Incremental Model Predictive Control System Design And Implementation Using Matlab/Simulink

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Incremental Model Predictive Control System Design And Implementation Using Matlab/Simulink
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Lin, Xin
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University of Florida
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Master's ( M.S.)
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University of Florida
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Chemical Engineering
Committee Chair:
Crisalle, Oscar Dardo
Committee Members:
Svoronos, Spyros A

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integral -- mpc -- offset-free
Chemical Engineering -- Dissertations, Academic -- UF
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Chemical Engineering thesis, M.S.
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Abstract:
The integral and model predictive controller (MPC) drive controlled outputs to their desired targets, and this thesis addresses the problem of integral controller, incremental and integral MPC when tracking the constant or inconstant references. Design and implementation of the MPC under MATLAB/Simulink environment are discussed both in incremental and integral form. Also one CSTR example is presented to compare the control performances among different integral controller and MPCs.
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In the series University of Florida Digital Collections.
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by Xin Lin.
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Thesis (M.S.)--University of Florida, 2013.
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Adviser: Crisalle, Oscar Dardo.
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RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2013-11-30

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INCREMENTALMODELPREDICTIVECONTROLSYSTEMDESIGNAND IMPLEMENTATIONUSINGMATLAB/SIMULINK By XINLIN ATHESISPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF MASTEROFSCIENCE UNIVERSITYOFFLORIDA 2013

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2013XinLin 2

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Toallchemicalengineers 3

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ACKNOWLEDGMENTS TheauthorissincerelythankfultoProf.OscarD.Crisalle, Distinguished TeachingScholarandProfessor intheChemicalEngineeringDepartmentof UniversityofFlorida,forhelpfuladvicethroughoutthestageofthisworkand doctoralcandidatesM.RafeBiswasandShyamP.Mudirajfortheirvaluable feedbackontherevisionsofthisthesis. 4

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TABLEOFCONTENTS page ACKNOWLEDGEMENTS.............................4 LISTOFTABLES..................................7 LISTOFFIGURES.................................8 ABSTRACT.....................................10 CHAPTER 1INTRODUCTION...............................11 1.1Background...............................11 1.1.1IntegralControllerandOffset-freePerformance.......11 1.1.2OverviewofMPC........................12 1.2DescriptionofContents........................14 2LITERATUREREVIEW...........................16 2.1DiscreteIntegralController......................16 2.2StandardMPCController.......................17 2.3OffsetPerformanceinMPCControlSystems............20 2.4Offset-freePerformanceConsideringInconstantSetpointTracking20 2.5Challenges...............................22 3CONTROLSYSTEMDESIGNANDIMPLEMENTATIONUSINGDISCRETEINTEGRALCONTROLLER....................24 3.1IntegralControllerStructure......................24 3.2Offset-freePerformanceAnalysis...................25 3.3ControlValidation............................27 3.3.1Example1:DiscreteSystemfromOgata[1].........27 3.3.2Example2:CSTRPlantfromSeborg[2]...........31 3.3.2.1Modelingequations.................31 3.3.2.2Linearizationofthesystem.............33 3.3.2.3Offset-freeperformance...............35 3.3.2.4ControlPerformanceonthelinearcontinuous andnonlinearCSTRsystems............36 4CONTROLSYSTEMDESIGNANDIMPLEMENTATIONUSINGINCREMENTALMPC..............................41 4.1BasicEquationsforIncrementalMPC................41 4.2ControlValidation............................44 4.2.1Example1:CSTRPlantfromSeborg[2]...........44 4.2.2Example2:QuadrupleTankSystem.............51 5

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5CONTROLSYSTEMDESIGNANDIMPLEMENTATIONUSINGINTEGRALMPC................................55 5.1BasicEquationsforIntegralMPC...................55 5.1.1FirstIntegralState z k ....................55 5.1.2SecondIntegralState w k ..................57 5.1.3ControlLawfortheIntegralMPCController.........58 5.2ControlValidation............................58 6CONCLUSIONSANDFUTUREWORK..................63 6.1Conclusions...............................63 6.2FutureWork:ConstrainedIncrementalMPC.............63 APPENDIX ADERIVATIONOF K 1 AND K 2 FORTHEDISCRETEINTEGRALCONTROLLER...................................66 REFERENCES...................................70 BIOGRAPHICALSKETCH............................71 6

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LISTOFTABLES Table page 3-1PlantparametersoftheCSTRmodel...................34 4-1Parametervaluesforthequadrupletanksystem.............52 4-2Initialvaluesofthequadrupletanksystem.................52 7

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LISTOFFIGURES Figure page 1-1Atypicalblockdiagramoftheclosed-loopsystemwithastatefeedbackintegralcontroller............................12 1-2AblockdiagramofanMPCcontroller....................14 3-1Closed-loopMATLAB/SimulinkmodelfromOgata'sbook,twodifferent discretecontrollerscontrollerimplementedbyconnectedSIMULINK blocksandbyS-Functionareused....................30 3-2Offset-freecontrolperformanceoftheclosed-loopsystemwithadiscreteintegralcontroller.Themodelisexample6-12inOgata[1].....32 3-3ThediagramofCSTRsysteminthebook.................33 3-4Offset-freeperformanceofthelineardiscreteclosed-loopCSTRsystem.Therearestepchangesinsetpoint,statedisturbanceandoutput disturbanceatdifferenttime.........................37 3-5ControlperformanceonthelinearcontinuousCSTRsystemusingidenticalstepchangesinsetpoint,statedisturbanceandoutputdisturbance tothosegraphsingure3-4.........................39 3-6ControlperformanceonthenonlinearCSTRsystemusingtheidenticalstepchangesinsetpoint,statedisturbanceandoutputdisturbance assubsection3.3.2.3.............................40 4-1Controldiagramofaclosed-loopsystemwiththeincrementalMPC controller...................................45 4-2MATLAB/Simulinkdiagramoftheclosed-loopsystemwiththeincrementalMPCcontroller............................47 4-3SimulationresultsforthelineardiscreteCSTRsystemwiththeincrementalMPCcontroller............................48 4-4SimulationresultsforthelinearcontinuousCSTRsystemwithincrementalMPCcontroller............................49 4-5SimulationresultsforthenonlinearCSTRsystemwithincremental MPCcontroller................................50 4-6Processdiagramforthequadrupletanksystemfromkesson[3]....52 8

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4-7SimulationresultsofincrementalMPConthelinearizedquadruple tanksystem.Thecontrolvariablewithitssetpointwhichispresented inthetopgraphisthelevelfortank1.Twoinputvariablesdenotedas u 1 and u 2 areshowninthebottomgraph..................54 5-1Closed-loopsimulationresultforthelineardiscreteCSTRsystemwith anintegralMPCcontroller.TheCSTRisgivenbysubsection3.3.2 andthestepchangesarepresentedinsubsection3.3.2.3........59 5-2Closed-loopsimulationresultforthelinearcontinuousCSTRsystem withanintegralMPCcontroller.TheCSTRisgivenbysubsection3.3.2 andthestepchangesarepresentedinsubsection3.3.2.3........61 5-3Closed-loopsimulationresultforthenonlinearCSTRsystemwithan integralMPCcontroller.TheCSTRisgivenbysubsection3.3.2and thestepchangesarepresentedinsubsection3.3.2.3..........62 9

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AbstractofaThesisPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofMasterofScience INCREMENTALMODELPREDICTIVECONTROLSYSTEMDESIGNAND IMPLEMENTATIONUSINGMATLAB/SIMULINK By XinLin May2013 Chair:Oscar.D.Crisalle Major:ChemicalEngineering TheintegralandmodelpredictivecontrollerMPCdrivecontrolledoutputs totheirdesiredtargets,andthisthesisaddressestheproblemofintegralcontroller,incrementalandintegralMPCwhentrackingtheconstantorinconstant references.DesignandimplementationoftheMPCunderMATLAB/Simulinkenvironmentarediscussedbothinincrementalandintegralform.AlsooneCSTR exampleispresentedtocomparethecontrolperformancesamongdifferent integralcontrollerandMPCs. 10

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CHAPTER1 INTRODUCTION Thischapterisaguidetothetopicscoveredbythisthesis.Theobjectives ofthisworkincludethedesignandimplementationofintegralandMPCcontrollersusingS-functionsundertheMATLAB/Simulinkenvironment.Verication andcomparisonofthecontrolperformanceamongdifferentcontrollersare presented. SeveralcontrollersandthechallengesassociatedwiththeirdesignaredescribedinSubsection1.1.Anexamplefortheplantanditsdetailedinformation isprovidedinSubsections1.2and1.3,andtheorganizationofthisthesisis explainedinthedescriptionofcontentssection. 1.1Background 1.1.1IntegralControllerandOffset-freePerformance Aclosed-loopcontroldiagramofanintegralcontrollerisshowningure1-1. Theintegralcontrollerutilizestheplantstatesandplantoutputstocalculate itsoutput.Suchinformationcanbedirectlymeasuredorestimatedbyusing anobserver.Thedesignofanintegralcontrollerissimpleandstraightforward. Sinceitusesanintegralstate,itcantrackconstantsetpointswithoutoffset.If alltheclosed-looppolesareplacedproperlyandtheplantmodelisaccurate enough,theintegralcontrollerachievesthegoalsofbothsetpointtrackingand disturbancerejection. Adisadvantageoftheintegralcontrolleristhatwhenthereisstronginteractionamongdifferentcontrolloops,itsperformancewilldegradesignicantly. Moreover,sinceitisdesignedbasedonalinearmodel,itmayfailtoobtain satisfactorycontrolperformanceontheplantifthereisseriousmodelmismatch ornonlinearityinsidetheplant. 11

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Figure1-1.Atypicalblockdiagramoftheclosed-loopsystemwithastate feedbackintegralcontroller. 1.1.2OverviewofMPC MPCisanoptimalcontrollerbasedonreal-timenumericaloptimization.A typicalMPCcontroldiagramisgiveningure1-2.Theplantoutputispredicted byusinganestimatedsystemmodel.Theplantinputisoptimizedateachtime instantanceaccordingtopenaltyfunctionandconstraints.Themainideasof MPCoriginallycomefromacomputationaltechniqueusedtoimprovecontrol performanceinprocessindustries.Sincethen,predictivecontrolhasbecamethe mostwidespreadadvancedcontrolstrategyinchemicalengineering.AnMPC controllercanachievedesiredcontrolperformanceinlarge-scalemultivariable systems,andprovideasystematicmethodofdealingwithstatesandinputs constraintswithsimpledesignandtuning. ThegeneralgoalofMPCistocalculateatrajectoryoffuturemanipulated variable u tooptimizethefuturebehaviouroftheplantoutput y .Theoptimization iscarriedoutwithinalimitedtimeinstancebyusingtheplantinformationatthe startofthetimeinterval. 12

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TherearethreefundamentalconceptinthedesignofMPC.Therstishowto predictthefuturestatesandoutputsmodel;thesecondisthewayonhowto obtainthecurrentinformationoftheplantmeasurementandthethirdisthe approachontheimplementationoffutureactivitiesrealizationofcontrol.The keyissuesinthedesignare: 1.Thetimeintervalforthedesignisaconstant; 2.Peopleneedtohaveaccesstothecurrentstatesbeforethecontroldesign; 3.Peopletaketheconstraintsintoconsideration,andtheoptimizationis performedinreal-timewithatimewindowthatmovesfowardandwiththe latestplantinformationavailable. OtherconceptsthatareusedfrequentlyinthedesignofMPCarethefollowing: themovinghorizonwindow,predictionhorizon,recedinghorizoncontrol,and controlobjective.Theyarediscussedhereasfollowing: 1.Themovinghorizonwindowisreferredtothetime-dependentwindowfrom anarbitrarytime t i to t i + T p .Thelengthofthewindow T p isaconstant. However, t i ,whichisthebeginningofthewindow,dependsontimeand increasesastimeevolves. 2.Thepredictionhorizondetermineshowlongintothefuturestatesand outputsaretobepredictedfor.Thisparameterisdenedasthelengthof themovinghorizonwindow, T p 3.Recedinghorizoncontrolisusedtodescribethecontrolstrategythat althoughtheoptimalseriesoffuturecontrolleroutputsarecalculatedwithin themovinghorizonwindow,theactualinputtotheplantisonlytherst valueoftheseries,andtherestofthecontrolleroutputsarediscarded. 4.Forallprocesses,peopleneedthestateinformationattime t i topredict thefuturebehaviorofthesystem.Thisinformationisdenedas x t i whichisavectorcontainingthecurrentprocessdata,andiseitherdirectly measuredorestimated. 5.Amodeldescribingthedynamicsofthesystemisextremelyimportantin predictivecontrol.Awell-designeddynamicmodelcanpredictthefuture performanceofthesystemaccurately. 13

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6.Toreachthebestdecision,amathematicalcriterionisusedtodescribethe controlobjective.Theobjectiveisusuallypresentedasafunctionbased onthedifferencebetweenthedesiredandtheactualresponses.This objectivefunctionisoftencalledthepenaltyorcostfunction, J ,andthe optimalcontrolleroutputisobtainedbyminimizingthepenaltyfunction withinthegivenoptimizationwindow. Figure1-2.AblockdiagramofanMPCcontroller. 1.2DescriptionofContents ThethesismaterialhasevolvedatUniversityofFloridaoverthelastoneand ahalfyears.Thethesisisdividedintovechaptersthataddressthedesignand implementationoftheintegralandIMPCcontrollers. Chapter1providesanintroductiontothisthesis,includingthebriefdescriptionaboutdiscreteintegralcontroller,andtheMPCcontrolmethod. Chapter2isconcernedwithaliteraturereview.Theconditionsforoffset-free performancewithdiscretecontrollersandMPCcontrollersobtainedbyprevious researchersareprovidedanddiscussed.Severaldisturbancemodelsusedby thepioneerstoeliminatetheinuenceofexistingdisturbancesandplantmodel mismatcharepresentedanddiscussed. 14

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Chapter3through5addressestheanalysisanddesignofdiscreteintegral controllersandIMPCcontrollersundertherequirementofoffset-freeperformanceforbothconstantandnon-constantsetpoint.Simulationsoftheplant systemareusedtoverifythetheorydiscussedinthispart. Chapter6presentsconclusionsandrecommendationsforfuturework. Finallyoneappendixincludesthecontrollergainderivationrelatedtothe thesis. 15

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CHAPTER2 LITERATUREREVIEW 2.1DiscreteIntegralController Chemicalengineershaveuseddiscreteintegralcontrollersforavery longtimetoachieveoffset-freeperformance.Ifalltheclosed-looppolesare properlyplaced,theintegralcontrollercanalsorejectanunmeasuredconstant disturbance.Ogata[1]considersadiscretesystemwithstateandoutput equation x k +1 = A d x k + B d u k y k = C d x k x k 2 R n isthestatevector, y k 2 R m isoutputvectorand u k 2 R p isthe manipulatedvariable.Thematrices A d 2 R n n B d 2 R n p and C d 2 R m n are thestatematrix,inputmatrixandoutputmatrix,respectively.Thepair A d B d is assumedtobecontrollable.Anintegralstateequationisdenedas 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 where r k 2 R m isthesetpointvector, z k 2 R m istheintegralstatevectorand thecontrolleroutput u k isgivenby u k = )]TJ/F44 11.9552 Tf 9.298 0 Td [(K 2 x k + K 1 z k Forthepurposeofdesign,itisconvenienttodenetheaugmentedstatevector 2 6 4 x k u k 3 7 5 Thesetpointisassumedtobeaconstantorastepfunction.Underclosed-loop condition u 1 and x 1 existifthefollowingconditionsaresatised: 16

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1.Thepair A d B d iscontrollable. 2.Thematrix A d )]TJ/F44 11.9552 Tf 11.955 0 Td [(IB d C d 0 isfullrank. 3.Alltheclosed-looppolesareproperlyplaced. Sinceaccordingtoequation2 u k makesuseofboththefeedbackstate x k andtheintegralstate,if z 1 existsforagiveninput,thevectors x k u k ,and z k willconvergetotheirsteadystatevaluesdenoted x 1 u 1 and z 1 respectively.Thenthefollowingequationatthesteadystatecanbe obtainedfrom2 z 1 = z 1 + r 1 )]TJ/F44 11.9552 Tf 11.955 0 Td [(y 1 whichimpliesthat r 1 = y 1 Therefore,theintegralcontrollercanachieveoffset-freeperformancewhenthe setpointisaconstantorastepfunction. 2.2StandardMPCController TheMPCcontrollerisusedinawidevarietyofindustrialsystems,including chemicalreactors,automobiles,robotsandsoon.Rawlings[4]discussesthe designofapredictivecontrolsystemintwoparts:predictionandoptimization. Forprediction,denotethefuturecontrolleroutputas u k i u k i +1,..., u k i + N c )]TJ/F26 11.9552 Tf 11.955 0 Td [(1, where N c isthecontrolhorizon.Alsodenotingthefuturestateas x k i +1 j k i x k i +2 j k i ,..., x k i + m j k i x k i + N p j k i 17

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where N p isthepredictionhorizonthatdeterminesthenumberofthefuturestate peoplewouldliketocalculate.Ingeneral x k i + m j k i iscalledthefuturestateat time k i + m predictedatinstant k i andbasedonthegivensystemstate x k i Forastatespacemodel,calculatethefuturestatesandoutputsasfollows x k i +1 j k i = Ax k i + B u k i x k i +2 j k i = Ax k i +1 j k i + B u k i +1 = A 2 x k i + AB u k i + B u k i +1 x k i + N p j k i = A N p x k i + A N p )]TJ/F27 7.9701 Tf 6.586 0 Td [(1 B u k i +...+ A N p )]TJ/F45 7.9701 Tf 6.587 0 Td [(N c B u k i + N c )]TJ/F26 11.9552 Tf 11.955 0 Td [(1 andthefutureoutputas y k i +1 j k i = CAx k i + CB u k i y k i +2 j k i = CA 2 x k i +1 j k i + CAB u k i +1+ CB u k i +1 y k i + N p j k i = CA N p x k i + CA N p )]TJ/F27 7.9701 Tf 6.587 0 Td [(1 B u k i +...+ CA N p )]TJ/F45 7.9701 Tf 6.586 0 Td [(N c B u k i + N c )]TJ/F26 11.9552 Tf 11.955 0 Td [(1 Dene Y = [ y k i +1 j k i y k i +2 j k i ... y k i + N p j k i ] T 2 R N p p p U = [ u k i u k i +1... u k i + N c )]TJ/F26 11.9552 Tf 11.955 0 Td [(1 ] T 2 R N c p p theoutputequationbecomes Y = Fx k i + U 18

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where F = 2 6 6 6 6 6 6 6 4 CA CA 2 CA N p 3 7 7 7 7 7 7 7 5 2 R N p m n = 2 6 6 6 6 6 6 6 4 CB 00...0 CABCB 0...0 CA N p )]TJ/F27 7.9701 Tf 6.587 0 Td [(1 BCA N p )]TJ/F27 7.9701 Tf 6.587 0 Td [(2 BCA N p )]TJ/F27 7.9701 Tf 6.586 0 Td [(3 B ... CA N p )]TJ/F45 7.9701 Tf 6.587 0 Td [(N c B 3 7 7 7 7 7 7 7 5 2 R N p m N c p Fortheoptimizationpart,dene R T s = 11...1 r k i thenthepenaltyfunctioniswrittenas J = R s )]TJ/F44 11.9552 Tf 11.955 0 Td [(Y T Q R s )]TJ/F44 11.9552 Tf 11.955 0 Td [(Y + U T RU = R s )]TJ/F44 11.9552 Tf 11.955 0 Td [(Fx k i T R s )]TJ/F44 11.9552 Tf 11.955 0 Td [(Fx k i )]TJ/F26 11.9552 Tf 11.955 0 Td [(2 U T T R s )]TJ/F44 11.9552 Tf 11.956 0 Td [(Fx k i + U T T + R U where Q and R aretheoutputweightmatrixandvelocityweightmatrix,respectively.Thesetwodiagonalmatricesareusedtotunethedesiredclosed-loop controlperformance. Tondoptimal U thatwillminimizethepenaltyfunction J ,calculatetherst derivativeof J withrespectto U @ J @ U = )]TJ/F26 11.9552 Tf 9.298 0 Td [(2 T R s )]TJ/F44 11.9552 Tf 11.956 0 Td [(Fx k i +2 T + R U 19

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Thenecessaryconditionforminimal J isthat2equalstozero.Assuming that T + R )]TJ/F27 7.9701 Tf 6.587 0 Td [(1 exists,then U canbesovledtominimize J ,shownas2 U = T + R )]TJ/F27 7.9701 Tf 6.586 0 Td [(1 T R s )]TJ/F44 11.9552 Tf 11.955 0 Td [(Fx k i Dene R s = 11...1 T then U canbedeterminedbythefollowingequation U = T + R )]TJ/F27 7.9701 Tf 6.586 0 Td [(1 T R s r k i )]TJ/F44 11.9552 Tf 11.956 0 Td [(Fx k i whichisthecontrollawofthestandardMPCcontroller. 2.3OffsetPerformanceinMPCControlSystems ForanMPCcontrollergivenbygure1-2,Maeder[5]demonstratesthatif thenumberofmesurementsisgreaterthanthenumberofmanipulatedvariables, notallcntrolledvairablescanhaveoffset-freeperformance.AccordingtoRawlings[4]MPCcontrollerscanachieveoffset-freecontrolperformancebyadding anintegraldisturbancetothesystem.Thisadditionaldisturbanceeliminates themismatchbetweenrealsystemandthemodelbyitscorrespondingintegral action.Sincetheintegrationroutineisindependentofthecontrolleritself,itmay causewindupproblemsinconstrainedsystem. Tosolvethisproblem,onecancombinethecontrollerwithadisturbance estimator.Thenthestateequationforthepredictionisaugmentedbythe setpointanddisturbanceasRawlings[4]describes.WiththisimprovementMPC controllercansolvethewindupproblem. 2.4Offset-freePerformanceConsideringInconstantSetpointTracking Qian[6]statesthattheexistingcontrolalgorithmsassumethedisturbance andsetpointareconstants.Hence,fornon-constantdisturbancesandsetpoints 20

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suchasrampsandsinusoidfunctions,thesealgorithmsfailtoachieveoffset-free performance.Maeder[7][8]provideageneralizedapproachtotrackanarbitrary setpoint. Thestateandreferenceequationforadiscretesystem 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.Ifthereis alinearsystemsuchthat s k = C s x s k x s k +1 = J p x s k k =0,1,... where J p = 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 2 R p p isaJordanblockmatrixfor withorder p ,theclosed-loopsystemwithapredictivecontrollercanattainoffset-freeperformancewiththefollowingassumptions: 1.Thestateestimatorisstable. 2.Thefollowingexpansionsfor y and u exist y = m X i =1 y i p i u = m X i =1 u i p i where i isthe i -theigenvalueof A withthemultiplicity p i 21

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However,thisapporachincreasesthecomputationalcostofthetrajectoryoptimization.Moreover,thereisverylimitedimprovementontheoffsetperformance forcertainreferencesiftheexpansions2and2abovedonotexist. 2.5Challenges Thecontrollawforadiscreteintegralcontrollerisdesignedas u k = )]TJ/F44 11.9552 Tf 9.298 0 Td [(K 2 x k + K 1 z k .Whenthesystemreachessteadystate,onecanhavethe followingequation 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.Theproblemisthatthislimitdoes notensurethataconstantvector z 1 exists. Thekeyisthatwhetherthesquarematrix 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 where K = K 2 K 1 2 R 1 n and G isthecharacteristicequationof G where 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 Assuming 2 6 4 A )]TJ/F44 11.9552 Tf 11.955 0 Td [(I n B CACB 3 7 5 isinvertible,solvefor K 2 K 1 toget 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 22

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Solvefor K 1 toobtain 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 6.586 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 6.586 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 Fromtheaboveequationitishardtoknowwhetherthematrix K 1 isfullrankor notsincetheinvertibilityofthematrix 2 6 4 A )]TJ/F44 11.9552 Tf 11.955 0 Td [(I n B CACB 3 7 5 isnotensured. InPannocchia[9],thisapproachisproveneffectiveforcertainsquares casesonlyinRawlings[10].Moreover,thisapproachaugmentstheoriginal statespaceequationswithaintegratingdisturbancemodelandhencethe computationalcosthasbeensignicantlyincreased.Anothertwoimproved methodsbyusingtheincrementalandintegralMPCcontrollersareappliedin thisthesisinChapter4and5. 23

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CHAPTER3 CONTROLSYSTEMDESIGNANDIMPLEMENTATIONUSINGDISCRETE INTEGRALCONTROLLER 3.1IntegralControllerStructure Adiscrete-time,linear,timeinvariantmodel x k +1 = A d x k + B d u k y k = C d x k x = x 0 isgiven,where x k 2 R n istheplantstatevector, y k 2 R m isoutputvector and u k 2 R p isthecontrolvector.Furthermore A 2 R n n B 2 R n p ,and C 2 R m n .Thepair A B isassumedtobecontrollable.Thepurposeisto designanintegralstate-spacecontroller.Theclosed-loopcontroldiagramis showningure1-1.ThecontrolleroutputadoptedisproposedinOgata[1]as follows: u k = )]TJ/F44 11.9552 Tf 9.298 0 Td [(K 2 x k + K 1 z k where z k = z k )]TJ/F26 11.9552 Tf 11.956 0 Td [(1 + r k )]TJ/F44 11.9552 Tf 11.955 0 Td [(y k z =0 isanintegralstatebelongingtothecontrollerandwherethe r k 2 R m isthe setpoint. K 1 2 R p n and K 2 2 R p m arethecontrollergains. Since u k isalinearcombinationofstatevectors x k and z k ,itis possibletodeneanaugmentedvectorconsistingof x k and u k insteadof x k and z k .Thenthefollowingaugmentedstateequationcanbeobtained. 24

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2 6 4 x k +1 u k +1 3 7 5 = 2 6 4 A d B d K 2 )]TJ/F44 11.9552 Tf 11.955 0 Td [(K 2 A d )]TJ/F44 11.9552 Tf 11.955 0 Td [(K 1 C d A d I m )]TJ/F44 11.9552 Tf 11.955 0 Td [(K 2 B d )]TJ/F44 11.9552 Tf 11.955 0 Td [(K 1 C d B d 3 7 5 2 6 4 x k u k 3 7 5 + 2 6 4 0 K 1 3 7 5 r k +1 Theoutputequationofsystem3thencanberewrittenas y k =[ C d 0 ] 2 6 4 x k u k 3 7 5 3.2Offset-freePerformanceAnalysis Theclosed-looppolesaredeterminedbythecharacteristicofthesystem itselfandisindependentofthesetpoint r k .Onlytheeigenvaluesofthe augmentedstatematrixareusedinplacingthepolesofthesystem. Assumetheconstantsetpoint r k isappliedtothesystemsuchthat r k = r Thentheaugmentedstateequationcanbewrittenas 2 6 4 x k +1 u k +1 3 7 5 = 2 6 4 A d B d K 2 )]TJ/F44 11.9552 Tf 11.955 0 Td [(K 2 A d )]TJ/F44 11.9552 Tf 11.955 0 Td [(K 1 C d A d I m )]TJ/F44 11.9552 Tf 11.955 0 Td [(K 2 B d )]TJ/F44 11.9552 Tf 11.955 0 Td [(K 1 C d B d 3 7 5 2 6 4 x k u k 3 7 5 + 2 6 4 0 K 1 r 3 7 5 25

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Assumingthat z k approachesaconstantvector,denotedas, z 1 ,atsteady state5becomes z 1 = z 1 + r )]TJ/F44 11.9552 Tf 11.955 0 Td [(y 1 or y 1 = r whichindicatesoffset-freeperformance.TheanalysisrelatedtothecontrollabilityoftheaugmentedstatematrixAisgiveninChapter7.Assumingthat thematrix 2 6 4 A d )]TJ/F44 11.9552 Tf 11.955 0 Td [(I n B d C d A d C d B d 3 7 5 isinvertible, K 1 and K 2 canbeobtainedwiththe followingform K 2 K 1 = K + 0 I m 2 6 4 A d )]TJ/F44 11.9552 Tf 11.955 0 Td [(I n B d C d A d C d B d 3 7 5 )]TJ/F27 7.9701 Tf 6.586 0 Td [(1 Itisalsofoundthatiftheinput u k isan m -vectorwith m > 1 ,thereare morethanonesolutionstothematrix K usingthepole-placementtechnique, whichmeansthatmorethanonepairofmatrices K 1 and K 2 canbefound.Inthis situation,choosethepairof K 1 and K 2 thatobtainstheoptimaloffset-freecontrol performance. Theintegralcontrollerdesignedabovecanalsobeusedforaplantmodel combinedwithadisturbancemodel.Thedynamicsandoutputequationofthe disturbancemodelareshownas 26

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x k +1 = A d x k + B d u k + d k d k +1 = d k y k = C d x k + p k p k +1= p k where d k and p k cannotbedirectlymeasuredinmostsituations. 3.3ControlValidation 3.3.1Example1:DiscreteSystemfromOgata[1] Anexamplehereusedtoshowtheoffset-freeperformanceoftheintegral controlleristheExample6-12inOgata[1].Thediscretesystemischaracterized withdiscretematrices A d = 2 6 6 6 6 4 010 001 )]TJ/F26 11.9552 Tf 9.299 0 Td [(0.12 )]TJ/F26 11.9552 Tf 9.299 0 Td [(0.011 3 7 7 7 7 5 B d = 2 6 6 6 6 4 0 0 1 3 7 7 7 7 5 and C d = 0.510 Thecontrollabilitymatrixforthepair A d B d is 27

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Q c = B d A d B d A n )]TJ/F27 7.9701 Tf 6.586 0 Td [(1 d B d = 2 6 6 6 6 4 001 011 110.99 3 7 7 7 7 5 whichisfullrank. Theaugmentedstateandoutputequationsdenedinchapter7are A = 2 6 6 6 6 6 6 6 4 0100 0010 )]TJ/F26 11.9552 Tf 9.299 0 Td [(0.12 )]TJ/F26 11.9552 Tf 9.298 0 Td [(0.0110 0 )]TJ/F26 11.9552 Tf 9.298 0 Td [(0.5 )]TJ/F26 11.9552 Tf 9.298 0 Td [(11 3 7 7 7 7 7 7 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 Thecontrollabilitymatrixforthepair )]TJ/F26 11.9552 Tf 6.376 -7.027 Td [( A B oftheaugmentedstatespace systemis Q c = B A B A n )]TJ/F27 7.9701 Tf 6.586 0 Td [(1 B = 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 28

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whichisafull-rankmatrix.ThenusingthemethoddescribedinOgata[1],the gainmatrix K canbeobtained K = 0001 B A B A 2 B A 3 B )]TJ/F27 7.9701 Tf 6.587 0 Td [(1 A where A = A 4 Calculate K from3 K = )]TJ/F26 11.9552 Tf 9.299 0 Td [(0.12 )]TJ/F26 11.9552 Tf 9.298 0 Td [(0.130.991 andthus K 2 K 1 = K + 01 2 6 4 A d )]TJ/F44 11.9552 Tf 11.955 0 Td [(I 3 B d C d A d C d B d 3 7 5 )]TJ/F27 7.9701 Tf 6.587 0 Td [(1 = )]TJ/F26 11.9552 Tf 9.298 0 Td [(0.120.323320.6667 Thenobtaintheintegralandstatefeedbackgains K 1 and K 2 as K 1 =0.6667 K 2 = )]TJ/F26 11.9552 Tf 9.299 0 Td [(0.120.32332 Usingthecontrollaw3-3withthegainsgivenby3,offset-free controlperformancecanbeachieved. Aclosed-loopMATLAB/Simulinkmodelfortheexamplegureisshowningure 3-1.Therearetwodiscreteintegralcontrollersinthediagramlabelledascontroller1andcontroller2.Alloftheirparametersarethesameexcepttheimplementationisdifferent.Therstcontrollerisimplementedbyconnectedsimulink 29

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Figure3-1.Closed-loopMATLAB/SimulinkmodelfromOgata'sbook,two differentdiscretecontrollerscontrollerimplementedbyconnected SIMULINKblocksandbyS-Functionareused blocksandthesecondisimplementedbyanS-functionwiththeapproachintroducedinMathWorks[11].Theyhavetheidenticalcontrolperformanceforthe sameplant. Theoffset-freecontrolperformanceisshowningure3-2.Sincethesetwo integralcontrollerhavethesamecontrolperformance,onlytheperformanceof therstcontrollerisshown.Thetopgraphinthegurepresentsthesetpoint andplantoutputasafunctionoftime.Thex-axisisthetimeandthey-axis representsthevaluesforsetpointandplantoutput.Thesecondgraphisthe controlleroutputasafunctionoftime. 30

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Aconstantsetpoint r k =1 andintitialcondition x 1 =1 x 2 = )]TJ/F26 11.9552 Tf 9.299 0 Td [(0.5 x 3 =0 z =1 Offset-freeperformanceisobtainedat k =4 ,whichveriestheoffsetelimination propertyofthediscreteintegralcontroller. 3.3.2Example2:CSTRPlantfromSeborg[2] 3.3.2.1Modelingequations Acontinuousstirred-tankreactorCSTRasshowninFigure3-3isdiscussedhere.TheCSTRmodelisputforwardbySeborg[2].Anirreversible, rst-orderreaction, A B iscarriedoutintheliquidphaseinthereactor,andthereactortemperatureis controlledbyexternalcooling.Itisassumedthattheliquidlevelisaconstant. Massandenergybalancesleadtothefollowingnonlinearstate-spacemodel: Molarbalanceequation: V dC A dt = q C Ai )]TJ/F44 11.9552 Tf 11.955 0 Td [(C A )]TJ/F44 11.9552 Tf 11.955 0 Td [(Vkc A Energybalanceequation: V C dT dt = wC T i )]TJ/F44 11.9552 Tf 11.955 0 Td [(T + )]TJ/F26 11.9552 Tf 9.299 0 Td [( H R VkC A + UA T c )]TJ/F44 11.9552 Tf 11.956 0 Td [(T where C A isthemolarconcentration, T isthereactortemperature, q istheoutlet owrate, T c isthecoolantliquidtemperature. 31

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Figure3-2.Offset-freecontrolperformanceoftheclosed-loopsystemwitha discreteintegralcontroller.Themodelisexample6-12inOgata[1]. 32

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Figure3-3.ThediagramofCSTRsysteminthebook Thecontrolledvariableisdenedastheoutletmolarconcentration C A Thestatevariablesarethereactortemperature T ,and C A ,whilethemanipulatedvariablesaretheinletmolarconcentration C Ai ,inlettemperature T i andthecoolantliquidtemperature, T c .Moreover,therearestateandoutput disturbancesinthesystem. 3.3.2.2Linearizationofthesystem Theopen-loopsteady-stateoperatingconditionsarethefollowing C Ai =10 mol = L T i =298.2 K T =339.4282 K C A =5.5 mol = L T c =298.05 K 33

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Usingasamplingperiodtimeof 1min ,alinearizeddiscretestate-spacemodelis developedintermsofthedeviationstates,inputs,andoutputs x = 2 6 4 T )]TJ/F44 11.9552 Tf 11.955 0 Td [(T s C A )]TJ/F44 11.9552 Tf 11.956 0 Td [(C s A 3 7 5 u = 2 6 6 6 6 4 C Ai )]TJ/F44 11.9552 Tf 11.956 0 Td [(C s Ai T i )]TJ/F44 11.9552 Tf 11.956 0 Td [(T s i T c )]TJ/F44 11.9552 Tf 11.955 0 Td [(T s c 3 7 7 7 7 5 y = 2 6 4 T )]TJ/F44 11.9552 Tf 11.955 0 Td [(T s C A )]TJ/F44 11.9552 Tf 11.955 0 Td [(C s A 3 7 5 TheparametersneededforprocessmodelingisgiveninTable3-1. Table3-1.PlantparametersoftheCSTRmodel ParameterNominalValue q i = V 1 min )]TJ/F27 7.9701 Tf 6.586 0 Td [(1 T i 298.2 K C Ai 10 mol = L k 0 34930800 min )]TJ/F27 7.9701 Tf 6.587 0 Td [(1 )]TJ/F26 11.9552 Tf 9.298 0 Td [( E = R )]TJ/F26 11.9552 Tf 9.299 0 Td [(5963.6 K UA = V C p 0.3 min )]TJ/F27 7.9701 Tf 6.587 0 Td [(1 H = C p )]TJ/F26 11.9552 Tf 9.299 0 Td [(11.92 L K = mol Afterlinearization,augmentthelinearstatespaceequationbystatedisturbance d k and p k .TheCSTRmodelbecomes x k +1 = A d x k + B d u k + d k andthelinearoutputequationisgivenby y k = C d x k + D d u k + p k where A d = 2 6 4 1.47659.7527 )]TJ/F26 11.9552 Tf 9.298 0 Td [(0.2329 )]TJ/F26 11.9552 Tf 9.298 0 Td [(1.8182 3 7 5 34

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B d = 2 6 4 010.3 100 3 7 5 C d = 2 6 4 10 01 3 7 5 D d = 2 6 4 000 000 3 7 5 d istheunmeasuredstatedisturbanceand p istheoutputdisturbance. 3.3.2.3Offset-freeperformance Thecontrollabilitymatrixofthepair A d B d isfull-rowrank,andasisthe controllabilitymatrixofthepair )]TJ/F26 11.9552 Tf 6.376 -7.027 Td [( A B ,where A = 2 6 4 A d 0 )]TJ/F44 11.9552 Tf 9.299 0 Td [(A d I 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 9.298 0 Td [(0.0110 0 )]TJ/F26 11.9552 Tf 9.298 0 Td [(0.5 )]TJ/F26 11.9552 Tf 9.298 0 Td [(11 3 7 7 7 7 7 7 7 5 B = 2 6 4 I n )]TJ/F44 11.9552 Tf 9.299 0 Td [(C 3 7 5 B d = 2 6 6 6 6 6 6 6 4 0 0 1 0 3 7 7 7 7 7 7 7 5 whichindicatesthatthesystemcanachieveoffset-freeperformancewithproper gainmatrices K 1 and K 2 .Relatedsimulationresultsaregiveningure3-4.Each graphinthegureshowscertainvariablesasafunctionoftime.Thesevariables 35

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includethesysteminput,setpointandoutput,stateandoutputditurbances. ThecontrolledvariableistheoutletconcentrationoftheCSTRanditssetpoint decreasesfrom0to-1at k =30min inordertoexaminethecontrolperformance undertheconditionofswitchingthereactionconversionfromalowratetoa relativelyhighone.Thereisastepchangefrom0to1instatedisturbance2at k =100min andachangefrom0to1inoutputdisturbance2at k =150min Figure3-4showstheoffset-freeperformancefortheclosed-looplinear discreteCSTRsystem.Thediscretecontrollerdesignedherecandeliver satisfactoryperformanceonbothsetpointtrackinganddisturbancerejection. 3.3.2.4ControlPerformanceonthelinearcontinuousandnonlinearCSTR systems Usethesamediscreteintegralcontrolleronthelinearcontinuousand nonlinearCSTRsystems.Applyingthesamestepchangesinsetpoint,stateand outputdisturbancesassubsection3.3.2.3,thesystemresponsesareshownin gure3-5.Thegraphsforstatedisturbanceandoutputdisturbancearethesame asingure3-4,soonlythegraphsontheCSTRinput,outputandsetpointare giveninthegure. Fromgure3-5itisreasonabletoconcludethatoffset-freecontrolperformanceforthelinearcontinuousCSTRsystemcanbeachievedusingthesame integralcontrollerwhichtracksthesetpointandrejectsdisturbanceverywell. Figure3-6isobtainedwhiletestingcontrolleronthenonlinearsystem.Usingthe identicalstepchangesonsetpointanddisturbanceassubsection3.3.2.3,only thegraphsontheCSTRinput,outputandsetpointareshowninthegure.The x-axisisthetime,andthey-axisrepresentstheCSTRinputintherstgraphand theoutputwithitssetpointinthesecondgraph. 36

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Figure3-4.Offset-freeperformanceofthelineardiscreteclosed-loopCSTR system.Therearestepchangesinsetpoint,statedisturbanceand outputdisturbanceatdifferenttime. 37

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Asisshowningure3-6,duetomodelmismatch,althoughthecontroller successfullyachievestheoffset-freecontrolperformance,strongoscillation existsintheclosed-loopsystem.Forsetpointtrackingthetransientprocess takesabout50 min ,whichistoolongforthesystemtobestabilized. 38

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Figure3-5.ControlperformanceonthelinearcontinuousCSTRsystemusing identicalstepchangesinsetpoint,statedisturbanceandoutput disturbancetothosegraphsingure3-4. 39

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Figure3-6.ControlperformanceonthenonlinearCSTRsystemusingthe identicalstepchangesinsetpoint,statedisturbanceandoutput disturbanceassubsection3.3.2.3. 40

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CHAPTER4 CONTROLSYSTEMDESIGNANDIMPLEMENTATIONUSINGINCREMENTAL MPC 4.1BasicEquationsforIncrementalMPC Crisalle[12]describesthedesignprocedureoftheincrementalMPC.The incrementalMPCisanoffset-freeMPCcontrollerbasedontheincremental state-spacepredictionmodel ^ x k + j +1= I + A ^ x k + j )]TJ/F44 11.9552 Tf 11.955 0 Td [(A ^ x k + j )]TJ/F26 11.9552 Tf 11.955 0 Td [(1+ B u k + j + ^ d k + j ^ y k + j +1= C ^ x k + j +1+^ p k + j +1 ^ x = x 0 where j =0,1,..., N p u k 2 R p ,^ x k 2 R n ,^ y k 2 R m A 2 R n n B 2 R n m and C 2 R m n Assumethatthecurrentstateandoutputvectorcanbedirectlymeasured. i.e. ^ x k := x k ^ y k := y k Forconstantstateandoutputdisturbance,thedisturbancemodelcanbewritten as d k +1= d k p k +1= p k d k =0 p k =0 TheincrementalMPCcontrolobjectiveisgivenby J k = r )]TJ/F26 11.9552 Tf 12.244 0 Td [(^ y T Q a r )]TJ/F26 11.9552 Tf 12.243 0 Td [(^ y + u T T a u 41

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where r = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 r k +1 r k +2 r k + N c r k + N p 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 and u = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 u k u k +1 u k +2 u k + N c )]TJ/F26 11.9552 Tf 11.955 0 Td [(2 u k + N c )]TJ/F26 11.9552 Tf 11.955 0 Td [(1 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 Byminimizing4,thefollowingaugmented-incrementalcontrollawisobtained. u = K r )]TJ/F26 11.9552 Tf 12.244 0 Td [(^ y cf where K = D T I Q a D I + T a )]TJ/F27 7.9701 Tf 6.587 0 Td [(1 D T I Q a 2 R pN c mN p isthecontrollergainand ^ y cf = U ^ y y k + C I [^ x k )]TJ/F26 11.9552 Tf 12.139 0 Td [(^ x k )]TJ/F26 11.9552 Tf 11.955 0 Td [(1]+ G I ^ d k + I I ^ p k istheconstant-forcingresponse.Thenotationin4isasfollows 42

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U ^ y = 2 6 6 6 6 6 6 6 6 6 6 4 I I I 3 7 7 7 7 7 7 7 7 7 7 5 I 2 R m m C I = 2 6 6 6 6 6 6 6 6 6 6 4 C 1 P i =0 A i )]TJ/F44 11.9552 Tf 11.955 0 Td [(I C 2 P i =0 A i )]TJ/F44 11.9552 Tf 11.955 0 Td [(I C N p P i =0 A i )]TJ/F44 11.9552 Tf 11.955 0 Td [(I 3 7 7 7 7 7 7 7 7 7 7 5 D I = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 4 CB 00...0 C 1 P i =0 A i BCB 0...0 . ...0 C N p )]TJ/F27 7.9701 Tf 6.586 0 Td [(2 P i =0 A i BC N p )]TJ/F27 7.9701 Tf 6.586 0 Td [(3 P i =0 A i BC N p )]TJ/F27 7.9701 Tf 6.587 0 Td [(4 P i =0 A i B ... C N p )]TJ/F45 7.9701 Tf 6.586 0 Td [(N c )]TJ/F27 7.9701 Tf 6.587 0 Td [(1 P i =0 A i B C N p )]TJ/F27 7.9701 Tf 6.586 0 Td [(1 P i =0 A i BC N p )]TJ/F27 7.9701 Tf 6.586 0 Td [(2 P i =0 A i BC N p )]TJ/F27 7.9701 Tf 6.587 0 Td [(3 P i =0 A i B ... C N p )]TJ/F45 7.9701 Tf 6.586 0 Td [(N c P i =0 A i B 3 7 7 7 7 7 7 7 7 7 7 7 7 7 5 G I = 2 6 6 6 6 6 6 6 6 6 6 6 6 4 C C A + I C 2 P i =0 A i C N p )]TJ/F27 7.9701 Tf 6.587 0 Td [(1 P i =0 A i 3 7 7 7 7 7 7 7 7 7 7 7 7 5 and 43

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I I = 2 6 6 6 6 6 6 6 4 I 2 I N p I 3 7 7 7 7 7 7 7 5 Dening k T I = U T u j u k )]TJ/F27 7.9701 Tf 6.587 0 Td [(1 K = I 0...0 K where U T u j u k )]TJ/F27 7.9701 Tf 6.587 0 Td [(1 2 R pN c p andtheaugmented-incrementalMPCcontrollawforthe currentincrementalvector u k is u k = k T I h r )]TJ/F44 11.9552 Tf 11.955 0 Td [(U ^ y y k )]TJ/F44 11.9552 Tf 11.955 0 Td [(C I [^ x k )]TJ/F26 11.9552 Tf 12.139 0 Td [(^ x k )]TJ/F26 11.9552 Tf 11.955 0 Td [(1] )]TJ/F44 11.9552 Tf 11.955 0 Td [(G I ^ d k )]TJ/F44 11.9552 Tf 11.955 0 Td [(I I ^ p k i Noting u k = u k )]TJ/F26 11.9552 Tf 11.955 0 Td [(1+ u k ,theincrementalMPCcontrollawis u k = k T I [ r )]TJ/F44 11.9552 Tf 11.955 0 Td [(U ^ y y k ] )]TJ/F44 11.9552 Tf 11.955 0 Td [(k T I C I [^ x k )]TJ/F26 11.9552 Tf 12.139 0 Td [(^ x k )]TJ/F26 11.9552 Tf 11.955 0 Td [(1] )]TJ/F44 11.9552 Tf 11.955 0 Td [(k T I G I ^ d k )]TJ/F44 11.9552 Tf 11.955 0 Td [(k T I I I ^ p k or u k = u k )]TJ/F26 11.9552 Tf 10.005 0 Td [(1+ k T I [ r )]TJ/F44 11.9552 Tf 11.955 0 Td [(U ^ y y k ] )]TJ/F44 11.9552 Tf 10.005 0 Td [(k T I C I [^ x k )]TJ/F26 11.9552 Tf 10.189 0 Td [(^ x k )]TJ/F26 11.9552 Tf 10.005 0 Td [(1] )]TJ/F44 11.9552 Tf 10.005 0 Td [(k T I G I ^ d k )]TJ/F44 11.9552 Tf 10.005 0 Td [(k T I I I ^ p k 4.2ControlValidation 4.2.1Example1:CSTRPlantfromSeborg[2] Usethelineardiscretesystemdiscussedinsubsection3.3.2todesign theincrementalMPC.TheincrementalMPCcontroldiagramisshownas4-1. TheinputstotheMPCcontrollersaretheoutputsandstateoftheplant.Such 44

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Figure4-1.Controldiagramofaclosed-loopsystemwiththeincrementalMPC controller informationisusedtominimizethecostfunctionandnallytondtheoptimal controlleroutputwhichisalsotheinputtotheplantsystem. DesignthecontrollerintheMATLAB/Simulinkenvironment,theclosed-loop controldiagramisgiveningure4-2.Theplantmodelisthelineardiscrete CSTRdescribedinsubsection3.3.2whichcontainsunmeasuredstatedisturbance d andoutputdisturbance p .TheincrementalMPCcontrollerutilizesthe currentstateandoutputtocalculatethecontrolleroutput.Theloopswitchinthe controldiagramisusedtosettheplanttoworkundereithertheopen-loopand theclosed-loopmode. Runtheclosed-loopmodel,usingthetheidentialstepchangesas3.3.2.3in setpointanddisturbance.Thesimulationresultsareshowningure4-3.Since allthestepchangesregardingunmeasuredstateandoutputdisturbanceare thesameasthoseinsubsection3.3.2.3,onlythegraphsrelatedtothesystem input,outputandsetpointaregiveninthegure.Thetopgraphshowstheinput totheCSTRsystemasafunctionoftimeandthesecondgraphpresentsthe systemoutputandsetpointasafunctionoftime.Itcanbeconcludedthatthe incrementalMPCcontrollercanachieveoffset-freeperformanceonthediscrete 45

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CSTRsystem.Thecontrollersuccessfullytracksthesetpointaswellasrejects disturbanceswithoutoffset. Usetheidenticalcontrolleronthelinearcontiunuoussystemtotestits controlperformance.Thesimulationresultsaregiveningure4-4.Asinthe caseofthesimulationofthelineardiscretesystem,onlythegraphsonthe CSTRsysteminput,outputandsetpointareshowninthegure.Thetopgraph depictssysteminput,wherethechangeontheinletconcentrationislargerthan theinputforthediscretesystemofgure4-3.Thebottomgraphshowsthe outletconcentrationwithitssetpointasafunctionoftime.Theoutputcontrol performanceisbetterthangure4-3sincethereisnooscillationontheoutlet concentrationinthesecondgraph.Italsocanbeseenfromgure4-4thatoffsetfreeperformanceisachievedbyusingthesameincrementalMPCcontroller,and theperformanceforsetpointtrackinganddisturbancerejectionissatisfactory. FurthertestoftheincrementalMPCcontrolleronthenon-linearCSTR systemiscarriedout.Thecontrolperformanceisshowningure4-5.The designedcontrollerisbasedonthelineardiscreteCSTRmodelandnowit isappliedtothenonlinearmodel.However,thecontrolperformanceisstill satisfactory.Comparedtogure3-6,theincrementalMPCtracksthesetpoint andrejectsdisturbancewithmuchbetterperformancethanthediscreteintegral controller. 46

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Figure4-2.MATLAB/Simulinkdiagramoftheclosed-loopsystemwiththe incrementalMPCcontroller 47

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Figure4-3.SimulationresultsforthelineardiscreteCSTRsystemwiththe incrementalMPCcontroller 48

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Figure4-4.SimulationresultsforthelinearcontinuousCSTRsystemwith incrementalMPCcontroller 49

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Figure4-5.SimulationresultsforthenonlinearCSTRsystemwithincremental MPCcontroller 50

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4.2.2Example2:QuadrupleTankSystem Thequadruple-tanklaboratoryprocessisusedtotestthecontrolperformanceofMPCdesignedbykesson[3].Thesystemconsistsoffourtanksand theirpositionsarearrangedasatwo-by-twomatrix,wherewaterfromthetwo toptanksowsintothetwobottomtanks.Twopumpsareused.Oneisadopted totransportwaterintothetoplefttankandthebottomrighttank.Theotherone isusedtotransportwaterintothetoprighttankandbottomlefttank.Avalve isusedtoadjustpumpcapacitytothetopandbottomtankrespectively.The controlvariablesarethepumpvoltages.Letthestatesofthesystembedened bythewaterlevelsofthetanksexpressedincm x 1 x 2 x 3 and x 4 respectively. Themaximumlevelofeachtankis 20cm .Systemdynamicsaregivenas x 1 = )]TJ/F44 11.9552 Tf 11.589 8.088 Td [(a 1 A 2 p 2 gx 1 + a 3 A 1 p 2 gx 3 + 1 k 1 A 1 u 1 x 2 = )]TJ/F44 11.9552 Tf 11.589 8.088 Td [(a 2 A 2 p 2 gx 2 + a 4 A 2 p 2 gx 4 + 2 k 2 A 2 u 2 x 3 = )]TJ/F44 11.9552 Tf 11.589 8.088 Td [(a 3 A 3 p 2 gx 3 + )]TJ/F29 11.9552 Tf 11.955 0 Td [( 2 k 2 A 3 u 2 x 4 = )]TJ/F44 11.9552 Tf 11.589 8.088 Td [(a 4 A 4 p 2 gx 4 + )]TJ/F29 11.9552 Tf 11.955 0 Td [( 1 k 1 A 4 u 1 wherethe A i sandthe a i srepresentthecrosssectionareaofthetanksandthe tubes,respectively. i saredenedasthepositionofthevalveswhichcontrolthe owratetothetopandbottomtanksrespectively.Thecontrolvariablesarethe u i s.Theobjectiveistocontrolthelevelofthetwobottomtanksdenotedas x 1 and x 2 .Theprocessdiagramispresentedingure4-6.Theparametervalues areshownintable4-1andtheinitialvaluesofthesystemaregivenintable4-2. IncrementalMPCisusedheretotestitscontrolperformanceonthe linearizedquadrupletanksystem.Astepchangeofthesetpointoflevelfor tank 1 of 6cm isappliedat t =60s ,whilethesetpointoftanklevel 2 isheld constant.At t =600s ,aunitstepdisturbanceisappliedtothesecondinput 51

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Figure4-6.Processdiagramforthequadrupletanksystemfromkesson[3]. Table4-1.Parametervaluesforthequadrupletanksystem ParameterNominalValue A 1 A 2 28 cm 2 A 3 A 4 32 cm 2 a 1 a 2 0.071 cm 2 a 3 a 4 0.057 cm 2 k 1 3.33 cm 3 = Vs k 2 3.35 cm 3 = Vs g 981 cm = s 2 Table4-2.Initialvaluesofthequadrupletanksystem ParameterInitialValue x 0 1 8.2444 cm x 0 2 19.0163 cm x 0 3 4.3146 cm x 0 4 8.8065 cm u 0 1 u 0 2 3 V 52

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channel.Thesimulationresultsareshowningure4-7.Thetopgraphpresents thelevelfortank 1 withitssetpointandthebottomgraphshowsthesystem input.Thecontrollerachievesoffset-freeperformanceonbothsetpointtracking anddisturbancerejectionwithsatisfactorytransientprocess.Thelevelfortank 1 dosenotexceed 20cm, whichensuresthatsystemisstable. 53

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Figure4-7.SimulationresultsofincrementalMPConthelinearizedquadruple tanksystem.Thecontrolvariablewithitssetpointwhichispresented inthetopgraphisthelevelfortank1.Twoinputvariablesdenotedas u 1 and u 2 areshowninthebottomgraph. 54

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CHAPTER5 CONTROLSYSTEMDESIGNANDIMPLEMENTATIONUSINGINTEGRAL MPC 5.1BasicEquationsforIntegralMPC Crisalle[12]indicatesthatthestandardstate-spaceequationandthe incrementalstate-spacestateequationcanbeaugmentedwithoneormore integral-stateequations.Inthisthesis,thersttwointegral-states z k 2 R m and w k 2 R m areintroducedas z k +1= z k + r k )]TJ/F44 11.9552 Tf 11.955 0 Td [(y k w k +1= w k + z k where y k 2 R m isthecurrentsystemoutputvectorand r k 2 R m isthecurrent setpointvector. 5.1.1FirstIntegralState z k Therstintegralstateequation5canbeusedtogenerateprediction equations z k +1= z k + r k )]TJ/F44 11.9552 Tf 11.956 0 Td [(y k z k +2= z k +1+ r k +1 )]TJ/F26 11.9552 Tf 12.244 0 Td [(^ y k +1 z k + N p )]TJ/F26 11.9552 Tf 11.955 0 Td [(1= z k + N p + r k + N p )]TJ/F26 11.9552 Tf 11.955 0 Td [(1 )]TJ/F26 11.9552 Tf 12.243 0 Td [(^ y k + N p )]TJ/F26 11.9552 Tf 11.955 0 Td [(1 z k + N p = z k + N p + r k + N p )]TJ/F26 11.9552 Tf 12.243 0 Td [(^ y k + N p 55

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Applyarecursivesubstitutionroutinetothepredictionequationstoobtain First IntegralStatePredictor as 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 z k +1 z k +2 z k +3 z k + N p )]TJ/F26 11.9552 Tf 11.955 0 Td [(1 z k + N p 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 I I I I I 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 z k + 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 I I I I I 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 [ r k )]TJ/F44 11.9552 Tf 11.955 0 Td [(y k ] + 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 000 00 I 00 00 II 0 00 . . III 00 III I 0 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 8 > > > > > > > > > > > > > > < > > > > > > > > > > > > > > : 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 r k +1 r k +2 r k +3 r k + N p )]TJ/F26 11.9552 Tf 11.955 0 Td [(1 r k + N p 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 )]TJ/F35 11.9552 Tf 11.955 74.242 Td [(2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 ^ y k +1 ^ y k +2 ^ y k +3 ^ y k + N p )]TJ/F26 11.9552 Tf 11.955 0 Td [(1 ^ y k + N p 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 9 > > > > > > > > > > > > > > = > > > > > > > > > > > > > > ; whichcanberewrittenasthefollowingcompactformthatisreferredasthe First IntegralStatePredictionEquation z = U ^ y [ r k )]TJ/F44 11.9552 Tf 11.955 0 Td [(y k ] + U ^ y z k + V ^ y r )]TJ/F44 11.9552 Tf 11.955 0 Td [(V ^ y ^ y where U ^ y isdenedas4and V ^ y istheaugmentedmatrixas V ^ y = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 000 00 I 00 00 II 0 00 . . III 00 III I 0 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 whichconsistsofthematrices I 2 R m m and 0 2 R m m 56

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5.1.2SecondIntegralState w k Thesecondintegralstateequation5canbeshiftedforwardtogenerate thepredictionequations w k +1= w k + z k w k +2= w k +1+ z k +1 w k + N p )]TJ/F26 11.9552 Tf 11.955 0 Td [(1= w k + N P + z k + N p )]TJ/F26 11.9552 Tf 11.955 0 Td [(1 w k + N p = w k + N p + z k + N p Applyarecursivesubstitutionroutinetothepredictionequationstoobtain SecondIntegralStatePredictor 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 w k +1 w k +2 w k +3 w k + N p )]TJ/F26 11.9552 Tf 11.955 0 Td [(1 w k + N p 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 I I I I I 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 z k + 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 I I I I I 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 w k + 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 000 00 I 00 00 II 0 00 . . III 00 III I 0 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 z k +1 z k +2 z k +3 z k + N p )]TJ/F26 11.9552 Tf 11.955 0 Td [(1 z k + N p 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 whichcanberewrittenasthefollowingcompactform w = U ^ y z k + U ^ y w k + V ^ y z where U ^ y and V ^ y aretheaugmentedmatricesdenedas4and5. Substitutetherstintegralstatevector z intherighthandsideof5by5 toobtainthefollowing SecondIntegralStatePredictionEquation w = V ^ y U ^ y [ r k )]TJ/F44 11.9552 Tf 11.955 0 Td [(y k ] + V ^ y U ^ y + U ^ y z k + U ^ y w k + V 2 ^ y r )]TJ/F44 11.9552 Tf 11.955 0 Td [(V 2 ^ y ^ y 57

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5.1.3ControlLawfortheIntegralMPCController ConsideringtheintegralMPCperformanceobjective J k = r )]TJ/F26 11.9552 Tf 12.244 0 Td [(^ y T Q a r )]TJ/F26 11.9552 Tf 12.243 0 Td [(^ y + u T T a u + u T R a u + r )]TJ/F26 11.9552 Tf 12.243 0 Td [(^ y T S a u thecontrollawfortheintegralMPCisgivenby u k = k T I [ r )]TJ/F44 11.9552 Tf 11.955 0 Td [(U ^ y y k ] + k z z + k w w or u k = u k )]TJ/F26 11.9552 Tf 11.955 0 Td [(1+ k T I [ r )]TJ/F44 11.9552 Tf 11.956 0 Td [(U ^ y y k ] + k z z + k w w asisdescribedinPeek[13],where z and w aregivenbyequation5and 5respectively.Thegain k T I isdenedas4.Thegains k z and k w for z and w aredenotedas k z = k T I Q )]TJ/F27 7.9701 Tf 6.586 0 Td [(1 a V T ^ y S a k w = k z S )]TJ/F27 7.9701 Tf 6.587 0 Td [(1 a V T ^ y T a 5.2ControlValidation DesigntheintegralMPCcontrollerbasedonthelineardiscreteCSTR system3.3.2thentestitsclosed-loopcontrolperformanceusingtheidentialstep changesonthesetpointanddisturbanceassubsection3.3.2.3.Thesimulation resultisshowningure3.TheintegralMPCcontroldiagramandSimulink diagramaresimilartotheincrementalMPCandhencetheyarenotgivenhere.It canbeconcludedfromthetopgraphinthegurethattheintegralMPCcontroller successfullytracksthesetpointaswellasrejectthestateandoutputdisturbance withoutoffset. ContinuetotesttheintegralMPCcontrolleronthelinearcontinuousCSTR systemusingthesamestepchanges.Thesimulationresultisgiveningure 5-2.ItcanbeseenfromthegurethattheintegralMPCcontrollersuccessfully 58

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Figure5-1.Closed-loopsimulationresultforthelineardiscreteCSTRsystem withanintegralMPCcontroller.TheCSTRisgivenbysubsection 3.3.2andthestepchangesarepresentedinsubsection3.3.2.3 59

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achievesthesatisfactoryoffset-freecontrolperformanceonbothsetpoint trackinganddisturbancerejection. FinallyrunthesimulationforthenonlinearCSTRsystemwiththeintegral MPCcontrollerusingthesamestepchangesonthesetpointanddisturbance. Thesimulationresultispresentedbygure5-3.Itcanbeconcludedthatthe integralMPCcontrollersuccessfullytracksthesetpointandrejectsdisturbance. Alsoitsoffseteliminationperformanceissatisfactory.Furthervalidationprocess canbeadoptedtotesttheintegralMPCcontrollerontherealCSTRsystem. 60

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Figure5-2.Closed-loopsimulationresultforthelinearcontinuousCSTRsystem withanintegralMPCcontroller.TheCSTRisgivenbysubsection 3.3.2andthestepchangesarepresentedinsubsection3.3.2.3 61

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Figure5-3.Closed-loopsimulationresultforthenonlinearCSTRsystemwithan integralMPCcontroller.TheCSTRisgivenbysubsection3.3.2and thestepchangesarepresentedinsubsection3.3.2.3 62

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CHAPTER6 CONCLUSIONSANDFUTUREWORK 6.1Conclusions Inthisthesisthecontrolperformanceofthreedifferentcontrollersisexaminedusingthesameplantmodelandstepchangesonboththesetpointand disturbance.Theyarediscreteintegralcontroller,incrementalMPCcontroller andintegralMPCcontroller.Allofthemcansuccessfullytrackthesetpointand eliminatethesetpointwithoutoffsetwithsatisfactoryclosed-loopcontrolperformanceontheplantsystemoflineardiscreteCSTRandlinearcontinuousCSTR. However,duetotheincreasedmodelmismatch,thecontrolperformanceofthe discreteintegralcontrollerdeterioratessignicantlywhenitisappliedtothenonlinearCSTRsystem.ItssimulationresultonthenonlinearCSTRshowsstrong oscillationonthecontrolledvariable.FortheincrementalandintegralMPCcontroller,satisfactoryoffset-freecontrolperformanceisachievedonthenonlinear CSTRsystem,whichindicatestheyaremorerobusttomodelmismatchthan thediscreteintegralcontroller.Intheapplication,whenthemodeluncertainty issignicant,itispreferredtousetheincrementalortheintegralMPCcontrol strategyinsteadofthediscreteintegralcontrol. FortheincrementalandintegralMPCcontrollersimplementedusingSfunctions,thereisaproblemonthesigularityofthealgebraicloop.Thisisdueto whenag=2call,theS-functionwillautomaticallyupdatethestatevector,which willneedthecurrentplantinput u k .Toeliminatethealgebraicloopsingularity, amemoryblockisneededattheentryofthecontroller. 6.2FutureWork:ConstrainedIncrementalMPC Thefutureobjectiveisthedesignandimplementationoftheconstrained incrementalMPCcontrollerundertheMATLAB/Simulinkenvironment.Inthis thesis,onlytheunconstrainedincrementalMPCisdiscussed. 63

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Wang[14]introducestheapproachtodesigntheconstrainedMPC,which indicatesthattheconstrainedcontrollprobleminthecontextofpredictivecontrol isactuallyaquadraticprogrammingproblem.Thefrequentlyusedconstraintsin applicationsarethecontrolvariable u k ,theoutput y k ,thestatevariable x k andtherateofchangeofthecontrolvariable u k .Theupperandlowerlimit maybeimposedintotheseconstraintsrepresentedas u min u k u max y min y k y max x min x k x max u min u k u max ThestepsfortheconstraintMPCdesignare: 1.Specifyingsystemoperationallimits,includingcontrolvariable u k ,output y k ,statevariable x k andrateofchangeofthecontrolvariable u k 2.Expressingthelimitsbyusingthenotationsofminimumandmaximumof u k y k x k and u k 3.Expressingtheseminimumandmaximumvaluesintheformofinequalities thatconsistoftheparameterizedfuturecontroltrajectory u k u k + 1,... u k + N c )]TJ/F26 11.9552 Tf 11.955 0 Td [(1 4.ThedesignoftheconstrainedMPCnowisconvertedtotheminimization oftheoriginalpenaltyfunctionsubjecttotheinequalities,wherethe parameters u k u k +1,... u k + N c )]TJ/F26 11.9552 Tf 12.725 0 Td [(1 becomethedecision variables. 5.Usingquadraticprogrammingmethodstosolvetheconstrainedoptimizationproblemateachsamplinginstanttoobtaintheoptimaldecision variables. Sincetheconstraintsareexpressedasinequalities,generallyspeakingthere isnoanalyticalsolutiontotheconstrainedcontrolproblem,unlesstheactive constraintsforeachsamplinginstantareknown.Undertheconditionthatthe activeconstraintsareknown,optimalsolutionofdecisionvariablescanbefound 64

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inanalyticalform.Forthemostofpracticalapplications,insteadofiteratively solvingforthedecisionvariables,Hildreth'sprogrammingroutinecanbeused toidentifytheactivesetofconstraintsviaLagrangemultipliersinCrisalle[15], whichwouldmaketheconstraintsmuchsimplerandthusonlyaneasyiterative routineisneededinndingtheoptimalsolutionofthemultipliers.Thisiscrucial fortheonlineimplementationoftheMPCcontroller. 65

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AppendixA DERIVATIONOF K 1 AND K 2 FORTHEDISCRETEINTEGRALCONTROLLER Thestateandtheintegralstateatthetime k +1 are 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 d [ A d x k + B d u k ] = )]TJ/F44 11.9552 Tf 9.299 0 Td [(C d A d x k + z k )]TJ/F44 11.9552 Tf 11.955 0 Td [(C d B d u k + r k +1 x k +1 = A d x k + B d u k = A d x k + B d [ )]TJ/F44 11.9552 Tf 9.299 0 Td [(K 2 x k + K 1 z k ] Inthisservosystem,thedesignparametersarethegainmatrices K 1 and K 2 sothatthesystemhasthedesiredclosed-looppoles.Fromthestate-space equation,calculate u k +1 as u k +1= )]TJ/F44 11.9552 Tf 9.298 0 Td [(K 2 x k +1+ K 1 z k +1 = K 2 )]TJ/F44 11.9552 Tf 11.955 0 Td [(K 2 A d )]TJ/F44 11.9552 Tf 11.955 0 Td [(K 1 C d A d x k + I m )]TJ/F44 11.9552 Tf 11.955 0 Td [(K 2 B d )]TJ/F44 11.9552 Tf 11.955 0 Td [(K 1 C d B d u k + K 1 r k +1 A Consideringthestepinput, x k and u k convergetotheconstantvector, named x 1 and u 1 .Atsteadystate,equation3becomes 2 6 4 x 1 u 1 3 7 5 = 2 6 4 A d B d K 2 )]TJ/F44 11.9552 Tf 11.955 0 Td [(K 2 A d )]TJ/F44 11.9552 Tf 11.955 0 Td [(K 1 C d A d I m )]TJ/F44 11.9552 Tf 11.956 0 Td [(K 2 B d )]TJ/F44 11.9552 Tf 11.955 0 Td [(K 1 C d B d 3 7 5 2 6 4 x 1 u 1 3 7 5 + 2 6 4 0 K 1 r 3 7 5 Usingthedeviationvectors ~ x k = x k )]TJ/F44 11.9552 Tf 11.955 0 Td [(x 1 ~ u k = u k )]TJ/F44 11.9552 Tf 11.955 0 Td [(u 1 3canberewrittenas 66

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2 6 4 ~ x k +1 ~ u k +1 3 7 5 = 2 6 4 A d B d K 2 )]TJ/F44 11.9552 Tf 11.955 0 Td [(K 2 A d )]TJ/F44 11.9552 Tf 11.955 0 Td [(K 1 C d A d I m )]TJ/F44 11.9552 Tf 11.955 0 Td [(K 2 B d )]TJ/F44 11.9552 Tf 11.955 0 Td [(K 1 C d B d 3 7 5 2 6 4 ~ x k ~ u k 3 7 5 A Forconvenience,weusetheequivalentform 2 6 4 ~ x k +1 ~ u k +1 3 7 5 = 2 6 4 A d B d 00 3 7 5 2 6 4 ~ x k ~ u k 3 7 5 + 2 6 4 0 I m 3 7 5 w k A where w k = K 2 )]TJ/F44 11.9552 Tf 11.955 0 Td [(K 2 A d )]TJ/F44 11.9552 Tf 11.955 0 Td [(K 1 C d A d I m )]TJ/F44 11.9552 Tf 11.955 0 Td [(K 2 B d )]TJ/F44 11.9552 Tf 11.955 0 Td [(K 1 C d B d 2 6 4 ~ x k ~ u k 3 7 5 Given k = 2 6 4 ~ x k ~ u k 3 7 5 2 R n + m A = 2 6 4 A d B d 00 3 7 5 2 R n + m n + m B = 2 6 4 0 I m 3 7 5 2 R n + m m K = )]TJ/F35 11.9552 Tf 11.291 16.857 Td [( K 2 )]TJ/F44 11.9552 Tf 11.955 0 Td [(K 2 A d )]TJ/F44 11.9552 Tf 11.955 0 Td [(K 1 C d A d I m )]TJ/F44 11.9552 Tf 11.955 0 Td [(K 2 B d )]TJ/F44 11.9552 Tf 11.955 0 Td [(K 1 C d B d 2 R m n + m Abecomes k +1= A k + Bw k A w k = )]TJ/F26 11.9552 Tf 11.078 2.656 Td [( K k 67

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FortheaugmentedsystemA,thecontrollabilitymatrixisdendedas B A B ... A n + m )]TJ/F27 7.9701 Tf 6.586 0 Td [(1 B 2 R n + m m n + m A Intermsof A d and B d Acanbeexpressedas B A B ... A n + m )]TJ/F27 7.9701 Tf 6.587 0 Td [(1 B = 2 6 4 0 B d A d B d ... A n + m )]TJ/F27 7.9701 Tf 6.587 0 Td [(2 d B d I m 00...0 3 7 5 A Sinceitisassumedthat3iscompletelystatecontrollable,therankofmatrix B d A d B d ... A n )]TJ/F27 7.9701 Tf 6.586 0 Td [(1 d B d equals n .ThereforetherankofAequals n + m ,whichmakesitfeasible toarbitrarilyplacedesiredclosed-looppolesofAbyusingpole-placement method.Firstnoticethat K 2 K 1 2 6 4 A d )]TJ/F44 11.9552 Tf 11.955 0 Td [(I n B d C d A d C d B d 3 7 5 = K 2 A d )]TJ/F44 11.9552 Tf 11.955 0 Td [(K 2 + K 1 C d A d K 2 B d + K 1 C d B d A andthussolveforAtoobtain K = K 2 A d )]TJ/F44 11.9552 Tf 11.956 0 Td [(K 2 + K 1 C d A d )]TJ/F44 11.9552 Tf 9.298 0 Td [(I m + K 2 B d + K 1 C d B d = K 2 K 1 2 6 4 A d )]TJ/F44 11.9552 Tf 11.955 0 Td [(I n B d C d A d C d B d 3 7 5 + 0 )]TJ/F44 11.9552 Tf 9.298 0 Td [(I m Tond K 1 and K 2 ,noticethat K 2 K 1 2 6 4 A d )]TJ/F44 11.9552 Tf 11.955 0 Td [(I n B d C d A d C d B d 3 7 5 = K )]TJ/F35 11.9552 Tf 11.955 16.857 Td [( 0 )]TJ/F44 11.9552 Tf 9.299 0 Td [(I m 68

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Then K 1 and K 2 areexpressedas K 2 K 1 = K + 0 I m 2 6 4 A d )]TJ/F44 11.9552 Tf 11.955 0 Td [(I n B d C d A d C d B d 3 7 5 )]TJ/F27 7.9701 Tf 6.587 0 Td [(1 69

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REFERENCES [1]K.Ogata, Discrete-timecontrolsystems .Prentice-HallEnglewoodCliffs, NJ,1995,vol.8. [2]D.E.Seborg,D.A.Mellichamp,T.F.Edgar,andF.J.DoyleIII, Process dynamicsandcontrol .Wiley,2010. [3]J.kesson, MPCtools1.0ReferenceManual ,DepartmentofAutomatic ControlLundInstituteofTechnology,Box118SE-22100LundSweden, January2006. [4]J.B.Rawlings, ModelPredictiveControl:TheoryandDesign .NobHill Publishing,2009. [5]U.Maeder,F.Borrelli,andM.Morari,Linearoffset-freemodelpredictive control, Automatica ,vol.45,no.10,pp.2214,2009. [6]Y.Qian,Conditionsandmethodsforoffset-freeperformanceindiscrete controlsystems,Master'sthesis,UniversityofFlorida,2012. [7]U.MaederandMorari,Offset-freereferencetrackingforpredictivecontrollers,in DecisionandControl,200746thIEEEConferenceonDecision andControl .IEEE,2007,pp.5252. [8]U.MaederandM.Morari,Offset-freereferencetrackingwithmodel predictivecontrol, Automatica ,vol.46,no.9,pp.1469,2010. [9]G.PannocchiaandJ.B.Rawlings,Disturbancemodelsforoffset-free model-predictivecontrol, AIChEJournal ,vol.49,no.2,pp.426,2003. [10]J.B.Rawlings,E.S.Meadows,andK.R.Muske,Nonlinearmodelpredictivecontrol:Atutorialandsurvey,in PreprintsIFACSymposiumADCHEM, Kyoto,Japan ,1994,pp.203. [11]MathWorks, WritingS-Functions [12]O.D.Crisalle, ModelPredictiveControl-AState-SpaceAnalysisand DesignApproach ,2011. [13]C.S.Peek,Highperformancecontroltheory,design,andapplications, Ph.D.dissertation,UniversityofFlorida,2008. [14]L.Wang, ModelPredictiveControlSystemDesignandImplementation UsingMATLAB .Springer,2009. [15]O.D.Crisalle,Introductiontooptimalcontrol,Coursenotes,2011. 70

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BIOGRAPHICALSKETCH XinLin receivedhisMasterofSciencedegreeinchemicalengineering fromtheUniversityofFloridainthespringof2013andbachelor'sdegreein automationattheDepartmentofControlScienceandEngineeringfromZhejiangUniversityHangzhou,Chinain2011.InJanuary2011hejoinedthe processcontrolresearchgroupundertheguidanceofDr.OscarD.Crisalle.As amemberofthegroup,hisresearchconcentrationlayonthedesignandimplementationoftheincrementalandintegralcontrollersintheMATLAB/Simulink environment. 71