# Robust Control of Volatile Fatty Acids in Anaerobic Digestion Processes

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PROCESS DESIGN AND CONTROLRobust Control of Volatile Fatty Acids in Anaerobic Digestion ProcessesHugo O. Mendez-Acosta,*, Bernardo Palacios-Ruiz, Vctor Alcaraz-Gonzalez,Jean-Philippe Steyer, Vctor Gonzalez-Alvarez, and Eric LatrilleDepartamento de Ingeniera Qumica, CUCEI-UniVersidad de Guadalajara, BlVd. M. Garca Barragan 1451,C.P. 44430 Guadalajara, Jal, Mexico, and INRA, UR050, Laboratoire de Biotechnologie de lEnVironnement,AVenue des Etangs, Narbonne, F-11100, FranceThis paper is focused on the experimental implementation of a robust control scheme for the regulation ofvolatile fatty acids (VFA) in continuous anaerobic digestion processes. The robust scheme is made of anoutput feedback control, and an extended Luenberger observer is used to estimate the uncertain terms of theprocess (i.e., influent concentration and process kinetics). The control scheme is implemented in a pilot plantup-flow fixed-bed reactor that is treating industrial wine distillery wastewater. The performance of the robustscheme is tested over a period of 36 days, under different set-point values and several uncertain scenarios,including model mismatch, badly known parameters, and load disturbances. Experimental results show thatthe VFA concentration can be effectively regulated over a wide range of operating conditions. In addition, itis shown that the control scheme has a structure that improves its performance in the presence of noisymeasurements and control input saturations.1. IntroductionAnaerobic digestion (AD) has regained the interest of thewastewater treatment scientific and industrial community toreduce and transform the organic matter from industrial andmunicipal effluents into a gaseous mixture called biogas,1 whichis composed mainly by methane and carbon dioxide. Neverthe-less, its widespread application has been limited, because ofthe difficulties involved in achieving the stable operation of theAD process, which cannot be guaranteed by regulating tem-perature and pH, because the microbial community within theAD process is quite complex2 (composed of more than 500species). In addition, the behavior of such a process may beaffected by the substrate composition, inhibition by substratesor products, and the type of bioreactor. Moreover, it is well-known that, to guarantee the so-called operational stability3 andto avoid the eventual breakdown of the anaerobic digester, theorganic matter in the liquid phase must be kept in a set ofpredetermined values, depending on factors such as the reactorconfiguration and the characteristics of the wastewater to betreated.4 However, the complex nonlinear and nonstationarynature of the AD process, the feed composition overloads, andthe presence of toxic and inhibitory compounds enhance thecontrol problems that are associated with the regulation of theorganic matter and the compliance of the stringent environmentalpolicies.Over the past decade, the regulation of the organic matterhas been addressed by proposing many control techniques tokeep certain operating variables which are readily available (suchas the chemical oxygen demand (COD) and the biogas produc-tion) at a predetermined value.4-7 Classical proportional integral/proportional integral differential (PI/PID) control has beenrecognized as a good alternative for AD control when there islittle knowledge about the plant behavior and no mathematicalmodels are available.6,7 However, it is well-known that itsperformance is strongly dependent on the tuning parameters,which, in the application to nonlinear systems, are only validaround a given operating point. Moreover, the presence of inputconstraints (often called hard constraints) has been shown toseriously degrade the PI/PID performance limiting their practicalapplications. Nevertheless, the problem of the operationalinstability due to the accumulation of volatile fatty acids (VFA)remains open. The advent of reliable sensors8-10 for key ADvariables has brought about the possibility of implementing newcontrol alternatives to address the typical operating problemsin AD processes. In this context, the regulation of the VFAconcentration as a controlled variable seems to be very promis-ing, because the operational stability of the AD process is largelydependent on the accumulation of VFA.11,12 To our knowledge,only a few contributions among those that have addressed theregulation of VFA in AD processes13-16 have been experimen-tally implemented.13,15 Thus, the main motivation and contribu-tion of the present work is the experimental implementation ofa simple robust control scheme that is capable of regulatingthe VFA concentration in continuous AD processes in the faceof (i) uncertain load disturbances; (ii) model mismatch and badlyknown parameters due to the complex nonlinear nature of theprocess (i.e., uncertain kinetics), (iii) noisy measurements and(iv) control input constraints, because the dilution rate isbounded in practice to avoid undesired operating conditions,such as the washout condition.17 The paper is organized asfollows. First, a brief description of a mathematical model thatdescribes a typical continuous AD process is presented. Second,the control problem is stated in terms of the aforementionedmodel. Later, the robust control scheme is obtained and itsclosed-loop behavior is analyzed. The robust scheme then isexperimentally implemented in a pilot-plant up-flow fixed-bedreactor that is treating industrial wine distillery wastewater toevaluate the controller performance and robustness under the* To whom correspondence should be addressed. Fax: 52 3339425924. E-mail address: hugo.mendez@cucei.udg.mx. Departamento de Ingeniera Qumica, CUCEI-Universidad deGuadalajara. INRA, UR050, Laboratoire de Biotechnologie de lEnvironnement.Ind. Eng. Chem. Res. 2008, 47, 77157720 771510.1021/ie800256e CCC: $40.75 2008 American Chemical SocietyPublished on Web 09/13/2008influence of the aforementioned factors (i-iv). Finally, someconcluding remarks are given.2. Model DescriptionThe success of designing and applying most of the controltechniques to biological processes is dependent strongly on theaccuracy of the underlying dynamics and representation of theprocess, in terms of control relevant properties. In this regard,extensive research efforts are still being conducted on themodeling of AD processes18-25 for several purposes, includingprocess control. Nevertheless, many of the AD models availablein the current literature only describe particular aspects of theprocess resulting in complex highly dimensional models difficultto use for control purposes.26 Moreover, the identification andvalidation of several of these models have been restricted tostable operating conditions that are also called normal operatingconditions3 (NOC). Fortunately, simpler models have been alsodeveloped and they are still being used to synthesize efficientcontrollers for certain bioprocess variables, such as chemicaloxygen demand (COD) regulation and methane production.18-20Recently, a generic AD model was developed in 2001 byBernard et al.,23 which has been widely used for monitoringand control purposes, because of its simplicity and its capabilityto represent the dynamics of a various continuous AD biore-actors (e.g., continuous stirred tank, fixed-bed, expanded-bed,or fluidized-bed reactors).In this paper, a reduced version of the AD model proposedby Bernard et al.23 is used in the design of the control schemeand it is given by the following set of ordinary differentialequations (ODEs):X1 ) (1(S2)-RD)X1X2 ) (2(S2)-RD)X2S1 ) (S1,in - S1)D- k11(S1)X1S2 ) (S2,in - S2)D+ k21(S1)X1 - k32(S1)X2 (1)where X1, X2, S1, and S2 denote, respectively, the concentrationsof acidogenic bacteria (g/L), methanogenic bacteria (g/L),primary organic substrate (expressed as chemical oxygendemand (COD, g/L)), and volatile fatty acids (VFA, mmol/L).The subscript in denotes the influent concentration of eachcomponent. The dilution rate, D (h-1), is defined by the ratioD ) Q/V, where Q (L/h) is the feeding flow and V (L) thedigester volume, while k1, k2 (mmol/g), and k3 (mmol/g) areconstant yield coefficients. The introduction of R in Model (1)has made possible to describe the dynamic behavior of variouscontinuous bioreactor configurations. It is evident that by settingR ) 1, Model (1) describes the dynamics of the classicalcontinuous stirred tank reactor (CSTR) where the biomass iscompletely suspended in the liquid phase. Model (1) has beenalso used to describe the dynamics of fluidized-bed reactors orfixed-bed reactors (FBRs). Although it is well-known that FBRsare usually modeled by partial differential equations (PDEs), ithas been demonstrated that, under good mixing and recyclingconditions, together with generous biogas production, it ispossible to neglect the axial dispersion.27 Moreover, as aconsequence, one may use Model (1), with 0 < R < 1, todescribe the dynamics of the aforementioned bioreactors withbiomass suspended in the liquid phase. Finally, the biomassgrowth rates (1 and 2) are assumed to be described by theMonod and Haldane expressions, i.e.,1(S1)) 1,max( S1S1 +KS1)Table 1. Admissible Equilibrium Points of Model 1X1* X2* S1* S2*P1 0 0 S1,in S2,inP2 (S1,in - S1*)/(Rk1) 0 RD*KS1/(1,max - RD*) S2,in + {[k2(S1,in - S1*)]/k1}P3 0 (S2, - S2*)/(Rk3) S1,in see eq 2P4 (S1,in - S1*)/(Rk1) [(S2,in - S2*) + Rk2X1*]/(Rk3) RD*KS1/(1,max - RD*) see eq 2Figure 1. Block diagram of the robust control scheme described by eq 4.Figure 2. Diagram of a fully instrumented anaerobic up-flow fixed-beddigester.Figure 3. Response of the VFA concentration when the robust controlscheme that is described by eq 4 is implemented.Table 2. Set-Point Changes at Various Times during the ADExperimental Run0 h 127 h 192 h 242 h 533 h 600 h 868 hS2* (mg VFA/L) open loop 1400 1000 1800 2500 3500 15007716 Ind. Eng. Chem. Res., Vol. 47, No. 20, 20082(S2)) 2,max( S2S2 +KS2 + (S2 KI2)2)where 1,max (h-1), KS1 (g/L), 2,max (h-1), KS2 (mmol/L), andKI2 (mmol/L) are the maximum bacterial growth rate and thehalf-saturation constant associated to the substrate S1, themaximum bacterial growth rate in the absence of inhibition,and the saturation and inhibition constants associated to substrateS2, respectively.3. Controller Design3.1. Control Problem Statement. As previously noted, oneof the main control objectives when dealing with AD processesis to guarantee process stability.1-6 As a first contribution, thispaper focuses on the VFA regulation, because its behavior canbe directly linked to the digester stability. Thus, the controlproblem can be stated as follows: the proposal of a robust controlscheme capable of regulating the VFA concentration (S2) arounda desired setpoint (S2*) in the face of the aforementioned factors(i-iv) for AD processes operating under stable conditions(NOC) and whose dynamic behavior can be represented byModel (1). In other words, the objective of this work is theproposal of a robust scheme capable of preserving the processstability in continuous AD processes via the regulation of theVFA concentration, where the stable operating conditions (NOC)are defined in terms of Model (1) as those operating conditionswhere the following inequalities are fulfilled (see also section3.2):(a) X1(t), X2(t) > 0 tg 0(b) 0 0 eventually will be re-established. Moreover, evenin the situation at which S2,in(t) ) 0, for short periods of time,condition (b) establishes a sufficient condition to guarantee apermanent supply of substrate for methanogenic bacteria. Onthe other hand, condition (c) implies that the situation in whichS2,in(t) may be identically equal to zero (or even less than zero)but it will not prevail for long periods of time. Implications ofthese conditions on the steady-state and closed-loop robustnesswill be discussed in the following subsections.On the other hand, the controller design takes also intoaccount the following assumptions:(A1) The outlet VFA concentration (S2) is readily availablefrom online measurements.8-10(A2) For controller design purposes, the growth functionsthat are associated with the acidogenic and methanogenic stepsare assumed to be unknown but can be represented by 1( )and 2( ), respectively. Furthermore, based on biologicalevidence, it is nonrestrictive to assume that these functions aresmooth, bounded, and positive-definite.28(A3) The substrate influent concentration Sj,in for j ) 1, 2 isassumed to be unknown but bounded (i.e., Sj,inmin e Sj,in e Sj,inmax).(A4) R is assumed to be uncertain but does vary in the openinterval 0 < R < 1.(A5) The inlet flow rate Q, which is the manipulated variable,is constrained because of the capacity of the pumps used in theAD processes.28-30 Consequently, the manipulated variablegiven by the dilution rate is bounded by the following saturationfunction:sat(D)) {Dmax (if DgDmax)D (if Dmin loss of generality, that the influent VFA concentration can bedescribed as follows: S2,in ) Sj2,in + S2, where S2 is anuncertain and bounded function that is related to the variationof the influent composition around a well-known nominalvalue Sj2,in, which can be determined using a single VFA off-line measurement of the wastewater to be treated. A newfunction that lumps the system uncertainties then is definedas k21(S1)X1 - k32(S2)X2 + S2D. After this functionis defined, Model (1) can be rewritten in the followingextended state-space representation by means of a nonlinearcoordinate transformation:16z1 ) + (S2,in - z1)D)(z,D)zi ) i(z) (for i) 2, 3, 4) (3)where two important properties can be highlighted:33 (a) theextended state-space (eq 3) is equivalent to eq 1, and therefore,if a control law is applied to eq 3 to attain certain controlobjectives, such a law can also achieve the same controlobjectives when applied to eq 1 and, (b) the dynamics of theuncertain state can be reconstructed by using an ELO andreadily available information: the dilution rate, the VFAconcentration, its time derivative and its influent nominal value(i.e., ) z1 - Sj2,in - z1)D). Thus, one can devise the followingoutput feedback control, which has a linearizing-like structure:16z1 ) + (S2,in - z1)D+g1(z1 - z1))2g2(z1 - z1)D) sat{ 1(S2,in - z1)[--KC(z1 - S2/)]} (4)where is a tuning parameter and the constants g1 and g2 arechosen such that the polynomial s2 + g2s + g1 ) 0 is theHurtwitz polynomial, which is the characteristic polynomial ofthe linear part of the estimation error dynamics, and e is theestimation error vector, which is defined as e ) [z1 - z1, -]. In this way, it is possible to guarantee that e f as t f ,where is an arbitrarily small neighborhood around theorigin.31-33 Figure 1 shows the block diagram of the robustcontrol scheme described by eq 4, where DC ) 1/(Sj2,in - z1)[-- KC(z1 - S2*)] and D ) sat(DC) are the computed and actualcontrol inputs.3.4. Closed-Loop Behavior. Now let us analyze the closed-loop behavior of the robust control scheme that is described byeq 4 under the following two possible situations:(I) The control input does not saturate (i.e., D ) DC); then,z1 )g1(z1 - z1)-KC(z1 - S2/) (5)By taking eq 5 into the Laplace domain, the following transferfunction is obtained:z1(s)z1(s))g1s+g1 +KC(6)Clearly, in this situation the ELO acts as a first-order low-passfilter; in fact, the cutoff frequency of the low-pass filter (eq 6)depends on the observer and control gains , KC, and, as aconsequence, the effect of noisy measurements can be properlyhandled by selecting adequate values for both these parameters.(II) The control input does saturate (i.e., D ) Dmax, min, whereDmax, min is a constant value denoting the upper or/and lowerlimit of the saturation function defined in assumption A5; thus,z1 ) + (S2,in - z1)Dmax,min +g1(z1 - z1) (7)whose Laplace domain representation is given byz1(s)z1(s))g1s+2g2s2 + (g1 +Dmax,min)s+2g2(8)which shows that ELO has a structure of a second-order low-pass filter with an feed-forward action, which allows thecontinuous estimation of the uncertain state , even when thecontrol input saturates. In fact, by looking at the block diagramof the control scheme described by eq 4, one can clearly seethat this structure resembles an antiwindup bumpless transfer(AWBT) feedback scheme,34 which has been shown to suppressthe influence of significant external disturbances that typicallyrequires large control actions but limited by the presence ofinput constraints. In the particular situation in which S2,in(t)may become identically equal to zero (or even less than zero),it is clear that the control law that is described by eq 4 maysaturate; however, the antiwindup structure of the robust controlthat is described by eq 4 is designed to drive the process to thesetpoint as smoothly and quickly as possible, such that thecomputed control effort does not exceed their bounds.4. Experimental ImplementationThe performance and robustness of the control law that isdescribed by eq 4 was experimentally tested in a 0.528 m3 fullyinstrumented AD pilot-scale plant that was used to treat winedistillery wastewater (which is also known as vinasses; seeFigure 2). The experimental implementation of the proposedcontrol scheme was performed one month after the AD processwas restarted, following seven months of inactivity (that is,under the most highly uncertain conditions).The sensors and actuators information was supplied to aninput-output device that allowed the acquisition, treatment, andstorage of data in a personal computer (PC). The proposedscheme was implemented by using software developed inMatlab. The automatic titrimetric analyzer Anasense, com-mercialized by the Belgian company Applitek, which was placedat the output of the process, was used to measure the outletVFA concentration online every 30 min.10 This means that therobust control scheme that is decribed by eq 4 was calculatedevery 30 min, maintaining the control input constant until thenext measurement was available (see Figure 4b, given later inthis paper). However, because the sampling time was sufficientlyfast, compared to the process residence time (which was >20h), the assumption of continuous control was assumed to bevalid over the entire experimental time period. The influentconstraints were fixed as Dmin ) 0.0019 h-1 and Dmax ) 0.04167h-1. The tuned control parameters used in the real-time pilot-plant application were Kc ) 0.4 h-1, ) 0.7, g1 ) 2 h-1, andg2 ) 1 h-1. These parameters were determined from the apriori off-line numerical implementation of the controlscheme that is decribed by eq 4, using experimental data fromthe aforementioned AD pilot-scale plant. Different dilutedvinasses (i.e., H2O + vinasses) were used during theexperimental run. However, S2,in was fixed at 7500 mgVFA/Lwhich corresponds to a single VFA off-line measurement ofthe raw vinasses to be treated to induce a significant errorand to test the robustness of the proposed control law underthe influence of load disturbances. Finally, six set-pointchanges were induced during the experimental run to testthe output tracking capabilities of the proposed controlscheme. These changes are listed in Table 2.7718 Ind. Eng. Chem. Res., Vol. 47, No. 20, 2008The response of the robust scheme that is described by eq 4is depicted in Figure 3. As is seen, the proposed control schemesatisfactorily tracked the set-point changes while attenuating thedisturbances over the whole 36 days experimental run. Notethe effect of the open-loop operation at the startup of the ADprocess: the response of the VFA concentration deteriorated butdrastically changed when the control loop was closed at t )127 h as this variable reached the setpoint in a rather smoothand fast way. Two simultaneous disturbances were induced att ) 240 and 242 h: the first disturbance was a change on S2,infrom 5200 mg VFA/L to 6700 mg VFA/L, whereas the secondconsisted of a set-point change from 1000 mg VFA/L to 1800mg VFA/L. Clearly, the robust control scheme was able toregulate the VFA concentration around the setpoint. As a resultof a sensor failure, a major disturbance occurred at t ) 313 h,resulting in a rapid fall of the VFA concentration readings, whichled to the decision to operate the bioreactor in an open-loopmanner. However, after the sensor was fixed and the robustcontrol loop was again closed, the VFA concentration quicklyreturned to the setpoint. The cleaning maintenance of theAnasense sensor also introduced two additional operationdisturbances at t ) 624 h and t ) 731 h but were satisfactorilyhandled by the controller, which was able to regulate the VFAconcentration around the setpoint without retuning the controlparameters. It is evident that the control scheme performed quitewell, despite these disturbances and without the knowledge ofboth the inlet VFA concentration S2,in and the process kinetics.Figure 3 also shows that the estimation results of the extendedLuenberger observer was reasonably good, because the estimatedvalues (black line) were closer to those measured by the onlinesensor (gray line) during the experiment.Figure 4a illustrates the behavior of the control input D duringthe experimental run. Notice that, during the entire experimentalrun, the dilution rate calculated by the proposed control law(black line) was different from that measured at the entrance ofthe digester (gray line). This error was mainly due to acalibration problem in the feeding pump which caused the smalloff-set between the VFA concentration and the set-point value(see Figure 3). Also note that, after the introduction of most ofthe set-point changes, the control input saturated (from belowor from top, depending on the magnitude and direction of theset-point change) without serious consequences on the controllerperformance (see Figure 4b). Figure 4b finally depicts the effectof the calculated controller action on the control input. Althoughthe control law that is described by eq 4 was calculated every30 min, the behavior of D was quite smooth, despite saturation.This behavior is a desirable feature from a practical point ofview, to guarantee safety operating conditions and increase thelifetime of the pump.5. ConclusionsA model-based robust nonlinear control methodology foranaerobic digestion processes with uncertain dynamics, inputconstraints, parameter uncertainty, and load disturbances wasdeveloped. This methodology was obtained from the extensionof the previously reported ideas by Alvarez-Ramirez et al.31,32by the combination of an output feedback control with alinearizing-like structure and an extended Luenberger observerused to estimate the uncertain terms of the process. The proposedmethodology was experimentally implemented over a periodof 36 days under different uncertain scenarios and loaddisturbances in an anaerobic digester (AD) pilot-scale plant thatwas used for the treatment of industrial wine distillery waste-water. It was shown that the controller yields robustness in theface of parameter uncertainty, load disturbances, and variablesetpoints. The performance of the proposed robust controlscheme is particularly encouraging to scale it up to real-lifeindustrial applications: (i) its simple structure is easy toimplement and tune (there are only two parameters to adjust: and KC) and (b) it does not require the knowledge of the influentcomposition nor the process kinetics nor biogas productionmeasurements. Both of these features are among the mainadvantages of the proposed scheme, with regard to the previ-ously reported approaches.13-16 In other words, the proposedmethodology includes the advantages of proportional integraland proportion integral differential (PI/PID) control and therobustness of the nonlinear control schemes but requires a VFAonline analyzer, which is not very restrictive, because theFigure 4. Behavior of the control input during the experimental run.Ind. Eng. Chem. Res., Vol. 47, No. 20, 2008 7719technical and economical availability to acquire reliable VFAonline sensors has increased drastically over the past few years.AcknowledgmentThis work was partially supported by Project Nos. CONA-CyT/J50282/Y, PROMEP/103.5/05/1705 and PROMEP/103.5/02/2354. B.P.-R. thanks CONACyT for financial support (underGrant No. 185382).Literature Cited(1) Henze, M.; Harremoes, P.; Jansen, J. L. C.; Arvin, E. 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A PI Controller withdisturbance estimation. Ind. Eng. Chem. Res. 1997, 36, 36683675.(32) Alvarez-Ramrez, J.; Femat, R. Robust PI stabilization of a classof chemical reactors. Syst. Control Lett. 1999, 38, 219225.(33) Femat, R.; Alvarez-Ramrez, J.; Rosales-Torres, M. Robust as-ymptotic linearization via uncertainty estimation: Regulation of temperaturein a fluidized bed reactor. Comput. Chem. Eng. 1999, 23, 697708.(34) Kothare, M. V.; Campo, P. J.; Morari, M.; Nett, C. N. A unifiedframework for the study of anti-windup designs. Automatica 1994, 30, 18691883.ReceiVed for reView February 13, 2008ReVised manuscript receiVed June 25, 2008Accepted July 11, 2008IE800256E7720 Ind. Eng. Chem. Res., Vol. 47, No. 20, 2008

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