbytheeigenvaluesofthestatematrix,and thereforerelatedtocontrollabilityofthepair G H 3.1.2Controllability Considerthecontrollabilitymatrix HGHG 2 H G n + m )]TJ/F27 7.9701 Tf 6.586 0 Td [(1 H ThePBHtestrequires G )]TJ/F44 11.9552 Tf 11.955 0 Td [(sI H tobefullrowrankforall s ,and G )]TJ/F44 11.9552 Tf 11.955 0 Td [(sI H = 2 6 4 2 6 4 A 0 )]TJ/F44 11.9552 Tf 9.298 0 Td [(CAI m 3 7 5 )]TJ/F44 11.9552 Tf 11.955 0 Td [(sI m + n 2 6 4 B CB 3 7 5 3 7 5 21

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When s =1 ,thematrixbecomes 2 6 4 A )]TJ/F44 11.9552 Tf 11.955 0 Td [(IB C 0 3 7 5 andfullrowrankasassumed.When s 6 =1 rank 2 6 4 A )]TJ/F44 11.9552 Tf 11.955 0 Td [(sI n 0 B )]TJ/F44 11.9552 Tf 9.298 0 Td [(CA 1 )]TJ/F44 11.9552 Tf 11.955 0 Td [(s I m CB 3 7 5 = rank 2 6 4 A )]TJ/F44 11.9552 Tf 11.955 0 Td [(sI n 0 B 0 I m 0 3 7 5 = rank A )]TJ/F44 11.9552 Tf 11.956 0 Td [(sI n B + m sincetheknowledgeofthecontrollabilityofthepair A B ,theresultis m + n ,which indicatesthepair G H = 0 B @ 2 6 4 A 0 )]TJ/F44 11.9552 Tf 9.299 0 Td [(CAI m 3 7 5 2 6 4 I n )]TJ/F44 11.9552 Tf 9.299 0 Td [(C 3 7 5 B 1 C A iscontrollableaslongas Thepair A B iscontrollable Thematrix 2 6 4 A )]TJ/F44 11.9552 Tf 11.956 0 Td [(I n B C 0 3 7 5 isfullrank. 3.1.3OffsetPerformanceofSystem Thestateofthediscretesystemabovecanbewrittenas x k +1 = Nx k + Mr 22

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where N = 2 6 4 A )]TJ/F44 11.9552 Tf 11.955 0 Td [(BK 2 BK 1 )]TJ/F44 11.9552 Tf 9.298 0 Td [(CA + CBK 2 I m )]TJ/F44 11.9552 Tf 11.955 0 Td [(CBK 1 3 7 5 M = 2 6 4 0 I m 3 7 5 Atthetime k =1,2,..., n ,thesystemyields x k +1 = Nx k + Mr k =0 x 1 = Nx 0 + Mr k =1 x 2 = Nx 1 + Mr = N [ Nx 0 + Mr ] + Mr = N 2 x 0 + NMr + Mr k =2 x 3 = Nx 2 + Mr = N 3 x 0 + N 2 Mr + NMr + Mr Any kx k +1 = N k x 0 + P N k )]TJ/F45 7.9701 Tf 6.587 0 Td [(j )]TJ/F27 7.9701 Tf 6.587 0 Td [(1 Mr Alleigenvaluesof N aredesignedtobeinsidetheunitcircle,sothe 1 sttermofEquation 3with k !1 lim k !1 N k x 0 =0 the 2 ndtermofEquation3with k !1 lim k !1 X N k )]TJ/F27 7.9701 Tf 6.586 0 Td [(1 Mr = I )]TJ/F44 11.9552 Tf 11.955 0 Td [(N )]TJ/F27 7.9701 Tf 6.586 0 Td [(1 Mr 23

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thesteadyoutputwith k !1 y s = C 0 m x s = C 0 m I )]TJ/F44 11.9552 Tf 11.955 0 Td [(N )]TJ/F27 7.9701 Tf 6.587 0 Td [(1 Mr = C 0 m 0 B @ I )]TJ/F35 11.9552 Tf 11.955 27.617 Td [(2 6 4 A )]TJ/F44 11.9552 Tf 11.955 0 Td [(BK 2 BK 1 )]TJ/F44 11.9552 Tf 9.299 0 Td [(CA + CBK 2 I m )]TJ/F44 11.9552 Tf 11.955 0 Td [(CBK 1 3 7 5 1 C A )]TJ/F27 7.9701 Tf 6.587 0 Td [(1 2 6 4 0 I m 3 7 5 thus C 0 m 0 B @ I )]TJ/F35 11.9552 Tf 11.955 27.617 Td [(2 6 4 A )]TJ/F44 11.9552 Tf 11.955 0 Td [(BK 2 BK 1 )]TJ/F44 11.9552 Tf 9.298 0 Td [(CA + CBK 2 I m )]TJ/F44 11.9552 Tf 11.955 0 Td [(CBK 1 3 7 5 1 C A )]TJ/F27 7.9701 Tf 6.587 0 Td [(1 = Vy s where V A B C K 1 K 2 isamatrixfunctionof A B C K 1 K 2 .Then )]TJ/F44 11.9552 Tf 9.298 0 Td [(C 0 m = Vy s 2 6 4 A )]TJ/F44 11.9552 Tf 11.955 0 Td [(BK 2 )]TJ/F44 11.9552 Tf 11.955 0 Td [(I m BK 1 )]TJ/F44 11.9552 Tf 9.298 0 Td [(CA + CBK 2 )]TJ/F44 11.9552 Tf 9.298 0 Td [(CBK 1 3 7 5 = V )]TJ/F44 11.9552 Tf 11.955 0 Td [(y s C A )]TJ/F44 11.9552 Tf 11.955 0 Td [(BK 2 )]TJ/F44 11.9552 Tf 11.955 0 Td [(V V )]TJ/F44 11.9552 Tf 11.955 0 Td [(y s C BK 1 Since B isanarbitrarymatrix,and C isfullrowrank,thus V = C y s = r Hencethesystemworkswellwithzero-offsetperformance. 3.1.4RankoftheIntegralGain K 1 Noticethedesignoftheoutput u k = )]TJ/F44 11.9552 Tf 9.298 0 Td [(K 2 x k + K 1 z k 24

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Thecurrentproblemistocheckwhetheranon-full-rankintegral K 1 existswhenthe eigenvaluesofthematrix 2 6 4 A )]TJ/F44 11.9552 Tf 11.955 0 Td [(BK 2 BK 1 )]TJ/F44 11.9552 Tf 9.299 0 Td [(CA + CBK 2 I m )]TJ/F44 11.9552 Tf 11.955 0 Td [(CBK 1 3 7 5 canbearbitrarilysetbymatrices K 1 and K 2 .Thematrixcanbewrittenas 2 6 4 EF )]TJ/F44 11.9552 Tf 9.298 0 Td [(CEI )]TJ/F44 11.9552 Tf 11.955 0 Td [(CF 3 7 5 where E = A )]TJ/F44 11.9552 Tf 11.955 0 Td [(BK 2 F = BK 1 Andthematrices K 1 and K 2 aredesignedsothat 1 isnotoneeigenvalueofthematrix, thusthematrix I )]TJ/F35 11.9552 Tf 11.955 27.617 Td [(2 6 4 EF )]TJ/F44 11.9552 Tf 9.299 0 Td [(CEI )]TJ/F44 11.9552 Tf 11.955 0 Td [(CF 3 7 5 = 2 6 4 I )]TJ/F44 11.9552 Tf 11.955 0 Td [(E )]TJ/F44 11.9552 Tf 9.298 0 Td [(F CECF 3 7 5 isfullrank.Withrowoperations,thenthematrixisalsofullrank 2 6 4 I )]TJ/F44 11.9552 Tf 11.955 0 Td [(E )]TJ/F44 11.9552 Tf 9.299 0 Td [(F C 0 3 7 5 whichalsomeans C I )]TJ/F44 11.9552 Tf 11.955 0 Td [(E )]TJ/F27 7.9701 Tf 6.586 0 Td [(1 F isfullrank.Thus,thematrix F shouldbefullcolumnrank,andthusthematrix BK 1 is alsofullcolumnrank.Theintegralgain K 1 needstobefullcolumnrank. 25

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3.2SystemwithaConstantDisturbanceModel 3.2.1IntroductionoftheSystemwithaConstantDisturbanceModel Theplantmodelisaugmentedwithadisturbancemodelinordertocapturethea constantdisturbancemodel.Thedynamics,disturbancemodelsandoutputequation aregenerated: 8 > > > > > > < > > > > > > : x k +1 = Ax k + Bu k + B d d k d k +1 = d k = d y k = Cx k + C d d k where d k cannotbedirectlymeasuredgeneralinthepractice. 3.2.2ProblemDescription Fromtheintegralstateequation, z k = z k )]TJ/F26 11.9552 Tf 11.955 0 Td [(1 + r k )]TJ/F44 11.9552 Tf 11.955 0 Td [(y k = z k )]TJ/F26 11.9552 Tf 11.955 0 Td [(1 + r k )]TJ/F44 11.9552 Tf 11.955 0 Td [(Cx k )]TJ/F44 11.9552 Tf 11.955 0 Td [(C d d k usingthecontrollerstructure u k = )]TJ/F44 11.9552 Tf 9.298 0 Td [(K 2 x k + K 1 z k onecanobtaintheexpressionfortime k z k +1 = z k + r k +1 )]TJ/F44 11.9552 Tf 11.955 0 Td [(y k +1 = z k + r k +1 )]TJ/F44 11.9552 Tf 11.955 0 Td [(C [ Ax k + Bu k ] )]TJ/F44 11.9552 Tf 11.955 0 Td [(C d d k = )]TJ/F44 11.9552 Tf 9.298 0 Td [(CAx k + z k )]TJ/F44 11.9552 Tf 11.955 0 Td [(C d d k )]TJ/F44 11.9552 Tf 11.955 0 Td [(CBu k + r x k +1 = Ax k + Bu k + B d d k = Ax k + B [ )]TJ/F44 11.9552 Tf 9.298 0 Td [(K 2 x k + K 1 z k ] + B d d k 26

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Thenthesetwoparametersareusedtoobtainthestateequation 2 6 4 x k +1 z k +1 3 7 5 = 2 6 4 A )]TJ/F44 11.9552 Tf 11.955 0 Td [(BK 2 BK 1 )]TJ/F44 11.9552 Tf 9.298 0 Td [(CA + CBK 2 I m )]TJ/F44 11.9552 Tf 11.955 0 Td [(CBK 1 3 7 5 2 6 4 x k z k 3 7 5 + 2 6 4 0 B d I m C d 3 7 5 2 6 4 r d 3 7 5 3.2.3TheOffsetPerformanceoftheSystem FromEquation3,thestatevectorattime k +1 canbewrittenas x k +1 = Nx k + Mr where N = 2 6 4 A )]TJ/F44 11.9552 Tf 11.955 0 Td [(BK 2 BK 1 )]TJ/F44 11.9552 Tf 9.298 0 Td [(CA + CBK 2 I m )]TJ/F44 11.9552 Tf 11.955 0 Td [(CBK 1 3 7 5 Mr 0 = 2 6 4 0 B d I m C d 3 7 5 2 6 4 r d 3 7 5 r 0 = 2 6 4 r d 3 7 5 Atthetime k =1,2,..., n ,thesystemyields x k +1 = Nx k + Mr 0 k =0 x 1 = Nx 0 + Mr 0 k =1 x 2 = Nx 1 + Mr 0 = N [ Nx 0 + Mr ] + Mr 0 = N 2 x 0 + NMr 0 + Mr 0 k =2 x 3 = Nx 2 + Mr 0 = N 3 x 0 + N 2 Mr 0 + NMr 0 + Mr 0 Any kx k +1 = N k x 0 + P N k )]TJ/F45 7.9701 Tf 6.587 0 Td [(j )]TJ/F27 7.9701 Tf 6.587 0 Td [(1 Mr 0 27

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Forbrevity,wedonotrepeattheprocessinSubsection3.1.3here,oneoftheresultsof theprocessisthat y s = r 0 whichmeansthesystemalsoworksunderthezero-offsetperformance. 3.3Example 3.3.1Example1 Aexamplehereisusedtoshowtheoffset-freeperformancewith A = 2 6 6 6 6 4 010 001 )]TJ/F26 11.9552 Tf 9.299 0 Td [(0.12 )]TJ/F26 11.9552 Tf 9.298 0 Td [(0.011 3 7 7 7 7 5 B = 2 6 6 6 6 4 0 0 1 3 7 7 7 7 5 C = 0.510 B d = 2 6 6 6 6 4 0.1 )]TJ/F26 11.9552 Tf 9.299 0 Td [(0.1 0.05 3 7 7 7 7 5 and C d =0.01 withtheconstantdisturbance d = h )]TJ/F44 11.9552 Tf 11.955 0 Td [(h s =0.5 28

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Thecontrollabilitymatrixofthepair A B is ctrb A B = BAB A n )]TJ/F27 7.9701 Tf 6.587 0 Td [(1 B = 2 6 6 6 6 4 001 011 110.99 3 7 7 7 7 5 whichisfullrowrank. G = 2 6 4 A 0 )]TJ/F44 11.9552 Tf 9.298 0 Td [(CAI 3 7 5 = 2 6 6 6 6 6 6 6 4 0100 0010 )]TJ/F26 11.9552 Tf 9.298 0 Td [(0.12 )]TJ/F26 11.9552 Tf 9.299 0 Td [(0.0110 0 )]TJ/F26 11.9552 Tf 9.299 0 Td [(0.5 )]TJ/F26 11.9552 Tf 9.299 0 Td [(11 3 7 7 7 7 7 7 7 5 H = 2 6 4 I n )]TJ/F44 11.9552 Tf 9.298 0 Td [(C 3 7 5 B = 2 6 6 6 6 6 6 6 4 0 0 1 0 3 7 7 7 7 7 7 7 5 29

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Thecontrollabilitymatrixofthepair G H ctrb G H = GGH G n )]TJ/F27 7.9701 Tf 6.586 0 Td [(1 H = 2 6 6 6 6 6 6 6 4 0011 0110.99 110.990.86 0 )]TJ/F26 11.9552 Tf 9.299 0 Td [(1 )]TJ/F26 11.9552 Tf 9.298 0 Td [(2.5 )]TJ/F26 11.9552 Tf 9.298 0 Td [(3.99 3 7 7 7 7 7 7 7 5 thenusetheplacecommand place G H ,[ )]TJ/F26 11.9552 Tf 9.298 0 Td [(1 )]TJ/F26 11.9552 Tf 9.299 0 Td [(2 )]TJ/F26 11.9552 Tf 9.298 0 Td [(3 )]TJ/F26 11.9552 Tf 9.298 0 Td [(4 ] obtain K = )]TJ/F26 11.9552 Tf 9.299 0 Td [(24.12 )]TJ/F26 11.9552 Tf 9.298 0 Td [(34.0112 )]TJ/F26 11.9552 Tf 9.298 0 Td [(80 thus K 1 = )]TJ/F26 11.9552 Tf 9.298 0 Td [(80 K 2 = 24.1234.01 )]TJ/F26 11.9552 Tf 9.299 0 Td [(12 thematrixgives 2 6 4 A )]TJ/F44 11.9552 Tf 11.955 0 Td [(BK 2 BK 1 )]TJ/F44 11.9552 Tf 9.299 0 Td [(CA + CBK 2 I m )]TJ/F44 11.9552 Tf 11.955 0 Td [(CBK 1 3 7 5 eig K =[ )]TJ/F26 11.9552 Tf 9.299 0 Td [(1 )]TJ/F26 11.9552 Tf 9.298 0 Td [(2 )]TJ/F26 11.9552 Tf 9.299 0 Td [(3 )]TJ/F26 11.9552 Tf 9.298 0 Td [(4 ] Thustheoffset-freeperformanceisobtainedunderthisdesign. 30

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3.3.2Example2:CSTRPlantSystem 3.3.2.1Modelingequations Acontinuousstirred-tankreactorasshowninFigure3-1isconsideredinthisthesis. Anirreversible,rst-orderreaction, A B occursintheliquidphaseintheplant,andthereactortemperatureisregulatedwith externalcoolingwiththeknowledgethatthelevelisnotconstant.Massandenergy balancesleadtothefollowingnonlinearstate-spacemodel: Molarbalanceequation: dc dt = F 0 c 0 )]TJ/F44 11.9552 Tf 11.956 0 Td [(c r 2 h )]TJ/F44 11.9552 Tf 11.955 0 Td [(k 0 cexp )]TJ/F44 11.9552 Tf 15.189 8.088 Td [(E RT Energybalanceequation: dT dt = F 0 T 0 )]TJ/F44 11.9552 Tf 11.955 0 Td [(T r 2 h + )]TJ/F26 11.9552 Tf 9.299 0 Td [( H C p k 0 cexp )]TJ/F44 11.9552 Tf 15.189 8.088 Td [(E RT + 2 U r C p T c )]TJ/F44 11.9552 Tf 11.955 0 Td [(T Massbalanceequation: dh dt = F 0 )]TJ/F44 11.9552 Tf 11.955 0 Td [(F r 2 where h isthelevelofthetank, c arethemolarconcentration, T isthereactortemperature, F istheoutletowrate, T c isthecoolantliquidtemperature. 31

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Figure3-1.ThediagramofCSTRtanksystem. Intheplantcontrolsystemabove,thecontrolledvariablesarethelevelofthetank, h ,andthemolarconcentration c .Thethirdstatevariableisthereactortemperature, T ,whilethemanipulatedvariablesaretheoutletowrate, F ,andthecoolantliquid temperature, Tc .Moreover,theinletowrateisconsideredtoactasanunmeasured disturbance. 3.3.2.2Linearizationofthesystem Theopen-loopstablesteady-stateoperatingconditionsarethefollowing h s =0.659 m c s =0.877 mol = L T s =324.5 K F s =100 L = min T s c =300 K Usingasamplingperiodtimeof 1 min,alinearizeddiscretestate-spacemodelis developedintermsofthederivativestates,derivativeinputsandderivativeoutputs. 32

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x = 2 6 6 6 6 4 c )]TJ/F44 11.9552 Tf 11.955 0 Td [(c s T )]TJ/F44 11.9552 Tf 11.955 0 Td [(T s h )]TJ/F44 11.9552 Tf 11.955 0 Td [(h s 3 7 7 7 7 5 u = 2 6 4 T c )]TJ/F44 11.9552 Tf 11.956 0 Td [(T s c F )]TJ/F44 11.9552 Tf 11.955 0 Td [(F s 3 7 5 d = F 0 )]TJ/F44 11.9552 Tf 11.956 0 Td [(F s 0 y = 2 6 6 6 6 4 c )]TJ/F44 11.9552 Tf 11.955 0 Td [(c s T )]TJ/F44 11.9552 Tf 11.955 0 Td [(T s h )]TJ/F44 11.9552 Tf 11.955 0 Td [(h s 3 7 7 7 7 5 Table3-1.TheparametersoftheCSTR. ParameterNominalValue F 0 100 L = min T 0 350 K C 0 1 mol = L r 0.219 m k 0 7.2 10 10 min )]TJ/F27 7.9701 Tf 6.587 0 Td [(1 E = R 8,750 K U 915.6 W = m 2 K 1 kg = L C p 0.239 J = g K H )]TJ/F26 11.9552 Tf 9.298 0 Td [(5 10 4 J = mol ThelinearstatespaceequationfortheCSTRsystem: x k +1 = Ax k + Bu k + B d d k anditslinearoutputequation y k = Cx k + Du k inwhich 33

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A = 2 6 6 6 6 4 0.2511 )]TJ/F26 11.9552 Tf 9.299 0 Td [(3.368 10 )]TJ/F27 7.9701 Tf 6.587 0 Td [(3 )]TJ/F26 11.9552 Tf 9.298 0 Td [(7.056 10 )]TJ/F27 7.9701 Tf 6.587 0 Td [(4 11.060.3296 )]TJ/F26 11.9552 Tf 9.298 0 Td [(2.545 001 3 7 7 7 7 5 B = 2 6 6 6 6 4 )]TJ/F26 11.9552 Tf 9.298 0 Td [(5.425 10 )]TJ/F27 7.9701 Tf 6.587 0 Td [(3 1.530 10 )]TJ/F27 7.9701 Tf 6.587 0 Td [(5 1.2970.1218 0 )]TJ/F26 11.9552 Tf 9.298 0 Td [(6.592 10 )]TJ/F27 7.9701 Tf 6.586 0 Td [(2 3 7 7 7 7 5 C = 2 6 6 6 6 4 100 010 001 3 7 7 7 7 5 D = 2 6 6 6 6 4 000 000 000 3 7 7 7 7 5 where d istheunmeasureddisturbanceintheplantsystem.Theentriesofthematrix C varyindifferentexamplesinthisthesis,andallofthemarepresented. 3.3.2.3Offsetperformance Thecontrollabilitymatrixofthepair A B is ctrb A B = BAB A n )]TJ/F27 7.9701 Tf 6.587 0 Td [(1 B = 2 6 6 6 6 4 000 )]TJ/F26 11.9552 Tf 9.298 0 Td [(0.00620.800.002 )]TJ/F26 11.9552 Tf 9.299 0 Td [(1.1320 0.03500 )]TJ/F26 11.9552 Tf 9.298 0 Td [(0.0064.300 )]TJ/F26 11.9552 Tf 9.299 0 Td [(0.0600 0 )]TJ/F26 11.9552 Tf 9.298 0 Td [(6.640000000 3 7 7 7 7 5 34

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whichisfullrowrank.However,withthematrices G = 2 6 4 A 0 )]TJ/F44 11.9552 Tf 9.298 0 Td [(AI 3 7 5 = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 )]TJ/F26 11.9552 Tf 9.298 0 Td [(0.0191 )]TJ/F26 11.9552 Tf 9.298 0 Td [(0.1703 )]TJ/F26 11.9552 Tf 9.298 0 Td [(3.1330000 0.0005 )]TJ/F26 11.9552 Tf 9.298 0 Td [(0.0161 )]TJ/F26 11.9552 Tf 9.298 0 Td [(0.6495000 000000 0.01910.17033.1330100 )]TJ/F26 11.9552 Tf 9.298 0 Td [(0.00050.01610.6495010 000001 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 H = 2 6 4 I n )]TJ/F44 11.9552 Tf 9.298 0 Td [(C 3 7 5 B = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 000 0.035000 0 )]TJ/F26 11.9552 Tf 9.298 0 Td [(6.63700 000 )]TJ/F26 11.9552 Tf 9.298 0 Td [(0.035000 06.63700 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 35

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Thecontrollabilitymatrixofthepair G H ctrb G H = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 00 )]TJ/F26 11.9552 Tf 9.298 0 Td [(62021 )]TJ/F26 11.9552 Tf 9.299 0 Td [(113 )]TJ/F26 11.9552 Tf 9.299 0 Td [(6161 )]TJ/F26 11.9552 Tf 9.299 0 Td [(218721839 )]TJ/F26 11.9552 Tf 9.298 0 Td [(4389 40 )]TJ/F26 11.9552 Tf 9.298 0 Td [(643 )]TJ/F26 11.9552 Tf 9.299 0 Td [(2032135 )]TJ/F26 11.9552 Tf 9.299 0 Td [(605 )]TJ/F26 11.9552 Tf 9.299 0 Td [(2481767 )]TJ/F26 11.9552 Tf 9.298 0 Td [(668673 0 )]TJ/F26 11.9552 Tf 9.299 0 Td [(70000000000 006 )]TJ/F26 11.9552 Tf 9.299 0 Td [(211592 )]TJ/F26 11.9552 Tf 9.299 0 Td [(9 )]TJ/F26 11.9552 Tf 9.299 0 Td [(69209 )]TJ/F26 11.9552 Tf 9.298 0 Td [(791 )]TJ/F26 11.9552 Tf 9.298 0 Td [(6303598 )]TJ/F26 11.9552 Tf 9.298 0 Td [(4024322 )]TJ/F26 11.9552 Tf 9.298 0 Td [(75 )]TJ/F26 11.9552 Tf 9.298 0 Td [(113530125 )]TJ/F26 11.9552 Tf 9.298 0 Td [(1237803 )]TJ/F26 11.9552 Tf 9.298 0 Td [(1911 070707070707 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 isnotfullrowrank,whichindicatesthatthesystemperformancehasoffset. 36

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CHAPTER4 ANALYSISANDDESIGNOFTHEOBSERVER Considerthediscrete-timetime-invariantsystem x k +1 = f x k u k y k = g x k inwhich x k 2 R n istheplantstate, y k 2 R m isoutputvectorand u k 2 R m isthe controlvector. ALTIsystemmodelisapplied x k +1 = Ax k + Bu k y k = Cx k where A 2 R n n B 2 R n m and C 2 R m n ,assumedthatthepair A B iscontrollable, andthepair C A isobservable. Themodelisaugmentedwithadisturbancemodeltocapturethemismatch betweenEquation4andEquation4.Acommonconstantdisturbancemodelis used x k +1 = Ax k + Bu k + B d d k d k +1 = d k y k = Cx k + C d d k where d k 2 R n d 37

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Thestatedisturbanceestimatorisdesignedbasedontheaugmentedmodelas follows: 2 6 4 ^ x k +1 ^ d k +1 3 7 5 = 2 6 4 AB d 0 I 3 7 5 2 6 4 ^ x k ^ d t 3 7 5 + 2 6 4 B 0 3 7 5 u k + 2 6 4 L x L d 3 7 5 )]TJ/F44 11.9552 Tf 9.298 0 Td [(y k + C ^ x k + C d ^ d k inmostpapers,theywouldrequirethegain L x and L d chosentomaketheestimator stable.Theproperties L x and L d isdiscussedinthischapter. 4.1StabilityofRegulationProblem Thestateestimationerrorvectorsdenedtostudythestabilityoftheregulation problemofthesystem, ~ x k = x k )]TJ/F26 11.9552 Tf 12.139 0 Td [(^ x k ~ d k = d k )]TJ/F26 11.9552 Tf 12.48 2.657 Td [(^ d k therefore,subtracttheequationEquation4fromtheequationEquation4, 2 6 4 ~ x k +1 ~ d k +1 3 7 5 = 2 6 4 AB d 0 I 3 7 5 2 6 4 ~ x k ~ d t 3 7 5 )]TJ/F35 11.9552 Tf 11.955 27.617 Td [(2 6 4 L x L d 3 7 5 C ~ x k + C d ~ d k = 0 B @ 2 6 4 AB d 0 I 3 7 5 )]TJ/F35 11.9552 Tf 11.955 27.616 Td [(2 6 4 L x L d 3 7 5 CC d 1 C A 2 6 4 ~ x k ~ d k 3 7 5 theperformanceoftheoffsetcompletelydependsonthedesignof 2 6 4 L x L d 3 7 5 38

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ThetransposeofthestatematrixinequationEquation4showsas 0 B @ 2 6 4 AB d 0 I 3 7 5 )]TJ/F35 11.9552 Tf 11.955 27.617 Td [(2 6 4 L x L d 3 7 5 CC d 1 C A T = 2 6 4 AB d 0 I 3 7 5 T )]TJ/F35 11.9552 Tf 11.955 27.617 Td [(0 B @ 2 6 4 L x L d 3 7 5 CC d 1 C A T = 2 6 4 A T 0 B T d I 3 7 5 )]TJ/F35 11.9552 Tf 11.955 27.616 Td [(2 6 4 C T C T d 3 7 5 L T x L T d Theexistanceoftheanswertothestabilityquestionisdeterminatedbythecontrollability ofthepair 0 B @ 2 6 4 A T 0 B T d I 3 7 5 2 6 4 C T C T d 3 7 5 1 C A AsimpleanddirectmethodtoconsideristhePBHtestagain 2 6 4 A T )]TJ/F44 11.9552 Tf 11.956 0 Td [(sI 0 C T B T d 1 )]TJ/F44 11.9552 Tf 11.955 0 Td [(s IC T d 3 7 5 undertwosituations, s =1 sothatthematrix 2 6 4 A T )]TJ/F44 11.9552 Tf 11.955 0 Td [(IC T B T d C T d 3 7 5 needtobefullrowrank,whichindicatesthatthetransposeform 2 6 4 A )]TJ/F44 11.9552 Tf 11.955 0 Td [(IB d CC d 3 7 5 isfullcolumnrank. 39

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s 6 =1 sothatthematrix rank 2 6 4 A T )]TJ/F44 11.9552 Tf 11.955 0 Td [(sI 0 C T B T d 1 )]TJ/F44 11.9552 Tf 11.955 0 Td [(s IC T d 3 7 5 = rank A T )]TJ/F44 11.9552 Tf 11.955 0 Td [(sIC T + m isfullrowrankaslongastheoriginalsystemisobservable. Thus,thezerooffsetmaybeachievedbytheproperdesignofthepair 2 6 4 L x L d 3 7 5 4.2Example TheplantcontrolsystemdiscussedinSection3.3.2 A = 2 6 6 6 6 4 )]TJ/F26 11.9552 Tf 9.298 0 Td [(1.912 10 )]TJ/F27 7.9701 Tf 6.587 0 Td [(2 )]TJ/F26 11.9552 Tf 9.299 0 Td [(1.703 10 )]TJ/F27 7.9701 Tf 6.586 0 Td [(1 )]TJ/F26 11.9552 Tf 9.298 0 Td [(3.133 4.889 10 )]TJ/F27 7.9701 Tf 6.587 0 Td [(4 )]TJ/F26 11.9552 Tf 9.299 0 Td [(1.614 10 )]TJ/F27 7.9701 Tf 6.586 0 Td [(2 )]TJ/F26 11.9552 Tf 9.299 0 Td [(6.495 10 )]TJ/F27 7.9701 Tf 6.587 0 Td [(1 000 3 7 7 7 7 5 B = 2 6 6 6 6 4 000 3.499 10 )]TJ/F27 7.9701 Tf 6.587 0 Td [(2 00 0 )]TJ/F26 11.9552 Tf 9.298 0 Td [(6.6370 3 7 7 7 7 5 B d = 2 6 6 6 6 4 1239 256.8 6.637 3 7 7 7 7 5 C = eye 3 C d = zeros 40

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Thecontrollabilitymatrixis 2 6 4 C T A T C T )]TJ/F44 11.9552 Tf 5.48 -9.683 Td [(A T n + n d )]TJ/F27 7.9701 Tf 6.587 0 Td [(1 C T C T d B T d A + I C T + C T d B T d P n + n d )]TJ/F27 7.9701 Tf 6.587 0 Td [(1 i =0 A i C T + C T d 3 7 5 = 2 6 4 C T A T C T )]TJ/F44 11.9552 Tf 5.48 -9.684 Td [(A T n + n d )]TJ/F27 7.9701 Tf 6.587 0 Td [(1 C T 0 B T d A + I C T B T d P n + n d )]TJ/F27 7.9701 Tf 6.587 0 Td [(1 i =0 A i C T 3 7 5 = 2 6 4 C T A T C T )]TJ/F44 11.9552 Tf 5.479 -9.684 Td [(A T n + n d )]TJ/F27 7.9701 Tf 6.586 0 Td [(1 C T 0 A + I B T d C T P n + n d )]TJ/F27 7.9701 Tf 6.586 0 Td [(1 i =0 A i B T d C T 3 7 5 = 2 6 6 6 6 6 6 6 4 1000 010 )]TJ/F26 11.9552 Tf 9.299 0 Td [(0.2 001 )]TJ/F26 11.9552 Tf 9.298 0 Td [(0.31 0001151 WithSomething 3 7 7 7 7 7 7 7 5 whichisfullrowrank.Thusanappropriateestimatorcanbedesignedforthissystemto obtainzero-offsetperformance. G = 2 6 4 A T 0 B T d I 3 7 5 = 2 6 6 6 6 6 6 6 4 0000 0.2000 )]TJ/F26 11.9552 Tf 9.298 0 Td [(0.31 )]TJ/F26 11.9552 Tf 9.298 0 Td [(0.600 12392576.61 3 7 7 7 7 7 7 7 5 41

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H = 2 6 4 C T C T d 3 7 5 = 2 6 6 6 6 6 6 6 4 100 010 001 000 3 7 7 7 7 7 7 7 5 thenusetheplacecommand L = place G H ,[ )]TJ/F26 11.9552 Tf 9.298 0 Td [(1 )]TJ/F26 11.9552 Tf 9.299 0 Td [(2 )]TJ/F26 11.9552 Tf 9.298 0 Td [(3 )]TJ/F26 11.9552 Tf 9.299 0 Td [(4 ] obtainthetheestimatorset L = 2 6 6 6 6 4 )]TJ/F26 11.9552 Tf 9.299 0 Td [(99.1531 )]TJ/F26 11.9552 Tf 9.298 0 Td [(21.2744 )]TJ/F26 11.9552 Tf 9.298 0 Td [(0.5367 )]TJ/F26 11.9552 Tf 9.299 0 Td [(0.2957 509.3341109.10232.73101.4792 )]TJ/F26 11.9552 Tf 9.299 0 Td [(0.2301 )]TJ/F26 11.9552 Tf 9.298 0 Td [(0.05081.01560.0083 3 7 7 7 7 5 whichmeetstherequirementofoffset-freeperformancewiththepair L x = 2 6 6 6 6 4 )]TJ/F26 11.9552 Tf 9.299 0 Td [(99.1531 )]TJ/F26 11.9552 Tf 9.298 0 Td [(21.2744 509.3341109.1023 )]TJ/F26 11.9552 Tf 9.298 0 Td [(0.2301 )]TJ/F26 11.9552 Tf 9.298 0 Td [(0.0508 3 7 7 7 7 5 L d = 2 6 6 6 6 4 )]TJ/F26 11.9552 Tf 9.299 0 Td [(0.5367 )]TJ/F26 11.9552 Tf 9.299 0 Td [(0.2957 2.73101.4792 1.01560.0083 3 7 7 7 7 5 42

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CHAPTER5 ANALYSISANDDESIGNOFTHEMPCCONTROLLER MostmethodsinvolvingMPCtoachievethezero-offsetperformancearecomparabletothePIDmethodofintegrationoftheerror.Thetrackingerrorisfedintoaservo compensatorofthecontroller,whichcontainsamodelofthedisturbanceandreference dynamics.Theexistingmethodsaredesignedtorejecttheconstantdisturbanceand references. 5.1Offest-freeMPCDesignConditionsforConstantReference Consideranitediscrete-time,linear,timeinvariantsystemdescribedasfollows: x k +1 = Ax k + Bu k + B d d k + w k y k = Cx k + C d d k + v k inwhich x k 2 R n istheplantstate, y k 2 R m isoutputvector,and u k 2 R m isthe controlvector,and A 2 R n n B 2 R n m C 2 R m n B d 2 R n p ,and C d 2 R m p are matrices. Letthesystembecontrolledbyanobserver-basedcontrollerwithstatefeedback 2 6 4 ^ x k +1 ^ d k +1 3 7 5 = 2 6 4 AB d 0 I 3 7 5 2 6 4 ^ x k ^ d k 3 7 5 + 2 6 4 B 0 3 7 5 ^ u k + 2 6 4 L x L d 3 7 5 y k )]TJ/F44 11.9552 Tf 11.955 0 Td [(C ^ x k )]TJ/F44 11.9552 Tf 11.955 0 Td [(C d ^ d k wherethegains L x and L d areselectedtoensurethestabilityoftheestimator.Both estimated L x and L d andestimated B d and C d inuencethepoles,andthusaffectthe trackingperformance.Therelationship ^ d k +1 = ^ d k + L d y k )]TJ/F44 11.9552 Tf 11.955 0 Td [(C ^ x k )]TJ/F44 11.9552 Tf 11.955 0 Td [(C d ^ d k iscalledoutputdisturbancemodel. 43

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Theerrorvectorsaredenedas ~ x k = x k )]TJ/F26 11.9552 Tf 12.14 0 Td [(^ x k ~ d k = d k )]TJ/F26 11.9552 Tf 12.481 2.657 Td [(^ d k ~ u k = u k )]TJ/F26 11.9552 Tf 12.198 0 Td [(^ u k WhentheequationEquation5issubtratedfromtheequationEquation5,one obtainstheerrordynamicsoftheform 2 6 4 ~ x k +1 ~ d k +1 3 7 5 = 2 6 4 A )]TJ/F44 11.9552 Tf 11.955 0 Td [(BKB d )]TJ/F44 11.9552 Tf 9.298 0 Td [(CA + CBKI m )]TJ/F44 11.9552 Tf 11.955 0 Td [(CB d 3 7 5 2 6 4 ~ x k ~ d k 3 7 5 )]TJ/F35 11.9552 Tf 11.955 27.616 Td [(2 6 4 L x L d 3 7 5 C ~ x k + C d ~ d k = 0 B @ 2 6 4 A )]TJ/F44 11.9552 Tf 11.955 0 Td [(BKB d )]TJ/F44 11.9552 Tf 9.298 0 Td [(CA + CBKI m )]TJ/F44 11.9552 Tf 11.956 0 Td [(CB d 3 7 5 )]TJ/F35 11.9552 Tf 11.955 27.616 Td [(2 6 4 L x L d 3 7 5 CC d 1 C A 2 6 4 ~ x k ~ d k 3 7 5 = 0 B @ 2 6 4 A 0 )]TJ/F44 11.9552 Tf 9.298 0 Td [(CAI m 3 7 5 )]TJ/F35 11.9552 Tf 11.955 27.616 Td [(2 6 4 I n )]TJ/F44 11.9552 Tf 9.299 0 Td [(C 3 7 5 )]TJ/F44 11.9552 Tf 9.298 0 Td [(BKB d )]TJ/F35 11.9552 Tf 11.955 27.616 Td [(2 6 4 L x L d 3 7 5 CC d 1 C A 2 6 4 ~ x k ~ d k 3 7 5 ttheoffsetperformancedependsonthedesignof 2 6 4 L x L d 3 7 5 and )]TJ/F44 11.9552 Tf 9.299 0 Td [(BKB d Forconvenience,thefollowingnotationisintroduced ~ A = 2 6 4 AB d 0 I 3 7 5 ~ B = 2 6 4 B 0 3 7 5 ~ C = CC d ~ L = 2 6 4 L x L d 3 7 5 44

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Thecontrolgain K ischosenasthesolutionofthefollowingunconstrainedinnite horizonquadraticoptimizationproblem min = X y T k Qy T k + u T k Ru k Theestimatorparameter ~ L isdesignedtokeepthesystemstable,andthefollowing formulaensuresthenon-biasforwhitenoisedisturbance ~ L = ~ A ~ C T ~ C ~ C T )]TJ/F27 7.9701 Tf 6.586 0 Td [(1 wherethepredictorgainmatrix isthesymmetricpositivesemi-denitematrixsolution ofthediscretealgebraicRiccatiequation, = ~ A ~ A T )]TJ/F26 11.9552 Tf 12.852 2.657 Td [(~ A ~ C T ~ C ~ C T )]TJ/F27 7.9701 Tf 6.587 0 Td [(1 ~ C ~ A T Sincetheestimatormatrix ~ A )]TJ/F26 11.9552 Tf 12.271 2.657 Td [(~ L ~ C isstableandusingtheconditionsinSubsection3.1.2 andSection4.1,thestateandthedisturbanceestimatesaretoreachsteadyvaluesfor time k !1 .Vectors x s and y s areusedtodenotesteady-statevaluesofthemodel stateandoutput,respectively. Thesteady-stateoutputerrorisdenedas e s = y s )]TJ/F44 11.9552 Tf 11.955 0 Td [(Cx s )]TJ/F44 11.9552 Tf 11.955 0 Td [(C d d s fromtheequationsEquation5at k !1 x s = Ax s + Bu s + B d d s + L x e s Noticethattheexistenceof e s = e 1 and d s = d 1 45

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followsthediscussioninSubsection3.1.2andtheymustmeetthesamerequirementin Subsection3.1.2,inthiscase B d needstobefullrank.Thenwiththeoutputequationatsteadystate, 0= L d e s combiningtheequations,obtain x s )]TJ/F44 11.9552 Tf 11.955 0 Td [(x = I )]TJ/F44 11.9552 Tf 11.955 0 Td [(A )]TJ/F44 11.9552 Tf 11.955 0 Td [(BK )]TJ/F27 7.9701 Tf 6.587 0 Td [(1 L x e s inwhich I )]TJ/F44 11.9552 Tf 11.955 0 Td [(A )]TJ/F44 11.9552 Tf 11.955 0 Td [(BK )]TJ/F27 7.9701 Tf 6.587 0 Td [(1 existssince A + BK isastrictlystablematrix.Thentheoffset Hy s )]TJ/F44 11.9552 Tf 11.955 0 Td [(r = H e s + Cx s + C d d s )]TJ/F44 11.9552 Tf 11.955 0 Td [(Cx s )]TJ/F44 11.9552 Tf 11.955 0 Td [(C d d s = H I + C I )]TJ/F44 11.9552 Tf 11.955 0 Td [(A )]TJ/F44 11.9552 Tf 11.955 0 Td [(BK )]TJ/F27 7.9701 Tf 6.586 0 Td [(1 L x e s followsthediscussioninSection4.1anditmustmeetthesamerequirement 2 6 4 A )]TJ/F44 11.9552 Tf 11.955 0 Td [(IB d CC d 3 7 5 Soifthedesignofthecontrollersatisestheconditions,andalsohas null L d j null \002 I + C I )]TJ/F44 11.9552 Tf 11.955 0 Td [(A )]TJ/F44 11.9552 Tf 11.955 0 Td [(BK )]TJ/F27 7.9701 Tf 6.586 0 Td [(1 L x thereiszero-offsetinthecontrolledvariable. 5.2TheOffsetintheMPCWhenTrackingNonconstantReference 5.2.1Zero-offsetinOutputwithConstantReference Theexistingmethodsaredesignedtorejecttheconstantdisturbanceandreferences.Formoregeneralsignals,suchasrampsandsines,thesemethodswillfailto 46

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removetheoffset.Bydening x t k = x k y t k = y k + r where x t k and y t k aretheabsolutevalueofthestateandoutputofthesystem. Now,theproblemisturnedfromtrackingproblemtoregulationproblem. Considerthediscrete-timetime-invariantsystemforregulationproblem, x k +1 = f x k u k y k = g x k inwhich x k 2 R n istheplantstate, y k 2 R m isoutputvectorand u k 2 R m isthe controlvector. ALTIsystemmodelisapplied, x k +1 = Ax k + Bu k y k = Cx k where A 2 R n n B 2 R n m and C 2 R m n ,assumethatthepair A B iscontrollable, andthepair C A isobservable. Themodelisaugmentedwithadisturbancemodeltocapturethemismatch betweenEquation5andEquation5.Acommonconstantdisturbancemodelis used x k +1 = Ax k + Bu k + B d d k d k +1 = d k where d k 2 R n d 47

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Thestatedisturbanceestimatorisdesignedbasedontheaugmentedmodelas follows: 2 6 4 ^ x k +1 ^ d k +1 3 7 5 = 2 6 4 AB d 0 I 3 7 5 2 6 4 ^ x k ^ d t 3 7 5 + 2 6 4 B 0 3 7 5 u k + 2 6 4 L x L d 3 7 5 )]TJ/F44 11.9552 Tf 9.299 0 Td [(y k + C ^ x k + C d ^ d k y k = Cx k + C d ^ d k inmostpapers,theywouldrequirethegains L x and L d chosentomaintainstabilityof theestimator. Theerrorstatesaredenedas ~ x k = x k )]TJ/F26 11.9552 Tf 12.14 0 Td [(^ x k ~ d k = d k )]TJ/F26 11.9552 Tf 12.481 2.656 Td [(^ d k ~ u k = u k )]TJ/F26 11.9552 Tf 12.199 0 Td [(^ u k ThedifferencebetweenequationEquation5andequationEquation55is 2 6 4 ~ x k +1 ~ d k +1 3 7 5 = 2 6 4 A )]TJ/F44 11.9552 Tf 11.955 0 Td [(BKB d 0 I 3 7 5 2 6 4 ~ x k ~ d k 3 7 5 )]TJ/F35 11.9552 Tf 11.955 27.617 Td [(2 6 4 L x L d 3 7 5 C ~ x k = 2 6 4 A )]TJ/F44 11.9552 Tf 11.955 0 Td [(BK )]TJ/F44 11.9552 Tf 11.955 0 Td [(L x CB d L d CI 3 7 5 2 6 4 ~ x k ~ d k 3 7 5 48

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Andtheoffsetstates y k +1 = Cx k +1 + C d ^ d k +1 = C ~ x k +1 + C ^ x k +1 + C d ^ d k +1 = C ~ x k +1 + CC d 0 B @ 2 6 4 A )]TJ/F44 11.9552 Tf 11.956 0 Td [(BKB d 0 I 3 7 5 2 6 4 ^ x k ^ d k 3 7 5 )]TJ/F35 11.9552 Tf 11.955 27.616 Td [(2 6 4 L x L d 3 7 5 C ~ x k 1 C A = C 0 2 6 4 A )]TJ/F44 11.9552 Tf 11.955 0 Td [(BK + L x CB d L d CI 3 7 5 2 6 4 ~ x k ~ d k 3 7 5 + )]TJ/F35 11.9552 Tf 11.291 16.857 Td [( CC d 2 6 4 L x L d 3 7 5 C ~ x k + CC d 2 6 4 A )]TJ/F44 11.9552 Tf 11.955 0 Td [(BKB d 0 I 3 7 5 2 6 4 ^ x k ^ d k 3 7 5 = CA )]TJ/F44 11.9552 Tf 11.955 0 Td [(CBK )]TJ/F44 11.9552 Tf 11.955 0 Td [(C d L d CCB d 2 6 4 ~ x k ~ d k 3 7 5 + CA )]TJ/F44 11.9552 Tf 11.955 0 Td [(CBKCB d + C d 2 6 4 ^ x k ^ d k 3 7 5 Theoffset-freeperformanceisobtainedwhenthesystemistrackingaconstantreferenceandsatisestheconditions, 2 6 4 A )]TJ/F44 11.9552 Tf 11.955 0 Td [(IB d CC d 3 7 5 and null L d j null \002 I + C I )]TJ/F44 11.9552 Tf 11.955 0 Td [(A )]TJ/F44 11.9552 Tf 11.956 0 Td [(BK )]TJ/F27 7.9701 Tf 6.586 0 Td [(1 L x 49

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Theterms lim k !1 ~ x k =0 lim k !1 ~ d k =0 lim k !1 ^ x k =0 lim k !1 ^ d k =0 5.2.2CauseoftheOffsetwithNonconstantReference Ifthecontrolleristrackinganonconstantreference, r = r k TheMPCcontrollersforthetrackingproblemsareappliedseparatelyforreferenceat eachtime, Thediscrete-timetime-invariantsystem 0 x k +1 = f x k u k y k = g x k withreference r 0 k =0 k 0 r 0 k = r 0 k > 0 Thenalsteadystateswithzero-offsetperformanceisobtainedas y s 0 and x s 0 Thediscrete-timetime-invariantsystem 1 x k +1 = f x k u k y k = g x k 50

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withreference r 1 k =0 k 1 r 1 k = r 1 )]TJ/F44 11.9552 Tf 11.955 0 Td [(r 0 k > 1 Thenalsteadystateswithzero-offsetperformanceisobtainedas y s 1 and x s 1 ... Thediscrete-timetime-invariantsystem i x k +1 = f x k u k y k = g x k withreference r i k =0 k i r i k = r i )]TJ/F44 11.9552 Tf 11.955 0 Td [(r i )]TJ/F26 11.9552 Tf 11.956 0 Td [(1 k > i Thenalsteadystateswithzero-offsetperformanceisobtainedas y si and x si ... Withconstantreferences r = r k ,andtheequilibriumconditions, y e k = k X i =0 y si = r k x e k = x sk Thedenitionofthestates x k isthedifferencebetweentheabsolutevalueofthe statesandthelocalsteadystates x e k = x sk .Inaddition,thedenitionoftheoutput states y k isthedifferencebetweentheabsolutevalueoftheoutputstatesandthe localsteadystates y e k = P k i =0 y si = r k 51

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Thuswhentrackinganonconstantreference,theequationEquation5ischanged to y k +1 = CA )]TJ/F44 11.9552 Tf 11.955 0 Td [(CBK )]TJ/F44 11.9552 Tf 11.955 0 Td [(C d L d CCB d 2 6 4 ~ x k +~ x s k +1 )]TJ/F26 11.9552 Tf 12.139 0 Td [(~ x sk ~ d k + ~ d s k +1 )]TJ/F26 11.9552 Tf 12.48 2.656 Td [(~ d sk 3 7 5 + CA )]TJ/F44 11.9552 Tf 11.955 0 Td [(CBKCB d + C d 2 6 4 ^ x k +^ x s k +1 )]TJ/F26 11.9552 Tf 12.139 0 Td [(^ x sk ^ d k + ^ d s k +1 )]TJ/F26 11.9552 Tf 12.481 2.657 Td [(^ d sk 3 7 5 + r k +1 )]TJ/F44 11.9552 Tf 11.955 0 Td [(r k ThedifferencebetweentheequationEquation5andtheequationEquation5,the changeinthenalsteadystates,causesthefailuretorejectingtheoutputoffset.The errorwouldbuildupforsomenonconstantreference,suchasrampswithhighslope. 5.2.3NewControlStructureandMethodSummary NoticeintheequationEquation5.Sincetheoutputstatesatthesteadystates, y s = C ^ x sk + C d ^ d sk =0 thesteadystatesandsteadydisturbancesarelineardependent.Moreover,since y sk = r k ^ x sk islineardependentonthe r k .Thusifthecontrol u hastheformlike u k = )]TJ/F44 11.9552 Tf 9.298 0 Td [(K 1 x k )]TJ/F44 11.9552 Tf 11.955 0 Td [(K 2 [ r k +1 )]TJ/F44 11.9552 Tf 11.955 0 Td [(r k ] theoutputresponseisexpectedtohaveasignicantlybetteroffsetperformance. Thefollowingmethodisusedtooffset-freetrackingunderthenonconstantreference tracking. TheSubsection5.2.3.1andSubsection5.2.3.2brieysummarizethemainstepsof theprocedureproposedinthisthesis.TwotypesofrevisedMPCmethodsareused. 52

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5.2.3.1Method1 1.Computetheestimatorgains L x L d 2.UseMPCmethod min N )]TJ/F27 7.9701 Tf 6.586 0 Td [(1 X k =0 y k )]TJ/F44 11.9552 Tf 11.955 0 Td [(r k T Q y k )]TJ/F44 11.9552 Tf 11.955 0 Td [(r k + u T k Ru k + y N )]TJ/F44 11.9552 Tf 11.956 0 Td [(r N T P y N )]TJ/F44 11.9552 Tf 11.955 0 Td [(r N s.t.Constraintshold. wherethematrices Q R P S areweightmatricestocalculatetheoriginalcontrol u 0 k = )]TJ/F44 11.9552 Tf 9.298 0 Td [(K 1 x 3.Optimizethethegain K 2 undertheconstraints. 4.Therevisedcontrol1isused u k = )]TJ/F44 11.9552 Tf 9.299 0 Td [(K 1 x k )]TJ/F44 11.9552 Tf 11.955 0 Td [(K 2 [ r k +1 )]TJ/F44 11.9552 Tf 11.955 0 Td [(r k ] 5.2.3.2Method2 1.Computertheestimatorgains L x L d 2.TherevisedMPCproblemisgivenby min N )]TJ/F27 7.9701 Tf 6.587 0 Td [(1 X k =0 y k )]TJ/F44 11.9552 Tf 11.955 0 Td [(r k T Q y k )]TJ/F44 11.9552 Tf 11.955 0 Td [(r k + u T k Ru k + y N )]TJ/F44 11.9552 Tf 11.955 0 Td [(r N T P y N )]TJ/F44 11.9552 Tf 11.955 0 Td [(r N + u T k S r N )]TJ/F44 11.9552 Tf 11.955 0 Td [(r N )]TJ/F26 11.9552 Tf 11.955 0 Td [(1 s.t.Constraintshold. wherethematrices Q R P S areweightmatrices. 3.Therevisedcontrol2isobtained 5.3ExampleandComparison 5.3.1RampReference Forrampreference,thedifferenceinreference r k +1 )]TJ/F44 11.9552 Tf 11.955 0 Td [(r k = 4 53

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where 4 isaconstant.FromtheequationsEquation5andEquation5,thetracking offset y k +1 = CA )]TJ/F44 11.9552 Tf 11.956 0 Td [(CBK )]TJ/F44 11.9552 Tf 11.955 0 Td [(C d L d CCB d 2 6 4 ~ x k ~ d k 3 7 5 + CA )]TJ/F44 11.9552 Tf 11.956 0 Td [(CBKCB d + C d 2 6 4 ^ x k ^ d k 3 7 5 + r k +1 )]TJ/F44 11.9552 Tf 11.955 0 Td [(r k C )]TJ/F44 11.9552 Tf 5.479 -9.683 Td [(x s k +1 )]TJ/F44 11.9552 Tf 11.955 0 Td [(x sk + C d ^ d s k +1 )]TJ/F26 11.9552 Tf 12.481 2.656 Td [(^ d sk Sincetheoutputvectorareatsteadystatesforregulationproblem y s = C ^ x sk + C d ^ d sk =0 thesteadystatesandsteadydisturbancesarelinearlydependent.Thusthetracking offsetfortherampreference y k +1 = CA )]TJ/F44 11.9552 Tf 11.955 0 Td [(CBK )]TJ/F44 11.9552 Tf 11.955 0 Td [(C d L d CCB d 2 6 4 ~ x k ~ d k 3 7 5 + CA )]TJ/F44 11.9552 Tf 11.955 0 Td [(CBKCB d + C d 2 6 4 ^ x k ^ d k 3 7 5 + r k +1 )]TJ/F44 11.9552 Tf 11.955 0 Td [(r k Ifthecontrolisdesignedas u k = )]TJ/F44 11.9552 Tf 9.298 0 Td [(K 1 x k )]TJ/F44 11.9552 Tf 11.955 0 Td [(K 2 [ r k +1 )]TJ/F44 11.9552 Tf 11.955 0 Td [(r k ] 54

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theoffsetismodiedto y k +1 = CA )]TJ/F44 11.9552 Tf 11.955 0 Td [(CBK 1 )]TJ/F44 11.9552 Tf 11.956 0 Td [(C d L d CCB d 2 6 4 ~ x k ~ d k 3 7 5 + CA )]TJ/F44 11.9552 Tf 11.955 0 Td [(CBK 1 CB d + C d 2 6 4 ^ x k ^ d k 3 7 5 + I )]TJ/F44 11.9552 Tf 11.955 0 Td [(CBK 2 r k +1 )]TJ/F44 11.9552 Tf 11.955 0 Td [(r k whichimpliesthattheadditionalconditionforthezero-offsetperformanceforramp referenceis CBK 2 = I Theconvergencerateisthesameasthatintrackingaconstantreference.Figure5-1is thesimulationresultfortheMPCprobleminSubsection5.2.3.2. min N )]TJ/F27 7.9701 Tf 6.587 0 Td [(1 X k =0 y k )]TJ/F44 11.9552 Tf 11.955 0 Td [(r k T Q y k )]TJ/F44 11.9552 Tf 11.955 0 Td [(r k + u T k Ru k + y N )]TJ/F44 11.9552 Tf 11.955 0 Td [(r N T P y N )]TJ/F44 11.9552 Tf 11.955 0 Td [(r N + u T k S r N )]TJ/F44 11.9552 Tf 11.956 0 Td [(r N )]TJ/F26 11.9552 Tf 11.955 0 Td [(1 s.t.Constraintshold. where Q R P S areweightmatrices. 55

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Figure5-1.Comparisonofcloseloopresponsetoanrampsignal. TheoffsetisreducedandtheresultsshowgreatadvantageoftherevisedMPC methodintrackingthenonconstantreference,especiallyfortherampsignal. 5.3.2OscillatingReference Anothersimpledampedspring-masssystemisstudiedtodemonstratetheabilityof theproposedcontrolschemetohandleanunstableandoscillatingreference: d dt x t = 2 6 4 01 )]TJ/F44 11.9552 Tf 9.298 0 Td [(k = m )]TJ/F29 11.9552 Tf 9.298 0 Td [( 3 7 5 x t + k = m 2 6 4 0 1 3 7 5 u t y t = 10 x t where k =1 m =1 ,and =0.1 .Therealplanthowevershallbeslightlyperturbedwith k =1.2 and =0.09 .Thegoalistotrackanoscillatingreferencewithzerooffset.The referencemodelis r k = 10 2 6 4 0.1 )]TJ/F29 11.9552 Tf 9.298 0 Td [( 0.1 3 7 5 k r 0 56

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Figure5-2.Comparisonofcloseloopresponsetoanunstablesignal. Figure5-2showstheclosedloopresponsetotheunstablereference.Itcanbe observedthattheproposedcontrollersMPC1andMPC2bothbehavewithlower offsetintrackingofthereference.ThecontrollerMPC1takesadvantageoftheterm )]TJ/F44 11.9552 Tf 9.298 0 Td [(K 2 [ r k +1 )]TJ/F44 11.9552 Tf 11.955 0 Td [(r k ] thatdoesnotaccountinthecontrolpenalty.Thusithasaslightly betterperformancethantheoneofthecontrollerMPC2. 57

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CHAPTER6 FUTUREWORK 6.1TrackingNonconstantReferencebyModifyingtheDisturbanceModel U.Maederprovidesageneralizationofthedisturbanceestimationapproach toarbitraryunstabledynamics.Thereferencesignalisgeneratedbyananautonomous dynamicsystem x r k +1 = A r x r k +1 r k = C r x r k where A r 2 R n r n r and C r 2 R n y n r ,andthematrix A r maybeunstable.Thesignalis generatedbymode withorder p ,ifthereexistsalinearsystemsuchthat s k = C s x s k x s k +1 = J p x s k k =0,1,... where J p isaJordanblockmatrixfor withorder p J = 2 6 6 6 6 6 6 6 6 6 6 4 10 0 0 1 0 00 . . 1 0000 3 7 7 7 7 7 7 7 7 7 7 5 Offset-freeperformanceisachievedundertheassumptionthattheobserverisstable andthefollowingdecompositionsexist: y = m X i =1 y i p i u = m X i =1 u i p i 58

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where i isthe i theigenvalueof A withthelongestJordanchainoflength p i .This crucialpointsofthemethodarethechoiceofdisturbancemodelsatisfyingtheinternal modelcondition,andtheadditionoftargettrajectoryconditionstotheMPCproblems. However,thedecompositionsabovedon'tholdforcertainreference. Anotherproblemisthatsomedisturbancemodelswouldjeopardizethestabilityof thesystem.Theestimator L x L d isdesignedtokeepthesystemstablebasedonthe conditionsthematrix 2 6 4 A T )]TJ/F44 11.9552 Tf 11.955 0 Td [(sI 0 C T B T d 1 )]TJ/F44 11.9552 Tf 11.955 0 Td [(s A d C T d 3 7 5 isfullrowrankforall s IntheChapter3,thematrix A d = I sothatwhen s 6 =1 rank 2 6 4 A T )]TJ/F44 11.9552 Tf 11.955 0 Td [(sI 0 C T B T d 1 )]TJ/F44 11.9552 Tf 11.956 0 Td [(s IC T d 3 7 5 = rank A T )]TJ/F44 11.9552 Tf 11.955 0 Td [(sIC T + m thusaslongastheoriginalsystemisobservable,thezerooffsetmaybeachieved. Inthiscase,ifthematrix A d isnotfullrank,theconditionbecomesthatthematrix 2 6 4 A T )]TJ/F44 11.9552 Tf 11.956 0 Td [(sI 0 C T B T d A d C T d 3 7 5 mustbefullrowrank,for f s : s iseigenvalueofthematrix A ands 6 = 1 g 6.2FutureWork ThefutureobjectistocombinemymethodwithU.Maeder'sdisturbancemodel. Thebriefsummaryofthemainstepsoftheprocedureproposedinthisthesis: 1.Chooseareferencemodel A r and C r 59

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2.Chooseadisturbancemodel A d B d C d .Thedynamicsmatrix A d mustincorporate aninternalmodelof A r .Itmaycontainadditionaldynamicsofexpecteddisturbances.Theparameters B d and C d arechosensuchthattheaugmentedsystem isobservable. 3.Computertheestimatorgain L x L d 4.UseMPCmethod min N )]TJ/F27 7.9701 Tf 6.586 0 Td [(1 X k =0 y k )]TJ/F44 11.9552 Tf 11.955 0 Td [(r k T Q y k )]TJ/F44 11.9552 Tf 11.955 0 Td [(r k + u T k Ru k + y N )]TJ/F44 11.9552 Tf 11.955 0 Td [(r N T P y N )]TJ/F44 11.9552 Tf 11.955 0 Td [(r N s.t.Constraintshold. where Q R P S areweightmatricestocalculatetheoriginalcontrol u 0 k = )]TJ/F44 11.9552 Tf 9.299 0 Td [(K 1 x 5.Optimizethethegain K 2 undertheconstraints. 6.Therevisedcontrol1isused u k = )]TJ/F44 11.9552 Tf 9.299 0 Td [(K 1 x k )]TJ/F44 11.9552 Tf 11.955 0 Td [(K 2 [ r k +1 )]TJ/F44 11.9552 Tf 11.955 0 Td [(r k ] or 1.Chooseareferencemodel A r and C r 2.Chooseadisturbancemodel A d B d C d .Thedynamicsmatrix A d mustincorporate aninternalmodelof A r .Itmaycontainadditionaldynamicsofexpecteddisturbances.Theparameters B d and C d arechosensuchthattheaugmentedsystem isobservable. 3.Computertheestimatorgain L x L d 4.TherevisedMPCproblemisgivenby min N )]TJ/F27 7.9701 Tf 6.587 0 Td [(1 X k =0 y k )]TJ/F44 11.9552 Tf 11.955 0 Td [(r k T Q y k )]TJ/F44 11.9552 Tf 11.955 0 Td [(r k + u T k Ru k + y N )]TJ/F44 11.9552 Tf 11.955 0 Td [(r N T P y N )]TJ/F44 11.9552 Tf 11.955 0 Td [(r N + u T k S r N )]TJ/F44 11.9552 Tf 11.955 0 Td [(r N )]TJ/F26 11.9552 Tf 11.955 0 Td [(1 s.t.Constraintshold. 5.Therevisedcontrol2isobtained 60

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APPENDIXA RANKPRESERVINGTHEOREM Incontroltheory,therankofamatrixcanbeusedtodeterminewhetheralinear systemiscontrollableorobservable. A isassumedtobean m by n matrixovereither therealnumbersorthecomplexnumbers.If B isany n k matrixwithrank n ,then rank AB = rank A A IfCisany l m matrixwithrank m ,then rank CA = rank A A 61

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APPENDIXB THECAYLEY-HAMILTONTHEOREM TheCayleyHamiltontheorem,whichisnamedafterthemathematiciansArthur CayleyandWilliamHamiltom,asgivenbelow. Letthecharacteristicpolynomialof A bedenedas p = det I n )]TJ/F44 11.9552 Tf 11.955 0 Td [(A where A isagiven n n matrix, I n isthe n n identitymatrix,and" det "indicatesthe determinantoperation.Sincetheentriesofthematrix I m )]TJ/F44 11.9552 Tf 12.176 0 Td [(A arepolynomialsin ,the determinantisalsoapolynomialin .TheCayleyHamiltontheoremstatesthatifone substitutesthematrix A for inpolynomial p ,oneobtainsthezeromatrix: p A =0 Thepowersof thatbecomepowersof A bythesubstitutionshouldbecomputed byrepeatedmatrixmultiplication,andtheconstanttermin p shouldbemultiplied bytheidentitymatrix,whichcanalsobetreatedasthezerothpowerof A ,sothatit canbeaddedtotheotherterms.Thetheoremleavestotheconclusionthat A n canbe expressedasalinearcombinationofthelowermatrixpowersof A .Especiallywhen theringisaeld,theCayleyHamiltontheoremisequivalenttothestatementthatthe minimalpolynomialofasquarematrixdividesitscharacteristicpolynomial. 62

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REFERENCES [1]GabrielePannocchiaandJamesB.Rawlings,DisturbanceModelsfor Offset-FreeModel-PredictiveControl, AIChEJournal ,Vol.49,No.2 [2]P.HenrlkWallman,JamesM.Sllva,andAlanS.Foss,Multivariable IntegralControlsfortheFixedBedReactors, Ind.Eng.Chem.Fudam ., Vol.18,No.4,1979 [3]UrbanMaederandManfredMorari,Offset-freereferencetrackingwith modelpredictivecontrol, Automatica, 461469-1476 [4]GabrielePannocchia,Robustdisturbancemodelingformodelpredictive controlwithapplicationtomultivariableill-conditionedprocesses, Journal ofProcessControl 13693 [5]UrbanMaederandManfredMorari,Offset-FreeReferenceTracking forPredictiveControllers, Proceedingsofthe46thIEEEConferenceon DecisionandControl, NewOrleans,LA,USA,Dec.12-14,2007 [6]UrbanMaedera,FrancescoBorrellibandManfredMoraria,Linear offset-freeModelPredictiveControl, Automatica 452214 [7]B.Shafai,S.Beale,H.H.Niemann,andJ.L.Stoustrup,LTRDesignof Discrete-TimeProportional-IntegralObservers, IEEETransactionson AutomaticControl ,Vol.41,No.7,1056-1062,1996 [8]JohanAkesson,PerHagander,IntergalAction-ADisturbanceObserver Approach, ProceedingsofEuropeanControlConference, Cambridge, UK,September2003. [9]FrancescoBorrelli,ManfredMorari,OffsetFreeModelPredictive Control, Proceedingsofthe46thIEEEConferenceonDecisionand Control ,NewOrleans,LA,USA,Dec.12-14,2007 [10]P.Lancaster,JordanChainsforLambdaMatrices,II, AEQ.MATH ,292, 1970 [11]GabrielePannocchiaandEricC.Kerrigan,Offset-freecontrolofconstrainedlineardiscrete-timesystemssubjecttopersistentunmeasured disturbances, Proceedingsofthe42ndIEEEConferenceonDecision andControl ,Maui,HawaiiUSA,December2003 [12]LinhVuandDanielLiberzon,SupervisoryControlofUncertainLinear Time-VaryingSystems, IEEETransactionsonAutomaticControl ,VOL. 56,NO.1,JANUARY2011 [13]KatsuhikoOgata, Discrete-TimeControlSystem ,Prentice-HallInternationalEditions,1987 63

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[14]E.F.CamachoandC.Bordons, ModelPredicitiveControlintheProcess Industry ,Springer,1995 [15]DaleE.Seborg,ThomasF.Edgar, DuncanA.Mellichamp,Process DynamicsandControl ,Wiley,SecondEdition [16]KatsuhikoOgata, ModernControlEngineering ,Prentice-Hall,Third Editions,1997 [17]DonaldE.Kirk, OptimalControlTheory-AnIntroduction ,Dover,2004 [18]OscarD.Crisalle, ECH6326IntroductiontoAdvancedProcessDynamicsandControl ,Couresnotes,2011 [19]OscarD.Crisalle, SystemModelsforMPC ,Coursenotes,2011 [20]OscarD.Crisalle, IntroductiontoOptimalControl ,Coursenotes,2011 [21]GeoffreyA.Williamson, LinearSystemTheory ,Coursenotes,2007 [22]MarkCannon, C21ModelPredictiveControl ,Coursenotes, www.eng.ox.ac.uk/~conmrc/mpc 64

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BIOGRAPHICALSKETCH YuzhouQian receivedhisM.S.DegreeinChemicalEngineeringfromtheUniversity ofFloridainthesummerof2012,Bachelordegreesinphysicsandinstatisticsform PekingBeijingUniversityin2010.HeiscurrentlyastudenttowardshisPh.D.degree atDepartmentofChemicalEngineeringatRensselaerPolytechnicInstituteRPI.His researchinterestsincludecontroltheoryresearchandthelatestapplicationsofcontrol methodsentireresearchprocess,modeling,optimizing,controlling,andmaking impossiblethingspossible. 65