delay Distributed of interconnected control Chemical Reaction Networkswith Journal of Process Control

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Journal of Process Control

jo u rn al h om ep age :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

Distributed control of interconnected Chemical Reaction Networks with delay

L ˝orinc Márton

^a,∗

, Gábor Szederkényi

^b,c

, Katalin M. Hangos

^b,d

aDepartmentofElectricalEngineering,SapientiaHungarianUniversityofTransylvania,TirguMures,Romania

bSystemsandControlLaboratory,InstituteforComputerScienceandControl,HungarianAcademyofSciences,Budapest,Hungary

cFacultyofInformationTechnologyandBionics,PázmanyPéterCatholicUniversity,Budapest,Hungary

dDepartmentofElectricalEngineeringandInformationSystems,UniversityofPannonia,Veszprém,Hungary

a r t i c l e i n f o

Articlehistory:

Received12March2018

Receivedinrevisedform25July2018 Accepted5September2018

Keywords:

Processsystems

ChemicalReactionNetworks Multi-agentsystems Passivity

Delaysystems Distributedcontrol Nonlinearsystems

a b s t r a c t

ThispaperintroducesacontrolapproachforaclassofChemicalReactionNetworks(CRNs)thatare interconnectedthroughadelayedconvectionnetwork.First,acontrol-orientedmodelisproposedfor interconnectedCRNs.Second,basedonthismodel,adistributedcontrolmethodisintroducedwhich assuresthateachCRNcanbedrivenintoadesiredfixedpoint(setpoint)independentlyofthedelayin theconvectionnetwork.Theproposedalgorithmisalsoaugmentedwithadisturbanceattenuationterm tocompensatetheeffectofunknowninputdisturbancesonsetpointtrackingperformance.Thecontrol designappliesthetheoryofpassivesystemsandmethodsdevelopedformulti-agentsystems.Simulation resultsareprovidedtoshowtheapplicabilityoftheproposedcontrolmethod.

©2018TheAuthor(s).PublishedbyElsevierLtd.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).

1. Introduction

Thecontrolofplant-wideindustrialprocessesisinthefocus of the researchers for decades [1,2]. The decentralized or dis- tributedcontrolapproachesareadvantageousinprocessnetwork applicationstoreducethecommunicationcostsandpossiblecom- municationhazards that couldarise in the case of centralized control[3].Becauseoftheirpracticalimportanceandchallenging nature,thedifferentnonlinearcontrolapproachesappliedinpro- cesscontrolhavedevelopedtheirownversionofdecentralized, distributedorhierarchicalcontrolarchitectures.

In thepaper [4] the authors proposeda distributed control approachforsuchinterconnectedprocessesthatcanbemodeledas lineartime-invariantsystemsbasedonpassivitytheory.Thiscon- trolmethodalsotakesintoconsiderationthetransportdelayinthe interconnectionsamongtheprocesses.

Apowerfulcontrolapproachofinterconnectedprocesssystems isbasedonthethermodynamiccharacterizationofsuchsystems.

Amodelingframeworkhasbeendevelopedin[5]fornetworksof chemicalprocessesconsideringalsothethermodynamiceffects.

∗Correspondingauthor.

E-mailaddress:martonl@ms.sapientia.ro(L.Márton).

Usingthetheoryofcascade-connectednonlinearsystemsandthe propertiesofMetzlerandHurwitzmatrices,astabilizingdecentral- izedcontrolapproachwasdesignedin[6]utilizingthehierarchical structureofconservationbasedprocessmodels.

Thepopularandpowerfulmodelpredictivecontrolapproach is alsoused in distributed and in hierarchicalframeworks (see thepaper[7]fora review).Thisapproachcanhandlenonlinear interconnectedprocess systems, aswell.A more recent review highlightingfutureresearchdirectionsinthisapproachisavailable in[8].Constrainedcontrolmethodscanbeappliedwheninputor stateboundsshouldalsobetakenintoconsideration,seee.g.[9]or [10].

ChemicalReactionNetworktheoryprovidesefficientmodels andtechniquestodescribeandanalyzenotonlythedynamicsof chemicalreactions[11],butamorewideclassofnonlinearprocess systems.Thestudy[12]dealswiththemodelingofinterconnected reactorsandshowsthatthetransportmechanismcanbedescribed byalinearCRNmodel.TheCRNmodelsdescribepositivesystems andcanefficientlycapturecomplexnonlineardynamicalphenom- ena. The paper[13] offersa modeling approach for CRNs with inflowsandoutflowsanddiscussestherelationofthesesystems withtheconsensusdynamics.Thestabilityofthesesystemsismost oftenanalyzedusingentropy-basedLyapunovfunctions[14].

Unfortunately,however, onlya fewof thenonlinearprocess controlapproachesthatweredevelopedforcontrollingintercon- https://doi.org/10.1016/j.jprocont.2018.09.004

0959-1524/©2018TheAuthor(s).PublishedbyElsevierLtd.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).

nectedsubsystemsareabletoexplicitlyhandledelaysindynamic analysisandcontrollerdesign.Someexceptionsincludethepaper [15],where thesynchronization problemofaclass ofintercon- nectednonlinearbiochemicalprocesseswasconsideredbytaking aninput-outputmodelingapproach.Importantmodelingandsta- bilityandcontrolrelatedresultsondelayedkineticsystemscanbe foundin[16],[17],[18],[19]and[20].

Inthisstudyacontrol-orientedmodelingapproachisproposed forinterconnectedCRNs,relyingbothoncompartmentalsystems [21], and CRNtheory. Themodel takesinto consideration both thenonlinearnatureofthechemicalprocessesandtheunknown transportdelaysintheconvectionnetwork.Adistributedsetpoint controlalgorithmisintroducedwhichassuresthattheconcentra- tionlevelsofthechemicalsineachreactorreachprescribedfixed points.Theproposedcontrolmethodcanalsoattenuatetheeffect ofunknowninputdisturbancesonthecontrolperformance.The resultingcontrolalgorithmhasaneasilyimplementableform,itis independentoftheCRNskineticsandthedelayintheconvection network.Thestabilityandtrackingperformanceoftheintercon- nectedCRNwiththeproposedcontrolisanalyzedusingtechniques borrowedfromthetheoryofmulti-agentsystems.

2. Interconnectedpassivesystems

2.1. Passivesubsystems

ConsideraninterconnectedsystemconsistingofCsubsystems inwhichtheinputofeachsubsystemmaydependontheoutputs oftheothersubsystems.

EachsubsystemismodeledusingODEs(OrdinaryDifferential Equations)intheform

˙

c^(j)=f^(j)(c^(j))+G^(j)(c^(j))u^(j), c^(j)(0)=c^(j)_◦ , (1) y^(j)=h^(j)(c^(j))

wherec^(j) ∈Rⁿ,y^(j),u^(j) ∈R^marethestate-,output-andinputvec- tors,f^(j)(·),h^(j)(·),G^(j)(·)aresmoothmappingswithappropriate dimensionssuchthatf^(j)(0)=0,h^(j)(0)=0,j=1...C.

Definition1. System(1)iscalledpassive,ifthereexistsacontin- uouslydifferentiablefunctionS^(j):Rⁿ→Rsuchthat

S^(j)(c^(j))≥0, ∀^{c(j), (2)}

S^(j)(0)=0, (3)

S˙^(j)≤y^(j)Tu^(j), ∀^{u(j),c(j). (4)} Siscalledthestoragefunctionof(1)(see,e.g.[22]).

Theinput-affinesystem(1)ispassiveiffthefollowingconditions hold

∂S^(j)

∂c^(j)f^(j)(c^(j))≤0, (5)

∂S^(j)

∂c^(j)G^(j)(c^(j))=

h^(j)(c^(j))

T

, (6)

seee.g.[23].

Passivitytheoryplaysakeyrole inanalyzingthestabilityof nonlinearsystemsasitisshownthatpassivityof(1)involvesthe stabilityoftheautonomoussystem ˙c^(j)=f^(j)(c^(j))undermildcon- ditions[22].

2.2. Interconnections

Theunderlyinggraphoftheinterconnectedsystemisadirected graphwithCverticesinwhicheachvertexcorrespondstoasub- system.Thereisadirectededgefromthevertexktothevertexjif

theinputofthejthsubsystemdependsexplicitlyontheoutputof thekthsubsystem.

Neighborsetofthejthvertex(Nj):thekthvertexbelongstoNj

ifthereisadirectededgefromthevertexktothevertexj.

Alikethemodelingconceptsdevelopedforlarge-scalesystems [3],considertheinputofeachsubsystem(u^(j))asthesumofalocal controlinput(u^(j)_L )andantheinterconnectioninputtermwhich hastheform:

i^(j)(t)=i^(j)

y^(k1)(t−T_k1j),...,y^(kJ)(t−T_kJj)

, (7)

wherey^(kl)aretheoutputsofthesubsystemsoftheneighborsetNj

(dim(Nj)=J),and0≤T_kij<∞isaconstanttransportdelayfrom theagentk_itotheagentj.

Theoutputsofthesubsystemsintheinterconnectedsystemare synchronizediflim

t→∞|y^(j)(t)−y^(k)(t−T_kj)|→0,∀^j,k.

Assumetheinterconnectioninputsintheform(synchronization protocol)

i^(j)(t)=

k∈Nj

w_kj(y^(k)(t−T_kj)−y^(j)(t)), w_kj>0, (8)

andu^(j)_L =0element-wise.Fortheanalysisoftheinterconnected systemswithsuchsubsysteminputsthefollowingfunctionalcan beapplied:

S=

N i=1

S^(j)+

N j=1

k∈Nj

t t−T_kj

y^(j)T()y^(j)()d. (9)

Itisshownin[24,25]that,undercertainassumptionsontheunder- lying graph and the subsystems, the interconnection input (8) assuresthesynchronizationofthesubsystems.

3. Basicmodelingnotions

3.1. ChemicalReactionNetworksandtheirstability

ChemicalReactionNetworks(abbreviatedasCRNs) arecom- posedofelementaryirreversiblereactionsRk:C_i→C_j,k=1...,R, whereC_j, j=1,...,marethesocalledcomplexes.AcomplexC_j isformallyalinearcombinationofspeciesX_i, i=1,...,K,such thatC_j=

K

i=1ˇ_ijX_i,forj=1,...,m,whereˇ_ijisthenonnegative stoichiometriccoefficientcorrespondingtospeciesX_iincomplex C_j.

Theconcentrations ofthespecies arecollectedintoavector c ∈R^K sothatc_i=[X_i] for i=1,...,K.Thedynamics ofa CRN describingthetimeevolutionoftheconcentrationsofthespecies induced by thereactionscan bewritten in thefollowing form assumingconstantvolumeandtemperature(see,e.g.[26]):

˙

c=Mϕ(c)=YAϕ(c), c(0)=c_◦ (10)

where c_◦ is strictlypositiveelement-wise,and Y ∈R^K×m is the complexcompositionmatrixthejthcolumnofwhichcontainsthe stoichiometriccoefficientsofcomplexC_j,i.e.Y_ij=ˇ_ij, ∀i,j.More- over,ϕ_i(c)= ^K_i=1c^Y_i^ik isthemassactionvectorandA ∈R^m×mis theKirchhoffmatrix:

A(i,j)=

⎧ ⎪

⎨

⎪ ⎩

_ji, for j=/i

−

=/j

_j, if j=i. (11)

where_jiistherateconstantofthereactionRk:C_j→C_i.

The reaction vector of Rk is formed by the corresponding stoichiometricvectors, suchthate_k=Y_·i−Y_·j. Thespanof the reactionvectorsdefinesthestoichiometricsubspaceoftheCRN:

Sc=span

e_k

.Thepositivestoichiometriccompatibilityclasses ofaCRNarerepresentedbySc◦=(c_◦+Sc)∩R^nS+.

The generalCRN model (10) may have multiple (even infi- nitenumberof)steadystatesinthewholestatespace.Therefore, thestructureandnumberofequilibriaaremostoftenstudiedby restrictingthedynamicstothestoichiometriccompatibilityclasses correspondingtodifferentinitialconditions.

Example1(AsimpleCRN). LetusconsideraChemicalReaction Networkconsistingofthefollowingreactions:

R1:2X₃^k→¹²2X₁^k→²³2X₂, R2:2X₃^[13]

₃₁2X₂. (12)

Themodelcontainsthreespecies:X1,X2,X3,andthreecomplexes:

C₁=2X₃,C₂=2X₁,C₃=2X₂.Fromthese,thecomplexcomposition matrixcanbewrittenas

Y=

⎡

⎣

⁰0 ²0 ⁰2 2 0 0

⎤

⎦

(13)

Wecanseefrom(12)thatthenetworkcontainsfourelementary reactions.Theratecoefficientsofthesereactionsarethenon-zero off-diagonalelementsoftheKirchhoffmatrixwhichisgivenby

A=

⎡

⎢ ⎣

−(₁₂+₁₃) 0 ₃₁

₁₂ −₂₃ 0

13 23 −31

⎤

⎥ ⎦

⁽¹⁴⁾

Thepair(Y,A)iscalledarealizationofakineticsystemwitha givencoefficientmatrixMandreaction-monomialvectorϕ,where thecomplexcompositionmatrixYisdeterminedbyϕ.Itisimpor- tanttonotethatarealizationofakineticsystemmaynotbeunique, i.e.theremayexistmorethanoneKirchhoffmatrixAforakinetic dynamicsgivenbyMandϕ[27].

ThereareimportantstructuralpropertiesofaCRNrealization thatcanbeusedfor determiningthestability propertiesofthe dynamics,thatarethedeficiencyandthereversibilityproperties.

Thedeficiencyof aCRN realizationisdefined ası=dim(KerY∩ ImA).A CRN isweakly reversible iftheexistence of a directed path(i.e.reactionsequence)fromthecomplexC_itothecomplexC_j impliestheexistenceofadirectedpathfromC_jtoC_i.

Althoughweakreversibilityandzerodeficiencyisarealization propertyofaCRN,theyhaveimportantimplicationsonthestabil- ityoftheCRNsystem.IfaCRNisweaklyreversibleandhaszero deficiencythenthesystem(10)hasexactlyoneequilibriumpoint (c^∗)ineachpositivestoichiometriccompatibilityclass[11]thatis atleastlocallystablewiththefollowingLyapunovfunction:

S(c)˜ =

K i=1

c_i

lnc_i c^∗_i −1

+c_i^∗

(15)

Anequilibriumpointc^∗oftheCRN(10)iscalledcomplexbal- ancedif

Aϕ(c^∗)=0 (16)

Itisalsoknownthatifthereexistsa complexbalancedequilib- riuminaCRN,thenallotherequilibriaarecomplexbalanced,too [26].Therefore, complexbalanceis a systempropertyoncethe CRNstructureandparametersarefixed,andthusaCRNitselfcan becalledcomplexbalancedif(16)isfulfilled.However,akinetic differentialequationmayhaveseveralcomplexbalancedandnon- complex-balancedrealizations[27].Itisimportantthatcomplex balanceimpliesweakreversibility. Complexbalanceis strongly relatedtothestabilityofkineticsystems.Accordingtothewell- knownGlobalAttractorConjecture(GAC),complexbalancedCRNs aregloballystablewiththeLyapunovfunction(15).TheGACwas

provedforseveralspecialcases,mostremarkablyforCRNswith onereactiongraphcomponent[28],andaproofforthegeneral problemhasbeenreportedin[29].Asignificantresultinthethe- oryofCRNsisthatanydeficiencyzeroweaklyreversiblenetwork iscomplexbalancedindependentlyofthevaluesoftheratecon- stants[30].Thisensuresarobuststabilitypropertywhichcanbe importantinthegeneraltheoryofnonnegativesystems[31].

Inordertohavepassiveagentsfortheanalysis,weconsiderthe followingsub-classofCRNs.

Assumption1. TheCRN(10)iscomplexbalanced.

Assumption2. TheCRN(10)ispersistent.

Persistencemeansthatthetrajectoriesofthesystem(10)do notapproachtheboundary∂_R^K₊arbitrarilyclose,i.e.∀ⁱ=1...Kit stands:c_i(t)>0ifc_i◦>0andt≥0.Itisimportanttonotethat Assumptions1and2arestronglyrelated.InthecaseofCRNswith onegraphcomponent,complexbalanceimpliespersistence[28].

Werecalltheimportantspecialcasethatdeficiencyzeroweakly reversiblenetworksare complexbalanced for anypositiverate coefficients[30].Moreover,accordingto[29](whichisnotpub- lishedofficiallyatthetimeofwriting)thedynamicsofanycomplex balancedCRNispersistent.

3.2. NetworkofCRNsconnectedbyconvectionwithdelay

ItisconsideredthatthemassactionCRNsarelocatedincon- tinuouslystirredtankreactors(CSTRs)thatareconnectedthrough staticconnections.InordertohaveausualCRNmodelconsidered inSection3.1,weassumeconstantvolume,constanttemperatureand constantphysico-chemicalpropertiesineachCSTR.

Aswe shall seelater in Section4.1,the above assumptions togetherwithAssumptions1and2ensurethatthelocalCSTRsare passivewithacertaininput-outputpair.Inpracticetheconstant temperatureassumption–thatimpliesconstantphysico-chemical propertieswithconstantpressure–isapproximatelyvalidformost ofthebiochemicalapplications,wherealsothecomplexbalanced andpersistentnatureofthereactionnetworkisalsovalid.Thecon- stantvolumeassumption,however,putssevererestrictionsonthe convectionnetworkasitwillbedescribedlaterinSection3.2.2.

EachCSTRhasaninletandanoutletportwithvolumetricflow ratesv_Iiandv_isuchthatv_Ii=v_i, i=1...C,wherethenumberof CSTRsisdenotedbyC.

We also introduce a pseudo-CSTR (CSTR0) for describing the environment.Becauseoftheconstantvolumeassumptionofeach internalCSTR,thisassumptionalsoholdsfortheenvironment,such thatvI0=v0.

Assumption3. InCSTR0theconcentrationvector(c⁽⁰⁾)isconstant andstrictlypositiveelement-wise.

Intheusualpracticallyimportantcasesthevariationoftheinlet feed–usuallyitscompositionbutsometimesevenitsflowrate ischanging–isthemajordisturbancetoaplant,therefore,the constantconcentrationsassumptioninCSTR0thatrepresentsthe environmentdoesnotalwaysholdinpractice.However,onecan relaxthis assumptionifadisturbancerejectiveextensionofthe controlschemeisdeveloped:thistechniqueisusedforourpro- posedcontrolmethodinSection4.3.

3.2.1. OpenCRNmodel

Anumber ofC mass-actionChemical ReactionNetworksare consideredinthesystemunderinvestigation.LetY^(j)andA^(j) be thestoichiometricandKirchhoffmatricesoftheODEmodelofthe CRNthattakesplaceinthejthCSTR:

˙

c^(j)=Y^(j)A^(j)ϕ^(j)(c^(j)). (17)

Thenthecomponentmassbalancecontainingthein-andout- flowconvectivetermsofthejthCSTRreadsas

dc^(j) dt = 1

V_j

_C

=0

˛_jvc⁽⁾−vjc^(j)

+Y^(j)A^(j) ϕ^(j)(c^(j)) (18)

wherej=1...C.

3.2.2. Theconvectiveconnections

Connectionsaresetupbetweenthereactorssuchthattheoutlet oftheithreactorisdividedintofractionswiththefractioncoeffi- cients˛_ijthatarefedintothejthreactor.Thismeansthat

C

=0

˛_i=1, i=0,...,C, (19)

vIj=vj=

C =0

˛_jv, j=0,...,C. (20)

Becauseconstantvolumeisassumedineveryregion,thesum ofconvectiveinflows

v₀₌

C

=0

˛₀v (21)

isequaltothesumoftheconvectiveoutflowsoftheprocesssys- tem.Itisimportanttonotethatthecompositesystemconsistingof theoriginalprocesssystemanditsenvironmentisclosedwithC+1 regionseachofconstantvolume.

With thenotations above we can formulate theconvection matrixasfollows:

C_C=

⎡

⎢ ⎢

⎢ ⎣

−(1−˛₀₀)v₀ ˛₁₀v₁ ˛₂₀v₂ ... ˛_C0v_C

˛01v0 −(1−˛11)v1 ˛21v2 ... ˛_C1v_C

···

˛_0Cv₀ ˛_1Cv₁ ˛_2Cv₂ ... −(1−˛_CC)v_C

⎤

⎥ ⎥

⎥ ⎦

(22) ThismatrixwillbetermedKirchhoffconvectionmatrix.

TheconstantvolumeassumptionimpliesthatC_C1=0which is also a consequence of Eq. (19).Here 1=(11...1)^T and 0= (00...0)^T.Moreover,Eq.(20)impliesthat1^TC_C=0^T.

TheabovetwoequationsandthesignpatternofC_Cshowsthat Kirchhoffconvectionmatricesarebothrowandcolumnconserva- tionmatrices.WhenthereisnoflowfromtheithCSTRtothejth CSTR,theparameter˛_ij=0inC_C.TheKirchhoffmatrixdescribes thestructureoftheunderlyingdirectedgraphGCoftheconvection network:theweightedLaplacianofGCis−C_C.OnGCthefollowing assumptionismade:

Assumption4. GC contains adirectedspanningtreewithroot CSTR0.

Thisassumptionmeansthatthesupplyfromtheenvironment reacheseachindividualreactor,therearenoreactorswhichare unreachablefromtheCSTR0intheprocessnetwork.

Thisassumptioniseasytoverifyinpracticebasedontheflow sheetoftheplant,butitmaynotholdinallpracticalcasesforthe overallplant.Atthesametime,ifthesupplyisreallynecessaryfor theproduction,thentheplantcanbenaturallydecomposedinto sub-plantsthatobeyAssumption4individually,andthecontroller designcanbedoneforthemseparately.

3.2.3. Connectedmodelwithconvectiondelays

Considerthattransportdelaysarepresentintheinterconnec- tionsamongtheCSTRs.Denotethedelayvaluebetweenthethand

jthCSTRasT_j.Thenthestateequationof(18)obtainsthemodified form:

V_jdc^(j)_i dt =

C

=0

˛_jvc⁽⁾_i (t−T_j)−v_jc^(j)_i +V_jY^(j)A^(j)ϕ(c^(j)), (23)

c⁽⁾_i ()=⁽⁾_i (), −T_j≤≤0.

Herej=1...Cand_i⁽⁾()∈C⁺isaninitialconditionfunction.

4. Distributedcontrollerdesign

Thedesignofthedistributedcontrolschemeisbasedonthe passivityanalysisoftheinterconnectedCSTRsthatisgiveninthe nextsection. Thereafter the distributed setpoint control design willbeintroducedandfinally,itsdisturbancerejectiveversionis described.

4.1. StoragefunctionandpassivityanalysisofopenCRNs

LetusconsiderthattheinputofthejthopenCRN(23)isthe vectoru^(j) ∈R^Ksothat:

dc^(j)

dt =Y^(j)A^(j)ϕ^(j)(c^(j))+ 1

V_ju^(j). (24)

NotethattheabovegeneralformisderivedfromEq.(18)bycon- sideringthedifferencebetweentheconvectivecomponentmass in-andoutflowtermasaninput.However,becauseoftheconstant volumeassumptionineachCSTRonecanarbitrarilymanipulate onlythelocalinletconcentrationsfromtheenvironment,butnot theflowrates.

ChoosethestoragefunctionforthejthCRNastheweightedform oftheLyapunovfunction(15):

S^(j)= V_j

Lnc^(j)−Lnc^(j)∗

T

c^(j)−1^T

c^(j)−c^(j)∗

. (25)

Here ∈R+isapositivefiniteconstant,Lnisthenaturallogarithm appliedelement-wisetoavector,andc^(j)∗ isa(generallyinitial conditiondependent)equilibriumpointofthejthCRN.

Forthepassivitytheoutputmappingofthesubsystemhasto satisfytherelation(6).Fromthemodel(24)itresultsthattheoutput ofthejthsubsystemhastohavetheform:

y^(j)= 1 V_j

∂S^(j)

∂c^(j)

T

. (26)

ByEqs.(25)and(26)yieldsthepassiveoutputvectorofthejth CRN:

y^(j)=

Lnc^(j)−Lnc^(j)∗

. (27)

Lemma1. TheopenCRNsystem(24)withAssumption1ispassive fromtheinputu^(j)totheoutputy^(j)inEq.(27).

Proof. Considerthestoragefunction(25).Thetimederivativeof S^(j)readsas:

S˙^(j)(c^(j))= V_j

Lnc^(j)−Lnc^(j)∗+1

T

˙

c^(j)−1^Tc˙^(j)

, (28)

S˙^(j)(c^(j))= V_j

Lnc^(j)−Lnc^(j)∗

T

1

V_ju^(j)+Y^(j)A^(j)ϕ(c^(j))

. (29) ByAssumption1,ifu^(j)=0, ˙S^(j) isnon-increasing,seee.g.the study[11].Hence,

Lnc^(j)−Lnc^(j)∗

T

Y^(j)A^(j)ϕ(c^(j))≤0.Ityields:

S˙^(j)(c^(j))≤y^(j)Tu^(j), (30)

i.e.thesystem(24)ispassive.䊐

Remark1. ForpassivityanalysisofopenCRNsasimilarapproach was taken in [32]: the input was chosen proportional to the positive input flow rate and the passive output was taken as (Lnc−Lnc^∗)^T(c−c_IN),wherec_INistheinletconcentration.How- ever,forcontrolpurposesitismorebeneficialtochoosetheinput u^(j)asin(24)sincethecontrolcanbeimplementedbymodifying theinletconcentration.

4.2. Distributedsetpointcontrol 4.2.1. Controlproblemstatement

ConsideraprocessnetworkconsistingofCsubsystems(CRNs) thatareinterconnectedthroughadelayedconvectionnetwork.The reactionsthattakeplaceinthereactorsaredescribedbytheopen CRNmodel(24).

Controlobjective:LetthesetpointofthejthCSTRbec^(j)_SP that belongstotheequilibriumpointsetof(17).Designadistributed controllerforeachCRNsuchtoassurethatc^(j)→c^(j)_SPast→∞,∀^j= 1...C.

4.2.2. Localcontrolinput

Toimplementthecontrol,augmenttheinputofeachCRNwith alocalcontrolinflow(v_Ljc^(j)_L )inwhichthespecies’concentrations (c^(j)_L )arespecifiableandtheyareusedaslocalcontrolinputs,and v_Lj ∈R+istheconstantlocalinputflowrate,j=1...C;vL0=0.

As it has already been noted before, the constant volume assumptionineachCSTRimpliesthattheflowratescannotbefreely manipulated,buttheyarerelated.Therefore,thelocalinletspecies concentrations,thataredifferentfromtheoverallfeedconcentra- tionassumedtobeconstantinAssumption3,arethemanipulable localinlet concentrations. In practical schemes,the volume(or overallmass)ofeachCSTRisheldconstantonalowerlevelofa hierarchicalcontrolschemebymanipulatingtheoutputflowrates [6,33]whiletheconcentrationsarecontrolledonahigherlevel:

thatisconsideredinthispaper.

Inthecontrolledprocessnetworktheinputvectorofthejth openCRNhastheform:

u^(j)=

C =0

a_j˜vc⁽⁾(t−T_j)+a_Lj˜vjc^(j)_L −v˜jc^(j) (31)

wherev˜_j=v_j+v_Lj,a_Lj=v_Lj/˜v_j.

Inthecontrolledprocessnetworkthebalanceequationsinthe convectionnetwork(similartoEq.(19))readas:

C =0

a_j=1, j=0,...,C, (32)

˜v_j=

C

=0

a_j˜v₊a_Lj˜v_j, j=0,...,C. (33)

Letusdistinguishintheinflowvectoru^(j)theinterconnection term(i^(j))andthelocalcontrolterm(u^(j)_L )asfollows:

u^(j)=i^(j)+u^(j)_L , (34)

i^(j)=

C =0

a_jv˜c⁽⁾(t−T_j)−(1−a_Lj)˜vjc^(j), (35)

u^(j)_L =a_Lj˜v_j

c^(j)_L −c^(j)

. (36)

Theinterconnectiontermcontainstheflowsfrom-andtotheother openCRNs.Thevalueoflocalcontrolflowcanbespecifiedthrough c^(j)_L .

Fig.1.Interconnectionsbetweentheenvironmentandprocessnetwork.

Theflowsbetweenthecontrolledprocessnetworkandtheenvi- ronmentareshowninFig.1.

4.2.3. Theconceptofsynchronization-basedcontrol

Thepropertiesofthebalanceequations(32)and (33)canbe exploitedtoachieve thedistributedcontrolobjectivebytracing backtheformulatedcontrolproblemtothesynchronizationprob- lem,discussedinSection2.

Ifthesynchronizationcanbereachedintheprocessnetwork,the steady-stateoutputsoftheCRNstakethesamevalue,i.e.y^(j)=y^(k) ast→∞∀j,k=1...C.

Asthesupplyhasconstantconcentrationvector,i.e.c⁽⁰⁾=c^(0)∗ itcanbeconsideredthaty⁽⁰⁾(t)=0,see(27).

IftheoutputsofallCRNsaresynchronized(y^(j)=y⁽ⁱ⁾ ∀^i,j),it yieldsthaty^(j)=y⁽⁰⁾=0 ∀j∈N0,i.e.c^(j)=c^(j)∗ ∀j∈N0.More- over,ifAssumption4holds,y^(j)=0,i.e.c^(j)=c^(j)∗ ∀^j=1...C.

Accordingtothesynchronizationprotocol(8)theinputu^(j)of thesubsystemhastobedesignedasafunctionofthepassiveoutput (y^(j))ofthesubsystemandtheneighboringsubsystems.

However,intheprocessnetworktheinterconnectioninput(i^(j)) dependsonthestatevectors(c^(j)),seeEq.(35).

Definethedesiredinputvectoru^(j)_y oftheCRNsasafunctionof thepassiveoutputs:

u^(j)_y =

C

=0

a_jv˜y⁽⁾(t−T_j)−(1−a_Lj)˜v_jy^(j). (37)

Herey^(j)=

Lnc^(j)−Lnc^(j)_SP

wherec^(j)_SPistheprescribedequilib- riumpoint(setpoint)∀j=1...C.

Thelocalcontrolinputsu^(j)_L oftheCRNsin(34)havetobefor- mulatedsuchthattheycompensatethedifferencesbetweenthe desiredinterconnection(necessarytoreachthesynchronization withtheprescribedsetpoint)andtherealinterconnections.

Fromtherelations(35),(36),(37)wecancomputetheexplicit formofthelocalcontrolinputwhichsatisfiesu^(j)=i^(j)+u^(j)_L =u^(j)_y :

c^(j)_L =y^(j)+ 1 a_Lj˜v_j

_C

=0

a_j˜v

y⁽⁾(t−T_j)

−c⁽⁾(t−T_j)

−v˜j

y^(j)−c^(j)

. (38)

It is visible from (38) that the computation of the control law of each subsystem requires the knowledge of the outputs oftheneighboring(connected)subsystems.Thisnecessitatesthe localcommunicationofthecontrollersinastraightforwardimple- mentation.Therefore,theproposedapproachcanbeclassifiedas distributedcontroldesignasitisdefinede.g.in[7,34].

4.2.4. Stabilityandsteady-statesoftheclosedloopsystem

Theorem1. ConsiderasystemofinterconnectedCRNsinwhicheach subsystemmodelisgivenby(24),(34),(35),(36)andtheintercon- nectionparametersaredefinedasin(32)and(33).IfAssumptions 1–4holdandc^(j)_L in(36)canbechosensuchthatu^(j)=u^(j)_y element- wise∀j=1...C,thenc^(j)(t)isboundedfor t≥0andc^(j)→c^(j)_SP as t→∞∀^j>0.

Proof. BasedonthestoragefunctionS^(j),givenin(25),definethe followingLyapunov-Krasovskiifunctional:

S=2

C j=0

S^(j)+

C j=0

C =0

a_j˜v

t

t−Tj

y^()Ty⁽⁾d. (39)

Thetimederivativeofitreadsas S˙=2

C j=0

S˙^(j)+

C j=0

C

=0

a_j˜v

y^()Ty⁽⁾−y^()T(t−T_j)y⁽⁾(t−T_j)

.

(40)

ByLemma1ityieldsthat ˙S^(j)≤y^(j)Tu^(j) ∀^j.Accordingly:

S˙≤2

C j=0

y^(j)Tu^(j)+

C j=0

C

=0

a_j˜v

y^()Ty⁽⁾

−y^()T(t−T_j)y⁽⁾(t−T_j)

. (41)

By choosing c^(j)_L such that u^(j)=i^(j)+a_Ljv˜j

c^(j)_L −c^(j)

=

u^(j)_y ∀j=1...C,andsincey⁽⁰⁾=0,ityields:

S˙≤2

C j=0

y^(j)T

_C

=0

a_jv˜y⁽⁾(t−T_j)−(1−a_Lj)˜vjy^(j)

(42)

+

C j=0

C =0

a_j˜v

y^()Ty⁽⁾−y^()T(t−T_j)y⁽⁾(t−T_j)

.

Asv˜_j=a_Lj˜v_j+

_C

=0a_j˜v₌

C

=0a_jv˜_j (the sum of inflows is equaltothesumoftheoutflows),itresultsthat

S˙≤

C j=0

C

=0

a_j˜v

−y^(j)Ty^(j)+2y^(j)Ty⁽⁾(t−T_j)

−y^()T(t−T_j)y⁽⁾(t−T_j)

,

S˙≤−

C j=0

C =0

a_jv˜

y^(j)−y⁽⁾(t−T_j)

T

y^(j)−y⁽⁾(t−T_j)

≤0.

(43)

Letusintroducethenotation

e^(j,)_i (t)=y_i^(j)(t)−y⁽⁾_i (t−T_j). (44) As ˙S≤0 it yieldsthat S(∞)=lim_t→∞S(t)<∞ for finite S(0).Accordingly,by(43)ityieldsthat:

C j=0

C =0

a_jv˜

K i=1

_∞

t=0

y^(j)_i ()−y_i⁽⁾(−T_j)

2

d≤S(0)

−S(∞)<∞. (45)

Hence,e^(j,)_i ∈L₂ ∀^i,j,.

SinceS(t)<∞andc_SPi^(j)isafinite,strictlypositiveconstant∀^i,j, ityieldsthatc^(j)_i andconsequentlye^(j,)_i ,y^(j)_i ∈L_∞ ∀^{i,j,.Hence,all} theentriesofc^(j)arebounded∀j.

c^(j)_i ∈L_∞andy^(j)_i ∈L_∞impliesthatu^(j)_i ∈L_∞.ByEq.(24)itcan beseenthat ˙c_i^(j) ∈L_∞ ∀^{i,j.Thetimederivativeofy(j)}_i ^{readsas ˙y(j)}_i =

˙

c_i^(j)/c^(j)_i .ByAssumption2ityieldsthat ˙y_i^(j) ∈L_∞.Hence ˙e^(j,)_i ∈ L_∞ ∀^i,j,.

Ase^(j,)_i ∈L2,e^(j,)_i ∈L_∞and ˙e^(j,)_i ∈L_∞,byBarbalat’slemma,it yieldsthatlim_t→∞e^(j,)_i =0∀i,j,.

SinceT_jarefinite,limt→∞y_i^(j)(t)=limt→∞y⁽⁾_i (t), ∀^i,j,.From Assumption3itresultsthaty⁽⁰⁾_i =0.Hence,byAssumption4,it yieldsthatlimt→∞c_i^(j)(t)=c_iSP^(j) ∀^i,j.䊐

It is importanttonotethat S in (39)and thelocalcontrol input(38)areindependentofthekineticparametersoftheCRN subsystems.

4.3. Distributedsetpointcontrolwithdisturbanceattenuation Thecontrollaw,presentedintheprevioussubsections,requires theknowledgeoftheconvectionnetwork parameters.Theaug- mented controlproposedin this subsectionfollows theideaof high-gaincontrolmethodstoattenuatetheeffectsofunmodelled disturbancesanduncertaintiesintheinterconnectionsonthecon- trolperformances:instablecontrolloops,withsufficientlyhigh feedbackgaintheeffectoftheboundeddisturbancesonsteady- stateperformancescanbemadearbitrarilysmall(seee.g.[35]).

ConsiderthattheCRNsintheinterconnectedsystem,originally describedbythemodel(24),aresubjecttoadditivedisturbances, i.e.theycanbemodelledas

dc^(j)

dt =Y^(j)A^(j)ϕ^(j)(c^(j))+ 1

V_ju^(j)+d^(j), j=1...C (46) wherethedisturbanceinputd^(j)(t)∈R^K.

It isimportant tonoticethat theabovegeneralformis also derivedfromEq.(18)byconsideringthedifferencebetweenthe convectivecomponentmassin-andoutflowtermasaninput,and byseparatingtheinflowtermoriginatingfromtheenvironmentas thedisturbance,i.e.d^(j)=˛_0jv0c⁽⁰⁾.Byassumingthedisturbance changingintime,wehaverelaxedtheconstantenvironmentcon- centrationconditionsinAssumption3inordertodesignacontroller thatrejectsitseffectonthecontrolledoutput.

Notethatthepassivityproperty,discussedinLemma1,ispre- servedfromd^(j)toy^(j)whenu^(j)=0.Itisbecausethedisturbance inputvectorhasthesamedimensionandthesamelinearinput structureasthelocalcontrolinputvector.

Assumption5. Thedisturbanceinputd^(j)iscontinuousand

d^(j)2≤d^(j)_M (47)

whered_M^(j) ∈R+isafiniteconstant.

Definetheinputandoutputvectorsofallthesubsystems

u=(u^(1)T...u^(C)T)^T, (48)

d=(d^(1)T...d^(C)T)^T, (49)

y=(y^(1)T...y^(C)T)^T. (50)

ByAssumption5ityields

d2≤d_M, where d_M=

^C

j=1

d^(j)_M

2

. (51)

Fig.2.ExampleofcontrolledinterconnectedCRNs.

Toattenuatetheeffect ofdisturbances,augmentthedesired input,originallygivenin(37),as

u^(j)_d =

C =0

a_j˜vy⁽⁾(t−T_j)−(1−a_Lj)˜vjy^(j)−

2y^(j) (52)

where ∈R+istheconstant,finitecontrolgain.

Theorem2. ConsiderasystemofinterconnectedCRNsinwhicheach subsystemmodelisgivenby(46),(34),(35),(36)andtheinterconnec- tionparametersaredefinedasin(32)and(33).IfAssumptions1–5 holdandc^(j)_L in(36)canbechosensuchthatu^(j)=u^(j)_d element-wise

∀^j=1...Cwith >1+dM

ε , 0<ε<∞, (53)

thenyconvergestowardtheset{y|y2≤ε}.

Proof. ConsidertheLyapunovfunctioncandidateSgivenin(39).

WiththeCRNmodel(46),byfollowingsimilarargumentsasin theproofofTheorem1,thetimederivativeofSreadsas:

S˙≤−

C j=0

C =0

a_jv˜e^(j,)Te^(j,)−y²₂+2y^Td (54)

wheree^(j,)=y^(j)(t)−y⁽⁾(t−T_j).

By applying that y^Td≤|y|^T|d|≤(y²2+d²2)/2, and by Assumption5ityields:

S˙≤−

C j=0

C

=0

a_jv˜e^(j,)Te^(j,)

+(dM+(−1)y2)(dM−(−1)y2). (55)

Since >1+^d_ε^M, it yields that d_M+(+1)y2>0 and, if y2>ε, dM−(−1)y2<0. Accordingly, ˙S<0 i.e S is a decreasingstoragefunctionify2>ε.

By(39)thedecreaseofSinvolvesthedecreaseofS^(j) orthe decreaseof

t

t−T_jy^()Ty⁽⁾dterms.

AsS^(j)=0iff y^(j)_i =0,i=1...K,j=1...C thedecreaseof S^(j) involvesthatsomey^(j)_i tendto0.

As T_j are finite and positive ∀^{,j, the decrease of}

t

t−T_jy^()Ty⁽⁾dalsoinvolvesthatsomey^(j)_i tendto0.

Theconvergenceofy^(j)_i towardszeropersistsuntily2≥ε,i.e.

untiltherelation(53)issatisfied.䊐

4.4. Restrictiononconvectionflowrates:positivity

During thecontrol design thephysical meaning of thecon- trolinputs,thatareconcentrations,shouldbetakenintoaccount.

Thisimplies,thatthecontrolshouldalwaysbepositive,i.e.c^(j)_L ≥0 element-wise∀^j=1...C.Thesepositivityconditionsimplyrestric- tionsontheachievablesetpointsthatwillbederivedbelow.

Considertheequationsusedforcontroldesignu^(j)=i^(j)+u^(j)_L = u^(j)_y,andi^(j)+u^(j)_L =u^(j)_d respectively.Insteady-stateu^(j)_y =u^(j)_d =0 sincey^(j)=0∀^j=1...C.Hence,thesteady-statevalueofthecontrol is

c^(j)_L ^∗= 1 a_Lj˜v_j⁽⁻ⁱ

(j)

SP+a_Ljv˜jc^(j)_SP)= 1 a_Ljv˜_j

−

C =0

a_j˜vc⁽⁾_SP+˜vjc^(j)_SP

. (56)

Thepositivenessofthecontrolinputinsteadystateisassured ifthefollowinginequalitiesholdelement-wise:

C =0

a_j˜vc⁽⁾_SP ≤˜vjc^(j)_SP, ∀^j,=1...C. (57)

Theaboveinequalityexpressesthecomponentmassconservation conditions,similarlytothosefortheoverallmassinEq.(33).Eq.

(57)putsaconditionontheachievablesetpointinthejthCSTR dependingonthesetpointintheCSTRsconnectedtoitsincoming flows.

Therelation(56)canalsobeappliedtoderiverelationsamong thesetpoints of thecontrolledsubsystem and thesteady state upperboundofthecontrolinputs.

5. Acasestudy

5.1. InterconnectedCRNmodelforsimulations

AninterconnectedCRNnetworkwasconsideredconsistingof threedifferentCRNsandtheenvironment.TheCRNsinthethree subsystemswerechosenas

CRN1: R11:2C→³¹2A→¹²2B, R12:2C³²

[23]2B. (58)

CRN₂:R2: 2A¹²

[₂₁]2B. (59)

CRN₃:R3: A+C⁴⁵

[54]

B. (60)

Itis easytoseethat allthreeCRNsarereversibleorweakly reversibledeficiencyzeronetworkswithasinglegraphcomponent (linkageclass).Therefore,basedon[28],Assumptions1and2(com- plexbalanceandthuspersistenceofthedynamics)arevalidfor themindependentlyoftheirreactionratecoefficients.Itisimpor- tanttoremarkthatcomplexbalanceisoftenfulfilledinpracticefor thermodynamicalreasons,especiallyinthecaseofpurechemical reactions[36].

For thesimulations, therate parameters werechosen tobe _ij=1mol/m³/s, ∀^{i,j.Thereactionstakeplaceinreactorshav-} ingconstantvolumes(V_j=1m³).ForeachCRNthestatevector, incorporatingtheconcentrationofspeciesA,BandC,isdefinedas c^(j)=(c_Ac_Bc_C)^T, ∀^j.

TheinterconnectionsamongtheCRNswereimplementedasit ispresentedinFig.2withthefollowingparametersv˜₌0.1m³/s, a₃₁=0.1. Thedelays in theinterconnectionswerechosen T_ij= 10s, i,j=1,2,3.

Theflowratesintheinterconnectionsbetweentheenvironment andtheprocessnetworkare: fromthesupplytoCRN1 theflow rateis(1−a₃₁)˜v;thecontrolflowratesoftheCRNsarev_L1,v_L2, v_L3;theproductflowratefromtheCRN₃ totheenvironmentis

˜vP=(1−a31)˜v₊vL1+vL2+vL3.

TheoutflowsoftheopenCRNsare:v˜1=v˜+v_L1,v˜2=v˜1+v_L2,

˜v3=˜v2+v_L3.

Theconstantsupplyconcentrationwaschosen c⁽⁰⁾=(0.02 0.02 0.02)^Tmol/m³.

TheinitialstatesoftheCRNswerechosenas c^(j)(0)=(0.05 0.1 0.15)^Tmol/m³,j=1,2,3.

Thesetpointswerechosenas:

c⁽¹⁾_SP =(√ 0.016 √

0.032 √

0.016)^Tmol/m³, c⁽²⁾_SP =(0.180.18√

0.016)^Tmol/m³, c⁽³⁾_SP =(0.875 0.875² 0.875)^Tmol/m³.

5.2. Simulationexperiments 5.2.1. Nocontrolcase

ThesimulationswereperformedinMatlab/Simulinkenviron- ment.First,thebehavioroftheCRNnetworkwasexaminedwithout control(u^(j)_L =0, j=1,2,3).Fig.3showsthetrajectoriesforthis case.ThebehaviorsoftheinterconnectedCRNsdependboth on thedynamicsoftheindividualchemicalreactionsandontheflows throughtheinterconnectionsamongthereactors.Thetrajectories oftheuncontrolledsubsystemsinterconnectedwiththeconvection networkconvergetoinitialvaluedependentequilibriumstates.

Fig.3.Simulationresults–interconnectedCRNswithoutcontrol.

Fig.4. Simulationresults–interconnectedCRNswithdistributedsetpointcontrol.

5.2.2. Distributedsetpointcontrol

Second,thedistributedcontrolwascomputedandimplemented based on the relation u^(j)_L =u^(j)_y −i^(j) with v_Lj=0.01m³/s j= 1,2,3,seethecontrolsignaldefinedin(38).Duringthissimulation experiment,nodisturbancewasconsidered.Thecontrolledtrajec- toriesoftheCRNsarepresentedinFig.4.Fig.4showsthat,withthe proposedcontrol,theprescribedsetpointsarereached.Theoutput calibrationconstantin(27)waschosen =0.8mol/m³.

Theparameter ,introducedintheoutputEq.(27),actsasagain inthecontroller.Itseffectonthecontroltransientperformances ispresentedinFig.5.Thisexamplepresentsthetrajectoryofthe controlledconcentrationc⁽¹⁾_A fordifferentvaluesof .

5.2.3. Comparisonwithamodelpredictivecontroller

ForthesameinterconnectedprocessnetworkaModelPredic- tiveController(MPC) wasdesigned usingthe MPC Controller Simulinkblock of MatlabMPC Toolbox [37]. Thisperforms the linearizationaroundthesetpoint,andthediscretizationofthelin- earizedmodel.Thesamplingperiodfordiscretizationwaschosen 0.1s.TheMPCblocksolvesaconstrainedoptimizationproblemin eachsamplingperiodtoobtainthecontrolinput,see[38].Allthe outputweightsandinputcontrolrateweightsofthecostfunction werechosen0.1.Thelowerboundconstraintsforalltheoutputs andcontrolinputswerechosen0.

Fig.5.Simulationresults–effectof ondistributedsetpointcontrol.

The simulation results withthe MPC is presented in Fig. 6.

By comparing the simulations results of the proposed control withtheMPC,itcanbeaffirmedthatsimilarsettlingtimesand trackingcontrolperformancescanbeobtained.However,forthe implementationoftheproposeddistributedcontrolalgorithmonly the convection network parameters should be known. For the implementationoftheMPC,besideit,theparametersoftheCRN subsystemsshouldalsobeknown.

Fig.6.Simulationresults–interconnectedCRNswithMPC.