Journal of Process Control 20 (2010) 375–383

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Journal of Process Control

journal homepage: www.elsevier .com/ locate/ jprocont

A robust control scheme to improve the stability of anaerobic digestion processes

H.O. Méndez-Acosta *, B. Palacios-Ruiz, V. Alcaraz-González, V. González-Álvarez, J.P. García-SandovalDepartamento de Ingeniería Química, CUCEI-Universidad de Guadalajara Blvd. M. García Barragán 1451, C.P. 44430, Guadalajara, Jal., Mexico

a r t i c l e i n f o

Article history:Received 26 December 2008Received in revised form 15 January 2010Accepted 25 January 2010

Keywords:Robust controlAnaerobic digestionWastewater treatment

0959-1524/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.jprocont.2010.01.006

* Corresponding author.E-mail address: hugo.mendez@cucei.udg.mx (H.O.

a b s t r a c t

In this paper, the robust regulation of volatile fatty acids (VFA) and total alkalinity (TA) is addressed toimprove the stability of continuous anaerobic digestion (AD) processes. The control scheme is derivedfrom an AD model that includes TA as state variable and it is composed of a model-based multiple-inputmultiple-output feedback control and an extended Luenberger observer which not only allows the esti-mation of the process kinetics and variations in the influent composition but also the introduction of anantiwindup structure. The wastewater flow rate is used to regulate the VFA concentration, whereas analkali solution is added directly to the digester to regulate TA. The controller performance is evaluatedvia numerical simulations showing excellent responses under the influence of control input saturations,noisy measurements, load disturbances and uncertain kinetics. Finally, it is shown that well-known prac-tical stability criteria for AD processes can be also fulfilled by the proposed control scheme.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Anaerobic digestion (AD) has regained the interest of the waste-water treatment scientific and industrial community to reduce theorganic matter from industrial and municipal effluents because ofits low initial and operational costs, smaller space requirements,high organic removal efficiency and low sludge production, com-bined with a net energy benefit through the production of biogas.However, its widespread application has been limited, because ofthe intrinsic difficulties involved in achieving the efficient opera-tion of these processes such as: (i) highly nonlinear behavior; (ii)load disturbances; (iii) system uncertainties; (iv) constraints onmanipulated and state variables; and (v) limited online measure-ment information. Recently, Steyer et al. [1] have reported a sum-mary of the advantages and drawbacks of several control schemeswhen applied to AD processes, and pointed out that classical con-trol methods have not proven to tide over such difficulties and nei-ther have the more sophisticated control strategies, which haveshown to yield unsatisfactory performances when the AD processis subject to large disturbances or significant set-point changes.These authors also pointed out that nonlinear control schemeshave had limited success since they largely rely on the availabilityof a good process model, which is not always easy to obtain. In theparticular case of AD models, these are complex and high dimen-sional with a large number of poorly defined parameters whichmakes them very difficult to use in online control strategies [2].Moreover, most of the model-based control schemes that havebeen applied for AD stabilization, may appeared to be limited be-

ll rights reserved.

Méndez-Acosta).

cause they have been focused on the regulation of a single sub-strate without taking into account that certain operatingconditions may impose limitations on the process physiologicalproperties [3,4].

From an operating safety viewpoint, anaerobic digestion isintrinsically a very unstable process, variations of the input vari-ables (hydraulic flowrate, influent organic load) may easily leadthe process to a washout, i.e. a state where the bacterial life disap-pears [5]. This phenomenon takes place under the form of volatilefatty acids (VFA) accumulation which induces an overflow of pro-tons that decompose the liquid phase biocarbonates to produceCO2 (increasing its composition in the gas phase) and decreasingthe digester pH. Experimental results have shown that in the eventof a drastic drop in the process pH caused by the VFA accumula-tion, it is not always possible to drive the AD process back to nor-mal operation just by regulating pH. In fact, pH can only be used asan indicator of the process stability in wastewaters with low buf-fering capacity, because high biocarbonate concentrations maycompensate the pH changes due to VFA’s accumulation [6]. Thus,instead of pH, it is preferable to monitor the process alkalinity,which allows the timely detection of changes of the process buffercapacity and, as a consequence, a more accurate information aboutthe risk of process failure by the VFA accumulation. In this regard,Pohland [7] emphasized the need for a balance between total alka-linity (TA) and VFA concentrations for normal digestion and im-plied that appreciable variations occurred only after consistentvariations of the ratio VFA/TA (in fact, this ratio gives an idea ofthe relative amount of the buffer capacity that is still available toneutralize VFA). Zickefoose and Hayes [8] suggested that such a ra-tio should be maintained within the range 0:1—0:35 mEq=L

mEq=L to im-prove the digester operational stability. Other practical criteria

376 H.O. Méndez-Acosta et al. / Journal of Process Control 20 (2010) 375–383

have been also reported to relate the AD operational stability, acid-ification degree and buffer capacity with VFA and partial (PA),intermediate (IA) and total alkalinity (TA = PA + IA) concentrations.For instance, Ripley et al. [9] have found that the successful diges-ter operation occurs when TA is greater than 60 mEq=L and the ra-tio IA/PA is less than 0:3 mEq=L

mEq=L, while Bernard et al. [10] havesuggested that the digester stability is preserved when the ratioIA/TA is less or equal to 0:3 mEq=L

mEq=L. Moreover, it is well known thatAD processes operate under stable conditions if the VFA concentra-tion is kept under 25 mEq=L and the buffer capacity of the system(given by the bicarbonate concentration) is enough to maintain thesystem pH close to neutral [4,11]. These conditions are known asthe normal operating conditions (NOC) [4].

From the above, it is clear that control strategies in AD pro-cesses should not only be focused on using the pollution level ascontrol variable but also on those variables related to the processoperational stability such as alkalinity. In this contribution, theoperational stability problem in AD processes is addressed from amultiple-input multiple-output (MIMO) control point of view bythe robust regulation of both, VFA and TA. The proposed robustregulator, derived from a mathematical model of a typical AD pro-cess which includes TA as a state variable, is composed of a MIMOfeedback controller and an extended Luenberger observer used toestimate the process kinetics and variations in the influent compo-sition. The control objectives are achieved by manipulating thewastewater flow rate to regulate the VFA concentration and an al-kali solution is added directly to the digester to maintain TA at adesired set-point (restoring the bicarbonate buffer capacity as itis destroyed by VFA). We will show that by controlling both thesevariables, it is possible to satisfy the before mentioned practicalcriteria while improving the operational stability of AD processes.The paper is outlined as follows. First, the model used in the syn-thesis of the MIMO robust scheme is briefly described. Then, TA,NOC and the control problem are stated in terms of this model.Thereafter, the robust scheme is proposed and evaluated vianumerical simulations under different operating conditions includ-ing load disturbances, noisy measurements, uncertain kinetics andcontrol input constraints. Finally, we close the paper with someconcluding remarks.

2. Model description

2.1. The physicochemical equilibrium

TA is formally defined as the sum of equivalents of all the basesthat can be titrated with a strong acid to the first equivalence pointof the system (i.e., pH 4.3) [9]. In AD processes, alkalinity is due tomany species but the most important are bicarbonates and VFA. Inthis paper, we propose the following expression to define TA [12]:

TA ¼ fTc HCO�3� �

þ fTa S�2� �

ð1Þ

based on the following assumptions: (a) only VFA and bicarbonatesare present as weak acids, (b) VFA are mainly composed by acetateand (c) the process is operated in the pH range 6 < pH < 8. Here,S�2� �

and HCO�3� �

denote respectively the concentrations of dissoci-ated VFA and bicarbonates, while fTc and fTa are given by

fTc ¼ 1� 10�pH þ Kc

10�4:3 þ Kc

!; f Ta ¼ 1� 10�pH þ Kac

10�4:3 þ Kac

!;

where Kc and Kac (mmol/L) are the ionization constants for the equi-libria HCO�3 =CO2 and S�2 =S2, respectively.

One of the most widely accepted expressions to represent thestrong ions concentration in AD process is the following [13–15]:

Z ¼ HCO�3� �

þ S�2� �

ð2Þ

which, without loss of generality can be related to TA by

TA ¼ Z � b; ð3Þ

where b stands for the non titrated fraction of both bicarbonatesand VFA beyond the first equivalence point which, for practical pur-poses, may be considered constant [6].

2.2. The dynamical model

A suitable mathematical model that describes the dynamics ofcontinuous AD processes can be obtained by performing the massbalance of the species involved in the process (including TA) and byintroducing the following additional assumption: (d) the process isoperating under isothermal conditions and (e) the wastewater flowrate ðQ1Þ is much greater than the alkali flow ðQ2Þ (i.e., Q 1 � Q 2),which means that the total dilution rate ðD ¼ D1 þ D2Þ can beapproximated by D � D1. Thus, the resulting model is given by

_X1 ¼ ðl1ðp1; S1;pH;TAÞ � aD1ÞX1;

_X2 ¼ ðl2ðp2; S2;pH;TAÞ � aD1ÞX2;

_S1 ¼ ðS1;in � S1ÞD1 � k1l1ðp1; S1;pH;TAÞX1;

_S2 ¼ ðS2;in � S2ÞD1 þ k2l1ðp1; S1;pH;TAÞX1 � k3l2ðp2; S2;pH;TAÞX2;

_TA ¼ ðTAin � TAÞD1 þ ðTA0in � TAÞD2;

ð4Þ

where X1;X2; S1; S2 and TA denote, respectively, the concentrationsof acidogenic bacteria (g/L), methanogenic bacteria (g/L), primaryorganic substrate (other than VFA) expressed as Chemical OxygenDemand (COD) (g/L), VFA (mmol/L) and TA (mmol/L). The subscriptin is used to identify the concentration of each component in thewastewater inlet flow rate. TA0in (mmol/L) represents the TA concen-tration of the alkali solution while D1 and D2, respectively, representthe dilution rates associated to the wastewater and alkali solutionflow rates (i.e., Diðh�1Þ ¼ QiðL=hÞ=VðLÞ for i ¼ 1;2 and V is the diges-ter volume). It has been shown that by introducing a in Model (4), itis possible to describe the dynamic behavior of various continuousbioreactor configurations [15]. It is evident that by setting a ¼ 1,Model (4) describes the dynamics of the classical Continuous StirredTank Reactor (CSTR) where the biomass is completely suspended inthe liquid phase. On the other hand, with 0 < a < 1, Model (4) hasbeen successfully used to describe the dynamics of fluidized-bedand fixed-bed bioreactors operating under good mixing and recy-cling conditions together with a generous biogas production, whichallow us to neglect the effect of the axial dispersion [16]. Finally,l1ðp1; S1;pH; TAÞ and l2ðp2; S2;pH;TAÞ, respectively, represent thespecific growth functions associated to the acidogenic and metha-nogenic populations, where p1 and p2 are kinetic parameters. In or-der to simplify the mathematical notation, from now on, the specificgrowth functions will be represented as l1ð�Þ and l2ð�Þ.

2.3. Normal operating conditions

As previously noted, the operating conditions where the diges-ter stability prevails are also known as the Normal Operating Condi-tions (NOC). In this section, these conditions are formally stated interms of Model (4) by taking into account the stability criteria de-scribed in Section 1. Thus, the AD process (4) operates under NOC ifthe following conditions are fulfilled [17]:

(I) The biomass within the digester remains active, which interms of the model implies that XðtÞ1;XðtÞ2 > 0 8t P 0. Inaddition, x0 ¼ xðt ¼ 0Þ > 0 8t P 0 where x ¼ ½X1;X2; S1; S2;

TA�0.

H.O. Méndez-Acosta et al. / Journal of Process Control 20 (2010) 375–383 377

(II) The VFA concentration ðS2Þ is less than 25 mEql/L, TA greaterthan 60 mEql/L and the ratio VFA’s/TA is maintained withinthe range 0:1—0:3 mEq=L

mEq=L.(III) 0 < b1 6

R tþd1t DS1;inðsÞds 8t P 0 where DS1;inðtÞ is defined as

DS1;inðtÞ � S1;inðtÞ � S1ðtÞ and, b1 and d1 are positiveconstants.

(IV) 0 < b2 6 jR tþd2

t DS2;inðsÞdsj 8t P 0 where DS2;inðtÞ is definedas DS2;inðtÞ � S2;inðtÞ � S2ðtÞ and, b2 and d2 are positiveconstants.

Condition (III) implies that even when it is possible in practiceto have load organic charges for which S1;inðtÞ 6 S1ðtÞ, conditionDS1;inðtÞ > 0 eventually will be re-established. Moreover, even inthe situation at which S2;inðtÞ ¼ 0, for short periods of time, condi-tion (III) establishes a sufficient condition to guarantee a perma-nent supply of substrate to the methanogenic bacteria. On theother hand, condition (IV) implies that the situation at whichDS2;inðtÞ may be identically equal to zero (or even less than zero)but it will not prevail for long periods of time. Implications of theseconditions on the steady-state and closed-loop robustness will bediscussed in the following section.

There are other important process features that must be consid-ered in the controller design when dealing with AD processes suchas:

A1 VFA and TA are available online [18,19].A2 l1ð�Þ and l2ð�Þ are unknown, but based on biological evi-

dence, it is nonrestrictive to assume that, for controllerdesign purposes, these functions are smooth, bounded andpositive-definite [20].

A3 a is assumed to be uncertain but does vary in the open inter-val 0 < a < 1.

A4 The wastewater composition Sj;in for j ¼ 1;2 and TAin areassumed unknown but bounded, while the composition ofthe alkali solution TA0in, is assumed constant and known.

A5 The wastewater ðQ 1Þ and alkali solution ðQ2Þ flow rates usedas the manipulated variables to regulate VFA and TA, respec-tively, are constrained in practice, because of the capacity ofthe pumps and to avoid undesired operating conditions suchas the washout condition [20,21]. Consequently, the dilutionrates D1 and D2 are bounded by the following saturationfunction:

8 9

satðDiÞ ¼

Dmaxi ; if Di P Dmax

i

Di; if Dmini < Di < Dmax

i

Dmini ; if Di 6 Dmin

i

><>:>=>;; i ¼ 1;2; ð5Þ

where the upper and lower bounds are well known and, froma practical point of view, Di 2 Rþ is a piecewise linearfunction.

3. Controller design

In the past decade, the control theory based on differentialgeometry emerged as a powerful tool to deal with a great varietyof dynamic nonlinear systems. However, to obtain input–outputlinear dynamic behavior, this tool requires the exact cancellationof the nonlinear terms and, as a consequence, the perfect knowl-edge of the system. This means that the presence of modeling er-rors, unmeasured disturbances, and parametric uncertaintiescannot be taken into account in such a typical controller design,preventing its application in a large number of complex dynamicalprocesses such as AD [22]. Here, in order to overcome the afore-mentioned difficulties associated to the design of geometric con-trol schemes, a robust control approach is proposed to regulate

VFA and TA in AD processes. Thus, model mismatch, unmeasureddisturbances and parameter uncertainties are taken into accountin the controller design by defining an uncertain but observablefunction vector, whose dynamic behavior is estimated from avail-able measurements by using a simple nonlinear state estimator: anextended Luenberger observer (ELO). Then, the robust scheme isobtained from the combination of the ELO and a MIMO outputfeedback control with a linearizing-like structure.

3.1. Control problem statement

As previously stated, the successful operation of AD processes isa difficult and delicate task that may be improved by the use of ad-vanced monitoring and control schemes. Particularly, the control ofVFA and TA is of paramount importance in AD processes becausethese variables are directly related to the process operational sta-bility. Hence, the control problem addressed in this paper can bestated as follows: the proposal of a MIMO control scheme capableto achieve the robust regulation of the VFA concentration and TA inorder to improve the stability of continuous AD processes operat-ing under NOC in the face of load disturbances, control input con-straints and uncertain kinetics. A common practice to maintain thedigester buffer capacity is the addition of an alkali solution [6,23],because the presence of strong basis together with VFA keep theproduced CO2 in the liquid phase as bicarbonate, increasing thebuffer capacity of the system and, consequently, its stability [6].This is why we propose to regulate TA by manipulating the inletflow rate of an alkali solution.

3.2. Linearizing control scheme

Here, the existence of a MIMO input–output linearizing controlscheme capable to achieve the regulation of VFA and TA is demon-strated as a first step in the design of the robust control approach.For this purpose, let us rewrite Model (4) in the affine form forMIMO nonlinear systems [24]:

_x ¼ f ðxÞ þXm

i¼1

giðxÞui; y1 ¼ h1ðxÞ; . . . ; ym ¼ hmðxÞ;

where m is the number of state variables to be regulated, f ðxÞ; giðxÞ’sare smooth vector fields, hiðxÞ’s are smooth functions defined onU � R5

þ and U the set of NOC. yi represents the output functionswhereas the control inputs are denoted by ui. Particularly, the inputand output vectors are given by y ¼ ½S2;TA�; u ¼ ½D1;D2�. Then, it canbe easily shown that Model (4) has a well-defined relative degreevector r ¼ ½1;1� under NOC (see Proposition 1 in Appendix A).Therefore, it is straightforward to demonstrate the existence of a lo-cal coordinate transformation z ¼ UðxÞ in a neighborhood U of x

(see Proposition 2 in Appendix A), such that Model (4) can be recastin the following normal form [24]:

_z1 ¼ ðS2;in � z1ÞD1 þ ðk2l1ð�Þz3 � k3l2ð�Þz4ÞðS2;in � z1Þa;_z2 ¼ ðTAin � z2ÞD1 þ ðTA0in � z2ÞD2 ð6aÞ

_z3 ¼ z3 l1ð�Þ �ak3l2ð�Þz4 � k2l1ð�Þz3

ðS2;in � z1Þ1�a

!;

_z4 ¼ z4 l2ð�Þ �ak3l2ð�Þz4 � k2l1ð�Þz3

ðS2;in � z1Þ1�a

!; ð6bÞ

_z5 ¼ z5 l1ð�Þ �ak1k3l2ð�Þz4

k2ðz3=z5Þ1=aðS2;in � z1Þ1�a

!;

where (6b) denotes the internal dynamics of the system. Then, fromthe geometric control theory, it is well known that the followingoutput feedback control guarantees the exponential convergenceof the output vector y ¼ ½z1; z2� towards its set-point y ¼ ½S2;TA�

378 H.O. Méndez-Acosta et al. / Journal of Process Control 20 (2010) 375–383

if the stability of the internal dynamics (6b) is also guaranteed (seeLemma 4 in Appendix A):

½D1;D2�0 ¼ A�1ðzÞ�Lf h1ðzÞ � v1ðzÞ�Lf h2ðzÞ � v2ðzÞ

� �; ð7Þ

where v1ðzÞ ¼ K1ðz1 � S2Þ;v2ðzÞ ¼ K2ðz2 � TAÞ are such that thepolynomials P1ðsÞ ¼ sþ K1 ¼ 0 and P2ðsÞ ¼ sþ K2 ¼ 0 are Hurwitzand K1;K2 are the control gains. Unfortunately, the output feedbackcontrol (7) cannot be directly implemented in practice, because itneeds the knowledge of both the inlet composition and the kineticterms, which is a condition difficult to satisfy under actual operat-ing conditions (see A2 and A4). Hence, in the next section, a robustcontrol approach is proposed to overcome these limitations from anextension of the previously reported ideas by Alvarez-Ramírez et al.[25].

3.3. The robust control approach

Now, without any loss of generality, let us consider that thewastewater concentration of VFA and TA can be described, respec-tively, by S2;in ¼ eS2;in þ DS2 and TAin ¼ ~TAin þ DTA, where DS2 andDTA are uncertain and bounded functions associated to thevariation of the wastewater composition around the well-knownnominal values eS2;in and ~TAin, which can be determined by a singleoff-line measurement of the wastewater to be treated. Thus, bydefining the uncertain functions g1 � ðk2l1ð�Þz3 � k3l2ð�Þz4ÞðS2;in�z1Þa þ DS2 D1 and g2 � DTAD1, Model (6) can be recast in the follow-ing extended state-space representation

_z1 ¼ g1 þ ðeS2;in � z1ÞD1;

_z2 ¼ g2 þ ð ~TAin � z2ÞD1 � ðTA0in � z2ÞD2;

_g ¼ NðzÞ; g ¼ ½g1;g2�0;

_zi ¼ � ðzÞ; for i ¼ 3;4;5;

ð8Þ

where it can be shown that the extended state-space (8) is equiva-lent to (6) [26]. Hence, if a control law is applied to (8) to satisfy cer-tain control objectives, such a law can also meet these objectiveswhen applied to (6) and consequently, to Model (4). Notice thatthe augmented state vector g can be reconstructed from onlinemeasurements of the output and input variables by means of anELO. Then, one can devise the following robust control scheme bycoupling the ELO to the output feedback control (7):_z1 ¼ g1 þ ðeS2;in � z1ÞD1 þ C1g11ðz1 � z1Þ;_z2 ¼ g2 þ ð ~TAin � z2ÞD1 þ ðTA0in � z2ÞD2 þ C2g21ðz2 � z2Þ; ð9aÞ_g1 ¼ C2

1g12ðz1 � z1Þ;_g2 ¼ C2

2g22ðz2 � z2Þ;

D1 ¼ sat � 1

ðeS2;in � z1Þg1 þ K1ðz1 � S2Þ� �( )

; ð9bÞ

D2 ¼ sat � 1ð TA0in � z2Þ

ð ~TAin � z2ÞD1 þ g2 þ K2ðz2 � TAastÞh i� �

:

ð9cÞEq. (9a) allows the estimation of the uncertain states g1 and g2,while (9b) and (9c) induce the desired behavior on VFA and TA,respectively. g11; g12; g21 and g22 are chosen such that the matrixassociated to the linear part of the estimation error vector e(ei

1 ¼ zi � zi and ei2 ¼ gi � gi for i ¼ 1;2) is Hurwitz, whereas Ci

and Ki for i ¼ 1;2 are the estimation and control gains (tuningparameters), respectively. In this way, it is possible to guaranteethat e! e as t !1, where e is an arbitrarily small neighborhoodaround the origin (see Proposition 3 in Appendix A). In addition,in order to avoid undesired effects in the controller performancedue to the control input saturations such as the windup phenomena[27], the constrained values of the dilution rates are fed back to the

observer as shown in Fig. 1, to induce a structure into the controlscheme that resembles an observer-based antiwindup scheme [28].

3.4. On the ELO closed-loop behavior

The aim of this section is to show the relationship between z1

and z1 when the system is under closed-loop in order to provideevidence of how the ELO structure allows the estimation of theuncertain functions g1 and g2 (see Proposition 3), even under thesaturation of the control inputs, inducing in this way, a kind of ant-iwindup structure on the proposed control scheme. Then, in whatfollows, let us analyze the closed-loop behavior of the robust con-trol scheme (9) under the following two possible situations:

S1 The control inputs are not saturated (i.e., D ¼ DC); then,

_z1 ¼ C1g11ðz1 � z1Þ � K1 z1 � S2�

;

_z2 ¼ C2g21ðz2 � z2Þ � K2ðz2 � TAÞ: ð10Þ

By taking (10) into the Laplace domain, one obtains the followingtransfer functions

z1ðsÞ ¼C1g11

sþ C1g11 þ K1z1ðsÞ þ

K1

ðsþ C1g11 þ K1ÞsS2;

z2ðsÞ ¼C2g21

sþ C2g21 þ K2z2ðsÞ þ

K2

ðsþ C2g21 þ K2ÞsTA: ð11Þ

Clearly in this situation, the ELO acts as a first-order low-pass filter,where the cutoff frequency depends on the observer and controlgains Ci, Ki, and, as a consequence, the effect of noisy measurementscan be properly handled by selecting adequate values for both theseparameters.

S2 The control inputs are saturated (i.e., D ¼ Dmin;max); thus,

_z1 ¼ g1 þ ðeS2;in � z1ÞDmin;max1 þ C1g11ðz1 � z1Þ; ð12Þ

_z2 ¼ g2 þ ð ~TAin � z2ÞDmin;max1 þ ð TA0in � z2ÞDmin;max

2 þ C2g21ðz2 � z2Þ;

whose Laplace transform representations are given by

z1ðsÞ ¼C1g11sþ C2

1g12

s2 þ ðC1g11 þ Dmin;max1 Þsþ C2

1g12

z1ðsÞ

þ S2;inDmin;max1

s2 þ ðC1g11 þ Dmin;max1 Þsþ C2

1g12

;

z2ðsÞ ¼C2g21sþ C2

2g22

s2 þ ðC2g21 þ Dmin;max1 þ Dmin;max

2 Þsþ C22g22

z2ðsÞ ð13Þ

þ ðTAinDmin;max1 þ TA0inDmin;max

2 Þs2 þ ðC2g21 þ Dmin;max

1 þ Dmin;max2 Þsþ C2

2g22

:

In this case, the ELO has a structure of a second-order low-pass filterwith a feed-forward action, which allows the continuous estimationof the uncertain state g, even when the control inputs saturate. Thismeans that even under the uncommon situations at whichS2;inðtÞ ¼ S2 or TA0in ¼ TA, the robust control scheme (9) will still becapable to steer the process to the desired set-points as smoothlyand quickly as possible, such that the computed control efforts donot exceed their bounds.

4. Numerical implementation

In this section, the performance of the robust approach (9) isevaluated via numerical simulations under different operating con-ditions and the most uncertain scenarios. The numerical imple-mentation was carried out by using the Matlab-Simulink�

software. The model parameters used during the simulation runswere those used by Bernard et al. [15], while the initial conditions

0 200 400 600 80020

30

40

50

60

70

TAin (

mEq

/L)

Time (h)

0 200 400 600 8000

10

20

30

40

S 1,in (

g/L)

0 200 400 600 80060

80

100

120

140

S 2,in (

mEq

/L)

(a)

(c)

(b)

Fig. 2. Wastewater influent composition: (a) organic substrate (COD, S1;in), (b)volatile fatty acids (VFA, S2;in) and (c) total alkalinity ðTAinÞ.

Fig. 1. Block diagram of the robust regulator described by Eq. (9).

H.O. Méndez-Acosta et al. / Journal of Process Control 20 (2010) 375–383 379

were: X1ð0Þ ¼ 0:5 g=L;X2ð0Þ ¼ 0:7 g=L;TAð0Þ ¼ 50 mEq=L;S1ð0Þ ¼2:0 g=l;S2ð0Þ ¼ 30 mEq=L;z1ð0Þ ¼ 20 mEq=L;z2ð0Þ ¼ 60 mEq=L;g1ð0Þ ¼0 mEq=L h and g2ð0Þ ¼ 0 mEq=L h. Note that there is a significanterror between the initial conditions of the measured ðziÞ and theestimated ðziÞ states, which was induced in order to test the ELOperformance. The nominal values used for the wastewater influentcomposition were eS2;in ¼ 80 mEq=L and ~TAin ¼ 30 mEq=L, and theconcentration of the alkali solution TA0in was fixed at 8000 mEq/Lbased on the concentration that industrial soda normally has,which can vary between 8000 and 10000 mEq/L. The upper andlower bounds used in (5) to constrain the dilution rates wereD1 ¼ ½0:002;0:05� h�1 and D2 ¼ ½0;0:0005� h�1 (clearly, assumption(d) stated in Section 2 is satisfied since D1 � D2). The controlparameters are listed in Table 1, where the control and estimationgains were selected based on a specific cutoff frequency for thelow-pass filter.

In order to test the controller performance under the influenceof noisy measurements, uniformly random white noises with anamplitude of ±2.0 mEq/L and ±10.0 mEq/L were added to the sim-ulated measurements of VFA and TA, respectively. Furthermore,to ensure the digester stability, the set-point values were chosenby considering the NOC conditions stated in Section 2.3 as wellas actual operating conditions for which the inlet compositionwas randomly varied around the values reported by Bernardet al. [15] (see Fig. 2). Several wastewater composition changeswere also introduced during the simulation runs to test the perfor-mance of the control law (9) in the presence of load disturbancesand set-point changes.

The response of the proposed robust scheme (9) on the regula-tion of the VFA concentration ðS2Þ is depicted in Fig. 3a, where itcan be noted that four set-point changes between 10 and18 mEq/L were induced to evaluate the controller set-point track-ing capability. As is seen, set-point changes were satisfactorilytracked under the influence of load disturbances while attenuatingthe noisy measurements. Notice also the dynamic response of theVFA concentration estimated by the ELO ðbS2Þ, which was smootherthan that measured ðS2Þ, demonstrating the effect of the low-passfilter structure of the ELO on the controller response. The dynamicresponse of the wastewater dilution rate, which is the control inputassociated to the VFA concentration is shown in Fig. 3b. It is clearthat D1 saturated at various times; nevertheless, the controller per-formance did not deteriorate due to its antiwindup structure. Inaddition, see that D1 has a smooth response to the set-pointschanges induced to the VFA concentration, which under a realimplementation will imply a minimum wearing down of the feed-ing pump. Finally, the effect of operating the digester in an open-loop manner is also shown in Fig. 3a. During such an operation,the process stability was on the edge of operational instability be-cause of a drastic increase in the VFA concentration, which reached

Table 1Control parameters used during the numerical implementation.

g11 g12 g21 g22 C1 ðh�1Þ C2 ðh�1Þ K1 ðh�1Þ K2 ðh�1Þ

2.0 1.0 2.0 1.0 0.3 0.3 0.3 0.3

values higher than 20 mEq/L. Nevertheless, the robust controllerwas able to rapidly drive such concentration toward its set-pointonce the control loop was closed.

Fig. 4a shows the excellent response of TA in the face of both theset-point changes and load disturbances. Notice that the controlscheme was capable to maintain TA above the stability limit(60 mEq/L) during the simulation run. The response of the alkalidilution rate D2 is depicted in Fig. 4b, where one can see the controleffort to keep TA around its set-point and the effect of using theantiwindup structure in the robust control scheme. Although D2

saturated at different times, it provided the appropriate control ac-tion to satisfy the controlled variable reference value. Observe thatD2 has also a smooth response to the set-point changes induced toTA. Fig. 4a also shows the effect of operating the digester in open-loop, which caused a drop of TA below the stability limit. However,this behavior was rapidly corrected once the robust control lawwas applied.

0 100 200 300 400 500 600 700 80020

40

60

80

100

TA (m

Eq/L

)

0 100 200 300 400 500 600 700 800

0

2

4

6

x 10−4

D2 (h

−1)

Time (h)

ReferenceEstimated

Closed−Loop

Real

(a)

(b)Closed−Loop

Fig. 4. (a) TA response during the simulation. (b) Dynamic response of the alkalidilution rate ðD2Þ.

0 100 200 300 400 500 600 700 8000

0.01

0.02

0.03

0.04

0.05

0.06

D1 (h

−1)

Time (h)

0 100 200 300 400 500 600 700 8005

10

15

20

25

30

35

VFA

(S2, m

Eq/L

)

Reference (a)EstimatedRealClosed−Loop

Closed−Loop

(b)

Fig. 3. (a) Set-point tracking performance of the VFA concentration ðS2Þ. (b)Dynamic response of the wastewater dilution rate ðD1Þ during the simulation.

0 100 200 300 400 500 600 700 800

0.1

0.2

0.3

0.4

0.5

0.6

S 2/TA

Time (h)

VFAControl

VFA and TAControl

Stabilitybounds

Fig. 5. Behavior of the VFA/TA ratio during the simulation.

380 H.O. Méndez-Acosta et al. / Journal of Process Control 20 (2010) 375–383

The behavior of the S2=TA ratio during the simulation run is de-picted in Fig. 5. At the start-up, the AD process was operated in anopen-loop manner and clearly the S2=TA ratio did not meet the re-quired bounded response. Then, at 150 h, we closed the VFA loop,which initially drove the S2=TA response to the desired stabilityinterval, but this response deteriorated and went off-bounds, de-spite the proper VFA control. The proposed MIMO robust controlscheme was finally implemented at 400 hours providing the prop-er control action to maintain the S2=TA ratio within bounds in theface of load disturbances, uncertain kinetics and set-point changes.Thus, this figure allow us to illustrate that, under certain circum-stances, the proper VFA control does not guarantee the operationalstability of continuous AD processes, and clearly states the benefits

of using the proposed MIMO robust control scheme over a SISOcontroller.

5. Conclusions

A model-based MIMO robust control scheme was proposed toregulate both the VFA concentration and TA in order to improvethe operational stability of continuous AD processes. This schemeis designed by using a previously reported AD model, which wasmodified to include TA as a state variable. The control scheme com-bines an output feedback control with linearizing-like structureand an extended Luenberger observer, which not only allowedthe estimation of the uncertain terms associated to the controlledstates but induced a low-pass filter and an antiwindup structurethat improves the controller performance in the presence of noisymeasurements and control input constraints. It was shownthrough numerical simulations that the proposed MIMO controlscheme was capable to maintain the desired set-points in the faceof the complete ignorance of either the processes kinetics or thewastewater influent composition while satisfying previously re-ported practical stability criteria. The experimental implementa-tion of the proposed control scheme is currently under way andthe results will be reported in the near future.

Acknowledgements

This work was partially supported by CONACyT Ref. 25927/J50282/Y. Bernardo Palacios-Ruiz thanks CONACyT for the financialsupport under Grant No. 185382.

Appendix A. Mathematical analysis useful in the controllerdesign

Proposition 1. Let y ¼ ½S2; TA� and u ¼ ½D1;D2� the output and inputvectors of (4), respectively. Then, Model (4) has a relative degreevector r ¼ ½1;1� under NOC for any xð0Þ ¼ xðt ¼ 0Þ 2 U, where x is thestate vector.

Proof. By computing the Lie derivative of the output vector y alongthe vector fields f ðxÞ and giðxÞ’s, one gets Lg1

L0f h1ðxÞ ¼ ðS2;in � S2Þ

and Lg2 L0f h2ðxÞ ¼ ð TA0in � TAÞ, which are different from zero under

NOC. Then, the relative degree matrix

H.O. Méndez-Acosta et al. / Journal of Process Control 20 (2010) 375–383 381

AðxÞ ¼ðS2;in � S2Þ 0ðTAin � TAÞ ðTA0in � TAÞ

� �ð14Þ

is nonsingular and therefore, Model (4) has a well-defined relativedegree vector r ¼ ½1;1� for all xð0Þ 2 U. h

Proposition 2. Let

z ¼ UðxÞ ¼

S2

TAX1=ðS2;in � S2Þa

X2=ðS2;in � S2Þa

X1=½ðS1;in � S1Þ þ k1=k2ðS2;in � S2Þ�a

0BBBBBB@

1CCCCCCA; ð15Þ

a dipheomorphism of Model (4) , which qualifies as a local coordinatetransformation in a neighborhood U of x.

Proof. By defining ST � S1 þ k1=k2S2 as the total substrate concen-tration, it is straightforward to see that ðST;in � STÞ ¼ ½ðS1;in � S1Þþk1=k2ðS2;in � S2Þ�. Thus, the resulting determinant of the Jacobianmatrix of (15) under NOC (see Section 2.3) is

detðJðzÞÞ ¼ � aX1

ðS2;in � S2Þ2aðST;in � STÞaþ1 – 0;

where it is clear that the Jacobian matrix JðzÞ is nonsingular for allx 2 U guaranteeing the existence of an inverse given by

x ¼ U�1ðzÞ ¼

z3ðS2;in � z1Þa

z4ðS2;in � z1Þa

S1;in þ k1k2ðS2;in � z1Þ � z3

z5

�1=aðS2;in � z1Þ

z1

z2

0BBBBBBB@

1CCCCCCCA;

which ends the proof. h

Proposition 3. Let e the estimation error vector whose componentsare ei

1 ¼ zi � zi and ei2 ¼ gi � gi for i ¼ 1;2. Then, it is possible to guar-

antee that the error vector dynamics is practically asymptotically sta-ble; i.e., e! e as t !1 where e is an arbitrarily small neighborhoodaround the origin.

Proof. By subtracting (9a) to (8), the estimation error dynamicscan be stated as follows

_e ¼WðtÞeþ b; ð16Þwhere

WðtÞ ¼

�ðC1g11 þ D1ðtÞÞ 1 0 0�C2

1g12 0 0 00 0 �ðD1ðtÞ þ D2ðtÞ þ C2g21Þ 10 0 �C2

2g22 0

2666437775 and

b ¼

0N1ðzÞ

0N2ðzÞ

2666437775;

while N1ðzÞ and N2ðzÞ are the time derivatives of g1 and g2, respec-tively. Since g1 and g2 are bounded and smooth functions (see A2 inSection 2.3), then a sufficient condition to guarantee the practicalasymptotical stability of system (16) is that the following non-autonomous system is asymptotically stable_e ¼WðtÞe: ð17Þ

Now, let us rewrite system (17) as follows

_e ¼W1ðtÞ 0

0 W2ðtÞ

� �e;

where

W1ðtÞ ¼�ða1 þ D1ðtÞÞ 1

�b1 0

� �; W2ðtÞ ¼

�ða2 þ D1ðtÞ þ D2ðtÞÞ 1�b2 0

� �;

0 < Dmin1 6 D1ðtÞ 6 Dmax

1 8t P 0;

0 < Dmin2 6 D2ðtÞ 6 Dmax

2 8t P 0;

0 < Dmin1 þ Dmin

2 6 D1ðtÞ þ D2ðtÞ 6 Dmax1 þ Dmax

2 8t P 0:

Since WðtÞ has a diagonal form and W1ðtÞ;W2ðtÞ have a similarstructure, the stability proof for system (17) is reduced to proofthe stability of the following system

_x ¼ wðtÞx; ð18Þ

where

wðtÞ ¼�ðaþ dðtÞÞ 1�b 0

� �and 0 < dmin < dðtÞ < dmax 8t P 0:

Thus, by defining wðtÞ in terms of dmin and dmax, system (18) can berecast as follows

_x ¼ ðh1ðtÞw1 þ h2ðtÞw2Þx;

where

wðtÞ ¼ 1dmax � dmin

�ðdmax � dminÞðaþ dðtÞÞ ðdmax � dminÞ�bðdmax � dminÞ 0

!

¼ dmax � dðtÞdmax � dmin

�ðaþ dminÞ 1�b 0

!þ dðtÞ � dmin

dmax � dmin

�ðaþ dmaxÞ 1�b 0

� �¼ h1ðtÞw1 þ h2ðtÞw2

and

h1ðtÞ ¼dmax � dðtÞdmax � dmin

> 0; h2ðtÞ ¼dðtÞ � dmin

dmax � dmin¼ 1� h1ðtÞ < 1:

Now, let us define the following candidate Lyapunov function (CLF)

VðxÞ ¼ xT Px; ð19Þ

where

P ¼1 � ð1�qÞðaþdminÞ

b

� ð1�qÞðaþdminÞb

1b þ 1

1�q 1þ ðdmax�dminÞ

2ðaþdminÞ

�ð1�qÞðaþdminÞ

b

�2

0B@1CA

is a symmetric and positive definite matrix for all q such that

11þ 16b

ðdmax�dminÞ2< q < 1:

Therefore, positive definite matrices Q1 and Q2 can be defined asfollows

wTi P þ Pwi ¼ �Q i < 0; for i ¼ 1;2;

such that it can be shown that the time derivative of the CLF (19)given by the following expression

_Vðx; tÞ ¼ xT ½h1ðtÞðwT1P þ Pw1Þ þ h2ðtÞðwT

2P þ Pw2Þ�¼ �xT ½h1ðtÞQ 1 þ h2ðtÞQ 2�x ¼ �xT QðtÞx

is a negative definite function (i.e., _Vðx; tÞ < 0) which completes theproof. h

Lemma 4. The internal dynamics of Model (4) is exponentially stableunder NOC.

382 H.O. Méndez-Acosta et al. / Journal of Process Control 20 (2010) 375–383

Proof. From Propositions 1 and 2, it can be seen that the unobserv-able dynamics from the input–output representation of Model (4)is given by the dynamics of the state variables X1;X2 and S1. Inaddition, from Proposition 3, it can be shown that

D1 !k3l2ð�ÞX2 � k2l1ð�ÞX1

S2;in � S2:

Thus, the internal dynamics of Model (4) is denoted by the follow-ing set of equations

_X1 ¼ l1ð�Þ � ak3l2ð�ÞX2 � k2l1ð:ÞX1

S2;in � S2

� �X1;

_X2 ¼ l2ð�Þ � ak3l2ð�ÞX2 � k2l1ð:ÞX1

S2;in � S2

� �X2;

_S1 ¼ ðS1;in � S1Þk3l2ð�ÞX2 � k2l1ð�ÞX1

S2;in � S2� k1l1ð�ÞX1:

Now, by introducing the following variable

r1 ¼ ak2 þS2;in � S2 � ak3X2

X1

� �;

one obtains

_r1 ¼ �l1ð�Þr1;

whose solution is given by

r1 ¼ r1ðt0Þ exp �Z t

t0

l1ð�Þdt� �

;

which proof the exponential stability of this variable. In this way,from the definition of r1, it can be seen that

k3X2 � k2X1 !S2;in � S2

a:

Now, by defining

r2 ¼1

X1þ ak2

S2;in � S2

� �;

the X1 dynamics can be recast in terms of r2 as follows

_r2 ¼ � l1ð�Þ � l2ð�Þ�

r2;

whose solution is given by

r2 ¼1

X1ðt0Þþ ak2

S2;in � S2

� �exp �

Z t

t0

ðl1ð�Þ � l2ð�ÞÞdt� �

from which one can find

X1 ¼1

1X1ðt0Þ

þ ak2S2;in�S2

�exp �

R tt0ðl1ð�Þ � l2ð�ÞÞdt

�� ak2

S2;in�S2

which implies that

X1 !� S2;in�S2

ak2if l1ð�Þ > l2ð�Þ

X1ðt0Þ if l1ð�Þ ¼ l2ð�Þ0 if l1ð�Þ < l2ð�Þ

8><>: as t !1:

Clearly, the case where l1ð�Þ > l2ð�Þ has no physical meaning underNOC, because this equilibrium point can only be reached under theacidification of the digester. Now, by analyzing the case wherel1ð�Þ < l2ð�Þ, it can be seen that the corresponding equilibriumpoint is unstable and can only be reached for the initial conditionsðX1; S

1Þ ¼ ð0; S1;inÞ, which are not included in the set of NOC. Thus,

the only attainable condition under NOC is that of the case wherel1ð�Þ ¼ l2ð�Þ, which proof the exponential stability of the state vari-ables X1 and X2.

Finally, since X1;X2 become constants as t !1, by introducingthe following variable

r3 ¼ S1;in � S1 � ak1X1;

one gets

_r3 ¼ �l1ð�Þa

r3

and, as a consequence, the solution for S1 is given by

S1 ¼ exp � 1a

Z t

t0

l1ð�Þdt� �

S1ðt0Þ

þ S1;in � ak1X1�

1� exp �1a

Z t

t0

l1ð�Þdt� �� �

:

Then, it is straightforward to show that

S1 ! S1;in � ak1X1;

which means that, under NOC, the COD influent concentration willbe always greater than that of the effluent (see Section 2.3) endingthe proof. h

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