Simulation Modelling Practice and Theory...1298 D. Selis teanu et al./Simulation Modelling Practice and Theory 18 (2010) 1297–1313 In order to obtaina dynamicalstate-space modelof

Simulation Modelling Practice and Theory 18 (2010) 1297–1313

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Simulation Modelling Practice and Theory

journal homepage: www.elsevier .com/ locate/s impat

Pseudo Bond Graph modelling and on-line estimation of unknownkinetics for a wastewater biodegradation process

Dan Selis�teanu *, Monica Roman, Dorin S�endrescuDepartment of Automatic Control, University of Craiova, A.I. Cuza No. 13, Craiova, Romania

a r t i c l e i n f o

Article history:Received 12 October 2009Received in revised form 5 May 2010Accepted 7 May 2010Available online 31 May 2010

Keywords:Wastewater treatmentBiotechnologyNonlinear modelsBond GraphsNonlinear observers

1569-190X/$ - see front matter � 2010 Elsevier B.Vdoi:10.1016/j.simpat.2010.05.004

* Corresponding author. Tel.: +40 251 438198.E-mail addresses: [email protected] (D.

a b s t r a c t

This paper deals with the problem of modelling and on-line estimation of kinetics for a bio-methanation process. This bioprocess is in fact a wastewater biodegradation process withproduction of methane gas, which takes place inside a Continuous Stirred Tank Bioreactor.The reaction scheme and the analysis of biochemical phenomena inside the bioreactor areused in order to obtain a nonlinear dynamic model of the bioprocess, by means of thepseudo Bond Graph method. Two nonlinear estimation strategies are developed for theidentification of unknown kinetics of the bioprocess. First, an estimator is developed byusing a state observer based technique. Second, an observer based on high-gain approachis designed and implemented. Several numerical simulations are performed in order toanalyse and compare the behaviour and the performance of the proposed estimators.

� 2010 Elsevier B.V. All rights reserved.

1. Introduction

Nowadays the use of advanced control for wastewater treatment plants is quite low, due to the lack of quality of the data,and the fact that modern control strategies must be based on a model of the dynamics of the process [2,9,10,27]. When mod-ern control strategies are used in wastewater treatment, the nonlinearity of the bioprocesses and the absence of cheap andreliable instrumentation require an enhanced modelling effort and on-line estimation strategies for the bioprocess kinetics.

The bioprocess modelling is a difficult task; however, using the mass balance of the components inside the process andobeying some modelling rules, a dynamical state-space model can be obtained. Many times, the dynamic models are high-order and nonlinear [2,10,27]. A viable alternative to the classical modelling is the Bond Graph method, introduced by Payn-ter in 1961, and further developed by Rosenberg and Karnopp [16], and Thoma [31]. Over the last four decades there havebeen a lot of publications regarding the theory and application of Bond Graphs for different kind of systems [4], such as elec-trical [14], mechanical [6], hydraulic [7], thermal and chemical [1,5,15,32], etc. This method provides a uniform manner todescribe the dynamical behaviour for all types of physical systems and illustrates the exchange power in a system, which isnormally the product between the effort and flow variables in the true Bond Graph [3]. Besides this representation there isanother one, in which the product effort-flow does not have the physical dimension of power, called pseudo Bond Graph[5,15].

Pseudo Bond Graphs are more suitable for chemical systems due to the physical meaning of the effort and flow variables.The Bond Graph modelling of some biological systems was reported in some works, such as [8,19,26]. However, the BondGraph modelling of biotechnological processes is not fully exploited yet [24,25,33].

The advantages of Bond Graph modelling are the following: offers a unified approach for all types of systems; allows todisplay the exchange of power in a system by its graphical representation; due to causality assignment it gives the possibility

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Selis�teanu), [email protected] (M. Roman), [email protected] (D. S�endrescu).

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1298 D. Selis�teanu et al. / Simulation Modelling Practice and Theory 18 (2010) 1297–1313

of localization of the state variables and achieving the mathematical model in terms of state space equations in an easier waythan using classical methods; provides information regarding the structural properties of the system, in terms of controlla-bility and observability, etc. Also, there exists the possibility to design observers and controllers via Bond Graph approach[11,21–23].

Beside the modelling difficulties, a lot of problems appear in the measurement of substrates, biomass and product con-centrations [10,27]. In many cases the state variables, i.e. the concentrations, are analysed manually and consequently thereis no on-line (and real-time) control. These problems can be solved using ‘‘software sensors”. A software sensor is a combi-nation between a hardware sensor and a software estimator. These software sensors are used not only for the estimation ofconcentrations (state variables), but also for the estimation of kinetic parameters.

Very important is the estimation of kinetic rates inside a bioreactor (the so-called kinetics of the bioprocess) – the esti-mates of these rates are used for the control strategies. Concerning this issue, an early approach was based on Kalman filterwhich leads to complex nonlinear algorithms. A well-known technique is the approach based on adaptive systems theory,which consists in the estimation of unmeasured states with asymptotic observers, and after that, the measurements andthe state estimates are used for on-line estimation of kinetics. We remark that the state asymptotic observers are designedwithout any knowledge of kinetics. Such kind of algorithm is, for example, the estimator based on state observer [2,10]. Thisuseful method was applied for bioreactors [2,10], but in some cases, when many reactions are involved, it requires the cal-ibration of too many parameters. For example, if we have n components’ concentrations used for the estimation of m kineticrates, it is necessary to calibrate 2n tuning parameters. In order to overcome this problem, a possibility is to design an esti-mator using a high-gain approach [12,13,17]. The gain expression of high-gain observers involves a single tuning parameterwhatever the number of components and reactions.

High-gain observers have evolved over the past two decades as an important tool for the design of output feedback con-trol of nonlinear systems [17]. The early work on high-gain observers appeared in the late 1980s, and afterwards the tech-nique was developed independently by two schools of researchers: a French school lead by Farza et al. [12] and Gauthieret al. [13], and a US school lead by Khalil [17].

Generally speaking, the design of stable and convergent state and parameters estimators is a complex task and good solu-tions are given only by studying each particular bioprocess.

This paper, which is an extended work of [24,29,30], deals with the Bond Graph modelling and on-line estimation ofkinetics for a wastewater biodegradation process that takes place inside a Continuous Stirred Tank Bioreactor (CSTB). In[30] the high-gain estimators were designed for activated sludge bioprocesses, which take place inside sequencing batchreactors. Also, in [28,29] the kinetic estimators were discussed without considering the use of some state observers and noisymeasurements; also, parameter disturbances are not taken into consideration. In the present work concerning the wastewa-ter biodegradation process inside a CSTB, all these aspects are studied, including comparisons and performance analysis.

The organization of the paper is as follows. In Section 2, the Bond Graph technique and some rules regarding the bioproc-ess modelling are applied in order to design the pseudo Bond Graph model of a wastewater biomethanation process. TheBond Graphs are depicted using the modelling and simulation environment 20sim (registered trademark of Controllab Prod-ucts B.V. Enschede, Netherlands). The obtained model is complex and the kinetics is uncertain; therefore, on-line estimationstrategies are developed in next sections. Section 3 deals with the design of some on-line estimation algorithms for theimprecise kinetics of the process. First, an estimator based on a state observer technique is designed. This estimator is ableto provide the estimates of unknown specific growth rates of the bioprocess. Second, an on-line estimation algorithm basedon high-gain technique is developed. The estimation scheme does not require any model for the kinetic rates. If the observersrequire on-line state estimates, these will be provided by asymptotic observers. In Section 4, in order to analyse the perfor-mances of the estimators, numerical simulations are provided. Finally, in Section 5, concluding remarks are presented.

2. Nonlinear dynamical model of the biodegradation process

2.1. The general structure of biotechnological process models

A process carried out in a bioreactor can be defined as a set of m biochemical reactions involving n components (withn P m). The concentrations of the physical components will be denoted with ni; i ¼ 1;n. The reaction rates will be denotedas uj; j ¼ 1;m. The evolution of ith component is described by the differential equation [2]:

dni

dt¼ _ni ¼

Xj�i

ð�Þkijuj � Dni þ Fi � Qi; ð1Þ

where ni (g/l) is the concentration and the notation j � i indicates that the sum is done in accordance with the reactions j thatinvolve the component i. The positive and dimensionless constants kij are yield coefficients. The sign of the first term of (1) isgiven by the type of the component ni: plus (+) when the component is a reaction product and minus (�) otherwise. D is thespecific volumetric rate (h�1), usually called dilution rate. Fi represents the rate of supply of the component ni (external sub-strate) to the bioreactor per unit of volume (g l�1 h�1). When this component is not an external substrate, then Fi � 0. Qi rep-resents the rate of removal of the component ni from the bioreactor in gaseous form per unit of volume (g l�1 h�1).

D. Selis�teanu et al. / Simulation Modelling Practice and Theory 18 (2010) 1297–1313 1299

In order to obtain a dynamical state-space model of the entire bioprocess, we denote n ¼ n1 n2 � � � nn½ �T , where n is then-dimensional vector of the instantaneous concentrations, also is the state of the model. The m-dimensional vector of thereaction rates (the reaction kinetics) is denoted u ¼ ½u1 u2 � � � um �

T . Usually, a reaction rate is represented by a non-negative rational function of the state n. The yield coefficients can be written as the (n �m) – dimensional yield matrixK = [Kij], i ¼ 1;n; j ¼ 1;m, where Kij = (±)kij if j � i and 0 otherwise. Next, we introduce the notationsF ¼ F1 F2 � � � Fn½ �T ; Q ¼ Q1 Q 2 � � � Qn½ �T , where F is the vector of rates of supply and Q is the vector of rates of re-moval of the components in gaseous form.

From (1), with the above notations, the global dynamics can be represented by following the dynamical state-space model[2]:

_n ¼ K �uðnÞ � Dnþ F � Q : ð2Þ

This model describes in fact the behaviour of an entire class of biotechnological processes and is referred to as the generaldynamical state-space model of this class of bioprocesses [2,10]. In (2), the term K �uðnÞ is in fact the rate of consumptionand/or production of the components in the reactor, i.e. the reaction kinetics. The term �Dn + F � Q represents the exchangewith the environment, i.e. the dynamics of the component transportation through the bioreactor. The strongly nonlinearcharacter of the model (2) is given by the reaction kinetics. In many situations, the yield coefficients, the structure andthe parameters of the reaction rates are partially known or unknown.

2.2. Bond Graph modelling rules and a prototype model

One of the advantages of Bond Graph method is that models of various systems belonging to different engineering do-mains can be expressed using a set of only nine elements: inertial elements (I), capacitive elements (C), resistive elements(R), effort sources (Se) and flow sources (Sf), transformer elements (TF), gyrator elements (GY), 0 – junctions (J0) and 1 –junctions (J1). I, C, and R elements are passive elements because they convert the supplied energy into stored or dissipatedenergy. Se and Sf elements are active elements because they supply power to the system and TF, GY, 0- and 1-junctions arejunction elements that serve to connect I, C, R, Se and Sf, and constitute the junction structure of the Bond Graph model.

A Bond Graph consists of subsystems linked together by lines representing power bonds. Each process is described by apair of variables, effort e and flow f. Besides the effort and flow variables, two other types of variables are very important indescribing dynamic systems; these variables, sometimes called energy variables, are the generalized momentum p as timeintegral of effort and the generalized displacement q as time integral of flow [16,31].

One of the simplest biological reactions is the micro-organisms growth process, with the reaction scheme [2,10] given by:

S!u X; ð3Þ

where S is the substrate, X is the biomass and u is the reaction rate.This simple growth reaction represents in fact a prototype reaction, which can be found in almost every bioprocess. The

dynamic of the concentrations of the components from reaction scheme (3) can be modelled considering the mass balance ofthe components; accordingly, a dynamical model of the form (2) can be obtained. The dynamical model of the bioprocess (3)is quite simple, but if the reaction scheme is more complicated, the achievement of the dynamical model is difficult. In suchcases, the Bond Graph method can be a suitable approach. In order to model bioprocesses, pseudo Bond Graph method ismore appropriate because of the meaning of variables involved – effort (concentration) and flow (mass flow).

Next, the pseudo Bond Graph model for a prototype bioprocess taking place into a CSTB is obtained. In this case, the sub-strate (the nutrient) is fed to the bioreactor continuously and an effluent stream is continuously withdrawn such that theculture volume is constant. From the reaction scheme (3) and taking into account the mass transfer through the CSTB, usingthe Bond Graph modelling procedure, the pseudo Bond Graph model of the continuous bioprocess is achieved and is given inFig. 1. The directions of half arrows correspond to the run of the reaction, going out from S to X. For the model presented inFig. 1, the mass balances of the species involved in the bioreactor are represented by two 0-junctions: 01,2,3,4 (mass balancefor S), 08,9,10 (mass balance for biomass X).

The constitutive relations of these junctions are characterized by the equality to zero of the sum of flow variables corre-sponding to the junction bonds; therefore, the next relations are obtained: f2 = f1 � f3 � f4, f9 = f8 � f10. A modulated two-portresistive element MR6,7 was used to model the reaction rate. From the constitutive relations of the 1-junction (15,6) and MR6,7

0

C

TF 1 MR TF

C

0Sf

fSfS

e1

e2

e3

e6

f7 f8

e9

e10

f1

e4

f4

e5

f5 f6

e7 e8

f2

f3

f9

f10

k4,5 k7,8

Fig. 1. Bond Graph model of the prototype bioprocess.


element we obtain: f5 = f6, f6 being proportional to the reaction rate u and V, i.e. f6 = l( � )e9V, where l is the specific growthrate; u = l( � )e9. The specific growth rate is usually an unknown or imprecisely known nonlinear function of the state (con-centrations). For example, l can be modelled as a Monod or Haldane law [2,10].

Mass flow of the component entering the reaction is modelled using a source flow element Sf1. The quantities exitingfrom the reaction are modelled by using flow sources represented by bonds 3 and 10, i.e. Sf3, Sf10. The equations describingthese two flow sources are:

f3 ¼ Sf3 � e3; f 10 ¼ Sf10 � e10: ð4Þ

The accumulations of substrate and biomass in the CSTB are represented by bonds 2 and 9, and are modelled using capac-itive elements C, with the constitutive equations:

e2 ¼1C2

q2 ¼1C2

Ztðf1 � f3 � f4Þdt; ð5Þ

e9 ¼1C9

q9 ¼1C9

Ztðf8 � f10Þdt: ð6Þ

The transformer elements TF4,5, TF7,8 were introduced to model the yield coefficients. From the constitutive relations oftransformer elements we obtain the relations for the flows f4 and f8: f4 = k4,5f6, f8 = f6/k7,8, where k4,5, k7,8 are the transformersmodulus in Bond Graph terms, which are in fact yield coefficients of the bioprocess k4,5 = k7,8 = 1.

The signification of Bond Graph elements is as follows: e2 is the substrate concentration S (g/l), e9 is the biomass concen-tration X (g/l), f6 is u � V, C2 = C9 = V (l) is the volume of the bioreactor, Sf3 = Sf10 = F0, where F0 is the output flow (l/h), andf1 = FinSin, where Fin is the influent substrate flow (l/h) and Sin is the influent substrate concentration (g/l).

Therefore, from (4)–(6) and taking into account the constitutive relations of junction elements, we will obtain the dynam-ical model of the continuous bioprocess:

V � _S ¼ FinSin � F0S�uV ;

V � _X ¼ �F0X þuV ;ð7Þ

where Fin is the influent substrate flow (l/h), Sin is the influent substrate concentration (g/l).From the equation of continuity Fin = F0 and using the so-called dilution rate D = Fin/V = 1/tr, with tr – medium residence

time, Eq. (7) become:

_S ¼ DSin � DS�u;_X ¼ �DX þu:

ð8Þ

2.3. The model of a wastewater biodegradation process inside a CSTB

One of the most important bioprocesses is the wastewater biodegradation with production of methane gas [2,28,29]. Thisanaerobic degradation process is a frequent method of wastewater treatment, which consists of four metabolic phases: twophases for acid production and two phases for methanation [2] – see Fig. 2. In the first phase, the glucose from the

GLUCOSE

PROPIONIC ACID

INORGANIC CARBON

HYDROGEN ACETATE

CARBON DIOXIDE

METHANE GAS

acidogenic bacteria

ionised hydrogen

methanogenic bacteria

methanogenic bacteria

Fig. 2. The structure of the wastewater biodegradation process.


wastewater is decomposed in fat volatile acids (acetates, propionic acid), hydrogen and inorganic carbon under action of theacidogenic bacteria. In the second phase, the ionized hydrogen decomposes the propionic acid CH3CH2COOH in acetates, H2

and carbon dioxide CO2. In the first methanogenic phase, the acetate is transformed into methane and CO2, and finally, in thesecond methanogenic phase, the methane gas CH4 is obtained from H2 and CO2.

The reaction scheme of this complex bioprocess involves four reactions and 10 components [2]. From the reaction schemewe can obtain the dynamical model of this bioprocess that takes place inside a CSTB, considering the mass balance of thecomponents [2,29]. The obtained model is very complex and the design of useful control strategies is hampered becauseof this large dimension (10) of the model. So, it is necessary to reduce the model order, taking into consideration some par-ticular aspects and using the singular perturbation theory [18]. A simplified reaction scheme and the corresponding modelcan be derived for this bioprocess [28,29]:

S1!u1 X1 þ S2:

S2!u2 X2 þ P:

(ð9Þ

In the reaction scheme (9) (which is a simply qualitative relation and does not include stoichiometric considerations), Si,i = 1, 2 are substrates: S1 represents the glucose, S2 the acetate; X1 is the acidogenic bacteria, X2 the acetoclastic methano-genic bacteria and P represents the product, i.e. the methane gas. We remark that X1, X2 are auto-catalysts. The reaction ratesare ui, i = 1, 2.

Next, using the Bond Graph method, we will derive a model of the bioprocess (9). From the reaction scheme (9), and con-sidering the mass transfer through the CSTB, using the modelling procedure previously described, the pseudo Bond Graphmodel of the bioprocess is achieved [24]. This model is presented in Fig. 3. The directions of the half arrows in the BondGraph correspond to the progress of the reactions, going out from the component S1 towards X1 and S2, for the first reaction,and from S2 towards X2 and P, for the second reaction.

In Bond Graph terms, the mass balances of the species involved in the bioreactor are represented by five 0-junctions:01,2,3,4 (mass balance for S1), 08,9,10 (mass balance for X1), 012,13,14,15 (mass balance for S2), 019,20,21 (mass balance for X2),023,24,25,26 (mass balance for P). The constitutive relations of these junctions are characterized by the equality to zero ofthe sum of flow variables corresponding to the junction bonds; therefore, the next relations are obtained:

f2 ¼ f1 � f3 � f4; f 9 ¼ f8 � f10; f 13 ¼ f12 � f14 � f15; f 21 ¼ f19 � f20; f 24 ¼ f23 � f25 � f26:

Thus, the accumulations of species S1, X1, S2, X2, and P in the bioreactor are represented by bonds 2, 9, 13, 21 and 24,respectively, and are modelled using capacitive elements C.

The constitutive equations of C-elements are as follows:

e2 ¼1C2

q2 ¼1C2

Ztðf1 � f3 � f4Þdt; e9 ¼

1C9

q9 ¼1C9

Ztðf8 � f10Þdt; e13 ¼

1C13

q13

¼ 1C13

Ztðf12 � f14 � f15Þdt; e21 ¼

1C21

q21 ¼1

C21

Ztðf19 � f20Þdt; e24 ¼

1C24

q24 ¼1

C24

Ztðf23 � f25 � f26Þdt: ð10Þ

where C2, C9, C13, C21, C24 are the parameters of C-elements:

C2 ¼ C9 ¼ C13 ¼ C21 ¼ C24 ¼ V :

0

C

TF 1 MR TF

C

0Sf

Sf

Sf

TF

C

0

Sf

CFTFT 0

Sf

CFT 0

Sf

1 MR

Sf

1

2

3 4 5

6 7 8

9 10

11

12 13

14

15 16 17 18

19

20 21

22

23 24

25

26

Fig. 3. The pseudo Bond Graph model of the wastewater biodegradation process.


The quantities exiting from the reaction are modelled by using flow sources represented by bonds 3, 10, 14, 20, and 25;the equations describing these elements are:

f3 ¼ Sf3 � e3; f 10 ¼ Sf10 � e10; f 14 ¼ Sf14 � e14; f 20 ¼ Sf20 � e20; f 25 ¼ Sf25 � e25;

where Sf3, Sf10, Sf14, Sf20, Sf25 are parameters of flow sources: Sf3 = Sf10 = Sf14 = Sf20 = Sf25 = F0, and F0 is the output flow (l/h).Mass flow of the component entering the reaction is modelled using a source flow element Sf1. The transformer elements

TF4,5, TF7,8, TF11,12, TF15,16, TF18,19, TF22,23 were introduced to model the yield coefficients. We used a modulated Sf element inorder to model the methane gas outflow rate f26, which will be denoted with Q (g/h).

For the modelling of the reaction rates we used two modulated two-port R elements: MR6,7 and MR17,18. From the con-stitutive relations of the two 1-junction elements, 15,6,11 and 116,17,22, we obtain: f5 = f6 = f11, f16 = f17 = f22, where the consti-tutive relations of MR elements imply that f6 = l1e7V and f17 = l2e18V, where l1 and l2 are the specific growth rates, andu1 = l1e7, u2 = l2e18.

From the constitutive relations of transformer elements we obtain the relations for the flows f4, f8, f12, f15, f19, f23:

f4 ¼ k4;5f6; f 8 ¼ f6=k7;8; f 12 ¼ f6=k11;12; f 15 ¼ k15;16f17; f 19 ¼ f17=k18;19; f 23 ¼ f17=k22;23

with k4;5 ¼ k1; k11;12 ¼ 1=k2; k15;16 ¼ k3; k22;23 ¼ 1=k4; k18;19 ¼ 1=k5; k7;8 ¼ 1=k6 the transformers modulus in Bond Graphterms, which are in fact yield coefficients of the bioprocess.

The signification of Bond Graph elements is as follows: e2 is the glucose concentration S1 (g/l), e9 – the acidogenic bacteriaconcentration X1 (g/l), e13 – the acetate concentration S2 (g/l), e21 is the acetoclastic methanogenic bacteria concentration X2

(g/l), e24 is the product concentration P (g/l), f1 = FinSin, where Fin is the influent glucose flow (l/h) and Sin is the influent glu-cose concentration (g/l). Using these notations, from (10) we will obtain the following dynamical model of the wastewatertreatment bioprocess:

V � _S1 ¼ FinSin � F0S1 � k4;5u1V ;

V � _X1 ¼ �F0X1 þ ð1=k7;8Þu1V ;

V � _S2 ¼ �F0S2 þ ð1=k11;12Þu1V � k15;16u2V ;

V � _X2 ¼ �F0X2 þ ð1=k18;19Þu2V ;

V � _P ¼ �F0P � Q þ ð1=k22;23Þu2V

ð11Þ

From the equation of continuity Fin = F0, using the dilution rate D = Fin/V = 1/tr, with tr – medium residence time, and tak-ing into account that k5 = k6 = 1, the above equations become:

_S1 ¼ DSin � DS1 � k1u1;

_X1 ¼ �DX1 þu1;

_S2 ¼ �DS2 þ k2u1 � k3u2;

_X2 ¼ �DX2 þu2;

_P ¼ �DP � Q þ k4u2:

ð12Þ

The dynamical model (11) (or (12)), obtained using the Bond Graph approach, is equivalent with the dynamical state-space model obtained using classical methods in [9,28,29].

The model (12) can be written in the following compact form:

ddt

X1

S1

X2

S2

P

26666664

37777775 ¼1 0�k1 00 1k2 �k3

0 k4

26666664

37777775|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}

K

u1

u2

� �� D

X1

S1

X2

S2

P

26666664

37777775þ0

DSin

00�Q

26666664

37777775: ð13Þ

The state vector of (13) is n ¼ X1 S1 X2 S2 P½ �T ¼ n1 n2 n3 n4 n5½ �T .The vector of feed rates and of rates of removal of components (in gaseous form) can be written as

F ¼ 0 DSin 0 0 �Q½ �T . The dynamical model (13) can be compactly written as

_n ¼ K �uðnÞ � D � nþ F: ð14Þ

This model belongs to the class (2); K is the normalized matrix of the yield coefficients, and the vector of reaction rates(the reaction kinetics) is uðnÞ ¼ ½u1ðnÞ u2ðnÞ �

T :


3. On-line estimation strategies for the unknown kinetics

3.1. The statement of the problem

In industry, the control purpose for the biomethanation process is to regulate the acetate S2, which is an intermediatemetabolite responsible for biomethanation inhibition and furthermore is a measure of pollution level. This control taskcan be better accomplished if the advanced control strategies are used (such as adaptive control, neural and hybrid controlstrategies and so on). Usually, these strategies require the knowledge of reaction kinetics.

The most difficult task for the construction of the dynamical model (14) is the modelling of the reaction kinetics. The formof kinetics is complex, nonlinear and in many cases unknown. A general assumption is that a reaction can take place only ifall reactants are presented in the bioreactor. Therefore, the reaction rates are necessarily to be zero whenever the concen-tration of one of the reactants is zero. Thus, the reaction rates can be expressed as u(n) = H(n) � a(n, t), whereaðn; tÞ ¼ a1ðn; tÞ a2ðn; tÞ½ �T is a vector of time varying parameters. Each ai(n, t), i = 1, 2 is called the specific reaction rate.H(n) is a 2 � 2 state dependent diagonal matrix, whose elements correspond to the reactions’ reactants:

HðnÞ ¼S1X1 0

0 S2X2

� �: ð15Þ

X1 and X2 appear because they are auto-catalysts. Then the model (14) becomes:

_n ¼ K � HðnÞ � aðn; tÞ � D � nþ F: ð16Þ

Another formulation for the reaction rates implies the following structure:

uðnÞ ¼ l1ðnÞ � X1 l2ðnÞ � X2½ �T ð17Þ

with l1(n), l2(n) the specific growth rates.Now we can reconsider the form of the matrix H(n) and we have:

HðnÞ ¼X1 00 X2

� �: ð18Þ

The model of the bioprocess can be written as:

_n ¼ K � HðnÞ � lðn; tÞ � D � nþ F ð19Þ

with lðn; tÞ ¼ l1ðn; tÞ l2ðn; tÞ½ �T . For the specific growth rates l(n, t) different models exist, like Monod’s law or Haldanekinetic model [28]:

l1ðS1Þ ¼ l1S1

KM1 þ S1; l2ðS2Þ ¼ l0

2S2

KM2 þ S2 þ S22=Ki

ð20Þ

with l02 ¼ l2 1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiKM2=Ki

p� �. The first specific growth rate in (20) is modelled by the Monod law and the second specific

growth rate by the Haldane kinetic model, which takes into account the substrate inhibition on the growth. KM1 ; KM2 areMichaelis–Menten constants; l1; l2 represent the maxim specific growth rates and Ki is the inhibition constant.

However, in practice, the analytical models of the specific growth rates l(t) are difficult to obtain. Therefore, these uncer-tain rates need to be on-line estimated. If we consider the vector of unknown parameters qðtÞ ¼ q1ðtÞ q2ðtÞ½ �T

¼ l1ðtÞ l2ðtÞ½ �T , then the model can be written as:

_n ¼ K � HðnÞ � qðtÞ � D � nþ F: ð21Þ

3.2. The design of an asymptotic state observer

In many practical applications, only a part of the concentrations of the components involved are on-line measurable. Insuch cases, an alternative is the use of state observers. An important difficulty when applying state observers to bioprocessesis related to the uncertainty of models describing their dynamics. Presently two classes of state observers for bioprocessescan be found in the literature [2,9]. The first class of observers (including classical observers like Luenberger and Kalmanobservers, nonlinear observers) are based on a perfect knowledge of the model structure. A disadvantage of this class is thatthe uncertainty in the model parameters can generate possibly large bias in the estimation of the unmeasured states. A sec-ond class of observers, called asymptotic observers, is based on the idea that the uncertainty in process models lies in theprocess kinetics models. The design of these observers is based on mass and energy balances without the knowledge ofthe process kinetics being necessary.

The design of an asymptotic observer is based on some useful changes of coordinates, which lead to a submodel of (14)which is independent of the kinetics. In order to achieve the change of coordinates, a partition of the state vector n in two


parts is considered. This partition denoted (f1, f2) induces partitions of the yield matrix K: (K1, K2), and of F: (F1, F2). Wesuppose that a state partition is chosen such that the submatrix K1 is full rank and dim(f1) = rank(K1) = rank(K). Then a linearchange of coordinates can be defined as follows [2] z = G � f1 + f2, with z an auxiliary state vector and G the solution of thematrix equation G � K1 + K2 = 0. In the new coordinates, the model (14) can be rewritten as

_f1 ¼ K1uðf1; z� Gf1Þ � D � f1 þ F1;

_z ¼ �D � zþ G � F1 þ F2:ð22Þ

The main achievement of the change of coordinates is that the dynamics of the auxiliary state variables is independent ofthe reaction kinetics. Now z can be rewritten as a linear combination of the vectors of measured states, denoted fm, and ofunmeasured states, denoted fu:

z ¼ G1 � fm þ G2 � fu ð23Þ

with G1 and G2 well defined matrices. If the matrix G2 is left invertible, the asymptotic observer equations for (14) derivefrom the structure of Eqs. (22) and (23):

_z ¼ �D � zþ G � F1 þ F2

fu ¼ Gþ2 � ðz� G1fmÞð24Þ

where Gþ2 ¼ ðGT2G2Þ�1GT

2.The asymptotic observer is indeed independent of the kinetics. The asymptotic observer (24) has good convergence and

stability performances [2,9].For the particular case of our biodegradation process, usually the biomass concentrations are not on-line measurable, and

we will suppose that the substrate and product concentrations are available. Then we will have fu = [X1 X2]T and fm = [S1 S2

P]T. In order to design an asymptotic observer to reconstitute fu from the measurements fm, first we will select a partition ofthe state variables f1 = [S1 S2]T and f2 = [X1 X2 P]T. Then the following partitions are obtained:

K1 ¼�k1 0k2 �k3

� �; K2 ¼

1 00 10 k4

264375; F1 ¼

DSin

0

� �; F2 ¼

00�Q

264375:

After some straightforward calculations, the matrices G, G1 and G2 are obtained:

G ¼1=k1 0

k2=ðk1k3Þ 1=k3

k4k2=ðk1k3Þ k4=k3

264375; G1 ¼

1=k1 0 0k2=ðk1k3Þ 1=k3 0

k4k2=ðk1k3Þ k4=k3 1

264375; G2 ¼

1 00 10 0

264375:

Consequently, the auxiliary variables are obtained as:

z1 ¼ ð1=k1ÞS1 þ X1; z2 ¼ S1k2=ðk1k3Þ þ S2=k3 þ X2; z3 ¼ S1k4k2=ðk1k3Þ þ S2k4=k3 þ P

and the final equations of the asymptotic observer are found as:

_z1 ¼ �Dz1 þ DSin=k1;

_z2 ¼ �Dz2 þ DSink2=ðk1k3Þ;_z3 ¼ �Dz3 þ DSink4k2=ðk1k3Þ � Q ;

X1est ¼ z1 � S1=k1;

X2est ¼ z2 � S1k2=ðk1k3Þ � S2=k3:

ð25Þ

The asymptotic observer (25) provides the estimates of X1 and X2, and furthermore is independent of the kinetics. Theseestimates can be used in the kinetic parameters estimators design.

Remark 1. Another possibility is to design an asymptotic observer directly from the partition of the state vector in measuredand unmeasured states, but the calculations are more complicated.

3.3. On-line estimation algorithm based on a state observer technique

The design of the on-line estimation strategies will be done considering that the model of bioprocess fulfils next hypoth-eses [2,20,28]:

H1. All state variables are measurable in real-time (otherwise, a state observer is needed).H2. The vector of feed rates, the dilution rate and the methane gas outflow rate are measurable.H3. The matrix of yields coefficients K is known.


The basic idea behind the design of an estimator based on a state observer technique is to use a state observer, not for thestate estimation, but in order to provide the information needed for updating the estimates of the parameters. The on-lineestimation algorithm for the biodegradation process (21) can be written as [2,20]:

_n ¼ K � HðnÞ � qðtÞ � D � nþ F �X � ðn� nÞ; ð26Þ_q ¼ ½K � HðnÞ�T � C � ðn� nÞ: ð27Þ

In (26) and (27), q is the on-line estimate of the unknown vector of parameters (in fact, the unknown kinetics – the spe-cific growth rates). The first equation of this algorithm is a state observer, used for updating the estimate q, and not forstate estimation. The update is generated by the estimation error e ¼ ðn� nÞ, where n is the on-line estimation of the statevector. The error q� q is directly reflected by the estimation error e. X is a gain matrix; in the second equation of thealgorithm, the injection matrix C is chosen such that the matrix XTC + CX is negative defined, with dim(X) =dim(C) = n � n.

The design parameters of the estimator (26) and (27) are the matrices X and C. The choice of these matrices must be donesuch that the algorithm to be stable and convergent. The properties of stability and convergence for this estimator have beendiscussed at length in [2,20]. A typical choice for the matrices X and C is of diagonal form:

X ¼ diagi¼1;...;n

f�xig; C ¼ diagj¼1;...;n

fcjg; xi; cj 2 Rþ: ð28Þ

After some basic calculation, by using (21) (26)–(28), the detailed equations of the on-line estimator based on state observerfor the unknown specific rates are obtained as follows [28]:

_bX1 ¼ l1ðtÞ � X1 � DX1 þx1ðX1 � bX1Þ; ð29Þ_bS1 ¼ �k1l1ðtÞ � X1 � DS1 þ DSin þx2ðS1 � bS1Þ; ð30Þ_bX2 ¼ l2ðtÞ � X2 � DX2 þx3ðX2 � bX2Þ; ð31Þ_bS2 ¼ k2l1ðtÞ � X1 � k3l2ðtÞ � X2 � DS2 þx4ðS2 � bS2Þ; ð32Þ_bP ¼ k4l2ðtÞ � X2 � DP � Q þx5ðP � bPÞ; ð33Þ_l1 ¼ c1X1ðX1 � bX1Þ � c2k1X1ðS1 � bS1Þ þ c4k2X1ðS2 � bS2Þ; ð34Þ_l2 ¼ c3X2ðX2 � bX2Þ � c4k3X2ðS2 � bS2Þ þ c5k4X2ðP � bPÞ: ð35Þ

In the observer-based estimator (29)–(35), the biomass concentrations X1, X2 are considered measurable. Otherwise,these concentrations can be replaced with the estimates X1est, X2est provided by the asymptotic observer (25).

Remark 2. The estimation algorithm (29)–(35) provides good results; however, it requires the calibration of ten tuningparameters. In order to overcome this problem, a possibility is to design an estimator using a high-gain approach.

3.4. High-gain observer for the unknown kinetics of the bioprocess

The high-gain observers are simple yet efficient nonlinear observers, which can be designed by using a factorization of theprocess model [12,13,30]. The high-gain observer design is based on the work of Gauthier et al. [13], Farza et al. [12], focusedon deriving global results under global Lipschitz conditions.

For the model (21), the yield matrix K, and the matrix H(n) respectively are of full rank. This assumption is true for ourparticular model, and for general case of the class (2) is a generic property. We shall suppose that all states are measured(contrarily, a state estimator can be used).

Remark 3. In some cases, the matrix K is not left invertible. These situations appear for example when the number ofunknown parameters is greater than the number of state variables. In these particular cases, the proposed observers cannotbe implemented.

The high-gain observer will be designed in order to reconstitute the imprecisely known specific growth rates l(t). Since Kis of full rank, i.e. is left invertible, a full rank arbitrary submatrix Ka (2 � 2) of K can be considered. Let Kb be the remainingsubmatrix of K. The system (21) can be written as follows:

_na ¼ Ka � Hðna; nbÞ � qðtÞ � D � na þ Fa;

_nb ¼ Kb � Hðna; nbÞ � qðtÞ � D � nb þ Fb;ð36Þ

where (na, nb) and (Fa, Fb) are partitions induced by the factorization of K. We suppose nb(t) a known (measured) signal, de-noted r(t) = nb(t). Then consider the system [12]:


_na ¼ Ka � Hðna;rÞ � qðtÞ � D � na þ Fa;

_q ¼ gðtÞ;y ¼ na

ð37Þ

where g(t) is a bounded unknown function, which may depend on na, r, inputs, noise. The hypothesis of boundedness of thekinetics is in accordance with industrial practice.

The design of nonlinear high-gain observers is done in [12,13] with supplementary assumptions regarding the bounded-ness of H(n) diagonal elements’ away from zero. The high-gain based observer equations for a general bioprocess describedby the model (2) are [12,13]:

_na ¼ Ka � Hðna;rÞ � q� D � na þ Fa � 2 � h � ðna � naÞ; ð38Þ_q ¼ �h2 � ½Ka � Hðna;rÞ��1 � ðna � naÞ; ð39Þ

The estimator (38) and (39) is in fact a copy of the bioprocess model, but with state replaced by its estimate, and with acorrective term. The tuning of this observer is very simple because a single parameter is involved: h.

0 5 10 15 20 25 30 35 400

5

10

15

Time (h)

(g/l)

S1

S2

P

X1

X2

Fig. 4. Time profiles of concentrations.

0 5 10 15 20 25 30 35 400

1

2

3

4

5

6

Time (h)

(g/l)

X1

X2

X1est

X2est

Fig. 5. The estimates of biomass concentrations.


Remark 4. Note that in (38) and (39), na is an ‘‘estimate” of na, provided by the algorithm in order to be compared with thereal state (na is measured or provided by a state observer), and the resulting error to be used in the estimator (38) and (39).

Remark 5. Usually, the gain of the observers is calculated by solving some Ricatti or Lyapunov differential equations [12,13].The high-gain observer design consists in a simple calculation of the gain by using algebraic Lyapunov equation.

For the biomethanation process there are several possibilities of the factorization of yield matrix K. For simplicity we choose

Ka ¼1 00 1

� �ð40Þ

and consequently we obtain:

na ¼X1

X2

� �; nb ¼ r ¼

S1

S2

P

264375; Kb ¼

�k1 0k2 �k3

0 k4

264375; Fa ¼

00

� �; Fb ¼

D � Sin

0�Q

264375: ð41Þ

It is clear that Ka is full rank matrix. From Eqs. (38), (39) with H(n) of the form (18) and with the factorization (40) and(41), the equations of the high-gain observer are:

0 5 10 15 20 25 30 35 400

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Time (h)

(h-1)

1µ1µ

Fig. 6. Evolution of l1 and l1 – free noise data.

0 5 10 15 20 25 30 35 400

0.1

0.2

0.3

0.4

0.5

Time (h)

(h-1)

2µ

2µ

Fig. 7. Evolution of l2 and l2 – free noise data.

0 5 10 15 20 25 30 35 400

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Time (h)

(h-1)

1µ

1µ

Fig. 8. Evolution of l1 and l1 – noisy data.


0 5 10 15 20 25 30 35 400

0.1

0.2

0.3

0.4

0.5

Time (h)

(h-1)

2µ

2µ

Fig. 9. Evolution of l2 and l2 – noisy data.

ddt

bX1bX2

" #¼

1 00 1

� ��bX1 00 bX2

" #�

l1

l2

� �� D �

bX1bX2

" #� 2 � h �

bX1 � X1bX2 � X2

" #ð42Þ

ddt

l1

l2

� �¼ �h2 �

1 00 1

� ��bX1 00 bX2

" #" #�1

�bX1 � X1bX2 � X2

" #ð43Þ

It can be seen that the estimator (42) and (43) needs only the measurements of X1 and X2. If these concentrations are not on-line available, they can be replaced with the estimates X1est;X2est provided by the asymptotic observer (25).

The nonlinear observer is quite simple and moreover the tuning of the gain can be done by modifying only one designparameter.

Remark 6. The structure of the high-gain observer (42) and (43) can be easily modified in order to obtain an estimator forthe specific reaction rates a1(n) and a2(n), by using the matrix H(n) of the form (15). Then the next estimator is obtained:

ddt

bX1bX2

" #¼

1 00 1

� �� S1

bX1 00 S2

bX2

" #�

a1

a2

� �� D �

bX1bX2

" #� 2 � h �

bX1 � X1bX2 � X2

" #; ð44Þ

0 5 10 15 20 25 30 35 400

0.1

0.2

0.3

0.4

0.5

Time (h)

(h-1)

2µ

2µ

Fig. 11. Evolution of l2 and l2 – high-gain observer (free noise data).

0 5 10 15 20 25 30 35 400

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Time (h)

(h-1)

1µ

1µ

Fig. 10. Evolution of l1 and l1 – high-gain observer (free noise data).


ddt

a1

a2

� �¼ �h2 �

1 00 1

� �� S1

bX1 00 S2

bX2

" #" #�1

�bX1 � X1bX2 � X2

" #: ð45Þ

4. Simulation results

Several numerical simulations were performed in order to analyse and compare the behaviour and the performance of theproposed estimators.

First, the on-line estimator based on a state observer technique (29)–(35) was implemented. In order to study the perfor-mances of this observer, the results are compared with data generated from simulation of the process model (19). The spe-cific growth rates for this simulation are of form (20) – these kinetic expressions are introduced only for simulation;therefore these models are not used in the process of the observer design. The main goal of the on-line estimation algorithmwas to reconstitute the time evolution of the specific growth rates l1 and l2. Simulation was performed for a sinusoidal formof the dilution rate, with amplitude at 15% of its nominal value (0.1 h�1), and for the following bioreactor parameters [29]:

Sin ¼ 30 g=l; k1 ¼ 5:4; k2 ¼ 1; k3 ¼ 14:7; k4 ¼ 10; Q ¼ 0; l1 ¼ 0:2 h�1; l0

2 ¼ 0:6 h�1;

KM1 ¼ 0:75 g=l; KM2 ¼ 1 g=l; Ki ¼ 21 g=l:

0 5 10 15 20 25 30 35 400

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Time (h)

(h-1)

1µ

1µ

Fig. 12. Evolution of l1 and l1 – high-gain observer (noisy data).

0 5 10 15 20 25 30 35 400

0.1

0.2

0.3

0.4

0.5

Time (h)

(h-1)

2µ

2µ

Fig. 13. Evolution of l2 and l2 – high-gain observer (noisy data).


Fig. 4 shows the time evolution of S1 and S2 – the glucose and acetate concentration, and of X1 – population of acidogenicbacteria, X2 – population of methanogenic bacteria and P – the methane gas. These ‘‘measurements” obtained from simula-tion are free-noise data. The performance of the proposed estimator is also analysed for noisy data of S1 and S2. The measure-ments of S1 and S2 are vitiated by an additive Gaussian noise. This noise is with zero mean and amplitude equal to 5% of thefree-noise values. Also, because the biomass concentrations are not on-line available in most practical situations, the esti-mates X1est and X2est are provided by the asymptotic observer (25), and are depicted in Fig. 5 together with their real profiles.It can be seen that the observer provides good estimates, and that the effect of noisy measurements of S1 and S2 is significantespecially for X1est.

The estimation and the simulation of the specific growth rates obtained with the estimator based on state observer tech-nique are depicted in Figs. 6 and 7. These pictures correspond to free-noise measurements. Figs. 8 and 9 show the resultsobtained when the noisy data are used, and furthermore the biomass concentrations X1est and X2est are utilised.

In all figures, the values obtained from simulation are depicted with solid line and the estimates with dashed line. Thevalues of the tuning parameters were set to x1 = x4 = x5 = 1, x2 = x3 = 5, c1 = c3 = 5 and c2 = c4 = c5 = 1. It can be seen fromthese time evolution diagrams that the on-line estimator provides good estimates for the unknown kinetics of the bioproc-ess. We can observe that the effect of noise is important especially for the estimates of l1.

Second, the high-gain observer was implemented for the biomethanation process, by using the same simulation param-eters and the same conditions like in the previous case. The main goal was to reconstitute the time evolution of l1 and l2.

0 5 10 15 20 25 30 35 40-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Time (h)

(h-1)

1~µ

1~µ - high-gain

- based on state observer

Fig. 14. Estimation errors ~l1 (based on state observer – solid line; high-gain – dashed line).

0 5 10 15 20 25 30 35 40-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

Time (h)

(h -1)

2~µ

2~µ - based on state observer

- high-gain

Fig. 15. Estimation errors ~l2 (based on state observer – solid line; high-gain – dashed line).

Table 1Performance criterion results.

On-line estimation strategies Performances

I1 I2

Estimator based on state observer strategy (free noise data) 0.56 10�4 1.21 10�4

High-gain observer (free noise data) 0.63 10�4 1.49 10�4

Estimator based on state observer strategy (noisy data) 5.22 10�4 3.32 10�4

High-gain observer (noisy data) 2.35 10�4 2.11 10�4


The estimation and the simulation of the specific growth rates obtained with the high-gain observer are depicted in Figs. 10and 11. These pictures correspond to free-noise measurements. Figs. 12 and 13 show the results obtained when the noisydata and the estimates of biomass concentrations are used in the high-gain algorithm.

The obtained results illustrate that the high-gain observer provides accurate estimates of the kinetic rates. It can be seenthat the measurement noise induces some noisy estimates of the kinetics, but the noise effect is limited (this effect can bereduced for lower values of the tuning parameter). The value of the tuning parameter was set to h = 3. Notice that the esti-mation error can be made as small as wished if we choose greater values of h. The problem for a large value of h is that the


observer becomes noise sensitive. The value of the tuning parameter is therefore a compromise between a good estimationand the noise rejection.

Several comparisons and comments regarding the behaviour and the performance of the two proposed on-line estimationstrategies can be achieved. Some remarks can be done by visualization of estimation errors ~l1 ¼ l1 � l1 and ~l2 ¼ l2 � l2.Figs. 14 and 15 present the estimation errors for both estimators (the free noise case).

However, accurate comparisons can be realized by considering a criterion, for example one based on averaged square esti-mation errors:

I1 ¼1TS

Z TS

0

~l21ðtÞdt; ð46Þ

I2 ¼1TS

Z TS

0~l2

2ðtÞdt; ð47Þ

where TS is the total simulation time.The values of I1 and I2, computed for the studied on-line estimation strategies, for both free noise and noisy data, are pre-

sented in Table 1. The obtained results show that the performances of the two estimators are comparable when free noisedata is used. However, the high-gain observer seems to be more efficient when the measurements are corrupted with noise.

5. Conclusion

In this paper, some modelling issues and on-line estimation strategies for the unknown kinetics of a complex biometh-anation process were widely analysed. The general dynamic nonlinear model of a class of bioprocesses was presented; also,the wastewater biomethanation process, which takes place inside a Continuous Stirred Tank Bioreactor, was analysed and adynamical model was achieved. The nonlinear model of this biodegradation process was obtained by using a novel modellingprocedure, based on pseudo Bond Graph technique. One of the main advantages of the Bond Graph modelling of bioprocessesis the possibility to reuse the models, for example in the interconnected bioreactors.

In order to overcome problems such as the modelling uncertainties and the lack of on-line measurements for the biodeg-radation process, two on-line estimation strategies were designed for the estimation of imprecisely known kinetic rates in-side the CSTB.

The first estimator design was based on a state observer technique, used in order to provide the information needed forupdating the estimates of the kinetics. The second estimator was designed using the high-gain approach. Several simulationswere conducted in order to analyse the behaviour and to compare the performances of the proposed estimation strategies.

The advantages of the estimator based on state observer technique are the simplicity of design, the good convergence andstability properties, and the accuracy of estimates (for free noise data). On the other hand, the number of tuning parametersand the behaviour for noisy measurements can be considered drawbacks of this strategy. An important advantage of thehigh-gain observer remains the fact that the tuning of one single design parameter is necessary. The high-gain observerneeds only the measurements of X1 and X2; the estimator based on state observer technique requires the measurementsof all state variables (in fact, this estimator requires the measurements of S1, S2 and P, and the estimates of X1 and X2 providedby the asymptotic observer), which in practice can be a serious drawback. The estimation results for the high-gain observercan be improved if the tuning parameter is chosen higher in value, but only if the measurements are free-noise. Contrarily,the observer becomes noise sensitive and it is possible that the estimates of kinetics cannot be utilized.

The obtained results are quite encouraging from simulation point of view. The proposed observers can be used for thedesign of advanced control strategies for the wastewater bioprocess.

The presented results can be extended to the Bond Graph modelling of interconnected bioreactors. Also, a further researchdirection is the design of nonlinear observers by using the structural properties associated with Bond Graph models ofbioprocesses.

Acknowledgment

This work was supported by CNCSIS–UEFISCSU, Romania, Project Number PNII – IDEI ID548/2008.

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