AdvancedChemicalProcessControl
PuttingTheoryintoPractice
MortenHovd
ToEllen
Contents
Preface xvii
Acknowledgments xxi
Acronyms xxiii
Introduction xxv
1MathematicalandControlTheoryBackground 1
1.1Introduction 1
1.2ModelsforDynamicalSystems 1
1.2.1DynamicalSystemsinContinuousTime 1
1.2.2DynamicalSystemsinDiscreteTime 2
1.2.3LinearModelsandLinearization 3
1.2.3.1LinearizationataGivenPoint 3
1.2.3.2LinearizingAroundaTrajectory 6
1.2.4ConvertingBetweenContinuous-andDiscrete-TimeModels 6
1.2.4.1TimeDelayintheManipulatedVariables 7
1.2.4.2TimeDelayintheMeasurements 9
1.2.5LaplaceTransform 9
1.2.6The z Transform 10
1.2.7SimilarityTransformations 11
1.2.8MinimalRepresentation 11
1.2.9Scaling 14
1.3AnalyzingLinearDynamicalSystems 15
1.3.1TransferFunctionsofCompositeSystems 15
1.3.1.1SeriesInterconnection 15
1.3.1.2ParallelSystems 16
1.3.1.3FeedbackConnection 16
1.3.1.4CommonlyUsedClosed-LoopTransferFunctions 17
1.3.1.5ThePush-ThroughRule 17
1.4PolesandZerosofTransferFunctions 18
1.4.1PolesofMultivariableSystems 19
1.4.2PoleDirections 19
1.4.3ZerosofMultivariableSystems 20
1.4.4ZeroDirections 22
2.2InputandOutputSelection 62
2.2.1UsingPhysicalInsights 63
2.2.2Gramian-BasedInputandOutputSelection 64
2.2.3Input/OutputSelectionforStabilization 65
2.3ControlConfiguration 66
2.3.1TheRelativeGainArray 66
2.3.2TheRGAasaGeneralAnalysisTool 68
2.3.2.1TheRGAandZerosintheRightHalf-Plane 68
2.3.2.2TheRGAandtheOptimallyScaledConditionNumber 68
2.3.2.3TheRGAandIndividualElementUncertainty 69
2.3.2.4RGAandDiagonalInputUncertainty 69
2.3.2.5TheRGAasanInteractionMeasure 70
2.3.3TheRGAandStability 70
2.3.3.1TheRGAandPairingofControlledandManipulatedVariables 71
2.3.4SummaryofRGA-BasedInput–OutputPairing 72
2.3.5PartialRelativeGains 72
2.3.6TheNiederlinskiIndex 73
2.3.7TheRijnsdorpInteractionMeasure 73
2.3.8Gramian-BasedInput–OutputPairing 74
2.3.8.1TheParticipationMatrix 75
2.3.8.2TheHankelInteractionIndexArray 75
2.3.8.3AccountingfortheClosed-LoopBandwidth 76
2.4TuningofDecentralizedControllers 76
2.4.1Introduction 76
2.4.2LoopShapingBasics 77
2.4.3TuningofSingle-LoopControllers 79
2.4.3.1PIDControllerRealizationsandCommonModifications 79
2.4.3.2ControllerTuningUsingFrequencyAnalysis 81
2.4.3.3Ziegler–NicholsClosed-LoopTuningMethod 86
2.4.3.4SimpleFittingofaStepResponseModel 86
2.4.3.5Ziegler–NicholsOpen-LoopTuning 88
2.4.3.6IMC-PIDTuning 88
2.4.3.7SimpleIMCTuning 89
2.4.3.8TheSetpointOvershootMethod 91
2.4.3.9Autotuning 95
2.4.3.10WhenShouldDerivativeActionBeUsed? 95
2.4.3.11EffectsofInternalControllerScaling 96
2.4.3.12ReverseActingControllers 97
2.4.4GainScheduling 97
2.4.5SurgeAttenuatingControllers 98
2.4.6MultiloopControllerTuning 99
2.4.6.1IndependentDesign 100
2.4.6.2SequentialDesign 102
2.4.6.3SimultaneousDesign 103
2.4.7ToolsforMultivariableLoop-Shaping 103
x Contents
2.4.7.1ThePerformanceRelativeGainArray 103
2.4.7.2TheClosed-LoopDisturbanceGain 104
2.4.7.3IllustratingtheUseofCLDG’sforControllerTuning 104
2.4.7.4UnachievableLoopGainRequirements 107 Problems 108 References 112
3ControlStructureSelectionandPlantwideControl 115
3.1GeneralApproachandProblemDecomposition 115
3.1.1Top-DownAnalysis 115
3.1.1.1DefiningandExploringOptimalOperation 115
3.1.1.2DeterminingWheretoSettheThroughput 116
3.1.2Bottom-UpDesign 116
3.2RegulatoryControl 117
3.2.1Example:RegulatoryControlofLiquidLevelinaDeaerationTower 118
3.3DeterminingDegreesofFreedom 121
3.4SelectionofControlledVariables 122
3.4.1ProblemFormulation 123
3.4.2SelectingControlledVariablesbyDirectEvaluationofLoss 124
3.4.3ControlledVariableSelectionBasedonLocalAnalysis 125
3.4.3.1TheMinimumSingularValueRule 127
3.4.3.2DesirableCharacteristicsoftheControlledVariables 128
3.4.4AnExactLocalMethodforControlledVariableSelection 128
3.4.5MeasurementCombinationsasControlledVariables 130
3.4.5.1TheNullspaceMethodforSelectingControlledVariables 130
3.4.5.2ExtendingtheNullspaceMethodtoAccountforImplementation Error 130
3.4.6TheValidityoftheLocalAnalysisforControlledVariableSelection 131
3.5SelectionofManipulatedVariables 132
3.5.1VerifyingthattheProposedManipulatedVariablesMakeAcceptable ControlPossible 133
3.5.2ReviewingtheCharacteristicsoftheProposedManipulated Variables 134
3.6SelectionofMeasurements 135
3.7MassBalanceControlandThroughputManipulation 136
3.7.1ConsistencyofInventoryControl 138 Problems 140 References 141
4LimitationsonAchievablePerformance 143
4.1PerformanceMeasures 143
4.1.1Time-DomainPerformanceMeasures 143
4.1.2Frequency-DomainPerformanceMeasures 145
4.1.2.1BandwidthFrequency 145
4.1.2.2PeaksofClosed-LoopTransferFunctions 146
5.2.3OptimizingDeviationsfromLinearStateFeedback 181
5.2.4ConstraintsBeyondtheEndofthePredictionHorizon 182
5.2.5FindingtheTerminalConstraintSet 183
5.2.6FeasibleRegionandPredictionHorizon 184
5.3Step-ResponseModels 185
5.4UpdatingtheProcessModel 186
5.4.1BiasUpdate 186
5.4.2KalmanFilterandExtendedKalmanFilters 187
5.4.2.1AugmentingaDisturbanceDescription 188
5.4.2.2TheExtendedKalmanFilter 189
5.4.2.3TheIteratedExtendedKalmanFilter 189
5.4.3UnscentedKalmanFilter 190
5.4.4RecedingHorizonEstimation 193
5.4.4.1TheArrivalCost 195
5.4.4.2TheFilteringFormulationofRHE 196
5.4.4.3TheSmoothingFormulationofRHE 196
5.4.5ConcludingCommentsonStateEstimation 198
5.5DisturbanceHandlingandOffset-FreeControl 199
5.5.1FeedforwardfromMeasuredDisturbances 199
5.5.2RequirementsforOffset-FreeControl 199
5.5.3DisturbanceEstimationandOffset-FreeControl 200
5.5.4AugmentingtheModelwithIntegratorsatthePlantInput 203
5.5.5AugmentingtheModelwithIntegratorsatthePlantOutput 205
5.5.6MPCandIntegratorResetting 208
5.6FeasibilityandConstraintHandling 210
5.7Closed-LoopStabilitywithMPCControllers 212
5.8TargetCalculation 213
5.9SpeedingupMPCCalculations 217
5.9.1Warm-StartingtheOptimization 218
5.9.2InputBlocking 219
5.9.3EnlargingtheTerminalRegion 220
5.10RobustnessofMPCControllers 222
5.11UsingRigorousProcessModelsinMPC 225
5.12Misconceptions,Clarifications,andChallenges 226
5.12.1Misconceptions 226
5.12.1.1MPCIsNotGoodforPerformance 226
5.12.1.2MPCRequiresVeryAccurateModels 227
5.12.1.3MPCCannotPrioritizeInputUsageorConstraintViolations 227
5.12.2Challenges 227
5.12.2.1ObtainingaPlantModel 228
5.12.2.2Maintenance 228
5.12.2.3CapturingtheDesiredBehaviorintheMPCDesign 228 Problems 228 References 231
7.4PlantwideOscillations 269
7.4.1GroupingOscillatingVariables 269
7.4.1.1SpectralPrincipalComponentAnalysis 269
7.4.1.2VisualInspectionUsingHigh-DensityPlots 269
7.4.1.3PowerSpectralCorrelationMaps 270
7.4.1.4TheSpectralEnvelopeMethod 271
7.4.1.5MethodsBasedonAdaptiveDataAnalysis 272
7.4.2LocatingtheCauseforDistributedOscillations 273
7.4.2.1UsingNonlinearityforRootCauseLocation 273
7.4.2.2TheOscillationContributionIndex 273
7.4.2.3EstimatingthePropagationPathforDisturbances 274
7.5ControlLoopPerformanceMonitoring 278
7.5.1TheHarrisIndex 278
7.5.2ObtainingtheImpulseResponseModel 279
7.5.3CalculatingtheHarrisIndex 280
7.5.4EstimatingtheDeadtime 281
7.5.5ModificationstotheHarrisIndex 282
7.5.6AssessingFeedforwardControl 283
7.5.7CommentsontheUseoftheHarrisIndex 285
7.5.8PerformanceMonitoringforPIControllers 286
7.6MultivariableControlPerformanceMonitoring 287
7.6.1AssessingFeedforwardControlinMultivariableControl 287
7.6.2PerformanceMonitoringforMPCControllers 288
7.7SomeIssuesintheImplementationofControlPerformance Monitoring 290
7.8Discussion 290 Problems 291 References 291
8EconomicControlBenefitAssessment 297
8.1OptimalOperationandOperationalConstraints 297
8.2EconomicPerformanceFunctions 298
8.3ExpectedEconomicBenefit 299
8.4EstimatingAchievableVarianceReduction 300
8.5Worst-CaseBackoffCalculation 300 References 301
AFourier–MotzkinElimination 303
BRemovalofRedundantConstraints 307 Reference 308
CTheSingularValueDecomposition 309
Preface
Halfacenturyago,AlanFoss[1]wrotehisinfluentialpaperaboutthegapbetween chemicalprocesscontroltheoryandindustrialapplication.Fossclearlyputthe responsibilityonthechemicalprocesscontroltheoriststoclosethisgap.Sincethen, severaladvancesincontroltheory,someoriginatingwithinacademia,whileothers originatedinindustryandwaslateradoptedandfurtherdevelopedbyacademia, havecontributedtoaddressingtheshortcomingsofchemicalprocesscontroltheory, asaddressedbyFoss:
● Theextensionoftherelativegainarray(RGA)tononzerofrequenciesand Graminan-basedcontrolstructureselectiontoolshaveextendedthetoolkitfor designingcontrolstructures.1 Self-optimalcontrol[4]providesasystematic approachtoidentifyingcontrolledvariablesforachemicalplant.
● Modelpredictivecontrol(MPC)hasgreatindustrialsuccess[3].
● Robustnesstomodelerrorreceivedsignificantresearchfocusfromthe1980s onward.
● ControlPerformanceMonitoringhas,sincetheseminalpaperbyHarris[2], resultedintoolsbothformonitoringanddiagnosingtheperformanceof individualloopsaswellasforidentifyingthecauseofplantwidedisturbances.
Theaforementionedlistnotwithstanding,manywouldagreetotheclaimthat thereisstillawidegapbetweencontroltheoryandcommonindustrialpracticein thechemicalprocessindustries.Thisbookistheauthor’sattempttocontributingto thereductionofthatgap.Thisbookhastwoambitiousobjectives:
1.Tomakemoreadvancedcontrolaccessibletochemicalengineers,manyofwhom willonlyhavebackgroundfromasinglecourseincontrol.Whilethisbookis unlikelytoeffortlesslyturnaplantengineerintoacontrolexpert,itdoescontain toolsthatmanyplantengineersshouldfinduseful.Itisalsohopedthatitwillgive theplantengineermoreweightwhendiscussingwithcontrolconsultants–either fromwithinthecompanyorexternalconsultants.
1Thisauthorisawarethatmanycolleaguesinacademiaareoftheopinionthatthe frequency-dependentRGAisnot“rigorous.”Thisbookwillinsteadtakethestancethatthe dynamicRGAhasprovedusefulandthereforeshouldbedisseminated.Itisalsonotedthatthe “counterexamples”wherethesteady-stateRGAisclaimedtoproposeawrongcontrolstructureare themselvesflawed.
2.Toincreasetheunderstandingamongcontrolengineers(studentsorgraduates movingintothechemicalprocesscontrolarea)ofhowtoapplytheirtheoretical knowledgetopracticalproblemsinchemicalprocesses.
Thethirdapproachtoreducingthegap,i.e.todevelopandpresenttoolstosimplifytheapplicationofcontrol,isnotafocusofthisbook–althoughsomecolleagues wouldsurelyclaimthatthisiswhatproportionalintegralderivative(PI(D))tuning rulesaredoing.
Thereadershouldnotethatthisbookdoesnotstart“fromscratch,”andsome priorknowledgeofcontrolisexpected.TheintroductiontotheLaplacetransform isrudimentaryatbest,andmuchmoredetailcouldbeincludedinthepresentationoffrequencyanalysis.Someknowledgeoffinite-dimensionallinearalgebrais expected.Readerswhohaveneverseenalinearstate-spacemodelwillfaceahurdle.Still,thebookshouldbeaccessibletoreaderswithbackgroundfromacoursein processcontrol.
Readerswithamoreextensiveknowledgeofcontroltheorymayfindthebook lacksrigor.Frequently,resultsarepresentedanddiscussed,withoutpresenting formalproofs.Readersinterestedinmathematicalproofswillhavetoconsultthe references.Thisisinlinewiththisauthor’sintenttokeepthefocusonissuesof importanceforindustrialapplications(withoutclaimingto“knowitall”).
ThestructureoftheBook
Thisbookhasgrownoutofmorethanthreedecadesoflearning,teaching,anddiscussingthecontrolofchemicalprocesses.Whathasbecomeclearisthatprocess controlengineersarefacedwithawidevarietyoftasksandproblems.Thechapters ofthisbookthereforeaddressarangeofdifferenttopics–mostofwhichhavebeen thesubjectofentirebooks.Theselectionofmaterialtoincludeisthereforenottrivial norobvious.
● Chapter1 presentssomemathematicalandcontroltheorybackground.Readers withsomeknowledgeofcontrolmaychoosetoskipthischapterandonlyreturn toittolookupunfamiliar(orforgotten)conceptsthatappearintherestofthe book.Thischapterdefinitelyhasamoretheoreticalandlesspracticalflavorthat muchoftherestofthebook.
● Chapter2 addressescontrollertuningforPI(D)controllers,aswellascontrol configuration.Thetermcontrolconfigurationherecoversboththecontrolfunctionalityoftenencounteredintheregulatorycontrollayer(feedback,feedforward, ratiocontrol,...)anddeterminingwhichinputshouldbeusedtocontrolwhich outputinamulti-loopcontrolsystem.
● Chapter3 focusesondeterminingwhatvariablestouseforcontrol.Typically, therearemorevariablesthatcanbemeasuredthancanbemanipulated,sothe mostfocusisgiventothechoiceofvariablestocontrol.
● Chapter4 presentslimitationstoachievablecontrolperformance.Clearly,ifit isnotpossibletoachieveacceptableperformance,itmakeslittlesensetryingto designacontroller.Understandingthelimitationsofachievableperformanceis
Preface xix alsoveryusefulwhendesigningcontrollersusingloopshaping,aspresentedin Chapter2.
● Chapter5 isaboutMPC,whichisthemostpopularadvancedcontroltypein thechemicalprocessindustries.2 InadditiontopresentingMPCproblemformulations perse,importantissuessuchasmodelupdating,offset-freecontrol,and targetcalculationarealsodiscussed.
● Chapter6 presentssomepracticalissuesincontrollerimplementation.Thislist isfarfromcomplete,someoftheissuesincludedarewellknownandmaybe consideredtrivial–butallareimportant.
● Chapter7 addressescontrolperformancemonitoring(CPM).Thenumber ofcontrollersinevenamodestlysizedchemicalprocessistoolargeforplant personneltofrequentlymonitoreachofthem,andautomatedtoolsaretherefore neededforthemaintenanceofthecontrol.Thechapteralsoincludessometools forfindingtherootcauseofdistributedoscillations–oscillationsoriginatingat onelocationintheplantwilltendtopropagateanddisturbotherpartsofthe plant,andhenceitisoftennontrivialtofindwheretheoscillationoriginatesand whatremedialactiontotake.
● Chapter8 addressescontrolbenefitanalysis,i.e.toolsto apriori assesstheeconomicbenefitthatcanbeobtainedfromimprovedcontrol.Thisauthoradmits thatthechapterisdisappointinglyshort.Developinggenerictoolstoestimatesuch economicbenefitisindeeddifficult.Ontheotherhand,theinabilitytoestimate economicbenefitwithsomecertaintyisalsooneofthemajorobstaclestomore pervasiveapplicationofadvancedcontrolinthechemicalprocessindustries.
What’sNotintheBook
Selectingwhattocoverinatextbookinvariablyrequiresleavingoutinterestingtopics.Sometopicsthatarerelevantinamoregeneralsetting,butwhichthisbookdoes notmakeanyattempttocover,include
● Nonlinearcontrol.Real-lifesystemsarenonlinear.Nevertheless,thisbookalmost exclusivelyaddresseslinearcontrol–withthemainexceptionbeingthehandling ofconstraintsinMPC.3 Linearizationandlinearcontroldesignsufficesformost controlproblemsinthechemicalprocessindustries,andinothercasesstaticnonlineartransformsofinputsoroutputscanmakemorestronglynonlinearsystems closertolinear.ThebookalsodescribesbrieflyapproachestononlinearMPC (withnonlinearityinthemodel,notonlyfromconstraints).Suchcomplications areneededmainlyforcontrolofbatchprocesses(wherecontinuouschangesinthe operatingpointareunavoidable),orforprocessesfrequentlyswitchingbetween differentoperatingregimes(suchaswastewatertreatmentplantswithanaerobic andaerobicstages).Althoughlinearcontroloftensuffices,itisclearlyprudentto verifyacontroldesignwithnonlinearsimulationand/orinvestigatethecontrolat differentlinearizationpoints(ifanonlinearmodelisavailable).
2And,indeed,thechemicalprocessindustriesiswhereMPCwasfirstdeveloped.
3Constraintsrepresentastrongnonlinearity.
Acknowledgments
Itisnotedintheprefacethatthisbookisaresultofmorethanthreedecadesof learning,teaching,discussing,andpracticingchemicalprocesscontrol.Therefore, alargenumberofpeoplehavedirectlyorindirectlyinfluencedthecontentsofthis book,anditisimpossibletomentionthemall.Thepersonwiththestrongestsuch influenceisProfessorSigurdSkogestadatNTNU,whomIhadthefortunetohave asmyPhDsupervisor.Atthattime,Ihadanextendedresearchstayinthegroup ofProfessorManfredMorari(thenatCaltech).DiscussionstheseoutstandingprofessorsaswellaswithmycontemporariesasaPhDstudent,bothinSkogestad’s groupandinMorari’sgroup,greatlyenhancedmyunderstandingofprocesscontrol.IwouldparticularlyliketomentionEllingW.Jacobsen,PetterLundström,John Morud,RichardD.Braatz,JayH.Lee,andFrankDoyle.
Inlateryears,Ihavebenefitedfromdiscussionswithanumberofpeople,includingKjetilHavre,VinayKariwala,IvarHalvorsen,KristerForsman,AlfIsaksson, TorA.Johansen,LarsImsland,andBjarneA.Foss.MyownPhDcandidatesand Postdocshavealsoenrichedmyunderstandingofthearea,inparticular,Kristin Hestetun,FrancescoScibilia,GiancarloMarafioti,FlorinStoican,Selvanathan Sivalingam,MuhammadFaisalAftab,andJonatanKlemets.
Specialthanksareduetomyhostsduringsabbaticals,whenIhavelearneda lot:BobBitmead(UCSD),JanMaciejowski(UniversityofCambridge),SorinOlaru (CentraleSupelec),andAndreasKugi(TUVienna).
M.H.
Acronyms
AKFaugmentedKalmanfilter
ARauto-regressive(model)
BLTbiggestlogmodulus(tuningrule)
CCMconvergentcrossmapping
CLDGclosed-loopdisturbancegain
CPMcontrolperformancemonitoring
DNLdegreeofnonlinearity
EKFextendedKalmanfilter
EnKFensembleKalmanfilter
FCCfluidcatalyticcracking
FOPDTfirst-orderplusdeadtime(model)
GMgainmargin
IAEintegral(of)absoluteerror
IEKFiteratedextendedKalmanfilter
IMCinternalmodelcontrol
ISEintegral(of)squarederror
KISSkeepitsimple,stupid
LHPlefthalf-plane(ofthecomplexplane)
LPlinearprogramming
LQlinearquadratic
LQGlinearquadraticGaussian
MHEmovinghorizonestimation
MIMOmultipleinputmultipleoutput
MPCmodelpredictivecontrol
MVmanipulatedvariable
MVCminimumvariancecontroller
MWBmovingwindowblocking
NGInon-Gaussianityindex
NLInonlinearityindex
OCIoscillationcontributionindex
OPcontrolleroutput
PCAprincipalcomponentanalysis
PIDproportionalintegralderivative
