Modeling and Estimation Strategies for a Fed-batch Prototype Bioprocess*

Monica Roman, Dan Selişteanu, Emil Petre and Cosmin Ionete Dorin Popescu

Department of Automatic Control Department of Mechatronics University of Craiova University of Craiova

Craiova, A.I. Cuza no. 13, 200585, Romania Craiova, A.I. Cuza no. 13, 200585, Romania {monica, dansel, epetre & cosmin}@automation.ucv.ro dorinp@robotics.ucv.ro

* This work was supported by CNCSIS–UEFISCSU Romania, project PNII – IDEI 548/2008 and by ANCS Romania, project SICOTIR 71-084/2007.

Abstract - This paper deals with the Bond Graph modeling and the design of estimation strategies for a nonlinear fed-batch prototype bioprocess. The proposed strategies are developed for an aerobic microbial growth process coupled with an enzyme-catalyzed reaction, which is a usual bioprocess that takes place in fed-batch bioreactors. The rules for the design of pseudo Bond Graph model are obtained by using the reaction schemes and the analysis of biochemical phenomena. Two kinds of on-line estimation strategies are approached. First, a general state observer is analyzed and the exponential observability of the bioprocess is tested; two state estimation algorithms are designed: an extended Luenberger observer and an asymptotic observer. Second, an observer-based estimator is derived for the estimation of unknown kinetics. In order to test the behavior of proposed strategies, numerical simulations are included.

Index Terms - Biotechnology, Bond Graphs, Observers.

I. INTRODUCTION

Despite the progress concerning the use of modern control for bioprocesses, there are several problems due to the lack of quality of the data, and to the fact that advanced control strategies must be based on a model of the dynamics of process [1], [2], [3], [4]. The nonlinearity of bioprocesses and the absence of cheap and reliable instrumentation require an enhanced modeling effort and on-line estimation strategies.

The bioprocess modeling is a difficult task; however, using the mass balance of the components inside the process and obeying some modeling rules, a dynamical state-space model can be obtained - see [1], [2]. A feasible alternative to classical modeling is the Bond Graph method, which was first introduced by Paynter in 1961, and further developed in [5] and [6]. The Bond Graph approach is a powerful tool for modeling, analysis and design of different systems: electrical, mechanical, hydraulic [7], thermal, chemical [8], etc. This method illustrates the exchange power in a system, which is normally the product between the effort and flow variables. Besides this representation there is another one, in which the product effort-flow does not have the physical dimension of power, called pseudo Bond Graph [8]. Pseudo Bond Graphs are more suitable for chemical systems due to the physical meaning of effort and flow variables. The Bond Graph approach gives the possibility of localization of states and achieving the model in terms of state space equations in an easier way than classical methods [7]. The Bond Graph

modeling of biological systems was reported in some works, such as [9], [10]. However, the Bond Graph modeling of bioprocesses is not well exploited yet [11], [12].

Beside the modeling difficulties, a lot of problems appear in the measurement of substrates, biomass and product concentrations [1], [2], [3]. These problems can be solved using “software sensors”, i.e. combinations between hardware sensors and software estimators. These software sensors are used not only for the estimation of concentrations (state variables), but also for the estimation of uncertain kinetic parameters.

Presently two classes of state observers for bioprocesses can be found in the literature [1], [3]. The first class of observers (e.g. Luenberger and Kalman observers) is based on a perfect knowledge of the model structure. A second class of observers, called asymptotic observers, is based on the idea that uncertainty in process models lies in the kinetics models. The design of these observers is based on mass and energy balances without the knowledge of kinetics being necessary.

For the design of control strategies, very important is the estimation of kinetic rates inside a bioreactor. Concerning this issue, an early approach was based on Kalman filter which leads to complex nonlinear algorithms. A well-known technique is the approach based on adaptive systems, which consists in the estimation of states with asymptotic observers, and after that, the measurements and the state estimates are used for on-line estimation of kinetics. Such kind of algorithm is the observer-based estimator (OBE) [1], [3].

In the present work, the pseudo Bond Graph model of an aerobic micro-organisms growth process combined with an enzyme-catalyzed process is obtained. This prototype process is widely used in Fed-Batch Bioreactors (FBB) [1], [3], [11]. The pseudo Bond Graph model presented in Section 2 is achieved by using the reaction scheme and analysis of biochemical phenomena. In Section 3, a general form of state observer is analyzed and the exponential observability of the process is tested. Then, two estimation algorithms for the states of the prototype bioprocess are designed: an extended Luenberger observer and an asymptotic observer. Section 4 deals with the design of an observer-based estimator for the unknown kinetics of the process. In order to test the behavior of proposed strategies, numerical simulations are included in Section 5. Finally, the conclusions are presented in Section 6.

103

Proceedings of the 2010 IEEE

International Conference on Automation and Logistics August 16-20 2010, Hong Kong and Macau

978-1-4244-8376-1/10/$26.00 ©2010 IEEE

II. PSEUDO BOND GRAPH MODEL OF THE BIOPROCESS

In this section the micro-organisms growth process, combined with an enzyme-catalysed reaction will be modelled using pseudo Bond Graphs. The reaction scheme is [1], [11]:

XPXSXOS +→+→+ϕϕ 21

, (1)

In the reaction scheme (1), S is the substrate, O is the dissolved oxygen, X is the biomass, P is the product, and

21,ϕϕ are the reaction rates. The aerobic micro-organisms growth process combined with enzyme-catalyzed bioprocess described by the reaction scheme (1) is a prototype process, widely used in bioindustry [1], [3], [11]. Usually, this process takes place into Fed-Batch Bioreactors. In order to model this kind of processes, pseudo Bond Graph method is more appropriate because of the meaning of variables involved – effort (concentration) and flow (mass flow). This offers a flexible way to manage the material balances in terms of differential equations without losing the advantages of true Bond Graphs. By considering the reaction scheme (1), and the mass transfer through the FBB, the pseudo Bond Graph model of process (1) can be derived (for details concerning the Bond Graph bioprocess modeling procedure see [11], [12]). The pseudo Bond Graph model of this process is depicted in Fig. 1. The directions of half arrows in the Bond Graph correspond to the progress of reactions, going out from the components S and O towards X for the first reaction, and from S and X towards X and P for the second reaction. In Bond Graph terms, the mass balances of species are represented by four 0-junctions: 01,2,3,4,16 (mass balance for S), 06,7,8,9 (for O), 013,14,15,24,25 (for X), and 020,21,22 (for P ). The constitutive relations of these junctions are characterized by the equality to zero of the sum of flow variables corresponding to junction bonds; therefore, next relations are obtained:

0164321 =−−−− fffff , 09876 =−−− ffff , 02524151413 =−+−− fffff , 0222120 =−− fff .

Thus, the accumulations of species S, O, X, and P in the bioreactor are represented by bonds 2, 7, 14, and 21, respectively, and are modeled using capacitive elements C. The constitutive equations of C-elements are as follows:

( )dtffffCqCet∫ −−−== 164312222 )/1()/1( (2)

( )dtfffCqCet∫ −−== 9867777 )/1()/1( (3)

( )dtffffCqCet∫ −+−== 2524151314141414 )/1()/1( (4)

( )dtffCqCet∫ −== 222021212121 )/1()/1( (5)

where C2, C7, C14, and C21 are the parameters of C-elements: VCCCC ==== 211472 , with V the bioreactor volume (l).

The unreacted masses are modeled using resistive elements R represented by bonds 3, 8, 15, and 22.

0

C

TF

1 TF C0

Sf

R

R0

CTF

Sf

R

TF

1 C0

R

TF

TF TF

MR

MR

1

2

3 4

5

6

7

8

9

10

11 12 13

14

15

16

17 18

19

25 24

23

20 21

22

26

Fig. 1 Pseudo Bond Graph model of the prototype fed-batch bioprocess. The constitutive equations of R elements are: 333 / Ref = ,

888 / Ref = , 151515 / Ref = , 222222 / Ref = , where 3R , 8R ,

15R , and 22R are the parameters of the R-elements:

inFRRRR /1221583 ==== ; inF is the input feed rate (l/h). Mass flows of components entering the reaction are

modeled using two flow sources elements Sf1 and Sf6, and the transformer elements TF4,5, TF9,10, TF12,13, TF16,17, TF19,20, TF23,24, TF25,26 were introduced to model the yield coefficients. For the modeling of reaction rates we used two modulated two-port R-element, MR11,12, MR18,23. The first elements was introduced to model the reaction rate 1ϕ , and the second one for the reaction rate 2ϕ . From the constitutive relations of the two 1-junction elements, 15,10,11 and 117,18,19,26 we obtain:

11105 fff == , 26191817 ffff === , where 11f is directly proportional to the reaction rate 1ϕ and V, and 18f is proportional to the reaction rate 2ϕ and V.

From the constitutive relations of transformer elements we obtain the relations for the flows:

55,44 fkf = , 1010,99 fkf = , 1213,12

131 f

kf = , 1717,1616 fkf = ,

1920,1920 )/1( fkf = , 2324,2324 )/1( fkf = , 2626,2525 fkf =

with 5,4k , 10,9k , 13,12k , 17,16k , 20,19k , 24,23k , 26,25k being the transformers modulus, which are in fact the yield coefficients.

The signification of Bond Graph elements is: 2e - the substrate concentration S (g/l), 7e - the oxygen concentration O (g/l), 14e - the biomass concentration X (g/l), 21e - the product concentration P (g/l), ininS SFf =1 , ininOOFf =6 where inSF is the influent substrate flow (l/h), inOF - the influent oxygen flow (l/h), inS - the influent substrate concentration (g/l), inO - the influent oxygen concentration (g/l). Then, from (2)-(5) we will obtain the dynamical model:

104

VkVkSFSFSV ininin 217,1615,4 ϕ−ϕ−−= (6)

VkOFOFOV ininin 110,9 ϕ−−= (7)

VkVkVkXFXV in 224,23226,25113,12 /1/1 ϕ+ϕ−ϕ+−= (8)

VkPFPV in 220,19/1 ϕ⋅+−= (9)

Taking into account that the dilution rate rin tVFD /1/ == , with rt - medium residence time, and

24,2326,25 kk = , if we annotate 15,4 kk = , 217,16 kk = , 310,9 kk = ,

1FDSin = , 2FDOin = , the equations (6)-(9) can be written:

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⋅−⎥⎦⎤

⎢⎣⎡ϕϕ⋅

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

−−

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

2

1

2

1

3

21

00

01001

F

F

OPXS

D

k

kk

OPXS

dtd (10)

In the model (10), the state vector and the yield matrix can

be defined as: [ ] ,TOPXS=ξ T

kkkK ⎥⎦⎤

⎢⎣⎡−

−−= 01001

2

31 . Then

the model (10) can be compactly written as follows

FDK +ξ−ξϕ=ξ )( (11)

where [ ] [ ]TT FFF 2121 00,)()( =ξϕξϕ=ϕ . 1F represents the substrate rate supply and 2F the oxygen rate supply.

If the dissolved oxygen is not limited and the product has not inhibitory action, then the reaction rates can be modeled as

XSOPXSXSOPXS ⋅ν=ϕ⋅μ=ϕ )(),,,(,)(),,,( 21 (12)

with )(Sμ the specific growth rate (Monod type), and )(Sν the specific accumulation product rate (Haldane type) [15]:

SKSS

M +μ=μ

1

*)( , iM KSSK

SS/

)( 202

++ν=ν (13)

In relations (13), 21

, MM KK are the Michaelis-Menten

coefficients, iK is the inhibition coefficient and 0* , νμ are

the maximum specific rates [15], [17].

III. STATE OBSERVERS DESIGN

A. The Extended Luenberger Observer Often, several states of process (11) cannot be on-line measurable. Then, a state observer must be designed to obtain a good control strategy. A general class of observers for processes of form (11) is proposed by Bastin and Dochain [1]:

)ˆ()ˆ(ˆ)ˆ(ˆmmFDK ξ−ξ⋅ξΩ++ξ⋅−ξϕ=ξ (14)

where ξ is the estimated state vector, )ˆ(ξΩ is a gain matrix and mξ is the vector of measurable state variables:

ξ⋅=ξ Lm , with L a selection matrix. The dynamic of the

estimation error ξ−ξ= ˆe is

[ ] eLeDeKe ⋅⋅ξΩ−⋅−ξϕ−+ξϕ= )ˆ()ˆ()ˆ( (15)

It is clear that e = 0 is an equilibrium point of (15). The linear approximation around e = 0 can be easily obtained:

[ ] eLAe ⋅⋅ξΩ−ξ= )ˆ()ˆ( , 4ˆ

)()ˆ( IDKA ⋅−⎥⎦

⎤⎢⎣

⎡ξ∂ξϕ∂=ξ

ξ=ξ

(16)

If it is possible to impose desired values for the

eigenvalues of matrix [ ]LA ⋅ξΩ−ξ )ˆ()ˆ( by choosing the gain matrix, then system (11) can be made exponentially obser-vable and the observer (14) is an exponential observer [1], [3]. A necessary condition of exponential observability is that the observability matrix O [ ]TALALALL 32 )()()( ξ⋅ξ⋅ξ⋅= is a full rank matrix, i.e. rank(O) = 4 along the state trajectories. When the bioprocess is exponentially observable, is possible to implement an extended observer, for example an extended Luenberger observer. The design of this exponential observer consists in the choice of gain matrix such that the equilibrium point e = 0 of (15) to be asymptotically stable. Therefore, two conditions must be fulfilled [1]: (i) the matrix

[ ]LA ⋅ξΩ−ξ )ˆ()ˆ( and his derivative are bounded; (ii) the real

parts of eigenvalues of [ ]LA ⋅ξΩ−ξ )ˆ()ˆ( are strictly negative:

{ }[ ] ξ∀>δ<δ≤⋅ξΩ−ξλ ˆ,0,0)ˆ()ˆ(Re LAi (17)

Finally, the state estimator consists in the system (14) where the gain matrix is obtained using the conditions (i) and (ii), with the design parameters iλ .

Concerning the bioprocess described by (11)-(13), in most practical applications, the biomass concentration cannot be measured. For example, it can be considered that the measured states are P and O, and the unmeasured states are X and S:

[ ] [ ] ⎥⎦⎤

⎢⎣⎡==ξ=ξ 1000

0100,, LXSOP Tnm

Tm .

Under the hypotheses that D, Sin and F2 can be on-line measurable, the structure and coefficients of kinetics, and the yield coefficients are known, the equations of observer are

⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡−⎥⎦⎤

⎢⎣⎡Ω+

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−⎥⎦

⎤⎢⎣

⎡

νμ

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

−−

=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

OP

OP

F

DS

OPXS

DXSXS

k

kk

OPXS in

ˆˆ

00

ˆˆˆˆ

ˆ)ˆ(

ˆ)ˆ(

101001

ˆˆˆˆ

23

21

(18) T

⎥⎦

⎤⎢⎣

⎡

ξωξωξωξωξωξωξωξω=ξΩ

)ˆ()ˆ()ˆ()ˆ()ˆ()ˆ()ˆ()ˆ()ˆ(

42322212

41312111 (19)

By using the derivatives ⎥⎦

⎤⎢⎣

⎡

νν′μμ′=⎥

⎦

⎤⎢⎣

⎡ξ∂ξϕ∂

ξ 00)ˆ(ˆ)ˆ(00)ˆ(ˆ)ˆ()(

ˆ SXSSXS ,

where )ˆ()(ˆ

SdS

Sd

S

μ′=⎥⎦⎤

⎢⎣⎡ μ Ν

, )ˆ()(ˆ

SdS

Sd

S

ν′=⎥⎦⎤

⎢⎣⎡ ν Ν

, one obtains:

105

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−′−−′

−′=

DSkXSkDSXS

DSXSaa

A

0)ˆ(ˆ)ˆ(0)ˆ(ˆ)ˆ(00)ˆ(ˆ)ˆ(00

)ˆ(

33

1211

μμνν

μμξ (20)

with ,ˆ)ˆ(ˆ)ˆ( 2111 DXSkXSka −ν′−μ′−= )ˆ()ˆ( 2112 SkSka ν−μ−= . After some calculations, by analyzing the observability matrix, it can be concluded that the rank of O is maximum (i.e. 4) if condition 0)()()()( 33 ≠μ′ν−⋅ν′μ XSSkXSSk is achieved, or

0)()()()( ≠μ′ν−ν′μ SSSS ( 3k , X are positive) along the trajectories of the system. If the specific rates have the particular form (13), then the above condition becomes

0)(2211

2 ≠−⋅+⋅+ MMiM KKKSKS (21)

Because S and X are concentrations (positive values), the analysis of relation (21) shows that the rank condition is achieved only if

21 MM KK > . While conditions 0,0 ≠≠ XS are in general accomplished,

21 MM KK > is not universally valid. Concluding, an exponential observer of form (14) cannot be always designed for our process. In such situations is preferable the design an asymptotic state observer. Yet, if the particular values of coefficients allow the construction of the exponential observer, another difficult problem is the design of gain matrix (19). The conditions (i) and (ii) must be accomplished by imposing 4 eigenvalues with negative real part for the matrix of linearized estimation error system.

B. Design of the Asymptotic Observer When the system is not exponentially observable, a possibility is to design an asymptotic state observer, without the knowledge of kinetic reaction. The design is based on some useful changes of coordinates, which lead to a submodel of (11) independent of the kinetics. In order to achieve the change of coordinates, a partition of state vector is considered. This partition denoted ),( ba ξξ induces partitions of the yield matrix K: (Ka, Kb) and of rate vector F: (Fa ,Fb). We suppose that the state partition is chosen such that aK is full rank and

)()()dim( KrankKrank aa ==ξ . Then a linear change of coordinates (a diffeomorphism) can be defined [1], [3], [4]:

baCz ξ+ξ⋅= (22)

with z the auxiliary state vector and C the solution of matrix equation 0=+⋅ ba KKC . In the new coordinates, the model (11) can be rewritten as

ba

aaaaaa

FFCzDzFDCzK

+⋅+⋅−=+ξ⋅−ξ−ξϕ=ξ ),( (23)

The aim of the change of coordinates is that the dynamics of z is independent of the kinetics. z can be written as a linear combination of the vectors of measured and unmeasured states

nmm CCz ξ⋅+ξ⋅= 21 (24)

with C1 and C2 well defined matrices. If the matrix C2 is left invertible, the asymptotic observer equations for (11) derive from the structure of equations (23) and (24):

)ˆ(ˆ

)(ˆˆ

12 mnm

bbaa

CzC

QFQFCzDz

ξ−⋅=ξ

−+−⋅+⋅−=+

(25)

where TT CCCC 21

222 )( −+ = . The asymptotic observer (25) has good convergence and stability performances [1], [3]. For the process described by the model (11), the design of an asymptotic observer will be achieved by considering that the measured states are S and O, and the unmeasured states are X and P. In order to achieve the change of coordinates, the next partition is considered: [ ] [ ]T

bT

a OPXS =ξ=ξ , . This partition induces the correspondingly partitions:

⎥⎦⎤

⎢⎣⎡=⎥⎦

⎤⎢⎣⎡=⎥⎦

⎤⎢⎣⎡−=⎥⎦

⎤⎢⎣⎡ −−=

23

21 0,0,010,01 FFDSFkKkkK b

inaba

The submatrix aK is full rank, so the transformation (22)

can be defined. The new coordinates are [ ]Ta XS=ξ ,

baCz ξ+ξ⋅= , 1−−= ab KKC ⎥⎦⎤

⎢⎣⎡=

3

2120

//1k

kkk , with

OXkzPXkkSkz +=++= 322121 ,)/()/1( (26)

In the new coordinates the process model is written as

222211

21

,)/1()(,)()(

FDzzDSkDzzXSXDSDSXSkXSkS

in

in

+−=+−=μ=+−ν−μ−= (27)

The auxiliary state vector z can be written as a linear combination of [ ] [ ]T

nmT

m PXOS =ξ=ξ , , with C1 and C2

obtained from (24) and (26): ⎥⎦⎤

⎢⎣⎡=⎥⎦

⎤⎢⎣⎡= 0

1/,00/1

3

212

3

11 k

kkCkkC .

Now the equations of asymptotic observer can be written:

)ˆ(ˆ

)/1(ˆˆ

ˆˆ

12

2

2

2

1

2

1

mnm

in

CzC

FDSk

zzDz

zdtd

ξ−=ξ

⎥⎦⎤

⎢⎣⎡+⎥⎦

⎤⎢⎣⎡−=⎥⎦

⎤⎢⎣⎡

+

(30)

where z , nmξ are the estimations of z , nmξ , and 122−+ = CC

⎥⎦⎤

⎢⎣⎡

−= )/(1/10

321

3kkk

k . Then the detailed equations of (30) are:

)ˆ))(/(()/1(ˆˆ)ˆ)(/1(ˆ

ˆˆ,)/1(ˆˆ

232111

23

222211

OzkkkSkzP

OzkX

FzDzSDkzDz in

−−−=

−=

+⋅−=⋅+⋅−=

(31)

The inspection of equations (31) shows that indeed the knowledge of reaction kinetics is not necessary for estimation of state variables X and P. For the implementation of the observer the knowledge of yield matrix K is required.

106

IV. ESTIMATION OF UNKNOWN KINETICS

In practice, the analytical models of the reaction rates (12) or of specific rates (13) are difficult to obtain. Therefore, these uncertain rates need to be on-line estimated. The design of the estimation strategies will be done considering that the model of bioprocess fulfils next hypotheses [1], [14]: H1. All state variables are measurable in real-time (otherwise, a state observer is needed); H2. The vector of feed rate and the dilution rate are measurable; H3. The matrix K is known. The basic idea behind the design of an observer-based estimator (OBE) is to use a state observer, not for state estimation, but in order to provide the information needed for updating the estimates of parameters. The on-line estimation algorithm for the bioprocess (11) can be written as [1], [14]:

)ˆ()(ˆ)(ˆ ξ−ξ⋅Ψ−+ξ⋅−ρ⋅ξ⋅=ξ FDtHK (32)

)ˆ()]([ˆ ξ−ξ⋅Γ⋅ξ⋅=ρ THK (33)

In (32) and (33), ρ is the on-line estimate of the unknown vector of parameters. The first equation is a state observer, used for updating the estimate ρ , and not for state estimation.

The update is generated by the estimation error )ˆ( ξ−ξ=ε ,

where ξ is the on-line estimation of the state vector. The error ρ−ρ ˆ is directly reflected by the estimation error ε . The

matrix Ψ is a gain matrix, and the injection matrix Γ is chosen such that ΓΨ+ΓΨT is negative defined. The estimator (32), (33) was designed by taking into consideration a full state vector. However, it is possible to design a reduced order OBE, by selecting a part of state-space equations, with the obvious condition that this subsystem contains the unknown kinetics which will be estimated. Because in practice the substrate and dissolved oxygen concentrations are measured, we will use these states in order to develop a reduced OBE: [ ]T

R OS=ξ . The equations of reduced estimator are derived from (34), (35):

)ˆ()(ˆ)(ˆRRRRRR FDtHK ξ−ξ⋅Ψ−+ξ⋅−ρ⋅ξ⋅=ξ (34)

)ˆ()]([ˆ RRTHK ξ−ξ⋅Γ⋅ξ⋅=ρ (35)

The choice of design matrices Ψ and Γ must be done such that the algorithm to be stable and convergent. The properties of stability and convergence for this estimator have been discussed at length in [1], [14]. A typical choice for the matrices Ψ and Γ is of diagonal form:

}{2,1

iidiag ψ−=Ψ

=, }{

2,1j

jdiag γ=Γ

=, +ℜ∈γψ ji , (36)

By using (11) and (34)-(36), two OBEs can be achieved:

(i) One OBE for the unknown specific rates: Tttt ])()([)( 21 ρρ=ρ Ttt ])()([ νμ= . In this case ⎥⎦

⎤⎢⎣⎡=ξ XXH 0

0)(

and the detailed equations of the estimator are as follows:

)ˆ(ˆ

)ˆ()ˆ(ˆ

)ˆ(ˆˆ

)ˆ(ˆˆˆ

12

2311

223

121

SSXk

OOXkSSXk

OOFDOXkO

SSDSDSXkXkS in

−γ−=ν

−γ−−γ−=μ

−ψ++−μ−=

−ψ++−ν−μ−=

(37)

(ii) If all reaction rate vector is considered as unknown, then an OBE for Tttt ])()([)( 21 ρρ=ρ Ttt ])()([ 21 ϕϕ= can be obtained (with 1)( =ξH ):

)ˆ(ˆ

)ˆ()ˆ(ˆ

)ˆ(ˆˆ

)ˆ(ˆˆˆ

122

23111

2213

12211

SSk

OOkSSk

OOFDOkO

SSDSDSkkS in

−γ−=ϕ

−γ−−γ−=ϕ

−ψ++−ϕ−=

−ψ++−ϕ−ϕ−=

(38)

Please note that the observer (37) needs the measurements of S and O, but also those of X (if the measurements of X are not available, then the asymptotic observer (31) can be used). The estimator (38) needs only the measurements of S and O for the reconstruction of unknown reaction rates.

V. SIMULATION RESULTS

Several numerical simulations were performed in order to analyze and compare the behavior of proposed estimators. The bioprocess parameters used in simulation are as follows [13]:

./2,/15,2.0,2,1

,/10,6,/20,/1,1

21

231

10

1*21

lhgFlgShDkkk

lgKhlgKlgKh

in

iMM

======

==ν===μ−

−−

Fig. 2 depicts the time evolutions of concentrations obtained by using model (11). The kinetic rates for this simulation are of form (12), (13) – these kinetic expressions are introduced only for simulation; therefore these models are not used in the process of observers design. First, the on-line OBE (37) was implemented. The main goal was to reconstitute the time evolution of specific rates μ and ν . The biomass concentration used by the estimator was provided by the asymptotic observer (31).

0 5 10 15 20 250

1

2

3

4

5

6

7

8

9

Time (h)

(g/l)

X

S

P

O

Fig. 2 Time profile of the concentrations for the prototype bioprocess.

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The tuning parameters were set to 1021 =ψ=ψ and

121 =γ=γ . In order to test the robustness of proposed observers, the simulations were performed for noisy data of S and O (vitiated by an additive Gaussian noise, with zero mean and amplitude equal to 5% of free-noise values). Also, a parametric disturbance for the dilution rate was introduced (75% of nominal value, for 5 h). Next, only the simulation results obtained for noisy data will be presented. Fig. 3 shows the estimates X and P versus their “real” profiles. Fig. 4 presents the profile of the estimated and real specific rates.

0 5 10 15 20 250

1

2

3

4

5

6

7

8

9

Time (h)

(g/l)

X X

P

P

Fig. 3 Evolution of concentrations X, P and of their estimates.

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

1.2

Time (h)

(h-1)

μ

μ

ν

ν

Fig. 4 Profiles of the specific rates and of their estimates.

0 5 10 15 20 250

0.5

1

1.5

2

2.5

3

3.5

4

Time (h)

(g/lh)

1ϕ

1ϕ

2ϕ

2ϕ

Fig. 5 Evolution of the reaction rates and of the corresponding estimates.

Second, the OBE (38) was implemented, in the same conditions as previously, but with 1021 =ψ=ψ and

1521 =γ=γ . The profiles of estimated and real reaction rates are depicted in Fig. 5, these results being obtained in the presence of parametric disturbance and noisy measurements.

VI. CONCLUSION

In this work, some modeling issues and on-line estimation strategies for the state and unknown kinetics of a prototype fed-batch bioprocess were widely analyzed. The nonlinear model of this bioprocess was obtained by using a novel modeling procedure, based on pseudo Bond Graph technique. In order to overcome problems such as the modeling uncertainties and the lack of on-line measurements, two state observers and two kinetics estimators were designed and discussed. The asymptotic state observer was easier to design and implement. The simulation results showed a good behavior of this observer. Two OBEs were designed for on-line estimation of unknown kinetics: specific rates and reaction rates, respectively. From the time evolution diagrams it can be seen that the OBEs provides good estimates. The measurement noise induces some noisy estimates, but the noise effect is limited, and can be reduced for lower values of tuning parameters. The choice of these values represents a compromise between a good estimation and the noise rejection. The advantages of OBEs are the simplicity of design, the good convergence and stability properties, and the accuracy of estimates.

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