International Conference on Control, Automation and Systems 2008 Oct. 14-17, 2008 in COEX, Seoul, Korea

1. INTRODUCTION

Nowadays, the development of advanced control strategies for bioprocesses is hampered by major difficulties [1-3]: nonlinearity and nonstationarity, the uncertainty of the process parameters, the absence of the cheap and reliable instrumentation for the biological state variables, i.e. the substrates, biomass, and product concentrations.

Several estimation strategies have been developed to provide accurate on-line estimations of state variables. Presently, two classes of state observers for bioprocesses can be found [1-3]. The first class of observers (including classical observers like Luenberger and Kalman observers, nonlinear observers) is based on a perfect knowledge of the model structure. A second class, called asymptotic observers, is based on the idea that the process uncertainty lies in the kinetics models. The design is based on mass and energy balances without the knowledge of kinetics being necessary.

Another important issue is the estimation of parameters and especially of kinetic rates (the so-called kinetics of the bioprocess). This problem can be solved using the so-called “software sensors”, which are combinations between a hardware sensors and software estimators [1, 3]. A well-known technique is the Bastin and Dochain approach based on the adaptive systems theory [1, 3]. This method is useful, but in some cases, when many reactions are involved, the implementation requires the tuning of too many parameters. Another possibility is to design a high gain estimator [4, 8]. The gain expression of these observers involves a single tuning parameter whatever the number of components and reactions.

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

In order to control the bioprocesses, some specific strategies were developed, such as sliding mode control [6], adaptive control [1-3], vibrational control, fuzzy and neural strategies and so on.

This paper, using the results of [2, 7-8], deals with the design of estimation and adaptive control strategies for a lipase production bioprocess that is carried out in a Fed-batch Bioreactor (FBB). For FBB, a typical problem is that of generating the substrate feed rate profile to optimize a performance criterion [1-2]. This optimal control problem can be solved when a state-space representation with known reaction kinetics is available.

The adaptive control can constitute a valuable alternative when the structure of the kinetics and the kinetic parameters are imprecisely known. The objective of FBB control is to maximize the final product quantity, even if some state variables are not measurable and the kinetics is imprecisely known, by using a closed-loop adaptive controller, as a suboptimal substitute of an open-loop optimal profile. The advantage is the rejection of disturbances.

The organization of the paper is as follows. In Section 2, the lipase production process is presented and the nonlinear dynamical model is analyzed. Section 3 deals with the design of some estimation algorithms. Two on-line state estimation strategies are proposed: an extended Luenberger observer and an asymptotic observer. Also, the unknown kinetic parameters of the bioprocess are estimated by using a regressive parameter estimator and a high-gain observer. In Section 4, an exact feedback linearizing control law and then adaptive linearizing control strategies are developed such that the final lipase production is maximized. The performance and the behavior of the estimation and control strategies are illustrated by numerical simulations. Finally, concluding remarks are collected in Section 5.

Estimation and Adaptive Control of a Fed-batch Bioprocess Dan Selişteanu, Emil Petre, Constantin Marin and Dorin Şendrescu

Department of Automatic Control, University of Craiova, Romania (Tel: +40-251-438198; E-mail: {dansel, epetre, cmarin, dorins}@automation.ucv.ro)

Abstract: This paper deals with estimation and adaptive control strategies for a biotechnological process, which is in fact a lipase production process that takes place inside a Fed-batch Bioreactor. The lipase production process is highly nonlinear and, furthermore, the available on-line measurements are lack and the reaction kinetics is not perfectly known. On-line state estimation strategies based on extended Luenberger observer and asymptotic observer approach are derived. The unknown kinetic parameters of the bioprocess are estimated by using nonlinear techniques, such as regressive parameter estimator and high-gain observer. The control goal is to maximize the lipase production by controlling the substrate feeding rate. A nonlinear feedback control law is obtained by means of exact linearization technique. By coupling this controller with the parameter estimation algorithms, a nonlinear adaptive controller is obtained. Numerical simulations are included in order to test the behavior and the performance of the proposed estimation and control strategies. Keywords: Nonlinear systems, Adaptive control, Estimation, Biotechnology.

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2. DYNAMICAL MODEL OF THE LIPASE PRODUCTION BIOPROCESS

A bioreactor is a tank in which several biological

reactions occur in a liquid medium [1]. The bioreactors can operate in three modes: continuous mode, fed-batch mode and batch mode [1, 3]. A FBB initially contains a small amount of substrates and micro-organisms and is progressively filled with the influent substrates. When the FBB is full the content is harvested.

The lipase is an enzyme able to split fats and to synthesize glycerides, so the large scale production of lipase is very useful in many applications. This process usually takes place inside FBB; the bioprocess is widely discussed in [2] and a nonlinear model is designed:

( )( ) ( ) ( )( )

( ) ( ) ( )( )( )( )⎪

⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

+μ=ν=

μ−ν−μν=

μ=

+⋅μ−η=

+η−=

.

,,

2

21

2

2212

11

XbSaCerXLL

LSLXSL

XSX

SYSSS

FXSS

inexex

ininexpin

(1)

1S (g/l), 2S (g/g), X (g/l), inL (u/mg), exL (u/ml) are the concentrations of external substrate, internal substrate, biomass, internal and external lipase, respectively, and Cer is the carbon dioxide excretion rate (a, b are excretion parameters) [2, 7]. F is the feeding rate of substrate (when 0=F , the operating mode is batch) and Y is a yield coefficient. η is the absorption rate of external substrate 1S , μ is the specific growth rate of biomass X, pν is the production rate of internal lipase inL and exν is the excretion rate of external lipase exL . The form of these rates is of Monod type for exνμη ,, , but very complex for pν - Haldane law plus influence of the specific growth rate of biomass [2]:

( ) ( )

( ) ( )( ) ( )

( )

( ) .

,//

/,,

,,

*

2211

1*

1

22

2*

211

1*

1

inex

inexinex

ip

pp

MM

LKL

L

SXSKXSK

XSXS

SKS

SSK

SS

+ν

=ν

μ⋅++

⋅ν=μν

+μ

=μ+

η=η

(2)

All coefficients in (1), (2) are strictly positive. expMM KKKK ,,, 21 are Michaelis-Menten constants,

**** ,,, exp ννμη are maximum specific rates, and iK is an inhibition constant. It must be noted that the model (1), (2) is valid for biomass and substrate concentrations between 0 g/l and 8 g/l [2, 7].

The strongly nonlinear character of this model is given by the reaction kinetics. In many situations, the yield coefficients, the structure and the parameters of the reaction rates are partially known or unknown.

The main purpose of FBB control is to maximize the final lipase product quantity. This goal can be achieved through an optimal control of the bioreactor, i.e. the

calculation of the optimal profile for the feeding rate, which is a common solution for FBB control [1, 3]. This optimal control is unsatisfactory when the kinetics is imprecisely known (the nonlinear expressions of the kinetic parameters (2) are not totally realistic; in fact, these parameters are imprecisely known). Two possible suboptimal alternatives are the sliding mode control [6] and the adaptive control [2, 7].

The dynamics of concentrations of external substrate, internal substrate and of the biomass (first three equations of (1)) are significant for the evolution of the internal and external lipase concentrations (given by the equations four and fifth of (1)) [2, 7]. Thus, the equations describing the dynamics of 21 , SS and X are used for the control design. Then, we consider the first three equations, compactly written as:

( ) ( ),ξ=ξ ft (3)

with ( ) ( ) ( ) ( )( ) ( )[ ] ,3222131TYuf ξξμξ+ξμ−ξη+ξξη−=ξ

[ ] [ ]TT XSS 21321 =ξξξ=ξ the state vector, and u = F (feeding rate is the control input).

In order to implement useful control strategies, it is necessary to design state observers and identification methods for the unknown kinetics.

3. ESTIMATION STRATEGIES 3.1 On-line state estimation strategies

In the lipase production process, often the only state variable that is on-line available is the biomass concentration X (obtained via the measurements of Cer), and it is necessary to reconstitute the external and especially the internal substrate concentrations. In order to estimate the state variables 1S and 2S , two solutions are discussed: an extended Luenberger observer and an asymptotic observer.

A general class of observers for bioprocesses is proposed by Bastin and Dochain [1]:

( ) ),ˆ()ˆ(ˆˆ11 ζ−ζξΩ+ξ=ξ f (4)

where ξ is the estimated state vector, )ˆ(ξΩ is a gain matrix and 1ζ is the vector of measurable state variables: ξ=ζ L1 , with L a selection matrix. The estimation error is ξ−ξ= ˆe .

In order to design an extended Luenberger observer, firstly we deduced that the system (3) is exponentially observable [1, 3]. Then, the design of the observer consists in the choice of gain matrix )ˆ(ξΩ such that the equilibrium point e = 0 of the linearized error system is asymptotically stable [1, 3].

The dynamic of the estimation error is:

.)ˆ()ˆ()ˆ( eLfefe ξΩ−ξ−+ξ= (5) It is clear that e = 0 is an equilibrium point of (5).

Then, the linear approximation around e = 0 can be easily obtained:

( ) ,)ˆ()ˆ()( eLAte ξΩ−ξ= ξ=ξ

Δ

⎥⎦

⎤⎢⎣

⎡ξ∂ξ∂=ξ

ˆ

)()ˆ( fA . (6)

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If it is possible to impose desired values for the eigenvalues of matrix LA )ˆ()ˆ( ξΩ−ξ by choosing the gain matrix, then the system (3) is exponentially observable and the observer (4) is an exponential observer [1]. A necessary condition of exponential observability is that the observability matrix

[ ]TnLALALALO 12 )()()( −ξξξ= (7)

is a full rank matrix: rank(O) = n along the state trajectories, where n is the dimension of state vector.

In the case of our lipase production bioprocess, X=ξ=ζ 31 , [ ]TL 100= . After some straightforward

calculations, the matrix ( )[ ]3,1,

ˆ=

ξ=jiijaA can be

obtained, and then the observability matrix is derived:

( ) ( )( ) ( ) ( ) ( ) ( )( ) ( )

.0100

2333222332132

3332

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

ξξ+ξξξξξξ=

aaaaaaaaO

Thus we obtain that ( ) ( ) ( ) 0det 21232 ≠ξξ−= aaO (for

03 >=ξ X ); therefore rank(O) = 3 and the necessary condition of exponential observability is achieved.

Now is possible to try the implementation of an extended observer for the system (3). The design of the extended Luenberger observer consists in the choice of gain matrix )ˆ(ξΩ such that the equilibrium point e = 0 of (6) is asymptotically stable. Therefore, the gain matrix must to obey two conditions [1]: (i) the matrix

LA )ˆ()ˆ( ξΩ−ξ and his derivative are bounded; (ii) the

real parts of eigenvalues of LA )ˆ()ˆ( ξΩ−ξ are strictly

negative: ( ) ( ){ }[ ] 3,1,ˆ,0,0ˆˆRe =ξ∀>δ<δ≤ξΩ−ξλ iLAi .

The characteristic polynomial of LA )ˆ()ˆ( ξΩ−ξ is

( ) ( )( )( ) .ˆˆdet 322

13 α+λα+λα+λ=ξΩ−ξ−λ LAI

The gain matrix (which in our particular case is a

vector) has the structure ( ) ( ) ( ) ( )[ ]Tξωξωξω=ξΩ ˆˆˆˆ321 .

Using the connection between 3,1, =α ii and the

eigenvalues 3,1, =λ ii , the components of )ˆ(ξΩ are:

( )( )( )[ ]

[ ( )]322232111332213322113213221

1

3332211221131322132

2

3213322113

1

1

ω−ω+++λλλ=ω

−ω++−λλ+λλ+λλ=ω

λ+λ+λ−++=ω

aaaaaaaaaaa

aaaaaa

aaa(8)

The on-line extended Luenberger state estimator consists of the system (4) where the gain matrix is given by (8), with the design parameters 3,1, =ℜ∈λ − ii .

The extended Luenberger observer is based on a perfect knowledge of the model structure, which is not a true assumption. Also, the design of the observer is quite complicated. In order to overcome these disadvantages, it is possible to design an asymptotic observer, without the knowledge of the process kinetics

being necessary. The asymptotic observer will be designed for the estimation of the internal substrate concentration 22 S=ξ , which it is most difficult to measure in practice. The design of an asymptotic observer for bioprocess (3) was based on some useful changes of coordinates, which leads to a submodel of (3) independent of the kinetics. For instance, if we define an auxiliary variable as

32 )( ξ+ξ= Yz , (9)

with the dynamics 31)( ξξη=z , then the estimate of the internal substrate concentration is Yz −ξ=ξ 32 /ˆ . The equations of the asymptotic observer are:

./ˆ,)( 3231 Yzz −ξ=ξξξη= (10)

The dynamics of auxiliary state variable is independent of the kinetics. The estimations of 2ξ obtained using this asymptotic observer can be used in the control laws. The asymptotic observer (10) is very simple and it has good convergence and stability performance [1, 3]. 3.2 Parameter estimation algorithms

Two parameter estimation algorithms are proposed. The kinetic parameters which are estimated are the absorption rate and the specific growth rate. These parameters are modeled by imprecisely known nonlinear functions (see equations (2)). The estimation problem is based on the model equations (3). The absorption and the specific growth rates are considered time-varying parameters which are on-line estimated. We suppose that all states are measurable or obtained from estimations. The kinetic expressions of the absorption and specific growth rates from (2) are introduced only for simulation, therefore these models are not used in the process of the observers' design.

The first implemented algorithm is a regressive parameter estimator. The equations of this estimator are based on the least squares technique [1]. The estimator will use only the first and the third equations of the model (3), which can be parameterized in this way:

( ) ( ) ,rrrr FtGK +θξ=ξ (11)

where [ ]Tr XS1=ξ is a reduced state vector of the

system, ⎥⎦⎤

⎢⎣⎡−= 10

01rK , ⎥⎦

⎤⎢⎣⎡= XXG 0

0 , ⎥⎦⎤

⎢⎣⎡= 0FFr , and

( ) ( ) ( )[ ] ( ) ( )[ ]TT ttttt μη=θθ=θ 21 is the vector of time-varying unknown parameters.

The equations of the regressive parameter estimator for the system (11) are [7]:

( ),0)0(,)(

ˆ)0(ˆ,ˆ)(ˆ)(

)()(

0

00

00

>Γ=ΓΓλ+ΓΓΨΨ−=Γ

θ=θθΨ−Ψ−ξΓΨ=θ

+ωξ+Ψω−=Ψ

ξ+Ψω−=Ψ

T

Tr

rr

rrTT

dttd

dttd

Ft

GKt

(12)

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where ( ) ⎥⎦⎤

⎢⎣⎡

ψψ−=Ψ

2

10

0t is the regressor matrix, θ is

the estimation of θ , and { }idiag γ=Γ , ( ) +ℜ∈γ 0i

2,1=i . The forgetting factor ( )1,0∈λ and +ℜ∈ω are the design parameters. The stability and convergence properties of the regressive parameter estimator are widely discussed and proven in [1, 3]. Even if this estimator is relatively difficult to implement in practice, the algorithm (12) provides good estimates for the unknown kinetics.

The second algorithm is based on a high-gain approach [4, 8]. The design of high gain observers is done in [4], with some assumptions regarding the rank of the matrix rK and the boundedness of )( rG ξ diagonal elements’ away from zero. For the model (11), the yield matrix rK is of full rank. This assumption is true for our model, and for the general case is a generic property. We shall suppose that all state variables are measured (contrarily, a state estimator can be used). With these assumptions fulfilled, the observer equations for the reduced system (11) are [4, 8]:

),ˆ(2)ˆ(ˆrrrrrr FGK ξ−ξρ−+θξ=ξ (13)

[ ] ).ˆ()ˆ(ˆ 12rrrr GK ξ−ξ⋅ξ⋅⋅ρ−=θ

− (14)

The high-gain observer (13), (14) is in fact a copy of the bioprocess model, with a corrective term. The tuning of this observer is very simple because a single parameter 0>ρ is involved. It can be seen that the estimator needs the measurements of 1S and X . Note that in (13) and (14) rξ is an “estimate” of the state vector, provided by the algorithm in order to be compared with the real state, and the resulting error to be used for the production of the estimates of unknown kinetic parameters.

3.3 Simulation results

In order to test the performance and the behavior of the proposed estimation algorithms, several simulation experiments were performed. The values of the bioprocess parameters are the following [2, 7-8]:

( )

.g/g16.1u/mg;5.19;h09.4

g/g;2.22g/g;26.0mol/g;0185.0;hmol/g000216.0u/mg;123g/l;25.0

;h25.0g/l;11.0;h21.0

1*

1*2

1*1

1*

===ν

===

==ν=

=μ==η

−

−

−−

YK

KKabK

K

exex

ip

pM

M

The bioprocess evolution was examined in open loop. This simulation is considered in order to analyze the lipase production and the performance of state observers and of parameter estimators when the control law is missing. The following simulation cases are considered:

1) The extended Luenberger estimator (4), (8) was implemented, with the design parameters ,11 −=λ

,82 −=λ 01.03 −=λ . The "measured" variable X=ξ3 is vitiated with 3% additive white noise. The estimated state variables 2211

ˆˆ,ˆˆ SS =ξ=ξ are compared with the

simulated state values provided by model equations (1). The asymptotic observer (10) was also implemented in order to reconstitute the internal substrate concentration.

Fig. 1 depicts the extracellular lipase concentration (open loop); from this figure it can be seen that after the consumption of external substrate, the growth of external lipase is limited. Fig. 2 depicts the internal substrate concentration versus its estimate for the both observers (up – the Luenberger observer, and down the asymptotic observer). The simulations show that the estimation algorithms are quite good. It can be seen that the measurement noise induces some noisy estimates of

2S . From Fig. 2 it can be observed that the convergence of the asymptotic observer seems to be better than the convergence of the extended Luenberger estimator. Several other simulations prove that the estimations of the external substrate concentration 1S are very good, and the effect of the noise is small.

2) Both regressive parameter estimator (12) and high- gain observer (13), (14), respectively were implemented and simulated, in the same conditions. The design parameters of the regressive estimator are: ,6.3=ω

,1.01 =γ ,52 =γ ,95.0=λ and the single tuning parameter of the high-gain observer is set to 5=ρ .

0 5 10 15 20 250.5

1

1.5

2

2.5

3

3.5

4

Time (h)

(u/ml)

L ex

exL

Fig. 1 External lipase concentration (open loop).

0 5 10 15 20 250

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Time (h)

(g/l)

S

S 2

2

0 5 10 15 20 250

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Time (h)

S 2

S 2

(g/l)

S2

LuenbergerS 2ˆ

S2

asymptoticS 2ˆ

Fig. 2 Time profiles of 2S and 2S .

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0 5 10 15 20 250

0.05

0.1

0.15

0.2

0.25(h-1)

η

regη

Time (h)

hgη

Fig. 3 Evolution of absorption rate and its estimates.

0 5 10 15 20 250

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16(h-1)

μ

regμ

Time (h)

hgμ

Fig. 4 The specific growth rate and the estimates. Fig. 3 depicts the absorption rate η (solid line) and

its estimates ( regη - the regressive estimator, dotted line; hgη - the high-gain observer, dashed line). Fig. 4 shows the specific growth rate μ (solid line) and the related estimates ( regμ - the regressive estimator, dotted line; hgμ - the high-gain observer, dashed line).

The simulation shows that the convergence of the high-gain observer is better than the convergence of the regressive estimator, but the high-gain estimates are affected by the noise (especially μ ).

4. CONTROL STRATEGIES

4.1 Control laws design

In FBB control, a typical problem is that of generating the substrate feed rate profile to optimize a performance criterion [1-3, 6]. For the lipase production bioprocess, the main control objective is to maximize the final lipase product quantity. This goal can be achieved through an optimal control, i.e. the calculation of the optimal profile for the feeding rate. This optimal control is unsatisfactory when the kinetics is imprecisely known. A possible suboptimal alternative is the adaptive control [2, 7]. The control strategy consists in finding out which process variables are to be regulated in order to maximize the lipase production. From the detailed analyze of the process, we have two possible strategies: to maintain constant the ratio

XS /1 or to maintain constant above a reference value the concentration 1S (for more details see [2, 7]). In

this section an exact linearizing control law is obtained when the output is 1S . Then, an adaptive version of this law is implemented, considering that some state variables and the absorption rate are unknown, and using the estimation algorithms previously described.

The exact linearizing control law for the model (3) is obtained in a classical three steps strategy (see [5] for general point of view; [1-3, 6] for bioprocess control). The control purpose for the fed-batch bioreactor is that the external substrate concentration ( ) ( ) ( )tStty 11 =ξ= to track the desired substrate trajectory ( ) ( )tSty *

1* = , with the feeding rate as control action: ( ) ( )tFtu = .

First, from (3) one obtains an input-output model for the bioprocess:

( ) .3 uty +ηξ−= (15)

Second, we consider a stable and linear reference model for the tracking error yy −* :

( ) ( ) .0,0** >β=−β+− yyyydtd (16)

Finally, in the third step, the exact linearizing feedback control law is obtained by calculus of ( )tu such that the input-output model (15) has the same behavior as the reference model (16):

( ) ( ) ( ) ( ),/ *13

**3 yyyKyyytu M −β++ξη=−β+ηξ= (17)

where we consider .* consty = The exact linearizing control (17) can be

implemented and can ensure the achievement of the control goal only if the concentrations 11 Sy =ξ= and

X=ξ3 are on-line measurable. Also, it is necessary to know the absorption rate η . Contrarily, the utilization of the state observers and parameter estimators is needed. If the estimations provided by the estimators are used in the exact linearizing control law, adaptive versions of the nonlinear law are obtained.

For example, if the estimations of the absorption rate η=θ ˆˆ

1 provided by the regressive parameter estimator (12) are used in the linearizing control law (17), an adaptive version of this law is obtained as follows:

( ) ( ).ˆ *3 yytu −β+ξη= (18)

The same form of the adaptive control law is obtained if the high-gain observer (13), (14) is used instead of the estimator (12).

When the output of the system cannot be measured (i.e. the state variable 11 S=ξ ), the state observer (4), (8) can be implemented, and the following adaptive control law is obtained:

( ) ( ).ˆˆ *3 yytu −β+ξη= (19)

In this case, the entire adaptive control algorithm consists of the equations (4), (8), (12) and (19).

Remark. There are several problems regarding the stability and convergence properties of the overall controlled system (bioprocess, exact linearizing controller, adaptive law); these issues are widely discussed for the general class of bioprocesses in [1, 3].

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4.2 Simulation results Various simulations were performed in order to test

the performance and the behavior of the controllers, but here we present the following two situations:

1) The exact linearizing law (17) was implemented, with 5=β . The closed loop system was tested for a step profile of external substrate reference. Firstly, *y is set to 0.6 g/l; then the reference is set to the optimal value g/l8.0* =y (the profile ensures the maximization of lipase production – see [2, 7]). In order to test the robustness properties of the system, the measurements of X are influenced by a 3% additive noise, and a parametric disturbance of η occurs twice (3 h each time) in the simulation time interval. The simulation for the closed-loop evolution is ended when X reach the limit value 8 g/l; after this time interval, the fed-batch operation is over and the external lipase is collected.

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9 *1

* Sy =(g/l)

Time (h)

exy ady

Fig. 5 Evolution of the output versus reference.

0 5 10 15 20 250

0.5

1

1.5

2

2.5 (g/lh)

Time (h)

exu adu

Fig. 6 Control input (exact and adaptive case).

0 5 10 15 20 250

20

40

60

80

100

120 (u/ml)

Time (h)

exL

Fig. 7 The profile of the external lipase (closed loop).

2) In the same conditions as in the previous case, the adaptive control law version (19) was implemented, using the Luenberger estimator for the external substrate concentration and the regressive estimator for the absorption rate, which is considered unknown.

Fig. 5 shows the output for the exact case (which can be considered as a benchmark, because all the states and parameters are known) and for the adaptive case. Fig. 6 depicts the input for the two cases (exact/adaptive). Finally, in Fig. 7 it can be seen that the production of the lipase (both for exact or adaptive case) is bigger than the production obtained in open loop (Fig. 1).

5. CONCLUSION

In this paper, some estimation and control strategies

were developed for a lipase production fed-batch process. Two state observers are designed: a Luenberger estimator and an asymptotic observer. The unknown kinetics is estimated using a regressive parameter estimator and a high-gain observer. The adaptive control laws are based on exact linearizing control law coupled with estimation algorithms. The adaptive controllers provide for the bioprocess the exponentially increase of lipase concentration. The main goal of feedback control is not to stabilize the process globally, but to keep an unstable nearly optimal trajectory under control. This fact is explainable because a FBB operates during a finite time, and the aim is to accumulate the synthesis product. The simulation results show that the adaptive controllers maintain a high level of the lipase production and can cope with noise and disturbances.

ACKNOWLEDGMENT

This work was created in the frame of the research

project CNCSIS ID 786, no. 358/2007, Romania.

REFERENCES [1] G. Bastin and D. Dochain, On-line estimation and

adaptive control of bioreactors, Elsevier, 1990. [2] S. Charbonnier and A. Cheruy, “Estimation and

control strategies for a lipase production process,” Contr. Eng. Practice, Vol. 4, pp. 1521-1534, 1996.

[3] D. Dochain (Ed.), Automatic control of bio- processes, ISTE Publ. and Wiley & Sons, 2008.

[4] M. Farza, K. Busawon and H. Hammouri, “Simple nonlinear observers for on-line estimation of kinetic rates in bioreactors,” Automatica, Vol. 34, No. 3, pp. 301-318, 1998.

[5] A. Isidori, Nonlinear Control Systems - The Third Edition, Springer-Verlag, Berlin, 1995.

[6] D. Selişteanu, E. Petre and V. Răsvan, “Sliding mode and adaptive sliding-mode control of a class of nonlinear bioprocesses,” Int. J. of Adapt. Contr. and Signal Proc., Vol. 21, pp. 795-822, 2007.

[7] D. Selişteanu and E. Petre, “On nonlinear adaptive control of a lipase production process,” Proc. of 12th Int. Conf. on Contr. Syst. and Comp. Science, Vol. 1, pp. 384-390, May 26-29, 1999, Bucharest.

[8] D. Selişteanu, D. Popescu and C. Barbu, “On-line state estimation and identification of a fed-batch bioprocess,” Proc. of 9th WSEAS Int. Conf. on Math. and Comp. Methods in Science and Eng., pp. 85-90, November 5-7, 2007, St. Augustine.

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