New methods for interaction analysis of complex processes using weighted graphs

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ContentslistsavailableatScienceDirect

Journal of Process Control

jo u r n al h om ep a ge :w w w . e l s e v i e r . c o m / l o c a t e / j p r o c o n t

New methods for interaction analysis of complex processes using weighted graphs

MiguelCasta ˜noArranz^∗,WolfgangBirk

LuleåUniversityofTechnology,SE-97187Luleå,Sweden

a r t i c l e i n f o

Articlehistory:

Received14January2011

Receivedinrevisedform9June2011 Accepted23July2011

Available online 19 September 2011

Keywords:

Complexsystems Interactionmeasures Structuralproperties Systemanalysis

a b s t r a c t

Theselectionofthestructureofacontrollerinlargescaleindustryprocessesusuallyrequiresextensive processknowledge.Theaimofthispaperistoreportnewresultsonrecentlysuggestedmethodsfor theanalysisofcomplexprocesses.Thesemethodsaidthedesignersincomprehendingaprocessby representingstructuralandfunctionalrelationshipsfromactuatorsandprocessdisturbancestomeasured orestimatedvariables.Themethodsareformulatedinaﬂexibleframeworkbasedongraphtheory,which canalsobeusedforclosed-loopanalysis.Additionally,thesensitivityofthemethodstoscalingandtime delaysarediscussedandresolved.Itisalsoproposedhowﬁlteringcanbeusedtorestricttheanalysisto afrequencyregionofinterest.

Thefeasibilityofthemethodsisshownbytheuseofthreecasestudies.Aquadrupletankprocessis usedtoexemplifythemethodsandtheiruse.Thenthemethodsareappliedonareal-lifeprocess,the stockpreparationplantofapulpandpapermill.Thethirdstudycaseanalyzesapreviouslypublished exampleinclosedloop.

Itisshownthatthemethodscanbeusedtotakeefﬁcientdecisionsondecentralizedandsparsecontrol structures,aswellasassessingthechannelinteractionsinaclosed-loopsystem.

1. Introduction

Largescaleindustrialprocessplantsarecharacterizedbyahigh degreeofinteractionbetweenprocessvariables,wherehundreds oreventhousandsofvariablesareconnectedthroughdynamicsys- tems.Examplesofsuchinterconnectionsarematerialﬂowsand reﬂows,thelattere.g.duetodiscardedmaterialbeingreturnedto previousprocessstepswhichgives risetolargefeedbackloops.

Otherexamplesareconnectionsthroughsupplygridsfore.g.pres- surizedair.Oneprocessstepconsumingpressurizedairmaygive risetoapressuredropthatpropagatestoeveryotherconsumerin theplant.Addingcontrolloopstotheprocessondifferentlevelsof hierarchymayresultinasystemwithunintelligiblecausalityand unpredictabledynamics.

Forthecontrol engineer,theseverycomplexinterconnected systemsareachallenge.Thequestionishowtorepresentthecom- plexityinacomprehensiblewayandhowtoanalyzeitregardinge.g.

dynamicbehaviorandcontrolstructuredesign.Traditionally,inter- actionmeasuresareusedforthecontrolstructuredesign,namely controlstructureselectionanddecisiononthecontrollerconﬁg- uration.Controlstructure selection isaboutdetermining which dynamicinterconnectionsshouldbeusedforthecontrollerdesign;

∗ Correspondingauthor.Tel.:+46920492328.

E-mailaddress:miguel.castano@ltu.se(M.Casta ˜noArranz).

itisalsoreferredtoasinput/outputselection,seee.g.[1].When thestructureisselected,theinterconnectionscanbeusedtocon- ﬁgure a controller, that is decisions on e.g. degree-of-freedom, feed-forwards,orcascadedconﬁgurationsneedtobetaken.Inthis paperthefocuswillbeonthecontrollerstructureselection.

Oneofthemostwidelyspreadcontrolstructuresisthedecen- tralizedcontrolstructureinaonedegreeoffreedomsetting,and themostwell-knownmethodstodeterminetheinterconnections thatareusedforthecontrollerdesignaretheRelativeGainArray (RGA)[2],andtheDynamicRGA[3].Anextensionofcontrolstruc- tureselectiontoblockdiagonalcontrolstructuresisdiscussedin [4],[5].Generally,themethodsaredesignedfromtheperspective thatallcontrolledandmanipulatedvariablesareconsideredduring thestructureselectionandthatthecontrolconfigurationisoften seta-priori.Inthesurvey[1],mostoftheavailablemethodsforthe controlstructuredesignarereviewed,apartfromgramianbased methods which weresuggested more recently,[6,7]. Thelatter havetheadvantagethatnoassumptiononthecontrolconfiguration isusedandvirtuallyanycontrolstructurecouldbededuced.But theirinterpretationbecomesdifficultwhenthenumberofvariables becomeslarge.

For large scale systems, where the amountof variables can beinthemagnitudeofthousands,usuallythecontrolstructure selectionneedtobeprecededbyastepwheremanipulatedand controlledvariablesaregroupedintosetswherethenumberof variablesisreducedtoacoupleofdozens.Thus,makingtheabove

doi:10.1016/j.jprocont.2011.07.011

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methodsmoreapplicable.Methodsforsuchadecompositionare proposedin[8,9],wherestatespacerepresentationsareusedand produceinterconnectedmultivariablesystems.Itshouldalsobe notedthatinterconnectedsystemscanalsoberepresentedbysig- nalﬂow graphs, which dates backto thework of Mason [10], andprovidesacomprehensivevisualinterpretationatthesame time.

Analternativetotheaboveistheconceptofplantwidecontrol (PWC)whichhasreceivedmuchmoreattentioninrecentyears.

Methodologieslikethenine-stepmethodproposedbyLuybenetal.

[11]ortheself-optimizingcontrolprocedurebySkogestad[12]are suggestedandareevaluatedonrealisticlargescalesystem,seee.g.

[13].Inthis,acompleteproceduretoaddressthecontrolproblem ofalargescalesystemisprovided.

Ashortcomingofthementionedmethodsiseithertheirinability toscaletolargescaleproblemsorthatcontrolstructureandcon- ﬁgurationarelimiteda-priori.Theaimofthispaperistosuggest newmethodsforthecontrolstructureselectionwithoutimposing limitationsontheeitherstructureorconﬁguration.Butforthetime being,itisassumedthatthelargescaleproblemisalreadydecom- posedin multivariable problems witha rather limited number ofmanipulatedandcontrolledvariables.Inordertodemonstrate theusefulnessofthesuggestedmethods,a2×2systemwithdis- turbances(Quadrupletanklaboratorysetup)anda5× 5real-life process(StockpreparationplantatSCAObbolaAB,Sweden)are discussed.

Themethodsareinspiredbytheworkwithinbrainconnectiv- itytheory[14],wheresignalﬂowgraphscanbeanalyzedfromthe perspectiveofthesignalenergyorpowerthatistransferredfrom aselectedinput/outputsetofnodes.Inthiscontext,structuraland functionalpropertiesaretreatedseparately,wherethestructural propertiescapturethestructural natureof a signalﬂow graph, whilethefunctionalpropertiesresolveallintermediateconnec- tionsinordertomapthepropertiestotheinput/outputsetalone.

Consequently,theanalysiscanbeadaptedtotheneedsforthetask athand,whichmakesthemethodsmoreversatile.

Asaresultquantitativemethodsarecombinedwiththesignal flowgraphsapproachbyintroducingweightsontheedges.Itwill beshownthattheH2-normcanbeeffectivelyusedtoassessthe significanceofedgesinrelationtoeachother,namelybytheassess- mentofthetransferofsignalenergyorpower.Itistheauthors believethatthiscombinationenablescontrolengineerstomake betterdecisionsonbothcontrolstructureselectionandcontroller configuration.But,itstillremainstoprovethescalabilitytolarge scalesystemsandifthemethodscanbeusedinthescopeofthe previouslymentionedPWCprocedures.Inapreparatorystepand tofacilitateapplicabilitythemethodshavebeenintegratedintoa prototypetoolcalledProMoVis,thatcanbeusedtobothmodel, visualizeandanalyzeindustrialprocesses,[15].

Thepaperlayoutisasfollows.Firstthepreliminariesforthe representationoflinearsystemsasasignalﬂowgraphisgivenin Section2.Section3introducesamethodologyforobtainingascal- ingindependentrepresentationoflinearprocessesasasignalﬂow graph.Section4describesthequadrupletankprocessandderives ascalingindependentrepresentationforaselectedworkingpoint.

Section5describesthenormsandnormalizationswhichwillbe usedforquantifyingandcomparingthesigniﬁcanceoftheprocess interconnections.Thequadrupletankprocess isusedinSection 6asanillustrativeexampleforintroducingthemethodsforthe structuralandfunctionalanalysisofcomplexprocesseswhichare thesubjectofthispaper.Section7describeshowtointerpretthe methodsandusethemforcontrolstructureselection,comparing theresultsobtainedintheanalysisofthequadrupletankwiththose oftheRGA.Section8analyzesthescalingsensitivityofthemeth- odsforthecasesinwhichascalingindependentrepresentationof theprocessisnotused.ThestockpreparationplantatSCAObbola,

Swedenisareal-lifeprocesswhichisanalyzedinthissectionto illustratehowthescalingissuespresentinthetraditionalgramian basedIMswhenanalyzingindustrialprocesscanberesolvedby usingthediscussedmethods.Section9introducestheusefulnessof theanalysismethodsfortheevaluationoftheinteractionbetween controlloops,andcontainsexampleswhichshowtheuseofﬁl- tersforselectingarangeof frequenciessubjecttoanalysis.The conclusionsareﬁnallygiveninSection10.

2. Signalﬂowgraphsrepresentingcomplexprocesses

Arepresentationofcomplexsystemsinrelationtographrepre- sentationswasintroducedin[16].Thesamerepresentationwillbe usedheredispensingwiththeoutputequation.Whenrepresent- inganopenloopprocess,itwillbeconsideredthattheinputvector uiscomposedbytheexogenousinputstotheprocess(actionson actuatorsandprocessdisturbances),andxcollectsalltheinternal statesandmeasurementswhicharesubjecttoanalysis,beingboth relatedinanexpressionlike:

x_i=i1· x1+···+in·xn+i1·u₁+···+ip·up

whereijandijarelineardynamicsystems.Thesignalﬂowgraph isthenformulatedas

x=x+u (1)

GivenamultivariablesystemH,theduple(,)iscalledavisual- izationofH.Theinput–outputmatrixofHcanbecomputedas

=(I−)⁻¹.

ThisformulationwillalsobeusedinSection9forrepresentinga closedloopsystem.Inthatcase,thereferencesignalsfortheclosed loopsareaggregatedtotheinputvectoru,andthecontrolactions whicharemanipulatedbyacontrollerarenowaggregatedtothe vectorxandremovedfromtheinputvectoru.

Thefollowingtransformationswillbehereusedforobtaining processrepresentationsdescribingthenodesandinterconnections whichareofinterestfortheanalysis.

2.1. Hidingofself-references

Anodedependingonitself isknownasa self-reference.For theworkdescribed inthispaperwe willassumethatthevisu- alizationrepresentingtheprocesshasnoself-references,thatis, thediagonalofmatrixiscomposedbyzeros.Ifavisualization withselfreferencesistobeanalyzed,theuserhastoﬁrstmakean operationwhichhidestheself-referencesandstillpreservesthe physicalstructureoftheplant.Thedetailsforsuchanoperation arediscussedin[16].

2.2. Hidingarbitrarynodes

Differentlevelsofhierarchyareusuallypresentwhenrepre- sentingandanalyzinglargescaleinterconnectedsystems,andthe hidingof nodesallowstodisregardthevariables whicharenot presentlyimportantwhilepreservingthephysicalstructureofthe plant.

Considerthepartitioningoftheinterconnectedsystem(,) as

x₁ x₂

=

11 12

21 22

x₁ x₂

+

1

2

u

where x₂=[0_m_×(m−n)| Im]·x represents the last m of the total n nodes. Assuming that x=22x+u is well-posed, then the

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visualizationofthesystemwhenthelastmnodesarehidden,and whichpreservesthesystemstructureis[16]:

x₁= ˆx1+ ˆu with

=ˆ 11+12(I−22)⁻¹21 , =ˆ 1+12(I−22)⁻¹2

WhichcanbeexpressedasafunctionF of,,andI_m,beingm thenumberofnodestobehidden.

[ ˆ ˆ]=[1 11]+12(1−22)⁻¹[2 12]=F([ ],Im) TohideanarbitrarysubsetNofnodes,letusdeﬁnethepermutation matrixT^T=[E^T E^T]suchthatx₁=Exisnowavectorcontainingthe nodestoberetainedandx₂=Excontainsnowthenodestobehid- den.TheoperationHNofhidinganarbitrarysetofnodes,isthen:

HN[ ]=F(T[ ·T^T],I_m)

3. Scalingindependentrepresentation

Scalingisanimportantissueinmanyapplications.Themeth- odsusedinthispaperarebasedoncomparingthesignalpower(or energy)thatcanbetransferredthroughtheprocessinterconnec- tions.Thispowertransferisquantiﬁedassignalpowertransfer,and thereforeitdependsonthescalesusedtorepresenttheanalyzed signals.

Usualmethodsforscalingsignalsinvolvedividingeachvariable byitsmaximumexpectedorallowedchange[17].

LetQ=(,)beavisualizationofasystemH.Whentheinput vectoruandtheoutputvectorxarescaledbythediagonalscaling matricesD_u andD_xrespectively,thenthenewinputandoutput vectorsareu=D⁻¹_u ·uandx=D⁻¹_x ·x.ThescaledvisualizationQ= (,)isthen:

=D⁻¹_x Dx; =D⁻¹_x Du

andthescaledinput–outputmatrixis:

=D⁻¹_x Du

Itcanbetedioustoﬁndanappropriatescalingforeachofthe processvariables.Therefore,wepresentarepresentationoflinear processesbasedonavisualization,whichis independentofthe selectedscaling.Forobtainingsucharepresentation,anestimation ofthestandarddeviationofthesignalsrepresentedbytheprocess variablesisneeded.

Letu,u,xandxbediagonalmatricescollectingthestandard deviationofeachoftheinputoroutputsignalsandoftheirscaled version.Then,u=uD⁻¹_u andx=xD⁻¹_x .

Lemma1. Thepair(⁻¹x ␴x,⁻¹x ␴u)isscalinginvariant.

Proof.

⁻¹x ␴x=⁻¹_x D⁻¹_x DxD⁻¹x Dxx=⁻¹_x ␴x

⁻¹x ␴u=⁻¹_x D⁻¹_x DxD⁻¹u Duu=⁻¹_x ␴u

GiventhevisualizationQ=(,)ofalinearmultivariablesystem H,thepair ˜Q=(⁻¹x x,⁻¹x ␴u)willbenamedscalinginvariant representationofH.

Remark 1. The matrix ˜ =(I−⁻¹x ␴x)⁻¹⁻¹x ␴u=⁻¹x ␴u

relatedtothescalinginvariantrepresentation ˜Q,is alsoscaling invariant.Thisfollowsfrom:

⁻¹x ␴u=⁻¹_x D⁻¹_x DxD⁻¹u Duu=⁻¹_x u (2)

Remark2. Thepremultiplicationofmatrix or by⁻¹x makes it invariantto outputscaling,and thepostmultiplicationby u

makesitinvarianttoinputscaling.Matrix isindependentofthe selectedinputscaling,andthepremultiplicationby⁻¹x withthe postmultiplicationbyxmakesitindependenttooutputscaling.

Therefore,applyinganyofthestructuralorfunctionalmethodsto thescalinginvariantrepresentation ˜Q =(⁻¹x ␴x,⁻¹x ␴u)with input–outputmatrix ˜ =⁻¹x ␴u, alwaysgivethesameresult independentlyofthechosenscaling.

Remark3. ThegramianbasedIMsaresensitivetothescalingof theprocessvariables[7,6,18].Thescalinginvariantrepresentation isalsousefulforcomputingascalinginvariantversionofanyof theseIMs,byapplyingthecorrespondingmethodtothematrix

˜ =⁻¹x ␴u.

The following section presents the quadruple tank process, and introduces an example of how to create a scaling inde- pendentrepresentationoftheprocess.Later,thisrepresentation will be used to apply the discussed structural and functional methods.

4. Thequadrupletankprocess

Thequadrupletankprocesshasbeenintroducedin[19],and is a well-known interactingprocess which hasbeen usedby a largenumberofauthorsasabenchmarktotestseveralcontroland analysismethods.TheprocessisdepictedinFig.1.Twoprocess disturbanceshavebeenaddedtotheprocessandmodeledasﬂow disturbancesintheuppertanks.Thedifferentialequationsofthis modiﬁcationweredescribedin[20].Theprocesslinearmodelcan beformulatedasavisualizationasfollows:

⎛

⎜⎜

⎝

h1

h2

h3

h4

⎞

⎟⎟

⎠⁼

⎛

⎜⎜

⎜⎝

0 0 A3

A1T3(s+1/T1) 0

0 0 0 A4

A2T4(s+1/T2)

0 0 0 0

⎞

⎟⎟

⎟⎠

⎛

⎜⎜

⎝

h1

h2

h3

h4

⎞

⎟⎟

⎠

+

⎛

⎜⎜

⎜⎝

1k1

A1(s+1/T1) 0 0 0

0 2k2

A2(s+1/T2) 0 0

0 (1−2)k2

A3(s+1/T3) − 1 A3(s+ 1

T3

) 0

(1−1)k1

A4(s+1/T4) 0 0 − 1

A4(s+1/T4)

⎞

⎟⎟

⎟⎠

⎛

⎜⎜

⎝

u1

u2

d1

d2

⎞

⎟⎟

⎠⁽³⁾

whereT_iarethetimeconstantsofthetanks

T_i= A_i a_i

2h^o_i

g , i={1,2,3,4}

ThevaluesoftheconstructionparametersaregiveninTable1.

The variables u_j are associated with the two actuators and expressthespeedsettingofthetwopumpsin%.Themeasured

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Fig.1.Interactingsystemwithwatertanks.Eachofthearrowsrepresentsadynamicmodelconnectingtwoprocessvariables.ProMoVisscreenshot.

Table1

Constructionparametersofthequadrupletankprocess.

Parameter

A1,A2,A3,A4 a1 a2 a3 a4 g k1 k2 1,2

Value 730cm² 2.05cm² 2.26cm² 2.37cm² 2.07cm² 981cm/s² 7.45cm³/(sV) 7.30cm³/(sV) 0.3

Table2

Selectedworkingpointforthequadrupletankprocess.

Variable

u1 u2 d1 d2 h1 h2 h3 h4

Value 50 50 15.5 15.5 14.77 12.36 5.11 7

Range 0–100 0–100 0–31 0–31 0–20 0–20 0–20 0–20

Units % % cm³/s cm³/s cm cm cm cm

variablesh_iarethelevelofthetanksexpressedin cm.Thetwo processdisturbancesd₁ andd₂ areﬂowperturbationsinthetop tanks,andareexpressedincm³/s.Therangeofvalueswhichthese variablescantakeissummarizedinTable2.

Theinput–outputrelationshipisgivenby:

⎛

⎜⎝

h1

h2

h3

h4

⎞

⎟⎠⁼

⎛

⎜⎜

⎝

1k1

A1(s+1/T1)

(1−2)k2

A1T3(s+1/T1)(s+1/T3) − 1

A1T3(s+1/T1)(s+1/T3) 0 (1−1)k1

A2T4(s+1/T2)(s+1/T4)

2k2

A2(s+1/T2) 0 − 1

A2T4(s+1/T2)(s+1/T4)

0 (1−2)k2

A3(s+1/T3) − 1

A3(s+1/T3) 0

(1−1)k1

A4(s+1/T4) 0 0 − 1

A4(s+1/T4)

⎞

⎟⎟

⎠

=(I−)⁻¹

⎛

⎜⎝

u1

u2

d1

d2

⎞

⎟⎠ ⁽⁴⁾

Theworkingpointselectedfortheanalysiscorrespondstoan openingof50%onbothpumps,andtheprocessdisturbancesare assumedtobeathalfoftheirpossiblemaximumvalue.Thevalues fortheworkingpointaresummarizedinTable2.