5 Finite-Dimensional Methods for Optimal Control of Autothermal Thermophilic Aerobic Digestion Ellina Grigorieva1, Natalia Bondarenko2, Evgenii Khailov2 and Andrei Korobeinikov3 1Texas Woman’s University 2Lomonosov Moscow State University 3University of Limerick 1USA 2Russia 3Ireland 1.Introduction Anactivatedsludgeprocess(ASP)isabiochemicalprocessemployedfortreatingsewageand industrialwaste-water,whichusesmicroorganismsandair(oroxygen)tobiologicallyoxidize organicpollutants,producingawastesludge(orfloc)containingtheoxidizedmaterial. The optimal operationof the biologicaltreatment processesisa challenging task because of the stringent effluent requirements, the complexityof processesas an object of control and the needtoreducetheoperationcost.USlawhasstrictrequirementsontheeffluentqualityofthe ASP;similarstrictrequirementswereadoptedduringthelastdecadeinEuropeandinSouth Africa(MaineDepartmentofEnvironmentalProtection,Augusta,ME,2010;Tzoneva,2007). Polluted water contains a wide range of inorganic and organic chemical species and microorganisms, including pathogens. Filtered concentrated sludge is produced at initial stagesofthetreatment. Theobjectivesofthekeysubsequentstageareeliminatingpotential pathogenic micro-organisms and reducing organic chemical content, which might act as a substrate for further microbial growth, to an acceptable level. Autothermal thermophilic aerobicdigestion(ATAD)isaprocessthatiswidelyusedtoachievethesegoals.ATADmakes useofbacterialgrowthwithinthesludgebothtoreduceorganicchemicalcontentandtokill pathogenic bacteria; aeration of the sludgepromotes the growthof aerobic bacteria, which feedonandreducetheorganicsubstratesinthesludge.AreviewoftheATADorigin,design and operationcan be found inBojarskietal. (2010);Capon-Garciaetal. (2010);Graellsetal. (2010);Layden(2007);Rojas&Zhelev(2009). 1.1Waste-watertreatmentplantoperation ThissubsectionisextractedfromRojasetal.(2010). ATADisoperatedasabatchorsemi-batchprocess.Alargereactorcontainingsludgereceives anadditionalvolumeofuntreatedsludgeatthestartofabatchviaafeedinlet(seeFigures1 www.intechopen.com 92 Industrial Waste 2 Will-be-set-by-IN-TECH and 2). During the batch reaction air is pumped continuously into the reactor providing boththeoxygenationrequiredforaerobicbacterialgrowthandthemechanicalmixingofthe sludge. Digestion of organic substrates proceeds with bacterial growth, predominantly of thermophilesduetoelevatedtemperature. Atthe endofthebatch period,afractionofthe treatedvolumeisremovedand isimmediatelyreplacedbythe nextbatch of intakesludge. Thus the outflow sludge has the same composition as the sludge in the reactor at the end of the batch reaction time t. Batch outflow and inflow cannot be opened at the same time topreventthe untreatedinflowsludgemightfrombedrawnoffwithand taintthe outflow sludge. Thetimebetweensuccessivebatchintakesistypicallyfixedat24hoursforstaffing Fig.1.Externalviewofareactor. Fig.2.Schemeofthereactorforwaste-watertreatment. reasons. In order to achieve the desiredtreatment outcomes, treatment plant designs may includeasingleATADreactorstage(asisshowninFigure1),ortworeactorsinserieswith outflowfromthefirstreactorbeinginflowintothesecond. www.intechopen.com Finite-Dimensional Methods for Optimal Control of Autothermal Thermophilic Aerobic Digestion 93 Finite-DimensionalMethodsforOptimalControlofAutothermalThermophilicAerobicDigestion 3 1.2ModelsofASPandATAD InordertoimproveefficiencyofATADprocess,inrecentdecadesvariouscontrolstrategies for the ASP have been developed by engineers. Simple strategies are typically limited to themaintenanceofsomedesiredvaluesforeasilydeterminableprocessparameters,suchas food-microorganismratio, sludgerecycleflowrate, oroxygenconcentration inthe aeration basin (Holmberg, 1982). More complex models take into consideration that the process also depends on a number of working conditions, such as air pump power to regulate the mean oxygen concentration (Rinaldietal., 1979). Optimal working conditions and control strategies are usually established with a support of mathematical models (Brune, 1985; Kobouris&Georgakakos, 1991; Moreno, 1999; Morenoetal., 2006; Tzoneva, 2007); relevant studies and comparison of control strategies have been done by Debuscheretal. (1999); Fikaretal.(2005);Lindberg(1998);Lukasseetal.(1998);Potteretal.(1996);Qinetal.(1997). Obviously, aparticularsolutiondependsonthe modelthat was employed; overthe last50 years avariety of modelsof differentlevelof complexitywereproposedforASP. It should be taken into consideration, however, that there is considerable uncertainty in inferring both the functional description of processes and the accuracy of parameterizations when applyinglaboratory-derivedmodelstoareal-lifesystem.Inparticular,manyexistingmodels of the waste-water treatment are motivated by microbiological and engineering interest and have been constructed with an intention to incorporate as much as possible of the apparent understanding of underlyingprocesses. As a result, such models tend to include toomanyvariables,processesandparameters,whichareknownwithasubstantialdegreeof uncertainty. Laboratory scale experiments provided understanding (to varying degrees) of many of the microbiological and chemical processes involved. By encapsulating much of this understanding inmathematical models, engineersprovidedtoolsforcomputer simulations thatcanbeusedinthedesignprocess. The "Activated Sludge Model 1" (ASM1) (Henzeetal., 2000) has achieved a broad level of acceptance inthewaste-water treatmentcommunity. BasedonthedominanceoftheASM1 modelinthefield,itsextensiontotheATADprocessassetoutbyGomezetal.(2007)appears to create a de facto standard for modeling this process. The drawbacks of the model are also obvious (Rojasetal., 2010). The full-scale ASM1 model incorporates large numbers of variables and functional responses, and consequently requires many parameters. The motivationforincludingtheseparticularitiesinthemodelispredominantlymicrobiological, and seems to arise from the urge to include all possible components about which there is some knowledge. However, details for many of these functional responses are unknown, and the accuracy of parameter values is uncertain: they are dependent on the chemical andmicrobiologicalmake-upoftheparticularsludge,andthereforetypicallyusedasfitting parameters. Withsuchahighlyparameterizedmodel,setsofparametervaluescanbefound to fit most data, but it is unclear whether the model remains reliable in such situations. Applying a parameterized model under different (but physically reasonable) operating conditionsmaygivespuriousmodelingartifacts.Thismayalsocausenumericaldifficultiesin attemptinganyoptimization. Moreover,amathematicalmodelisnotveryuseful,ifitcannot becompletelyinvestigatedusingexistingoptimizationmethods. Mathematicaloptimization canbeconductedwiththeuseofoptimalcontroltheory. www.intechopen.com 94 Industrial Waste 4 Will-be-set-by-IN-TECH 1.3Optimalcontrolofnonlinearmodels Optimal control problem for a system of ordinary differential equations is the problem of finding an extremum of a functional (objective function) under differential restrictions in the infinite dimensional functional space (for example, the space of all bounded piecewise continuous functions). This isachallenging mathematical problem. Thereareanumber of approachesthatallowtoreducetheproblemtotheanalysisofafinitedimensionalproblem; one sucheffectiveapproachisthe PontryaginMaximumPrinciple(PMP)(Pontryaginetal., 1962), which is a necessary condition for optimality in the original infinitely dimensional problem. It reduces the optimal control problem to the analysis of two point boundary valueproblemoftheMaximumPrinciple.Thisboundaryvalueproblemisfiniteinthesense thatitssolutions,phasevectorandadjointvariable,dependeitherontheinitialorterminal conditions,whichbelongtotheelementsofafinitedimensionalEuclideanspace.Theoptimal controlisuniquelydeterminedbythe so-calledmaximumcondition. Therefore,adesirable control depends on the variables selected as the initial or terminal conditions of the phase vectorandtheadjointvariable.Thus,theconsideredoptimalcontrolproblemcanbereduced toafiniteproblem,andthewell-knownmethodscanbeappliedtosolveit(Fedorenko,1978). Ofcourse,theseargumentsexcludefromconsiderationspossibilitiessuchassingulararcsin the correspondingoptimal control (Bonnard&Chyba, 2003), orsituations when the control hasaninfinitenumberofswitchingsonafinitetimeinterval(Bressan&Piccoli,2007). Undertheassumptionsabove,lookattheboundaryvalueproblemoftheMaximumPrinciple ontheotherside. Forasystemofordinarydifferentialequationswhichislinearincontrols, the maximumconditionalmosteverywhereuniquelydeterminesthe optimal controlinthe form of a piecewise constant function with a finite number of switchings. If we take the moments of switching as the corresponding variables, then the phase vector and adjoint variablebecomedependentontheseswitchings,andtheconsideredoptimalcontrolproblem becomesfiniteinthesensethattherequiredcontrolbelongstotheclassofpiecewiseconstant functionswithanestimatedfinitenumberofswitchingsdeterminedfromtheanalysisofthe maximumcondition. Grigorieva&Khailov(2010a;b),whichareassociatedwiththeproblem ofoptimizingthewaste-water treatmentprocess,supportthesearguments. Theconsidered model, which is due to Brune (1985), is sufficiently simple to be investigated analytically, while it adequately describes the principal features of the ASP and water cleaning control processBrune(1985)formulatedanoptimalcontrolproblemfortheminimizationofthewaste concentration in the ASP and offered the PMP for its solution. However, the analysis of thecorrespondingboundaryvalueproblemfortheMaximumPrinciplewasnotcompleted; insteadtheauthorofferedanumericalsolutiontotheproblemfordifferentpiecewiseconstant controls. Grigorieva&Khailov(2010a;b)dealwiththecompleteanalysisofthismodelfora different objective functions. Grigorieva&Khailov (2010a) formulated the optimal control problemofminimizingthepollutionconcentrationsattheterminaltimeandfoundoptimal solutions forthis. Authors also investigated how these optimal solutionsdepend oninitial conditions and conducted numerical simulation of the ASP for different parameter values. Grigorieva&Khailov (2010b) stated an optimal control problem of minimizing the water pollution concentrations on a given time interval. The optimal solutions for this problem is found in two stages: firstly, the authors investigated the optimal control problem using the PMP. Secondly, an approachbased onGreen’sTheorem(Hájek, 1991), was applied. As a result of this study the authors obtained possible types of optimal solutions. This study www.intechopen.com Finite-Dimensional Methods for Optimal Control of Autothermal Thermophilic Aerobic Digestion 95 Finite-DimensionalMethodsforOptimalControlofAutothermalThermophilicAerobicDigestion 5 reviled a possibility of the existence of singular arcs at corresponding optimal trajectories (Bonnard&Chyba, 2003); this fact makes an analytical study of the problem extremely difficult.However,theapproachbasedonGreen’sTheoremenabledtheauthorstoprovethe absenceofsingulararcsatoptimaltrajectories.Inthesametime,thisapproachoverestimates the number of switchings for optimal controls. Ultimately, a combination of the results obtainedapplyingthesedifferentapproachesallowedtosolvetheconsideredoptimalcontrol problemanalytically. The modelstudiedin these papers is not intended to be the finest orthe mostpreciseASP model; on the contrary, it is robust and rather basic. However, it includes all the essential featuresoftheprocess.Furthermore,themodelisnonlinearandassumesaboundedcontrol; thesefeaturesmakethemodelveryinterestingfromthemathematicalpointofview. An alternative approach, which also allows a reduction of the original optimal control problem to a finite dimensional problem, is a constructing of a finite dimensional parametrization by the moments of switching for some piecewise constant controls of the corresponding attainable set, which fully characterizes the behavior of the original control system.Thisapproachcanbeappliedtosolvearangeofproblems,including: estimatingofcontrolopportunity,thatisinvestigatingwheretrajectoriesofthesystemcan • leadunderdifferentadmissiblecontrols; solvingthecontrollabilityproblem; • solving optimal control problems with a terminal functional, such as the problem of • minimizingpollutionataterminaltime; solvingatimeoptimalcontrolproblem(thatis,howtomoveasystemfromaninitialstate • totheterminalstateforminimaltime). Methods for solving these problems, except the last one, with particular application to the optimalcontrolforATADwillbediscussedinthisChapter. Atimeoptimalcontrolproblem andoptimalcontrolproblemswithanintegralfunctional,suchastheproblemofminimizing pollutionforaspecifiedtimeintervalandtheproblemofminimizingenergyconsumptionfor aerationonagiventimeinterval,aresubjectsoffutureresearch. 1.4Rationalandbasicassumptionsformodeling The ATAD process is efficient but it can be costly, as it requires continuous aeration that generallyisenergy-consuming. Optimizingtheaerationcansignificantlyreducethecostof operation.ExistingATADmodelsarefartoolargeandcomplex,andthiscomplexityprevents the application of the usual optimization techniques. Our aim is, therefore, to construct a modelthatretainsthefundamentalpropertiesoftheprocess(aswellastheseofthelargescale existingmodels)whilebeingsufficientlysimpletoallowtheuseofoptimizationprocedures. With this intention in mind, we formulate and investigate a simplifiedmodelof the ATAD reactionbasedontheexistingASM1modelatthermophilictemperatures. InthisChapterweformulateamodelthat(incontrasttoarathercomplexASP1model)we are capable of investigating analytically and optimizing numerically. This model includes the essential mechanisms of the ATAD process while ignoring the issues, which, though www.intechopen.com 96 Industrial Waste 6 Will-be-set-by-IN-TECH microbiologicallyknown,arenotmaterialtotheprocesswithinthelikelyrangeofoperating conditions. It is our expectation that this model will describe the process with a sufficient degreeofaccuracywhileprovidingasuitablebasisforoptimization.Anumberofsimplifying assumptions permitus to reduce the complexity of the model while retaining the essential aspectsoftheprocesses.Theseassumptionsaresummarizedasfollows: Weassumethatthereactorinwell-stirred(asitisinpractice),andhenceconcentrationsof • thereagentsarehomogeneousthroughoutthevolume. Allthebiologicalactivitytakesplaceonlyinthereactor. • The batch outflow and inflow stages are of sufficiently short duration, and the biological • activityduringthesestagesisnegligible. Aerationissufficient,andanaerobicmetabolicactivityisnegligible. • We consider this process as a reaction with three reagents, namely: oxygen with • concentrationx(t),organicmatterofconcentrationy(t),andbacteriaofconcentrationz(t). Thereactionisgovernedbythelowofmassaction. • Theaerationrateu(t)istheonlycontrol.Thiscontrolfunctionisbounded. • 2.Mathematicalmodel Inordertodescribetheprocessofaerobicbiotreatment,weconsiderasimplemathematical model, which representthe process as a chemical reaction with three reagents, namely the concentration of oxygen x(t), organicmatter with concentration y(t), and the thermophilic aerobic bacteria with concentration z(t). It is assumed that the mass in the reactor is well stirred,andhencethereactantconcentrationsarehomogeneousinthevolume.Thechangesof concentrationsofthereagentsaredescribedbyathree-dimensionalnonlinearcontrolsystem x˙(t)= x(t)y(t)z(t)+u(t)(m x(t)), t [0,T], − − ∈ y˙(t)= x(t)y(t)z(t), ⎧ − (1) ⎪⎪⎨zx˙((t0))==xx(t,)yy((t0))z(=t)y−,bzz((0t)),=z , x (0,m), y >0, z >0. 0 0 0 0 ∈ 0 0 Its nonlinearity⎪⎪⎩is justified by the law of mass action (Krasnovetal., 1995) describing the dependenceofthe rateofchemicalreactionontheconcentrations ofinitialsubstances. The firstequationofsystem(1)representstheevolutionofoxygenconcentration: thefirstterm, x(t)y(t)z(t),describestheprocessofitsabsorptioninthereaction,whereasthesecondterm − describesinfluxofoxygen(bypumping)intothereactorfromoutside. Here,u(t)istherate of aeration, which at the same time is the control function. The second equation describes adecreaseofthe organicmatter inthe reaction. The thirdequationofsystem(1) showsan evolutionoftheactivebiomassconcentration;thebacteriamassgrowsattheratex(t)y(t)z(t) and decays at a rate b. The original system also includes positive initial conditions and a restrictionontherateofpumpingair.WeconsiderallpossibleLebesguemeasurablefunctions u(t),whichforalmostallt [0,T]satisfytheinequality ∈ 0 u(t) umax, (2) ≤ ≤ www.intechopen.com Finite-Dimensional Methods for Optimal Control of Autothermal Thermophilic Aerobic Digestion 97 Finite-DimensionalMethodsforOptimalControlofAutothermalThermophilicAerobicDigestion 7 astheadmissiblecontrolsD(T). Thephasevariablesx,y,zofsystem(1)satisfythefollowingproperties. Lemma 1. Let u( ) D(T) be an arbitrary control. Then the corresponding solution w(t) = · ∈ (x(t),y(t),z(t))⊤ofthesystem(1)isdefinedontheinterval[0,T],thecomponentsx(t),y(t),z(t)of whichsatisfythefollowinginequalities: 00forallt Δ. ∈ Next,byanalogy,weobtainthesolutionofthethirdequationoftheconsideredsystem z(t)=z0e 0t(x(ξ)y(ξ)−b)dξ, (5) (cid:6) fromwhichwehavethatz(t)>0forallt Δ. ∈ Finally,wecanwritethesolutionofthefirstequationofsystem(1)as x(t)=e− 0t(y(ξ)z(ξ)+u(ξ))dξ x0+m te 0s(y(ξ)z(ξ)+u(ξ))dξu(s)ds . (6) (cid:6) (cid:7) (cid:8)0 (cid:6) (cid:9) Fromthisformularegardinginequality(2)wefindthatx(t)>0forallt Δ. ∈ Therefore,weobtaintheinequalities: x(t)>0, y(t)>0, z(t)>0, thatarevalidforallt Δ. ∈ Further, considering positiveness of solutions x(t) and z(t), from formula (4) we find the inequalityy(t)0. (cid:10) 0 − 0 Using the variation of a parameter method (Hartman, 1964) we find the solution of this problemas ν(t)=e− 0t(y(ξ)z(ξ)+u(ξ))dξ ν0+m te 0s(y(ξ)z(ξ)+u(ξ))dξy(s)z(s)ds . (cid:6) (cid:7) (cid:8)0 (cid:6) (cid:9) ByLemma1fromthisexpressionwehaverelationshipν(t) >0forallt [0,T],fromwhich ∈ therequiredinequalityfollows. Lemmas1and2implythatforanycontrolu( ) D(T)solutionsx(t),y(t),z(t)ofthesystem · ∈ (1)retaintheirphysicalmeaningsforallt [0,T]. ∈ 3.Attainablesetanditsproperties Let X(T) ⊂ R3 be the attainable set for system (1) from an initial point w0 = (x0,y0,z0)⊤ at the moment of time T; that is, X(T) is the set of all ends w(T) = (x(T),y(T),z(T))⊤ of trajectoriesw(t)=(x(t),y(t),z(t))⊤ofsystem(1)underallpossiblecontrolsu( ) D(T).By · ∈ Lee&Markus(1967)andLemma1,itfollowsthatsetX(T)isacompactsetinR3locatedinto theregion w=(x,y,z)⊤ R3 :x>0,y>0,z>0 . ∈ (cid:11) (cid:12) We denote by ∂Q and intQ the boundary and the interior of the compact set, Q R3, ⊂ respectively. www.intechopen.com Finite-Dimensional Methods for Optimal Control of Autothermal Thermophilic Aerobic Digestion 99 Finite-DimensionalMethodsforOptimalControlofAutothermalThermophilicAerobicDigestion 9 Next, we study the boundary of the attainable set X(T). Consider a point w = (x,y,z)⊤, such that w ∂X(T). It corresponds to control u( ) D(T) and a trajectory w(t) = ∈ · ∈ (x(t),y(t),z(t))⊤, t [0,T], of system (1), such that w = w(T). Then it follows from ∈ Lee&Markus (1967) that there exists a non-trivial solution ψ(t) = (ψ1(t),ψ2(t),ψ3)⊤, t ∈ [0,T]oftheadjointsystem ψ˙ (t)=u(t)ψ (t)+y(t)z(t)(ψ (t)+ψ (t) ψ (t)), 1 1 1 2 − 3 ⎧ψ˙ (t)=x(t)z(t)(ψ (t)+ψ (t) ψ (t)), (7) ⎪⎪⎪⎨ 2 1 2 − 3 ψ˙ (t)=x(t)y(t)(ψ (t)+ψ (t) ψ (t))+bψ (t), 3 1 2 − 3 3 forwhich,byLemm⎪⎪⎪⎩a2,thefollowingrelationshipisvalid: 0, if L(t)<0, u(t)=⎧ u [0,umax], if L(t)=0, (8) ⎪⎪⎨u∀ma∈x, if L(t)>0. Here L(t) = ψ (t). Function L⎪⎪⎩(t) is the so-called switching function and its behavior 1 determinesthetypeofcontrolu(t). Forconvenienceinsubsequentarguments,weintroducethefollowingauxiliaryfunctions: G(t)=ψ (t)+ψ (t) ψ (t), P(t)= ψ (t), d(t)=y(t)z(t)+z(t)x(t) x(t)y(t), t [0,T]. 1 2 − 3 − 3 − ∈ Usingtheadjointsystem(7),wewriteforfunctions L(t), G(t), P(t)thefollowingsystemof lineardifferentialequations L˙(t)=u(t)L(t)+y(t)z(t)G(t), t [0,T], ∈ ⎧G˙(t)=u(t)L(t)+d(t)G(t)+bP(t), (9) ⎪⎪⎨P˙(t)= x(t)y(t)G(t)+bP(t). − ⎪⎪⎩ ThevalidityofthefollowingLemmaimmediatelyfollowsfromthenon-trivialityofsolution ψ(t)=(ψ1(t),ψ2(t),ψ3(t))⊤oftheadjointsystem(7). Lemma 3. The switching function L(t) and the auxiliary functions G(t) and P(t) are nonzero solutionsofsystem(9). Lemma3allowsustorewritetherelationship(8)intheform 0, if L(t)<0, u(t)= (10) (cid:13)umax, if L(t)>0. Atpointsofdiscontinuitywewilldefinefunctionu(t)byitslimitfromtheleft.Consequently, the control u(t), t [0,T], corresponding to a point w ∂X(T), is a piecewise constant ∈ ∈ function,takingvalues 0,umax .Suchtypeofcontrolisusuallycalledabang-bangcontrol. { } www.intechopen.com 100 Industrial Waste 10 Will-be-set-by-IN-TECH Next,weestimatethenumberofswitchingsofcontrolfunctionu(t),t [0,T].Itfollowsfrom ∈ (10) that it is sufficientto estimate the number of zeros of the function L(t) on the interval (0,T).Thefollowingimportantstatementisvalid. Lemma4. TheswitchingfunctionL(t)hasatmosttwozerosontheinterval[0,T]. Proof. Letusintroduceforsystem(9)newvariables: r(t)=L(t), v(t)=G(t), μ(t)=P(t)+q (t)L(t)+q (t)G(t), 1 2 wherefunctionsq (t),q (t)mustbedetermined. Usingnewvariablesr(t),v(t),μ(t)system 1 2 (9)hasthefollowingform r˙(t)=u(t)r(t)+y(t)z(t)v(t), ⎧v˙(t)=(u(t)−bq1(t))r(t)+(d(t)−bq2(t))v(t)+bμ(t), ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨μ˙(t)+=(cid:14)qq˙˙1((tt))++y((ut()tz)(−t)qb)(qt1)(+t)(+d(ut()t)qb2()tq)(−t)bq1b(tq)2q(2t()t)(cid:15)rx((tt))+y(t) v(t)+ (11) 2 1 − 2 − 2 − Wewillse⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩lectsuc+hb(cid:14)f(u1n+ctiqo2n(st)q)μ((t)t)a.ndq (t)thatinsystem(11)theexpressions(cid:15)insidebrackets 1 2 byr(t)and v(t)arezero. Now, wehave forfunctions q (t), q (t)the systemofdifferential 1 2 equations q˙ (t)+(u(t) b)q (t)+u(t)q (t) bq (t)q (t)=0, 1 − 1 2 − 1 2 (12) q˙ (t)+y(t)z(t)q (t)+(d(t) b)q (t) bq2(t) x(t)y(t)=0. (cid:10) 2 1 − 2 − 2 − Thensystem(11)willberewrittenas r˙(t)=u(t)r(t)+y(t)z(t)v(t), v˙(t)=(u(t) bq (t))r(t)+(d(t) bq (t))v(t)+bμ(t), (13) ⎧ − 1 − 2 ⎨μ˙(t)=b(1+q2(t))μ(t). ⎩ Insystem(13)wewillmakethefollowingsubstitutions: r(t)=r(t), v(t)=v(t)+q (t)r(t), μ(t)=μ(t), 3 whereq (t)isthefunctionthatmustbedetermined.Thensystem(13)willberewrittenas 3 (cid:16) (cid:16) (cid:16) r˙(t)=(u(t) y(t)z(t)q (t))r(t)+y(t)z(t)v(t), − 3 ⎧v˙(t)= q˙ (t) bq (t)+(u(t) d(t))q (t)+ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨(cid:16)(cid:16) +b(cid:14)q23(t)q−3(t)−1 y(t)z(t)q(cid:16)23−(t)+u(t3) r(t(cid:16))+ (14) (cid:15) +(d(t)+y(t)z(t)q (t) bq (t))v(t)+bμ(t), ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩μ˙(t)=b(1+q2(t))μ(t).3 − 2 (cid:16) (cid:16) (cid:16) (cid:16) (cid:16) www.intechopen.com

Finite-Dimensional Methods for Optimal Control of Autothermal Thermophilic Aerobic Digestion

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