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Biochemical Engineering Journal 91 (2014) 23–36

Contents lists available at ScienceDirect

Biochemical Engineering Journal

jo ur nal home page: www.elsev ier .com/ locate /be j

egular Article

icrobial production of enzymes: Nonlinear state and kinetic reactionates estimation

an Selis teanua,∗, Sihem Tebbanib, Monica Romana, Emil Petrea, Vlad Georgeanuc

Department of Automation, Electronics and Mechatronics, University of Craiova, A.I. Cuza 13, Craiova 200585, RomaniaDepartment of Automatic Control, SUPÉLEC, Plateau de Moulon, 3 rue Joliot Curie, Gif sur Yvette 91192, FranceFaculty of General Medicine, University of Medicine and Pharmacy “Carol Davila” Bucharest, Eroilor Sanitari 8, Bucharest 050474, Romania

r t i c l e i n f o

rticle history:eceived 12 April 2014eceived in revised form 25 June 2014ccepted 11 July 2014vailable online 21 July 2014

eywords:nzymesipaseodellingonlinear estimationbserversynamic simulation

a b s t r a c t

The nonlinearity of the biotechnological processes and the absence of cheap and reliable instrumentationrequire an enhanced modelling effort and estimation strategies for the state and the kinetic parameters.This work approaches nonlinear estimation strategies for microbial production of enzymes, exemplifiedby using a process of lipase production from olive oil by Candida rugosa. First, by using a dynamical math-ematical model of this process, an asymptotic observer which reconstructs the unavailable state variablesis proposed. The design of this kind of observers is based on mass and energy balances without the knowl-edge of kinetics being necessary; only minimal information concerning the measured concentrations isused. Second, a nonlinear high-gain observer is designed for the estimation of imprecisely known kineticsof the bioprocess. An important advantage of this high-gain estimator is that the tuning is reduced tothe calibration of a single parameter. Numerical simulations in various scenarios are provided in orderto test the behaviour and performances of the proposed nonlinear estimation strategies.

© 2014 Elsevier B.V. All rights reserved.

. Introduction

Enzymes are biological molecules that are produced by a living organism, which act as catalysts for specific biochemical reactions. Morerecisely, they are highly selective catalysts that can significantly accelerate the rate and specificity of metabolic reactions, which rangerom the digestion of food to the synthesis of DNA [1]. Lipases (triacylglycerol acylhydrolase, EC 3.1.1.3) are enzymes which are responsibleor the hydrolysis of triglyceride ester bonds into diglycerides, monoglycerides, fatty acids, and glycerol using a complex phenomenon ofnterfacial activation [2]. The most commercially important fields of application of lipases in the last three decades comprise the industryf additives, food industry, fine chemistry, detergents, cosmetics and perfumery, wastewater treatment, pharmaceutical applications, andedical area [1–4]. Microbial lipases have gained special industrial attention due to their stability, selectivity, and broad substrate specificity

5,6]. Candida rugosa has been called the most frequently used organism for lipase synthesis [1,6]. A comprehensive study regarding theipase production processes can be found, for example, in [6], and a particular review about the lipases production by C. rugosa is given in7].

Mathematical modelling is an important tool for optimization and control of bioprocesses, particularly for the microbial productionf enzymes. Despite the major interest for lipase production, few works dealt with the modelling of this enzyme production [8–11]. Forxample, in [10] it is proposed a reaction scheme for lipase production that uses the diauxic growth of C. rugosa on olive oil. A structuredathematical model for lipase production by C. rugosa in batch fermentation is elaborated in [12]. Montesinos et al. [13] simulated the best

onditions to produce lipase. The conclusion was that best lipase productivity was obtained in continuous culture, whereas the highest

redicted lipase activity was obtained in fed-batch cultures. Continuous production of lipase is a complex process because of the tetraphasicature of culture broth: a highly dynamic system of four phases (biomass, gas phase, aqueous phase, and organic phase) is required [7].

The advanced monitoring and control of microbial production of enzymes is a difficult task. Often, on-line measurements of someiological variables involved in these processes (such as biomass, substrate, and product concentrations) are not easily available. The

∗ Corresponding author. Tel.: +40 722541809.E-mail address: dansel@automation.ucv.ro (D. Selis teanu).

ttp://dx.doi.org/10.1016/j.bej.2014.07.010369-703X/© 2014 Elsevier B.V. All rights reserved.

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4 D. Selis teanu et al. / Biochemical Engineering Journal 91 (2014) 23–36

ndirect measurement or estimation of state variables and kinetic parameters, which constitutes the so-called software sensors, is a valuablelternative [8,14–16]. In fact, a software sensor is a combination between a hardware sensor and a software estimator, which can be usedot only for the estimation of concentrations (states), but also for the estimation of kinetic parameters. During the last decades, severalstimation strategies have been developed to provide accurate on-line estimations of state variables. Two main classes of state observersor bioprocesses were developed [15–19]. The first class (including classical observers like extended Luenberger and Kalman observers,onlinear observers) is based on the knowledge of model structure. Such state observers were used in some applications concerning lipaseroduction processes (extended Kalman observers [20,21], nonlinear observers [12,22]). A drawback of this class is that the uncertainty

n the model parameters can generate possibly large bias in the estimation of the unmeasured states. A second class, called asymptoticbservers, is based on the idea that the process uncertainty lies in the kinetics models. The design is based on mass and energy balancesithout the knowledge of kinetics being necessary. Some studies regarding the design of asymptotic observers for simple prototype lipaseroduction processes were reported [21,23]. The potential weakness of asymptotic observers is a low rate of estimation convergence, whichepends on the operating conditions. However, the key advantage is the independence of the kinetics. Besides these two classes, anotherpproach, used in the last decade for bioprocesses without a complete knowledge of inputs, is represented by the interval observers, whichllows the reconstruction of a guaranteed interval on unmeasured states instead of reconstructing their precise numerical values [24–27].

A problem of great importance is the estimation of kinetic rates, i.e. of the so-called kinetics of the bioprocess. One of the first approachesrom historically point of view is based on Kalman filter which leads to complex nonlinear algorithms. Another classical technique is theastin and Dochain approach based on adaptive systems theory [15,17]. The strategy consists in the estimation of unmeasured states withsymptotic observers, and after that, the measurements and the estimates of state variables are used for on-line estimation of kinetic rates.his technique was applied for numerous bioprocesses, including lipase production processes [21]. This approach is useful, but in someases, when many reactions are involved, the implementation requires the calibration of too many parameters. To overcome this problem,

possibility is to design an estimator using a high gain approach [21,28,29]. The gain expression of these observers involves a single tuningarameter whatever the number of components and reactions. High gain observers have evolved as a valuable tool for the design of outputeedback control of nonlinear systems [30,31]. The early work on high gain observers emerges in the late 1980s, and later the techniqueas developed mainly by French researchers (Gauthier, Hammouri, Farza and others [28,29]) and by U.S. researchers (see, for example,halil [31]). Other kinetic parameters estimation techniques include recursive prediction error algorithms, neural networks and hybridpproaches [12,32,33].

However, all the estimation strategies need a better understanding of biotechnological processes and a complex mathematical support.enerally speaking, due to specificity and nonlinearity of the lipase production processes, there is no universal solution to the estimationroblem, and good solutions are given only by studying each particular bioprocess.

In this paper, two correlated estimation issues concerning the processes used for the production of microbial enzymes are studied. Theeported results can be applied to all microbial productions of enzymes which hydrolyze primary substrate and use products of hydrolysisor microbial growth, for example various glysosidases. As exemplification, the proposed estimation techniques are implemented for arocess of lipase production from olive oil by C. rugosa [34,35]. First, by using a dynamical mathematical model of this process, an asymptoticbserver which reconstructs the unavailable state variables is proposed. More precisely, by using three measured concentrations (states)f the bioprocess, other four concentrations are estimated. This design is possible due to the structure of the dynamical nonlinear model.econd, because the kinetic rates of the process are nonlinear and highly uncertain, an on-line estimation strategy is designed. A nonlinearigh-gain observer is proposed for the estimation of three imprecisely known kinetic rates. This parameter estimator possesses certaindvantages concerning the robustness against disturbances and simple tuning (which consists in the calibration of a single parameter).he high gain estimation scheme does not require any model for the kinetics. The performances and the behaviour of the proposedstimation techniques are studied by using extensive numerical simulations. All these simulations are achieved by using the development,rogramming and simulation environment MATLAB (registered trademark of The MathWorks, Inc., USA).

. Materials and methods

.1. Lipase production by C. rugosa: a mathematical model

In this paper, we will consider a lipase production from olive oil by C. rugosa, which takes place into a continuous reactor [34,35].ore precisely, there is a growth of microorganisms on two substrates that are produced by the hydrolysis of a primary complex organic

ubstrate. The following three-step reaction network of this complex bioprocess has been assumed in the literature [34,35]:

a) The hydrolysis:

k1S1 + Eϕ1→S2 + k2S3 + E (1)

b) The growth on substrate S2 (glycerol):

k3S2 + k4Oϕ2→X + k5P (2)

c) The growth on substrate S3 (fatty acids):

k6S3 + k8Oϕ3→X + k7E + k9P (3)

In the above reaction scheme S1 is the primary substrate, i.e. the olive oil (which is made of several compounds, especially triglycerides),2 and S3 (the secondary substrates) are the glycerol and the fatty acids, respectively, E is the enzyme (lipase), X is the biomass (C. rugosa),

is the dissolved oxygen and P is the dissolved carbon dioxide. ϕ1, ϕ1 and ϕ3 are the reaction rates corresponding to the three reactionsf the lipase production process, and ki, i = 1, 9 are the yield coefficients.

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D. Selis teanu et al. / Biochemical Engineering Journal 91 (2014) 23–36 25

The bioprocess modelling is a quite difficult task; however, by using the mass balance of the components inside the process andbeying some modelling rules, a dynamical state-space model can be obtained starting from the reaction scheme [15,30]. Next, the Bastinnd Dochain [15] formalism is used in order to derive the dynamical state-space model of the lipase production process from the reactioncheme (1)–(3). The model expresses the mass balance of components from the reaction scheme and consists of seven differential equationsescribing the dynamics of components’ concentrations inside the continuous process [34,35]:⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

S1

S2

S3

E

X

O

P

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−k1 0 0

1 −k3 0

k2 0 −k6

0 0 k7

0 1 1

0 −k4 −k8

0 k5 k9

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

︸ ︷︷ ︸K

⎡⎢⎣ϕ1

ϕ2

ϕ3

⎤⎥⎦ − D

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

S1

S2

S3

E

X

O

P

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

+

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

DS1in

DS2in

DS3in

0

0

0

0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

−

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0

0

0

0

0

QO2

QCO2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(4)

In the dynamical model (4), the concentrations of components were denoted, for convenience, with the same symbols as the components.he left hand of system (4) contains the time derivatives of these concentrations. D is the so-called dilution rate (i.e. the influent flowate/volume of the medium in the reactor), and K is the matrix of yield coefficients (called also the pseudo-stoichiometric matrix). S1in, S2innd S3in are the concentrations of corresponding influent substrates, and QO2 , QCO2 are the gaseous flow rates (of the oxygen and carbonioxide, respectively). If not available through measurements, the oxygen and carbon dioxide transfer between liquid and gaseous phasean be modelled by QO2 = (kLa)O2

(O − Osat), QCO2 = (kLa)CO2(P − Psat), where (kLa)O2

, (kLa)CO2are mass transfer coefficients, and Osat, Psat

re the concentrations of dissolved oxygen and dissolved carbon dioxide at saturation given by the Henry’s law.

Next, the state vector of the system (4) will be denoted as � =[S1 S2 S3 E X O P

]T. As can be seen, this vector contains the

oncentrations of the components. Also, the following notations will be used: F =[DS1in DS2in DS3in 0 0 0 0

]Tis the vector of feed

ates, Q =[

0 0 0 0 0 QO2 QCO2

]Tis the vector of gaseous flow rates, and ϕ =

[ϕ1 ϕ2 ϕ3

]Tis the vector of reaction rates, which

re nonlinear functions of state variables (concentrations). By using these notations, the dynamical nonlinear model (4) can be compactlyritten as:

� = Kϕ(�) − D� + F − Q (5)

The dynamical model of the lipase production process (5) belongs to a large class of nonlinear bioprocesses carried out in continuouseactors and is referred as general dynamical state-space model of this class of bioprocesses [15–19,34,35]. The term Kϕ(�) is the ratef consumption and/or production of the components, i.e. the reaction kinetics. The term −D� + F − Q represents the exchange with thenvironment (transportation dynamics). The strongly nonlinear character of this model is given by the reaction kinetics.

In many practical situations, a complete description of the reaction network is a priori not available. More precisely, two issues arenvisaged: first, the structure and the parameters of yield matrix and second the structure and the parameters of reaction rates areartially known or even completely unknown. As a consequence, concerning the first problem, it is necessary to develop a procedure offf-line identification of the yield coefficients. This problem is beyond of the scope of present work, and it was approached in some papersuch as [34,35]. It was shown that for the lipase production process described by the model (4) (or equivalent (5)), the yield coefficientsdentification can be completely decoupled from that of the reaction rates by means of an appropriate transformation of the dynamical

odel. Beside the investigation of the structural identifiability of yield matrix, several methods were used to provide estimates of the yieldoefficients [35].

The second issue constitutes one of the most difficult tasks for the construction of dynamical model (5), i.e. the modelling of reactionates. The form of kinetics is complex, nonlinear and in many cases partial or completely unknown. A possible structure of the reactionates, modelled as combined and modified Monod laws, is given in [35]:

ϕ1(�) = ϕ1(S1, E, X) = �∗1

S1

Km1 + S1

E

Km2 + EX = �1(�)X (6)

ϕ2(�) = ϕ2(S2, O, X) = �∗2

S2

Km3 + S2

O

Km4 + OX = �2(�)X (7)

ϕ3(�) = ϕ3(S2, S3, O, X) = �∗3

S3

(Km5 + S3)(Km6 + S2)O2

Km7 + O2X = �3(�)X (8)

here �∗i, i = 1.3 represent maximum specific reaction rates, Kmi, i = 1, 7 are Michaelis–Menten constants, and �i(�), i = 1, 3 are the

pecific reaction rates. The vector of specific reaction rates will be denoted as � =[�1 �2 �3

]T.

emark 1. It should be noticed that the expressions of the reaction rates (6) and (7), as stated in [35], were not selected on a very realisticasis, but more in order to illustrate their approach on a broad variety of kinetics. Nevertheless, in order to obtain a correct and useful

ynamic model it is important to find accurate models of the reaction rates. Some improvements can be achieved by using hypothesesesulting from the technological process. For example, the first reaction rate, accounting for the hydrolysis of olive oil, is independent ofhe biomass and it depends on enzyme activity and olive oil. Therefore a simpler model of ϕ1 can be used, which consider proportionalityo E and saturation term for S1. In reaction rate for fatty acids, a simple Monod pattern can be used for O. However, regardless of any

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6 D. Selis teanu et al. / Biochemical Engineering Journal 91 (2014) 23–36

upposition, in practice the reaction rates given by the relations (6)–(8) are imprecisely known. Therefore these uncertain kinetic rateseed to be estimated on-line. �

If we consider the vector of unknown parameters denoted �(t) (with �(t) = ϕ(t) or �(t) = �(t)), the model of the lipase production process5) can be written as

� = KH(�)�(t) − D� + F (9)

ith H(�) = I3 (the identity matrix) if all the structure of reaction rates ϕ is unknown, or

H(�) =

⎡⎣X 0 0

0 X 0

0 0 X

⎤⎦ = X · I3 (10)

f the specific reaction rates are unknown.

.2. Design of an asymptotic observer for the estimation of unavailable concentrations

In order to implement advanced monitoring and control techniques for the C. rugosa lipase production process, it is necessary that theeasurements of concentrations to be on-line accessible. However, the on-line measurements of some biological variables of this process

re not easily available [7,35]. To enhance the operative procedures, there are two major possibilities. The first one is to increase the numberf sensing devices available [8]. The second consists in the estimation of concentrations by using the available measurements.

The advantage of asymptotic observers (i.e. the design independent of the kinetic models) represents an attractive motivation to designuch observers.

In the following we will consider that the concentrations of secondary substrates S2 and S3 (the glycerol and the fatty acids) and of theiomass X are available via on-line measurements. Usually, X is determined via on-line measurements of carbon dioxide evolution rate [7].he rest of the state variables, i.e. the concentrations of S1 (the primary substrate), E (lipase), O (the dissolved oxygen), and P (the dissolvedarbon dioxide) are unavailable and the observer should reconstitute them from the available measurements.

The design of the asymptotic observer can be achieved if the model (5) satisfies the next hypotheses: the yield coefficients are known, theeaction rates are unknown, the dilution rate, the feed rates and gaseous flow rates are known, and the dimension q of the measurable stateector fulfils the condition: q ≥ p = rank(K) [15,23]. The design is based on some useful changes of coordinates, which lead to a submodelf (5) which is independent of the kinetics. In order to achieve the change of coordinates, a partition of the state vector � in two partss considered: (�a, �b). This induces partitions of the yield matrix K: (Ka, Kb), also of the vectors F and Q: (Fa, Fb)(Qa, Qb) accordingly. Weuppose that a state partition is chosen such that the submatrix Ka is full rank and dim(�a) = rank(Ka) = rank(K). Then a linear change ofoordinates can be defined as follows:

z = ��a + �b (11)

here z is an auxiliary state vector and � the solution of matrix equation � Ka + Kb = 0. In the new coordinates, the model (5) can beewritten as

�a = Kaϕ(�a, z − ��a) − D�a + Fa − Qa

z = −Dz + � (Fa − Qa) + Fb − Qb(12)

As it can be seen, the main achievement of the change of coordinates is that the dynamics of auxiliary state variables is independent ofhe reaction kinetics. If the matrix of yield coefficients is full rank (which is the case for our lipase process), then the state vector can be

irectly partitioned in the vectors of measured states and unmeasured states [15]. Thus, we will consider that �a =[S2 S3 X

]T(the vector

f measured states) and �b =[S1 E O P

]T(the vector of unmeasured states). The partition (�a, �b) induces the next factorizations:

Ka =

⎡⎢⎣

1 −k3 0

k2 0 −k6

0 1 1

⎤⎥⎦ , Kb =

⎡⎢⎢⎢⎢⎣

−k1 0 0

0 0 k7

0 −k4 −k8

0 k5 k9

⎤⎥⎥⎥⎥⎦ ,

Fa =

⎡⎢⎣DS2in

DS3in

0

⎤⎥⎦ , Fb =

⎡⎢⎢⎣DS1in

0

0

0

⎤⎥⎥⎦ , Qa =

⎡⎣ 0

0

0

⎤⎦ , Qb =

⎡⎢⎢⎣

0

0

QO2

QCO2

⎤⎥⎥⎦ .

(13)

If the matrix Ka is left invertible (which is true in our case; moreover, it is invertible), then the asymptotic observer for (5) is derivedrom the structure of Eqs. (11) and (12):

˙

z = −Dz + � (Fa − Qa) + Fb − Qb

�b = z − ��a(14)

here � = −KbK−1a , and �b represents the vector of the estimated state variables.

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D. Selis teanu et al. / Biochemical Engineering Journal 91 (2014) 23–36 27

After the calculation of matrix � and taking into account the factorization (13), one can obtain the detailed equations of the asymptoticbserver (14) used for the estimation of unavailable state variables of the lipase production bioprocess:

⎡⎢⎢⎢⎢⎢⎢⎣

˙z1

˙z2

˙z3

˙z4

⎤⎥⎥⎥⎥⎥⎥⎦

= 1k6 + k2k3

⎡⎢⎢⎢⎢⎢⎣

k1k6 k1k3 k1k3k6

−k2k7 k7 −k2k3k7

k2(k8 − k4) k4 − k8 k4k6 + k2k3k8

k2(k5 − k9) k9 − k5 −k5k6 − k2k3k9

⎤⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎣DS2in

DS3in

0

⎤⎥⎥⎦ +

⎡⎢⎢⎢⎢⎣

DS1in

0

QO2

QCO2

⎤⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎣

S1

E

O

P

⎤⎥⎥⎥⎥⎦ =

⎡⎢⎢⎢⎢⎢⎣

z1

z2

z3

z4

⎤⎥⎥⎥⎥⎥⎦ − 1

k6 + k2k3

⎡⎢⎢⎢⎢⎢⎣

k1k6 k1k3 k1k3k6

−k2k7 k7 −k2k3k7

k2(k8 − k4) k4 − k8 k4k6 + k2k3k8

k2(k5 − k9) k9 − k5 −k5k6 − k2k3k9

⎤⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎣S2

S3

X

⎤⎥⎥⎦

(15)

emark 2. As it can be seen from (14) or (15), this observer is indeed independent of the kinetics, and as a consequence it can be usedithout the knowledge of the reaction rates. The asymptotic observer has good convergence performance [15,17,18] – see the Appendix. �

.3. On-line estimation of unknown kinetics by using high gain observers

As it was discussed previously, the estimation of kinetic rates, i.e. of the so-called kinetics of the bioprocess, has a key role in theodelling and later in the operation and control of microbial production of enzymes. Among the various estimation techniques, the high

ain observers have certain advantages regarding the robustness and simple tuning.The form of lipase kinetics is complex, nonlinear and in many cases partially or completely unknown. The expressions given by the

elations (6)–(8) are only some suppositions about the structure of these rates. In practice, these uncertain kinetic rates need to be estimatedn-line. Next we will consider the vector of unknown parameters �(t) and the model (9). If we suppose that the reaction rates are totallynknown, then �(t) = ϕ(t) and H(�) = I3. If the structure of the reaction rates is known ϕ(�) = H(�)�(�), but the specific reaction rates arenknown, then �(t) = �(t) and H(�) is the matrix given in (10).

The high gain observer design used hereinafter is based on the work of Gauthier et al. [28] and Farza et al. [29]. The design procedureecessitates a factorization of the model (9). Again, we will consider that the yield matrix K is of full rank, which is true for our particularodel, and for general class of bioprocesses’ models is a generic property. Another hypothesis requires that all state variables are measured

r are available via some observers (the asymptotic state observer can be used). Also, the same hypotheses from the previous subsectionemain valid. Since the yield matrix K is of full rank, then the partition (Ka, Kb) can be considered, such that the submatrix Ka has full rank.hus, a partition (�a, �b) of the state vector is obtained. For convenience and in order to exploit the available knowledge about the stateariables, the same partition as in previous subsection will be used. Therefore, the partitions (13) are obtained. Then, the system (9) cane written as follows:

�a = KaH(�a, �b)�(t) − D�a + Fa − Qa

�b = KbH(�a, �b)�(t) − D�b + Fb − Qb

(16)

By using this factorization, a high-gain observer can be implemented. The design of high-gain observers is done in [28,29], with supple-entary assumptions regarding global Lipschitz conditions, the boundedness of H(�) diagonal elements’ away from zero, etc. (for details

oncerning the high gain observers design, see Appendix B). The equations of the nonlinear high gain observer for (9) are obtained as [28]:

�aest = KaH(�aest, �b) � − D�aest + Fa − 2�(�aest − �a)

˙� = −�2[KaH(�aest, �b)

]−1(�aest − �a)

(17)

The observer (17) provides on-line estimates � for the unknown kinetics. This on-line estimation algorithm is a copy of the processodel, with a corrective term. The observer is simple and the gain tuning can be done by modifying only one design parameter: �.

emark 3. It should be noticed that �a est is an “estimate” of �a, provided by the algorithm in order to be compared with the real state

a, and the resulting error to be used in (17) (in fact, �a is known since it is measured). The notation “est” was used in order to avoid theonfusion with “’’, used for true estimates of unknown or unmeasurable entities. �

Then, for the lipase process, two high-gain estimators can be derived from (17):

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(

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o�˝

�

re

b

8 D. Selis teanu et al. / Biochemical Engineering Journal 91 (2014) 23–36

(a) An estimator for the specific reaction rates. In this case �(t) = �(t) and H(�) is the matrix given in (10). The equations of the high-gainobserver are:

⎡⎢⎣S2 est

S3 est

Xest

⎤⎥⎦ =

⎡⎢⎣

1 −k3 0

k2 0 −k6

0 1 1

⎤⎥⎦ ·

⎡⎢⎣Xest 0 0

0 Xest 0

0 0 Xest

⎤⎥⎦ ·

⎡⎢⎣�1

�2

�3

⎤⎥⎦ − D

⎡⎢⎣S2 est

S3 est

Xest

⎤⎥⎦ +

⎡⎢⎣DS2in

DS3in

0

⎤⎥⎦ − 2�

⎡⎢⎣S2 est − S2

S3 est − S3

Xest − X

⎤⎥⎦

⎡⎢⎢⎣

˙�1

˙�2

˙�3

⎤⎥⎥⎦ = −�2

⎡⎢⎣

⎡⎢⎣

1 −k3 0

k2 0 −k6

0 1 1

⎤⎥⎦ ·

⎡⎢⎣Xest 0 0

0 Xest 0

0 0 Xest

⎤⎥⎦

⎤⎥⎦

−1

·

⎡⎢⎣S2 est − S2

S3 est − S3

Xest − X

⎤⎥⎦

(18)

b) A second estimator can be obtained if the entire reaction rate vector is considered unknown. In this case �(t) = ϕ(t) and H(�) = I3. Theequations of high-gain observer are:

⎡⎢⎣S2 est

S3 est

Xest

⎤⎥⎦ =

⎡⎢⎣

1 −k3 0

k2 0 −k6

0 1 1

⎤⎥⎦ ·

⎡⎢⎣ϕ1

ϕ2

ϕ3

⎤⎥⎦ − D

⎡⎢⎣S2 est

S3 est

Xest

⎤⎥⎦ +

⎡⎢⎣DS2in

DS3in

0

⎤⎥⎦ − 2�

⎡⎢⎣S2 est − S2

S3 est − S3

Xest − X

⎤⎥⎦

⎡⎢⎢⎣

˙ϕ1

˙ϕ2

˙ϕ3

⎤⎥⎥⎦ = −�2

⎡⎢⎣

1 −k3 0

k2 0 −k6

0 1 1

⎤⎥⎦

−1

·

⎡⎢⎣S2 est − S2

S3 est − S3

Xest − X

⎤⎥⎦

(19)

Both observers (18) and (19) need the on-line measurements of S2, S3, and X.Next, a classical observer-based estimator (OBE) for the lipase production by C. rugosa will be derived, in order to use it to compare the

erformance of the proposed high gain observers. To estimate the reaction rates of bioprocesses of the form (9), Bastin and Dochain [15]roposed an OBE, considering that all state variables are measurable in real-time, F, Q and D are measurable, and the matrix K is known.he OBE is expressed as [15,36]:

�est = KH(�) �(t) − D� + F − Q − ˝(� − �est)

˙� = [KH(�)]T� (� − �est)(20)

The first equation of (20) is a state observer, used for updating the estimate �, and not for state estimation, similar with the high gainbserver case. The update is generated by the estimation error e = (� − �est), where �est is the on-line “estimation” of the state vector. The error

− � is directly reflected by the estimation error e. is a gain matrix; in the second equation, the injection matrix � is chosen such thatT� + � is negative definite. The design parameters are the matrices and � , with a typical choice of diagonal form: = diag

i=1,...,n{−ωi},

= diagj=1,...,n

{ j}, ωi, j ∈ � + [15,36].

The implementation of estimator (20) for the lipase production process leads to an observer with 14 tuning parameters. However, aeduced order OBE can be obtained by using a subset of the state equations provided that they involve all the kinetic rates that need to bestimated. This approach has the advantage of a reduced number of equations and tuning parameters. Therefore, a reduced order OBE will

e implemented, by using only the reduced state vector �r = �a =[S2 S3 X

]T. Then the equations of the reduced OBE are as follows:

�r est = KrH(�r) �(t) − D�r + Fr − Qr − ˝(�r − �r est)

˙� = [KrH(�r)]T� (�r − �r est)

(21)

Taking into account that Kr, Fr, Qr are Ka, Fa, Qa, two OBEs can be derived from (21):

(a) An OBE for the specific reaction rates. Then �(t) = �(t) and H(�) is given in (10).

⎡⎢⎣S2est

S3est

Xest

⎤⎥⎦ =

⎡⎢⎣

1 −k3 0

k2 0 −k6

0 1 1

⎤⎥⎦ ·

⎡⎣X 0 0

0 X 0

0 0 X

⎤⎦ ·

⎡⎢⎣�1

�2

�3

⎤⎥⎦ − D

⎡⎢⎣S2

S3

X

⎤⎥⎦ +

⎡⎢⎣DS2in

DS3in

0

⎤⎥⎦ +

⎡⎢⎣ω1 0 0

0 ω2 0

0 0 ω3

⎤⎥⎦ ·

⎡⎢⎣S2 − S2est

S3 − S3est

X − Xest

⎤⎥⎦

⎡˙�

⎤ ⎡⎡1 −k 0

⎤ ⎡ ⎤⎤T ⎡ 0 0

⎤ ⎡S − S

⎤ (22)

⎢⎢⎣1

˙�2

˙�3

⎥⎥⎦ = ⎢⎣⎢⎣3

k2 0 −k6

0 1 1

⎥⎦ · ⎣X 0 0

0 X 0

0 0 X

⎦⎥⎦ · ⎢⎣1

0 2 0

0 0 3

⎥⎦ · ⎢⎣2 2 est

S3 − S3 est

X − Xest

⎥⎦

D. Selis teanu et al. / Biochemical Engineering Journal 91 (2014) 23–36 29

Table 1Kinetic parameters values – lipase production process.

Kinetic parameter Value Unit

�∗1 0.0208 h−1

�∗2 0.125 h−1

�∗3 0.833 g/(L h)

Km1 2 g/LKm2 0.2 g/LKm3 1 g/LKm4 0.2 g/LKm5 1 g/LKm6 0.2 g/LKm7 2 g2/L2

(kLa)O20.208 h−1

(kLa)CO20.208 h−1

Osat 0.5 g/LPsat 15 g/L

Table 2Yield coefficients values – lipase production process.

Yield coefficient Value

k1 3k2 0.3k3 4.54k4 1.33k5 0.34k6 0.5k 0.19

(

3

wsi

Fo

7

k8 0.72k9 1.24

b) A second OBE can be obtained if the entire reaction rate vector is considered unknown. In this case �(t) = ϕ(t) and H(�r) = I3.⎡⎢⎣S2 est

S3 est

Xest

⎤⎥⎦ =

⎡⎢⎣

1 −k3 0

k2 0 −k6

0 1 1

⎤⎥⎦ ·

⎡⎢⎣ϕ1

ϕ2

ϕ3

⎤⎥⎦ − D

⎡⎢⎣S2

S3

X

⎤⎥⎦ +

⎡⎢⎣DS2in

DS3in

0

⎤⎥⎦ +

⎡⎢⎣ω1 0 0

0 ω2 0

0 0 ω3

⎤⎥⎦ ·

⎡⎢⎣S2 − S2 est

S3 − S3 est

X − Xest

⎤⎥⎦

⎡⎢⎢⎣

˙ϕ1

˙ϕ2

˙ϕ3

⎤⎥⎥⎦ =

⎡⎢⎣

1 −k3 0

k2 0 −k6

0 1 1

⎤⎥⎦T

·

⎡⎢⎣ 1 0 0

0 2 0

0 0 3

⎤⎥⎦ ·

⎡⎢⎣S2 − S2 est

S3 − S3 est

X − Xest

⎤⎥⎦

(23)

. Results and discussion

The behaviour and the performance of proposed estimation techniques were analyzed by using extensive simulations. The simulationsere performed in MATLAB environment (registered trademark of The MathWorks Inc., USA). The lipase production process has been

imulated by numerical integration of the basic model Eqs. (4) and (6)–(8). The values of kinetic parameters and of yield coefficients usedn simulations were obtained through identification procedures in [34,35]. These values are presented in Tables 1 and 2.

(a)

0 50 100 15 0 200 25 0 300

.045

0.05

.055

D (h-1)

Time (h )

(b)

0 50 100 15 0 20 0 250 3000

1

2

3

4

Time (h)

(g/L)

2S

3S

X

ig. 1. Simulation results – time profiles of the dilution rate and of on-line available concentrations. (a) The dilution rate. (b) The concentrations of secondary substrates andf biomass.

30 D. Selis teanu et al. / Biochemical Engineering Journal 91 (2014) 23–36

(a)

0 50 100 150 200 250 3000.8

1

1.2

1.4

1.6

1.8

2

Time (h)

1S

1S

(g/L)

(b)

0 50 10 0 150 200 25 0 30 00

.02

.04

.06

.08

Time (h)

E

E

(g/L)

(c)

0 50 100 150 200 250 3000

0.1

0.2

0.3

0.4

Time (h)

(g/L )

10 20 30 400.34 5

0.35

0.35 5

0.36

0.365

O O

(d)

0 50 100 150 200 250 3000

2

4

6

8

10

12

14(g/L)

Time (h)10 15 20 2511.5

11.6

11.7

11.8

11.9

12

P

P

Fig. 2. The asymptotic observer implementation – simulation results. (a) The estimated primary substrate concentration and its real profile. (b) Time evolution of the estimatedenzyme (lipase) concentration versus the “true” profile. (c) Profiles of the dissolved oxygen concentration and its estimation, including a zoom area. (d) The evolution of thedissolved carbon dioxide concentration and its estimation, with a zoom area. In all panels, estimations are represented with dashed lines, and the real values with solid lines.

Fig. 3. Estimation of reaction kinetics with high gain observers (HGO) vs. observer based estimators (OBE) – simulation results. (a) Profiles of specific rate �1 and its estimates.(b) Evolution of the specific rates �2, �3 and the corresponding estimations. (c) Evolution of reaction rate ϕ1 and of its estimates. (d) Profiles of reaction rates ϕ2, ϕ3 versustheir estimates. In all panels, estimations are represented with dashed and dotted lines (HGO and OBE), and the real values with solid lines.

D. Selis teanu et al. / Biochemical Engineering Journal 91 (2014) 23–36 31

Two simulation scenarios were taken into consideration:(i) The asymptotic observer (15) was implemented for the lipase production process, in order to reconstitute the concentrations S1

(primary substrate), E (lipase), O (dissolved oxygen), and P (dissolved carbon dioxide), by using the measurements of the concentrationsS2 and S3 (glycerol and fatty acids), and X (biomass).

In this simulation case, the measurements are considered free of noise, and are obtained from the integration of the model Eqs. (4)and (6)–(8). The kinetic expressions (6)–(8) were introduced only for simulation; thus, these models were not used in the process ofobserver design or implementation. The dilution rate and substrate inflow rates have been selected in order to guarantee that the systemis sufficiently excited; the profiles of these rates used in simulations are similar with those presented in [35].

The influent substrate concentration S1in is of square wave form (offset 1 g/L, amplitude 0.1 g/L), and the influent substrate concentrationsS2in and S3in are of sinusoidal form (offset 2 g/L, amplitude 0.2 g/L). The profile of dilution rate is given in Fig. 1, panel (a). Panel (b) showsthe evolution of measured concentrations S2, S2, and X.

The simulation results for the asymptotic observer are presented in Fig. 2. Panel (a) shows the estimated substrate concentration S1depicted versus its “true” profile S1. In panel (b), the enzyme concentration and its estimation are presented. Panel (c) shows the timeevolution of dissolved oxygen concentration together with its real profile. Panel (d) presents the carbon dioxide concentration and itsestimation. From these simulation results, obtained for free-noise measurements, it can be seen a very good behaviour of the asymptoticobserver, especially for the dissolved oxygen and carbon dioxide concentrations.

Next, the high gain observers (18) and (19) were implemented in the same conditions. The main goal of these estimators is to reconstitutethe time evolution of specific reaction rates �1, �2 and �3 – estimator (18), or of the reaction rates ϕ1, ϕ2 and ϕ3 – estimator (19), from themeasurements of S2, S3, and X. To evaluate and to compare the results, the classical OBEs (22) and (23) were used for the same purpose.Fig. 3 portrays the obtained results. Panel (a) depicts the evolution of specific reaction rate �1 and of its estimates (OBE versus high-gainobserver). Panel (b) presents the time profiles of specific reaction rates �2, �3 and their estimations. Panels (c) and (d) show the timeevolution of estimated and “true” reaction rates ϕ1 and ϕ2, ϕ3, respectively. The values of the tuning parameter of high gain observers werechosen by using a trial-and-error procedure; they were set to � = 0.5 for (18) and to � = 1 for (19).

As it was discussed, the OBEs necessitate much more tuning parameters; the values used are as follows: ωi = 20, i = 15, i = 1, 3for (22), and ωi = 10, i = 8, i = 1, 3 for (23).

As can be noticed from Fig. 3, the OBEs performance is inferior to the performance of the high gain observers: quite large estimationerrors for �1, ϕ1, and an oscillatory behaviour for ϕ2, ϕ3 (this kind of behaviour for OBEs was also reported in other works [36,37]).

(ii) In the second simulation scenario, in order to test the robustness of the proposed estimation algorithms to noisy measurements,the behaviour and the performance were analyzed for noisy data of S2, S3, and X. The results obtained for the estimation of state variablesS1, E, O, and P by using the asymptotic observer (15) in the same conditions like in case (i), but with noisy measurements, are presented inFig. 4. Panel (a) shows the profiles of the measured concentrations S2, S3, and X vitiated by an additive Gaussian noise, with zero mean and

(a)

0 50 10 0 15 0 20 0 25 0 3000

0.5

1

1.5

2

2.5

3

3.5

4 (g/L)

Time (h)

3S

2S

X

(b)

0 50 100 150 200 250 3000.8

1

1.2

1.4

1.6

1.8

2

Time (h)

1S

1S

(g/L)

(c)

0 50 10 0 150 200 250 30 00

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Time (h)

(g/L)

EE

OO

5 10 15 20 250.3

0.32

0.34

0.36

(d)

0 50 10 0 150 20 0 250 30 02

4

6

8

10

12(g/L)

Time (h )

10 15 20 25 3011.5

11.6

11.7

11.8

11.9

12 P P

Fig. 4. Asymptotic observer simulation results (noisy measurements). (a) The concentrations of secondary substrates and of biomass, affected by measurement noise. (b)The estimated primary substrate concentration and its real profile. (c) Time evolution of the estimated enzyme (lipase) and of the dissolved oxygen concentrations versustheir “true” profiles. (d) The evolution of the dissolved carbon dioxide concentration and its estimation.

32 D. Selis teanu et al. / Biochemical Engineering Journal 91 (2014) 23–36

(a)

0 50 100 150 200 250 3000

0.5

1

1.5

2

2.5x 10

-3

Time (h )

(h-1)

HGO1ˆ −μ

1μ

(b)

0 50 100 150 200 250 3000

0.01

0.02

0.03

0.04

0.05

0.06

Time (h)

(h-1)

HGO2ˆ −μ

HGO3ˆ −μ

2μ

3μ

2μ

3μ

(c)

0 50 100 15 0 200 250 30 00

0.5

1

1.5

2x 10

-3

Time (h)

(g/Lh )

HGO1ˆ −ϕ

1ϕ

(d)

0 50 10 0 15 0 20 0 250 3000

0.00 5

0.01

0.01 5

0.02

0.02 5(g/Lh )

HGO2ˆ −ϕ

2ϕ

HGO3ˆ −ϕ

3ϕ

Time (h )

2ϕ

3ϕ

Fs

aogp

Porg

kc

O

fbS

o

pphtc

Rm

ig. 5. Estimation of reaction kinetics with high gain observers (noisy measurements). (a) Profiles of specific reaction rate �1 and its estimates. (b) Time evolution of thepecific reaction rates �2, �3 and the corresponding estimations. (c) Evolution of reaction rate ϕ1 and of its estimates. (d) Profiles of reaction rates ϕ2, ϕ3 versus their estimates.

mplitude equal to 4% of the free-noise values. In panel (b), the time evolution of S1 and S1 is showed. Panels (c) and (d) show the profilesf E, E, O, O, and of carbon dioxide concentration and its estimation (P, P), respectively, all obtained by using the noisy data. From theseraphics, it can be seen a quite good behaviour of the asymptotic observer when some noisy measurements are used in the reconstructionrocess.

The kinetic parameters estimators (18) and (19) were tested in the case of noisy measurements. The results are presented in Fig. 5.anel (a) shows the evolution of specific reaction rate �1 and of its estimates provided by the high-gain observer. In panel (b), the profilesf specific reaction rates �2, �3 and their estimations are presented. Panels (c) and (d) depict the profiles of estimated and “true” reactionates ϕ1 and ϕ2, ϕ3, respectively. The tuning parameter value was set to � = 0.5 for (18) and to � = 1 for (19). The simulation results show aood behaviour of high gain observers implemented in noisy data conditions.

The behaviour and the performance of OBEs are deteriorated when the noisy measurement are used for the estimation of reactioninetics. As exemplification, Fig. 6 presents the evolution of specific reaction rates �1, �2, and �3, together with their estimates, in thease of noisy measurements. The tuning parameters of OBE (22) are: ωi = 10, i = 5, i = 1, 3.

Numerous supplementary simulations were performed for various values of the tuning parameters, for both high gain observer andBE algorithms. The results illustrate that the high gain observers provide accurate estimates of the kinetic rates.

The noisy measurement induces some noisy estimates of the kinetics, but the noise effect is limited (however, this effect can be reducedor lower values of tuning parameter �). Several comparisons and comments about the performance of on-line estimation strategies cane achieved. Qualitative remarks can be done by visualization of estimation errors (e.g. �1 = �1 − �1, �2 = �2 − �2 and �3 = �3 − �3).till, accurate comparisons can be achieved by considering a criterion, such as one based on averaged square estimation errors [30,36]:J1 = 1

TS

∫ TS0�2

1(t)dt, J2 = 1TS

∫ TS0�2

2(t)dt, J3 = 1TS

∫ TS0�2

3(t)dt, where TS is the total simulation time.The values of J1, J2, and J3 computed for different values of tuning parameters of high gain observer are given in Table 3, and for the

bserver-based estimator in Table 4, both for the estimation of specific reaction rates of the lipase production process.From the results presented in Tables 3 and 4, it can be seen that the accuracy of high gain observers can be increased if the tuning

arameter value is bigger. The problem for a large value of � is that the observer becomes noise sensitive. Therefore, the value of the tuningarameter is a compromise between a good estimation and the noise rejection. The obtained results concerning the noise sensitivity of theigh gain observers are similar with those discussed in several works, such as [38–41]. From the simulated experiments, the conclusion ishat the performance of the OBEs is inferior to that of the high-gain observers. The effect of noisy measurements is more significant in the

ase of OBE.

emark 4. Concerning the performed simulations, some additional comments should be given. The present work is focused only onodelling and estimation, without testing the model in actual fermentation experiments. Only simulations were performed, by using

D. Selis teanu et al. / Biochemical Engineering Journal 91 (2014) 23–36 33

(a)

0 50 10 0 15 0 20 0 25 0 300

0

0.5

1

1.5

2

2.5

3

3.5

4x 10

-3

Time (h)

(h-1)

OBE1ˆ −μ

1μ

(b)

0 50 10 0 15 0 20 0 25 0 3000

0.01

0.02

0.03

0.04

0.05

0.06

Time (h)

(h-1)

OBE2ˆ −μ

OBE3ˆ −μ

2μ

3μ

2μ

3μ

Fig. 6. Estimation of reaction kinetics with OBE (noisy measurements). (a) Profiles of specific reaction rate �1 and its estimates. (b) Time evolution of the specific reactionrates �2, �3 and the corresponding estimations.

Table 3Performance criterion results – high gain observer.

Tuning parameter Free noise measurements Noisy measurements

J1 J2 J3 J1 J2 J3

� = 0.5 5.51 × 10−8 8.95 × 10−5 1.71 × 10−5 1.22 × 10−4 2.31 × 10−4 5.46 × 10−5

� = 1 2.29 × 10−8 3.68 × 10−5 5.77 × 10−6 1.97 × 10−4 1.65 × 10−4 8.37 × 10−5

� = 5 4.33 × 10−9 5.61 × 10−6 1.11 × 10−6 0.024 0.009 0.009� = 10 2.84 × 10−9 3.49 × 10−6 7.01 × 10−7 0.601 0.216 0.217� = 15 7.76 × 10−9 9.45 × 10−6 1.88 × 10−6 Divergent Divergent Divergent

Table 4Performance criterion results – observer-based estimator.

Tuning parameters (i = 1, 2, 3) Free noise measurements Noisy measurements

J1 J2 J3 J1 J2 J3

ωi = 2, i = 0.5 7.25 × 10−5 4.14 × 10−4 1.02 × 10−4 4.55 × 10−4 3.37 × 10−4 7.11 × 10−4

ωi = 5, i = 2 5.27 × 10−5 2.22 × 10−4 7.98 × 10−5 6.71 × 10−4 4.91 × 10−4 8.24 × 10−4

ωi = 10, i = 5 5.25 × 10−5 1.82 × 10−4 6.54 × 10−5 9.54 × 10−3 9.08 × 10−4 7.36 × 10−4

ωi = 15, i = 10 5.21 × 10−5 1.38 × 10−4 5.07 × 10−5 0.002 0.004 9.10 × 10−3

tf(cv

4

ardub

wte

wsis

ωi = 20, i = 15 5.17 × 10−5 1.23 × 10−4 4.58 × 10−5 0.211 0.015 0.021ωi = 25, i = 20 Divergent Divergent Divergent Divergent Divergent Divergent

he available data from [34,35]. The model parameters used in simulations (Tables 1 and 2) are identified values that were obtainedrom simulated data by using a mathematical model neither calibrated nor validated. Thus, some model parameters are not very realisticbiochemical significance, orders of magnitude, etc.), and can lead to time evolutions (obtained in the performed simulations) of someoncentrations and reaction rates that have not fully experimental agreement. In conclusion, the estimation procedures should be finallyalidated by using real experiments. �

. Conclusions

This work dealt with the design and implementation of several estimation strategies for the unavailable state variables (concentrations)nd the unknown kinetics for microbial production of enzymes, exemplified by using a process of lipase production from olive oil by C.ugosa. The dynamical nonlinear model of the lipase production process was presented and a factorization in two submodels was used toesign the proposed observers. First, an asymptotic state observer was designed and the unavailable state variables were reconstructed bysing the measured concentrations of the lipase process. The design of this state observer was achieved without the knowledge of kineticseing necessary.

Second, high gain observers were designed and implemented for the on-line estimation of reaction kinetics. Two high-gain observersere developed for the estimation of three imprecisely known specific reaction rates and of three reaction rates, respectively. The advan-

ages of this kind of estimators are the simplicity of design, the good convergence and stability properties, and the accuracy of estimates,specially for free noise data. Another important benefit is the fact that the tuning of one single design parameter is necessary.

Several simulations were conducted to analyze the behaviour and the performance of estimation strategies. Two simulation scenariosere used in order to provide realistic tests for the proposed observers. In the case of kinetic parameters estimation, OBEs were implemented

o as to compare the performance of the high gain observers. The estimation results obtained with the high gain estimators can be improvedf the tuning parameter is chosen higher in value, but only if the measurements are free-noise; contrarily, the observer becomes noiseensitive and it is possible that the estimates of kinetics cannot be utilized.

3

e

A

B

A

A

o

T

P

w

b

A

gd

w

ww

4 D. Selis teanu et al. / Biochemical Engineering Journal 91 (2014) 23–36

The obtained results are quite encouraging from simulation point of view. These estimation strategies can be also applied to othernzymes production processes, and to some extent to processes belonging to the nonlinear class of bioprocesses considered in the study.

cknowledgements

This work was supported by UEFISCDI, Romania, project PACBIO no. 701/2013 (French-Romanian bilateral project), and project ADCOS-IO no. PN-II-PT-PCCA-2013-4-0544.

ppendix A.

.1. A. Convergence of the asymptotic observer

In order to analyze the convergence property of the asymptotic observer (14), we will define first the estimation error:

ε = �b − �b (A.1)

The dynamics of estimation error can be easily obtained from (11), (12) and (14):

ε = −Dε (A.2)

From the analysis of Eq. (A.1), it is clear that ε = 0 is an equilibrium point whose stability depends on the dilution rate. The convergencef the algorithm (14) can be proved by the means of the next theorem.

heorem A.1 ([15,18]). If the dilution rate D(t) is persistently exciting, i.e. if there exist two real and positive constants ı and such that

0 < <

t+ı∫t

D(�)d� (A.3)

then limt→∞

∥∥ε(t)∥∥ = limt→∞

∥∥∥�b(t) − �b(t)∥∥∥ = 0. (A.4)

�

roof. The solution of differential equation (A.2) is:

ε(t) = exp

⎛⎝−

t∫0

D(�)d�

⎞⎠ · ε(0) (A.5)

If we denote t = �ı, with �∈ ℵ, one obtains:

∥∥ε(�ı)∥∥ =

∥∥∥∥∥∥exp

⎛⎝−

�ı∫0

D(�)d�

⎞⎠

∥∥∥∥∥∥ ·∥∥ε(0)

∥∥ =

∥∥∥∥∥∥exp

⎛⎝−

ı∫0

D(�)d�

⎞⎠ exp

⎛⎝−

2ı∫ı

D(�)d�

⎞⎠ · · · exp

⎛⎝−

�ı∫(�−1)ı

D(�)d�

⎞⎠

∥∥∥∥∥∥ ·∥∥ε(0)

∥∥ (A.6)

Then, from (A.3) the next inequality is obtained:∥∥ε(�ı)∥∥ ≤ exp(−ˇ) exp(−ˇ)· · · exp(−ˇ) ·∥∥ε(0)

∥∥ = exp(−�ˇ) ·∥∥ε(0)

∥∥ (A.7)

hich implies lim�→∞

∥∥ε(�ı)∥∥ = lim�→∞

(exp(−�ˇ) ·∥∥ε(0)

∥∥) = 0. Finally, this equality implies that limt→∞

∥∥ε(t)∥∥ = 0, ∀ε(0). �

The hypothesis (A.3) implies in fact that the dilution rate to not be zero for a too long period of time. The convergence speed is determinedy the experimentally operational conditions, more precisely by the form of the dilution rate.

.2. B. Elements of high gain observers design

The high gain observers design is based on the approach given in [28,29]. Let’s consider the model of the lipase production processiven in (9) and the state partition (�a, �b) which leads to the system factorized in the form (16). We suppose �b to be a continuous signal,enoted = �b. Then consider the system [29]:

�a = KaH(�a, )�(t) − D�a + Fa − Qa, � = g(t), y = �a (B.1)

here g(t) is a bounded unknown function.The system (B.1) belongs to the general class of nonlinear systems [29]

x1 = ˚(x1(t), (t))x2(t) + b1(u(t), x1(t), (t))

x2 = b2(u(t), x1(t), (t)) + g(t) (B.2)

y(t) = x1(t)

ith x a 2n-dimensional state vector, x =[x1 x2

]T ∈ �2n; x1, x2 ∈ � n. In (B.2), u ∈ � p is the input, y is the output and is a n × n matrix,hose elements are real and scalar functions of Cq class (q ≥ 1) with respect to their arguments.

h

w

w

w

s

T‖

wi

d(o

atd

R

[[[

[

[[[

[[

D. Selis teanu et al. / Biochemical Engineering Journal 91 (2014) 23–36 35

Usually, the gain of observers is calculated by solving some Riccati or Lyapunov differential equations. Instead of this approach, theigh-gain observer design consists in a simple calculation of the gain by using algebraic Lyapunov equation [28,29,36].

The system (B.2) can be compactly written as

x = ˜ (x1(t), (t))x(t) + B(u(t), x1(t), (t)) + g(t)

y(t) = Gz(t)(B.3)

here ˜ (x1, ) =[

0 ˚(x1, )0 0

], B(u, x1, ) =

[b1(u, x1, )b2(u, x1, )

], g(t) =

[0 g(t)

]T, G =

[In 0

].

The following hypotheses will be considered [28,29,36]:H1. There exist ˇ1, ˇ2 ∈ � , 0 < ˇ1 ≤ ˇ2 such that ∀� ∈ � n, ∀ t ≥ 0, next inequality holds: ˇ2

1In ≤ ˚T (�, )˚(�, ) ≤ ˇ22In.

H2. The function g(t) is bounded.H3. The function (t) and its derivative with respect to time are bounded.H4. The function B is globally Lipschitz with respect to x1, locally uniformly Lipschitz with respect to u and , i.e. ∀ı1, ı2 > 0, ∃ ı3 > 0, ∀ u

ith ‖u‖ ≤ ı1, ∀ with ‖ ‖ ≤ ı2, ∀x1 ∈ � n, we have∥∥∥ ∂B∂x1

(u, x1, )∥∥∥< ı3.

H5. is globally Lipschitz with respect to x1, and locally uniformly with respect to .Consider now the system:

˙x = ˜ (x1, )x + B(u, x1, ) − �−1(x1, )S−1�ET (Ex − y) (B.4)

here u and y are the input and the output of the system (B.3) respectively; �(x1, ) =[In 00 ˚(x1, )

]; S� is a unique positive definite

ymmetric matrix.Then, the next theorem can be formulated:

heorem B.1 ([29]). Let be the system (B.3) that satisfies the hypotheses H1–H5. Then, ∀ı > 0, ∃�0 > 0, ∀� > �0, ∃�� , �� > 0, ∃ M� > 0, ∀ u withu‖∞ ≤ ı, ∀x(0) ∈ �2n, we have:∥∥x(t) − x(t)

∥∥ ≤ ��e−��t

∥∥x(0) − x(0)∥∥ + M�� (B.5)

here u is an admissible command, ‖u‖∞ is the upper bound of the command, x(t) is the trajectory of (B.3) associated to the input u, x(t)s any trajectory of (B.4) and � is the superior bound of ‖g‖. Furthermore, we have lim

�→∞�� = ∞ and lim

�→∞M� = 0. �

When g(t) = 0, the observer (B.4) has an exponential convergence of the estimation error x(t) = x(t) − x(t). Contrariwise, Theorem (B.1)emonstrates that the estimation error can become arbitrarily small choosing a sufficiently big value for the gain �. The gain of the observerB.4) can be obtained by solving a Lyapunov algebraic equation [16,28,29]. By using the gain thus calculated, the equations of high-gainbserver (B.4) are written as [29]:

˙x1 = ˚(x1, )x2 + b1(u, x1, ) − 2�(x1 − x1)

˙x2 = −�2˚−1(x1, )(x1 − x1)(B.6)

The high gain observer (17) applied to the lipase production process is obtained directly from (B.6). As a remark, the hypotheses H1–H5re generically fulfilled for the class of bioprocesses given by (5). However, the assumption H1 can generate some problems, but usuallyhe modelling rules for the bioprocesses impose that the diagonal elements of the matrix � are equal to zero if and only if in the reactoro not takes place any reaction [36].

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