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Contents lists available at ScienceDirectFueljournal homepage: /locate/fuelFull Length ArticleControllability and reachability of reactions with temperature and inowcontrolDniel Andrs Drexlera, Eszter Virghb, Jnos TthbaPhysiological Controls Research Center, buda University, Kiscelli u. 82, Budapest H-1032, HungarybDepartment of Analysis, Budapest University of Technology and Economics, Egri J. u. 1, Budapest H-1111, HungaryARTICLE INFOKeywords:Lie-algebraTemperature controlInow controlFuel cellABSTRACTKnowing the controllability properties of reactions may be critical in the design phase of controlled systems.Analysis of the controllability properties of reactions is done here with control inputs being the temperature ofthe reaction, and the inow rates of some species. The analysis is based on the Lie-algebra generated by thevector elds related to the dierential equation of the reaction. We prove that the chemical reactions arestrongly reachable on a subspace that has the same dimension as the number of independent reaction steps inevery point except where the concentration of reactant species is zero, and show that this result holds forreactions in continuously stirred tank reactors as well. Finally, we analyze the controllability of the anode andcathode reactions of a polymer electrolyte membrane fuel cell with the inow rates of hydrogen and oxygenbeing the control inputs and show that by using the inow rates as control inputs the dimension of the subspaceon which the system is controllable can be greater than the number of independent reaction steps.1. IntroductionControlling chemical reactions is a key issue in modern chemicalengineering science 13, see e.g. 4,5, where control of nonlinearchemical reactors is considered. The controllability properties of asystem can tell us how we can aect the system by using some controllaw, e.g. whether we can aect the resulting equilibrium state of asystem, that may be the case for a controllable system, or we can onlyaect the speed and certain qualities of the transient, as it may be thecase for a system that is only reachable, but not controllable. Thus,analysis of the controllability properties is desirable before the design ofcontrol systems.The controllability of chemical reactions is usually checked onlylocally using linear tests, i.e. approximation of the nonlinear dierentialequations with linear ones, and using results from the control theory oflinear systems, see one of the rst papers by 6 on the observabilityand controllability of continuously stirred tank reactors (CSTRs). Con-trollability of chemical reactions that have a positive equilibrium isestablished in 7 based on the linearized dynamics. In 8, controll-ability analysis of a liquid-phase catalytic oxidation of toluene to ben-zoic acid was done based on linearization at ve dierent operatingpoints, and controllability analysis of polymerization in dierent op-erating points was done in 9. Controllability analysis of proteinglycosylation was done based on a linear model identied from mea-surements in 10. The connection of controllability and structure ofchemical reaction was analyzed in 11, while controllability of che-mical processes was analyzed in 1216.A key issue in the formulation of the control problem is the controlinput. In 7 the reaction is considered as a multiple input system,where the rates of the reaction steps can be controlled independently,thus the control inputs were the reaction rates of the reaction steps. Thecontrollability of such a system was proved in a positive equilibrium,provided that such an equilibrium exists. This result was further gen-eralized in 17, and it was shown that the controllability is a globalproperty if we can control all the reaction rates independently, and thepossibility of reducing the required number of control inputs was alsoinvestigated. However, implementation of control by altering the re-action rates has some problems from the practical point of view. Herewe will consider the temperature of the reaction and inow rate ofsome species as control inputs.Controllability analysis in operating points, however, only giveslocal results. The application of Lie-algebra rank condition was sug-gested for the controllability analysis of chemical reactions in 18,however no general results were given. The Lie-algebra generated bythe vector elds of the reaction was used for controllability analysis ofreactions whose control inputs are the reaction rates in 17. A review/10.1016/j.fuel.2017.09.095Received 10 February 2017; Received in revised form 23 August 2017; Accepted 26 September 2017Corresponding author.E-mail addresses: drexler.danielnik.uni-obuda.hu (D.A. Drexler), viragh.eszter (E. Virgh), jtothmath.bme.hu (J. Tth).Abbreviations: PEMFC, polymer electrolyte membrane fuel cell; CSTR, continuously stirred tank reactorFuel 211 (2018) 906911Available online 07 October 20170016-2361/ 2017 Elsevier Ltd. All rights reserved.MARKof controllability analysis of chemical reactions was done in 19.Analternative method for controllability analysis is shown in 20.We give the denitions of the controllability properties and theconditions used for controllability analysis in Section 2. The mass actionkinetic model used to model reactions is discussed in Section 3. We givethe model of temperature dependence of the reaction rate coecients,and the dynamics of the temperature of the reaction.We analyze the controllability properties of chemical reactions inthe case if the controlled variable is the temperature of the reaction inSection 4. We extend the acquired results for reactions in CSTRs. Fi-nally, we analyze the controllability properties of a modied polymerelectrolyte membrane fuel cell (PEMFC) model with the control inputbeing the inow rate of the oxygen, and the inow rate of oxygen andhydrogen.2. Controllability and reachabilityConsider a nonlinear system whose dynamics is governed by thedierential equation=+x fx gxtt tut() () ()() (1)at time instant DoubleCapR+t , with DoubleCapRx t()Mbeing the state of the system attime instant ft, is a smooth vector eld (i.e. DoubleCapRDoubleCapRf (,)MMC ) called thedrift vector eld, g is a smooth vector eld (i.e. DoubleCapRDoubleCapRg (,)MMC ) calledthe control vector eld, and u t( ) is the control input at time instant t.Denition 1. 17 Dene the following sets, called the reachability (oraccessibility) sets:x T(,)0R =the set of the states of the system (1) at time T if theinitial condition is =x x(0)0with all possible inputs such thatDoubleCapRu T(0, , )L provided that the solution is dened on T0, .=xxt() (,)t000R R , i.e. all the states of the system that can bereached in arbitrary time with the initial condition =x x(0)0and anarbitrary bounded input.Denition 2. The system described with the dierential Eq. (1) iscalled strongly reachable (or strongly accessible) from the pointx ifthe setx T(,)R has an interior point for all T 0.Denition 3. The system described with the dierential Eq. (1) iscalled locally controllable in the pointx ifx is the interior point ofx()R .Denition 4. The system described with the dierential Eq. (1) iscalled small-time locally controllable in the pointx ifx is the interiorpoint ofx T(,)R for arbitrary T 0.The Lie-bracket of two smooth vector elds DoubleCapRDoubleCapRff,(,)MM12C isdened as= ff ff ff , .12 2 1 1 2 (2)We use the ad operator to denote the application of the Lie-bracketas=ffadf0221(3)=fffad , f12121(4)=ff fad ,ad .ii211211(5)Denition 5. The Lie-algebra generated by the vector elds f1and f2denoted by ffLie,12is the smallest Lie-algebra (subspace) that satisesthe following conditions:1. ff ffLie,12 12.2. For any two vector elds ffLie,12 1 2it is true that f fLie , ,12 1 2.Theorem 1. 21 The system dened by the dierential Eq. (1) is stronglyreachable from a point DoubleCapRxMif and only if the Lie-algebra generated bythe vector elds g and fg ,is M-dimensional at DoubleCapRxM.Note that it may happen that the dimension of gfgLie,is i 0=ffad .giTi(26)Let us introduce the notations=fk x xk k xr k f k() (,) , () / .rRrrrRrrr01(, )1(, )(27)Using these notations, the drift vector eld f becomes=fkfkkf()()().0(28)Note that f is written in this form as a function of k, and only thecoecients k depend on the temperature in the function f. Since f islinear in k, with the application of the chain rule we get =ffkkx kxf rk()() 1/ (, ).TiiiirRrri0()()()1()(29)Here we have used the notation ki()to denote the ith derivative of thefunctionk with respect to the temperature. Thus, the subspace spannedby the vector elds gff f,ad ,ad , ,adg ggR2is the range space of the matrix x k x k x k01, R(1) (2) ( )(30)where the stars denote general scalar elements that we do not write outfor the sake of simplicity. The rst column (related to the temperature)is trivially independent of the other columns, thus the subsystemrelated to temperature governed by the dierential Eq. (17) isreachable. However, we need to investigate the subspace spanned bythe columns of the matrix that we get after removing the rst columnand last row of the matrix (30), which results in =x k x k x k x D( ) R(1) (2) ( )(31)with D being the reaction dynamics matrix. Since the reactiondynamics matrix is of full rank if the activation energies of thereaction steps are dierent due to Lemma 1, the dimension of therange space of this matrix equals to the dimension of the range space ofxthat equals to the dimension of the range space of if xis of fullrank. However, if xis rank decient then it yields that theconcentration of some reactant species is zero, thus the conditions ofthe theorem exclude this case, meaning that xis of full rank. Since thedimension of the range space of equals to the dimension of thestoichiometric subspace (i.e. rank ), this concludes the proof. The consequence of Theorem 3 is that if the conditions of the the-orem are met, the reaction is strongly reachable, meaning that we canhave aect on the processes in the reaction by controlling the tem-perature. However, the reaction has the strong reachability propertyonly on a rank -dimensional subspace, that can be less than the numberof species in the reaction. If this is the case, then we can aect only theconcentration of rank number of species, while the concentration ofthe remaining M rank number of species are constrained by thestoichiometric equations. In Section 4.3 we will show on an examplethat in this case we may increase the dimension of the subspace (thenumber of species) we can aect by using the inow rates as controlinputs as well.The conditions of Theorem 3 state that the concentrations of thespecies that participate in reactant complexes must be nonzero. How-ever, it is enough if we choose rank number of independent reactionsteps, and the concentration of the species in these reaction steps mustbe nonzero. If there are species that are reactants in other reactionsteps, but products in at least one of the previously mentioned ranknumber of independent reaction steps, then their concentration may bezero, and the reaction is still strongly reachable.Note that if the concentration of some of the species participating insome of the independent reaction steps runs low, i.e. the species isconsumed, the strong reachability property of the reaction becomes ill-conditioned. This means that we can still aect the process, but it be-comes much harder (e.g. much larger amplitude is needed in the inputto have the same eect). Mathematically, the concentrations of reactantspecies can not become zero, if they were initially positive, but prac-tically their concentration may become so low that the strong reach-ability property becomes ill-conditioned. In this case the consumedspecies must be relled in order to regain the strong reachabilityproperty of the reaction.4.2. Strong reachability of CSTRs with temperature inputNow we show that Theorem 3 remains valid in the case of CSTRs.Consider a CSTR where the mth species enter the reactor withconstant ow rate am, and leaves the reactor with ow rate xb ( )mmthat depends on the concentrations, but is independent of the tem-perature and the reaction rate coecients. Let =a aa a(, )M12and=bx x x xb b b () ( (), (), ()MM1122. Then (11) is modied to=+=x x abxtkTrt() ( ) (,) () ().rRrr1(, )(32)The dierential Eq. (17) describing the dynamics of the temperatureis not aected directly by the inow and outow. The control vectoreld for the CSTR system is the same as in the previous subsection;however, the rst M components of the drift vector eld are modiedwith the term + abx( ). Note that these terms do not depend on thetemperature. This yields that the Lie-brackets used in the proof ofTheorem 3 are not aected, so Theorem 3 is true for CSTR models aswell.4.3. Controllability of a PEMFC with inow controlAbove we have given the condition for the strong reachability ofreactions if the input is temperature control. The theorem shows thatthis property can be guaranteed on a rank -dimensional subspace if theinput is the temperature. In certain cases this number can be small: inthis subsection we show an example of a PEMFC that is only stronglyreachable on a 2-dimensional subspace if we use the temperature ascontrol input. We show that the dimension of this subspace can be in-creased to four if we use the inow of some species as control inputs;moreover, we can ensure local controllability with these inputs that is astronger property than strong reachability.Consider a PEMFC with cathode reaction+12O2H2e HOk221(33)as its rst reaction step and anode reaction+ H 2H 2ek22(34)as its second reaction step. We will use the notations+X O ,X H ,X e ,X H O122 3 42and XH52and modify the stoi-chiometric coecients to get integers (thus modify the model) thatresults inD.A. Drexler et al. Fuel 211 (2018) 906911909+X4X4X2Xk123 41(35)+X2X2X.k5232(36)The matrices formed of the stoichiometric coecients are=1040400001,0002022000,1042422001.(37)Let xxxxx,12345denote the concentration of the species with theappropriate index, and let =x xxxxx(,)12345. The dierential equationof the reaction is=+xxxxxkxx xkxx x kxkxx x kxkxx xkx422fx12345112434112434251124342511243425() (38)without input. Now we suppose that the coecients k1and k2areconstant. Suppose that theu1control input is the inow of X1(O2), thusthe control vector eld g1is the rst unit vector, i.e. the dierentialequation of the system becomes=+xxxxxkxx xkxx x kxkxx x kxkxx xkxu42210000.fxgx12345112434112434251124342511243425()()11 (39)The Lie-bracket of f and g1is=fg kx x ,1442011 2434(40)that is independent from g1whenever x 02and x 03, thusgfgspan , , 11is two-dimensional if x 02and x 03. The Lie-bracketffg , , 1is=+ffg kxx kxx kxx kxx , , 14420(88)11 23331 2535252 253(41)that is a scalar multiple of fg ,1and vanishes at =x 02or =x 03as well,thus does not mean new direction. This yields that the system may onlybe controlled on a two-dimensional subspace of DoubleCapR5, if the sucientcondition for Theorem 2 holds. Since the Lie-bracket gfg , , 11is zero(and every iterated Lie-bracket containing the vector eld g1at leasttwice is zero, since fg ,1does not depend on x1), the sucient conditionof local controllability holds in every point. However, this system isonly controllable on a two-dimensional subspace of DoubleCapR5.Now consider a second input u2that is the inow of the species X5(H2), i.e. the control vector eld g2is the fth unit vector, and thedierential equation of the reaction becomes=+xxxxxkxx xkxx x kxkxx x kxkxx xkxuu4221000000001.fxgx gx12345112434112434251124342511243425()()1()212 (42)The Lie-bracket of f and g2is=fg k ,0220122(43)which is linearly independent from gg,12and fg ,1and every iteratedLie-bracket containing g2at least two times is zero, thus the sucientcondition for Theorem 2 holds, so the system is controllable on at leastgg fg fgspan , , , , , 12 1 2that is four-dimensional except when =x 02or=x 03. Now examine the Lie-bracket ffg , , 2that is=+ffgkkxxx x xkkxxxxx kkkxxxxx kkxxx x xk , , 8()2(16 ( ) )2(16 ( ) )16 ( )2121233323211233323 2211233323 21123332322(44)that can be written as the linear combination+fg fgkx x xxxk8( ), , 21 2 323122(45)thus it is not linearly independent from the vector elds fg ,1and fg ,2.Note that taking the iterated Lie-brackets with the drift vector eld fwill not result in new directions, since it was shown in 17 that the Lie-bracket of vector elds corresponding to reaction steps is the linearcombination of the columns of the stoichiometric matrix, and fg ,1andfg ,2already span the column space of . Consequently, the analysisshowed that the PEMFC is (small-time) locally controllable on a 4-di-mensional subspace of DoubleCapR5in every equilibrium where x 02andx 03and on every trajectory not passing through the lines =x 02or=x 03with controlling the inow of oxygen and hydrogen.Note that if we wanted to use the temperature as the control input tocontrol the PEMFC, by Theorem 3 we can only have strong reachabilityover a 2-dimensional subspace of DoubleCapR5.5. ConclusionOur analysis showed that every chemical reaction is stronglyreachable if the reactant species are present in the reaction. However,they are only strongly reachable on a rank -dimensional subspace ofDoubleCapRMthat may be a proper subspace of DoubleCapRM(i.e. have dimension less thanM). If the number of spe
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