A simple output-feedback strategy for the control of perfused mammalian cell cultures

Control Engineering Practice 32 (2014) 123–135

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Control Engineering Practice journal homepage: www.elsevier.com/locate/conengprac

A simple output-feedback strategy for the control of perfused mammalian cell cultures Mihaela Sbarciog a,n, Daniel Coutinho b, Alain Vande Wouwer a a b

Automatic Control Laboratory, University of Mons, Boulevard Dolez 31, B-7000 Mons, Belgium Department of Automation and Systems, Federal University of Santa Catarina, 476 Florianopolis, 88040-900, Brazil

art ic l e i nf o

a b s t r a c t

Article history: Received 12 February 2014 Accepted 6 August 2014 Available online 6 September 2014

This paper presents a framework for the multivariable robust control of perfusion animal cell cultures. It consists of a cascade control structure and an estimation algorithm, which provides the unmeasurable variables needed in the design of the control law, and ensures the regulation of the cell and glucose concentrations at imposed levels by manipulating the bleed and the dilution rates. The cascade control structure uses a feedback linearizing controller in the inner loop and linear (PI) controllers in the outer loops, and requires the measurement of the cell concentration and the glucose concentration in the bioreactor. Two approaches are provided: the first one assumes the availability of an approximate model of the process kinetics and uses an extended Kalman filter (EKF) to estimate the system states; the second approach does not require the prior knowledge of the process kinetics. These are estimated from the available measurements using sliding mode observers (SMO). A receding horizon optimization algorithm is employed to (periodically) tune the gains of the outer loop controllers. The proposed framework is easy to implement and tune, and may be applied to a general class of perfusion cell culture systems. Its effectiveness and robustness are illustrated by means of simulation results. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Animal cell cultures Multivariable feedback control cascade control Kalman filter Sliding mode observer robustness

1. Introduction Many of the important active pharmaceutical ingredients are produced by the cultivation of either genetically modified microorganisms or animal cells. These cells can grow in suspension in stirred tank reactors (Jain & Kumar, 2008), which may be operated in batch, fed-batch or perfusion mode. Among them, the operation in perfusion mode leads to higher cell density and higher productivity (Komolpis, Udomchokmongkol, Phutong, & Palaga, 2010). However, it also requires tight control to avoid nutrient limitation, accumulation of inhibitory metabolites, retardation in cell growth or even cell wash-out through the cell-containing flow (the bleed), which is necessary to maintain culture viability and to reach a steady state (Banik & Heath, 1995; Dalm, 2004; Ozturk, Thrift, Blackie, & Naveh, 1997). Control of animal cell cultures in perfusion mode is a delicate task as it usually requires the availability of a process model and of several on-line probes (Gnoth, Jenzsch, Simutis, & Lübbert, 2008). The former requires experimental data collection and model identification, whereas the latter is limited by investment and operational constraints. It is therefore of interest to develop n

Corresponding author. Fax: þ 32 65374136. E-mail addresses: [email protected] (M. Sbarciog), [email protected] (D. Coutinho), [email protected] (A. Vande Wouwer). http://dx.doi.org/10.1016/j.conengprac.2014.08.002 0967-0661/& 2014 Elsevier Ltd. All rights reserved.

control structures that require a minimum amount of prior process knowledge, and a minimum number of measured signals, while providing robust performance in terms of process variability and measurement errors. Additionally, the perfusion operation of a bioreactor is naturally a multivariable process, whose performance depends on both inputs: the dilution/perfusion rate and the bleed rate. However, it has been controlled for a long time in a suboptimal way as a single input single output system (Dowd, Kwok, & Piret, 2001a, 2001b; Ozturk et al., 1997). Recently, the potential of using the bleed flow (see Fig. 1) in multivariable control structures has been investigated in several simulation studies in view of a prospective practical implementation: Deschênes, Desbiens, Perrier, and Kamen (2006a, 2006b) have developed an adaptive backstepping strategy for a simple model to simultaneously control the cell and metabolite concentrations, while Sbarciog, Saraiva, and Vande Wouwer (2013) have designed a multivariable nonlinear predictive control strategy based on a more realistic model, for accelerating the growth of cells and controlling the substrate concentration in the effluent. A major issue in controlling cell cultures is the selection of the control criterion, which has to combine the knowledge about the process and the available on-line measurements. Several alternatives have been reported which can be mainly classified as (i) control of the growth rate, to increase system productivity; (ii) control of the metabolite concentrations, to avoid their negative effect such as growth inhibition or death enhancement; (iii) control of the

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Fig. 1. Schematic representation of the perfusion culture.

substrate concentrations, to minimize the formation of toxic byproducts and to avoid the waste of expensive nutrients via the effluent. Each approach has its own advantages and disadvantages. For example, the control of the growth rate, which has long been promoted by many researchers, seems to not have a great impact in the industrial practice. Gnoth et al. (2008) point out that this type of control is more affected by process disturbances and suggest the control of the cell concentration instead, which can be easily estimated in systems without this measurement. On the other hand, numerous (experimental) studies evidence that the control of glucose and glutamine, the two main nutrients, at low levels leads to reduced nutrient consumption as well as to reduced metabolite formation. Hence, the control of the nutrients at appropriate low levels is one of the most important criteria in cell cultures (Xu, Sun, Mathew, Jeevarajan, & Anderson, 2004). In this study, a cascade control structure (Seborg, Edgar, & Mellichamp, 1989) is proposed to simultaneously control the cell and glucose concentrations in a perfused bioreactor. The control structure consists of an inner loop, which uses a feedback linearization control law (Henson & Seborg, 1997) to cancel, as much as possible, the nonlinearity involved in the reaction kinetics, and two outer loops, which employ simple adaptive PI controllers (Åström & Wittenmark, 1995). It is assumed that only the cell and glucose concentrations are measured, the other variables needed for the controller implementation are estimated from these measurements. The feedback linearizing strategy is the basis of many works dedicated to the control of biological systems (see, for instance, Henson, 2006 and references therein). However, the feedback linearization heavily relies on the quality of available information (i.e., model accuracy and on-line measurements). To overcome these problems, several authors have employed some on-line adaptation schemes and nonlinear observers to deal respectively with model uncertainty and unpractical measurements (Smets, Claes, November, Bastin, & Van Impe, 2004). For instance, Coutinho and Vande Wouwer (2013) have proposed a robust approach for continuous bioreactors based on a cascaded-loop strategy. Zhu, Zamamiri, Henson, and Hjortso (2000) have proposed a linear model predictive control strategy based on a linear, discrete-time model for the stabilization of oscillating yeast cultures. Mjalli and Al-Asheh (2005) have compared two nonlinear neural networks (feedback linearizing and model predictive control) based algorithms for controlling an ethanol fermentation process. Farza, Nadri, and Hammouri (2000) have proposed nonlinear observers to estimate the specific growth rate which is the key parameter in bioreactor control. Here, two approaches are presented. The first approach assumes that some approximate knowledge on the process kinetics is available and the inner loop controller is tuned to minimize the effect of the unknown dynamics. An extended Kalman filter (EKF) (Jazwinski, 1970), which employs the approximate model of the inner loop, is designed to estimate the unknown process states required for the control law implementation. The second approach does not require any prior knowledge on the structure and parameters of the process kinetics and employs sliding mode observers (SMO)

(Friedman, Moreno, & Iriarte, 2011) to estimate the needed rates to render the controlled dynamics linear. The outer loop can in principle make use of any kind of controller (Sbarciog, Coutinho, & Vande Wouwer, 2013b). Here, we have chosen the PI controller for its simplicity and wide use in controlling cultivation processes (Gnoth et al., 2008). The novel contribution of this paper is twofold. On the one hand, the perfusion cell culture process is regarded as a multivariable system. Hence a multivariable control structure is proposed to simultaneously manipulate the two process inputs, i.e., the dilution rate and the bleed rate, which are equally important in the efficient operation of the process (presently, such studies are scarce). On the other hand, the potential complexity associated with the regulation of a nonlinear process is significantly reduced by combining principles and tools widely used in control engineering practice, such as the partial feedback linearization approach, and the cascade structure with a simple controller in the outer loop. In addition, state estimation schemes, either the classical EKF or the more fancy SMO, are showing excellent performance and robustness in combination with the proposed control scheme. Moreover, we propose a framework, which is not restricted to a specific process, but may be applied to a general class of cell culture systems. The paper is organized as follows. The next section introduces the class of perfused cell culture systems to which the framework applies, and presents a realistic model that will be used as a case study. Section 3 details the control structure. Firstly, the kineticsbased feedback linearizing controller and the EKF are developed and the accuracy of the proposed estimation scheme is illustrated in the presence of parametric uncertainty and measurement noise. Secondly, the sliding mode observers based on the super-twisting algorithm are introduced and their accuracy is illustrated, and the kinetics-independent feedback linearizing controller is presented. Thirdly, the implementation of the PI controllers, including an anti-windup mechanism, and the receding horizon tuning algorithm are discussed. Section 4 illustrates the effectiveness and robustness of the proposed control scheme by means of simulation results for the two approaches. In the end conclusions are drawn.

2. System dynamics The growth of animal cells is a complex process, which requires the supply of fresh medium rich in nutrients of which some components may be quite expensive. The dynamics of such systems have been modelled with different degrees of details, the existing models ranging from metabolic networks to macroscopic models. Independent of the cell line used, the components which play a crucial role in animal cell cultures are the cells, the two main nutrients glucose and glutamine and the two main byproducts lactate and ammonia. All macroscopic models, built on mass balance principles, include the concentrations of these components as state variables. In perfusion operation (Fig. 1), fresh medium is fed to replenish the consumed nutrients, while an equal volume of spent medium is continuously withdrawn, allowing for the removal of inhibitory components. Cells are retained or recycled back to the reactor by some type of retention device (for instance an acoustic filter). Hence, a general model for perfusion animal cell cultures may be written as

ξ_ ¼ D  ξin D  F  ξ þ K  rðξÞ

ð1Þ

where ξ A R (the set of positive real vectors of dimension 5) represents the state vector which is composed of the concentrations of cells, glucose, glutamine, lactate and ammonia; ξin A R þ 5 is the vector of component concentrations in the influent, which þ5

M. Sbarciog et al. / Control Engineering Practice 32 (2014) 123–135

assumes the form ½0 ξin2 ξin3 0 0T as generally only the nutrients glucose and glutamine are supplied to the bioreactor; D A R þ is the dilution rate, which is defined as the inlet flow scaled by the reactor volume; F A R55 , F ¼ diag½bl; 1; 1; 1; 1 is a diagonal matrix indicating at which rate the components leave the bioreactor: the cells leave the reactor at a rate determined by the bleed ratio bl A ½0; 1, while all the other components leave the reactor at a rate equal to the supply rate; K A R5m is the matrix of stoichiometric coefficients and rðξÞ A R þ m is the vector comprising the reaction rates, with m representing the number of reactions. Frequently, the cells growth, death and maintenance are included in the model, but the other phenomena such as the degradation of substrates into toxic products may also be modelled. The model employed here for illustrating the effectiveness and robustness of the control structure has been developed from batch and fed-batch hybridoma culture results (de Tremblay, Perrier, Chavarie, & Archambault, 1992) and has been long used to design and test control and optimization algorithms. The model expresses that the cell growth is activated by the presence of glucose and glutamine and their death is governed by lactate, ammonia and glutamine concentrations:

ξ_ 1 ¼  bl  Dξ1 þ r1 ðξÞ  r 2 ðξÞ

ð2Þ

ξ_ 2 ¼ Dðξin2  ξ2 Þ  ar1 ðξÞ  r 3 ðξÞ

ð3Þ

ξ_ 3 ¼ Dðξin3  ξ3 Þ  br 1 ðξÞ

ð4Þ

ξ_ 4 ¼  Dξ4 þ cr1 ðξÞ þ dr3 ðξÞ

ð5Þ

ξ_ 5 ¼  Dξ5 þ er 1 ðξÞ

ð6Þ

125

Table 1 Numerical values of the animal cell culture (as in de Tremblay et al., 1992). Y X v =Glc Y X v =Gln Y Lac=Glc Y Amm=Gln μmax kdmax V KGlc KGln kdLac kdAmm kdGln mGlc kmGlc

1:09  102 3:8  10 1.8 0.85 1.09 0.69 0.8 1 0.3 0.01 0.06 0.02

2

1:68  10  4 19

106 cells/mmol 106 cells/mmol mmol/mmol mmol/mmol d1 d1 L mmol/L mmol/L d  1 (mmol/L)  1 d  1 (mmol/L)  1 mmol/L mmol (106 cells)  1 d  1 mmol

where a; b; c; d; e 4 0 are the stoichiometric coefficients, defined as a¼

1 ; Y X v =Glc

b¼

1 ; Y X v =Gln

c¼

Y Lac=Glc ; Y X v =Glc

d ¼ Y Lac=Glc ;

e¼

Y Amm=Gln Y X v =Gln

Fig. 2. Control structure.

ð7Þ and r i ðξÞ, i¼1,2,3 are reaction rates, given by r 1 ðξÞ ¼ μmax 

ξ

ξ

2 3  ξ K Glc þ ξ2 K Gln þ ξ3 1 ¼ μ1 ðξÞ  ξ1

r 2 ðξÞ ¼

kdmax ðμmax  kdLac ξ4 Þðμmax  kdAmm ξ5 Þ kdGln   ξ ¼ μ2 ðξÞ  ξ1 kdGln þ ξ3 1

ξ2

r 3 ðξÞ ¼ mGlc 

kmGlc þ ξ2 ¼ μ3 ðξÞ  ξ1

ð8Þ

ð9Þ

 ξ1 ð10Þ

Hence, the stoichiometric matrix and the vector of reaction rates are given by 2 3 1 1 0 2 3 6 a r 1 ðξÞ 0 1 7 6 7 6 7 6 7 0 0 7 ð11Þ K¼6 6 b 7; rðξÞ ¼ 4 r 2 ðξÞ 5 6 7 r 3 ðξÞ 0 d 5 4 c e

0

0

The numerical values of the model parameters are given in Table 1. 3. Control structure The main objective in controlling a cell culture is to achieve and maintain a high cell density in the reactor. High amounts of

substrates supplied to the culture do not lead to a better and faster growth of the cells (Jain & Kumar, 2008). On the contrary, the medium is spent inefficiently, as large amounts of expensive nutrients are lost via the effluent. Moreover, toxic by-products causing cell death are produced. Therefore, many control implementations consider the regulation of the main nutrient glucose at a reasonable low level to minimize the formation of toxic metabolites (Ozturk et al., 1997). In this paper we design the control structure to achieve a similar goal, i.e., the regulation of cell and glucose concentrations at specified setpoints. The multivariable control structure used to simultaneously control the cell and the glucose concentrations is illustrated in Fig. 2. It consists of (i) a linearizing feedback controller, designed such as the inner loop has a decoupled linear dynamics and (ii) two PI controllers which compute the inputs of the nonlinear controller ξ 1 and ξ 2 . The autotuner determines the optimal parameters of each PI controller by minimizing a predictive control criterion via receding horizon optimization. As the controlled outputs are ξ1 ðtÞ and ξ2 ðtÞ and the manipulated inputs are the bleed rate Db ¼ bl  D and the dilution rate D, the general dynamics (1) for cell and glucose concentrations may be rewritten as

ξ_ 1 ¼  Db  ξ1 þ k1  rðξÞ

ð12Þ

ξ_ 2 ¼ Dðξin2  ξ2 Þ þ k2  rðξÞ

ð13Þ

where k1 and k2 respectively represent the first and second rows of the stoichiometric matrix K. Then the feedback control law, which ensures the full linearization of the dynamics (12), (13) is

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the following dynamics for the controlled outputs is obtained:

given by Db ¼

1

ξ1

ðk1  rðξÞ  λ1 ðξ 1  ξ1 ÞÞ

1 D¼ ðλ2 ðξ 2  ξ2 Þ k2  rðξÞÞ ðξin2  ξ2 Þ

ð14Þ ð15Þ

with ξ 1 , ξ 2 being the inputs of the feedback linearizing controller and λ1, λ2 being the design parameters. Unfortunately, the linearizing control law (14) and (15) cannot be implemented in practice as it requires the knowledge of all reaction rates (structures and parameters) and the measurement of the entire system state. Therefore, we propose two alternatives of the feedback linearizing controller below, which require only the measurement of the cell and glucose concentrations. 3.1. Kinetics-based control Partial feedback linearizing control: A partial linearizing control law may be derived from (14) and (15) under the following assumptions: A1

A model of the cell culture system, which may be not perfectly known, is available. A2 Aside the measurements of the cell and glucose concentrations, the estimates of the system states that are needed to compute the process kinetics are available. These assumptions can accommodate a large number of kinetics and model uncertainties. For a clear presentation, we develop the partially linearizing feedback controller for the system (2)–(6), assuming further that A3

The entire parametric uncertainty lies in the maximum specific growth μmax, which may vary 720% with respect to its nominal value, that is

μmax ¼ μmax ðδÞ ¼ μ max ð1 þ 0:2δÞ; δ A ½  1; 1;

ð16Þ

where μ max stands for the nominal value of μmax and δ is an uncertain parameter lying in the interval ½  1; 1. The death rate is close to zero and therefore it will be neglected in the design of the feedback linearizing control law.

A4

Note that the maximum specific growth μmax is one of the most influential parameters, and that it can be estimated with relatively good accuracy in most practical situations. The idea of associating a large uncertainty (720%) with μmax is to subsume the potential uncertainties on this factor as well as the uncertainties on the Monod factor, where the half saturation coefficient KGlc can be much more uncertain, but at the same time (and logically) is much less influential on the predictive capability of the model. Under Assumptions A1–A4, an approximation of the control law (14) and (15) for the system (2)–(6) is given by Db ¼ D¼

1

ξ1

ðr^ 1 ðξÞ  λ1 ðξ 1  ξ1 ÞÞ

ð17Þ

1 ða  r^ 1 ðξÞ þ r 3 ðξÞ þ λ2 ðξ 2  ξ2 ÞÞ ðξin2  ξ2 Þ

ð18Þ

where r^ 1 ðξÞ ¼ μ max 

ξ

ξ

2 3  ξ K Glc þ ξ2 K Gln þ ξ3 1

Using (17) and (18) in the model (2)–(6) and defining

χ 1 ¼ ξ 1  ξ1 ;

χ 2 ¼ ξ 2  ξ2 ;

ð19Þ

χ_ 1 ¼  λ1 χ 1  ðr 1 ðξ; δÞ  r^ 1 ðξÞÞ þ r 2 ðξÞ

ð20Þ

χ_ 2 ¼  λ2 χ 2 þ aðr1 ðξ; δÞ  r^ 1 ðξÞÞ

ð21Þ

Notice that the dynamics (20) and (21) is not linear since the term r 1 ðξ; δÞ  r^ 1 ðξÞ is not cancelled due to parameter uncertainty and r 2 ðξÞ is not considered in the feedback linearizing based controller. Therefore, we design λ1 and λ2 to minimize the effects of the non-cancelled nonlinearities in (20) and (21) on the state vector χ ≔½χ 1 χ 2 T in the H1 sense. To this end, we embed (20) and (21) into the following quasi-LPV representation (Leith & Leithead, 2000): ( " # " # 0  λ1  0:2δ 1 _ Gwz : χ ¼ χþ ð22Þ w; z¼χ 0  λ2 0 0:2aδ where the disturbance input w models the non-cancelled dynamics, that is 2 3

μ max ξ1 ξ2 ξ3

6 7 w≔4 ðK Glc þ ξ2 ÞðK Gln þ ξ3 Þ 5 r 2 ðξ Þ Then, the parameters λ1 and λ2 are designed to minimize an upper-bound on ‖Gwz ‖1 for all δ A ½  1; 1 using similar steps as in the approach proposed in Dewasme, Coutinho, and Vande Wouwer (2011). Notice that the overall feedback system aims at operating in set point regions such that the death rate r 2 ðξÞ is close to zero. In addition, if ‖Gwz ‖1 is relatively small, we may also assume that Δr 1 ≔r 1 ðξ; δÞ  r^ 1 ðξÞ C 0. Hence, the following simplified dynamics is considered as the model of the inner loop:

ξ_ 1 ¼ λ1 ðξ 1  ξ1 Þ

ð23Þ

ξ_ 2 ¼ λ2 ðξ 2  ξ2 Þ

ð24Þ

ξ_ 3 ¼ Dðξin3  ξ3 Þ br^ 1 ðξÞ

ð25Þ

ξ_ 4 ¼  Dξ4 þ cr^ 1 ðξÞ þ dr3 ðξÞ

ð26Þ

ξ_ 5 ¼  Dξ5 þ er^ 1 ðξÞ

ð27Þ

where D is given by (18). Extended Kalman filter: The nonlinear controller (17) and (18) requires the knowledge of the cell, glucose and glutamine concentrations. Cell and glucose concentrations can be measured online, however they may be affected by measurement noise which can affect the control loop. It is assumed that the glutamine concentration cannot be measured on-line, thus it needs to be estimated. Therefore an extended Kalman filter (EKF) is designed (Sbarciog, Coutinho, & Vande Wouwer, 2013a), which uses the model of the inner loop (23)–(27) to provide estimates of the system states from the (noisy) measurements of cell and glucose concentrations. In practice, the cell concentration can be measured on-line using plunging capacitance probes (as the ones provided by the company FOGALE, http://www.fogale.fr/biotech/pages/ home.php), whereas the glucose concentration can be measured at line with a BioPATsTRACE probe (from SARTORIUS, http:// www.sartorius.com/en/products/bioprocess/process-analysers/bio pat-trace/), which is based on sampling and analysis. In both cases, the information is provided – at the scale of the process dynamics – in almost continuous time and without significant delays. Fig. 3 shows the estimates provided by the EKF, when constant inputs are applied to the inner loop. The estimation is carried out in the scenario of a 20% increase in μmax with respect to the nominal value μ max . Noise free estimates of the cells and glucose concentrations are obtained. A small estimation error may be noticed for the cell and glutamine concentrations due to the

M. Sbarciog et al. / Control Engineering Practice 32 (2014) 123–135

14

2

12 3000

2000

0

5

Input feedback controller

ξ

1

Input feedback controller ξ

4000

Cells concentration: ξ1 (106 cells/L)

5500

5000

13

5000 4500 4000 3500 3000 2500 measured real state estimated

2000 1500

11 15

10

0

5

14.5

15

Glutamine concentration: ξ3 (mmol/L)

0.9

14 13.5

2

Glucose concentration: ξ (mmol/L)

10 Time (d)

Time (d)

13 12.5 12 11.5 measured real state estimated

11 10.5

127

0

5

10

15

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

real state estimated 0

Time (d)

5

10

15

Time (d)

Fig. 3. Estimation of biomass, glucose, and glutamine concentrations provided by the EKF: (a) inputs of the feedback linearizing controller; (b) measured, simulated, and estimated cell concentration; (c) measured, simulated, and estimated glucose concentration; (d) simulated and estimated glutamine concentration.

difference between the real inner loop model and the simplified version assumed by the EKF, which supposes that the cells death rate is negligible. Obviously, this estimation error is influenced by the uncertainty on μmax, as the cells death rate (9) depends on this parameter. Nevertheless, the overall performance will not be affected in steady state (assuming that μmax is time-invariant) by a constant estimation error as the integral action of the outer loop will reject in steady state constant errors.

3.2. Kinetics-independent control An alternative approach for the implementation of the feedback linearizing controller (14) and (15) is the use of growth/ consumption rates estimates in the computation of the control law. One way to estimate the growth rate is to consider it as an unknown input signal to the biomass dynamics. Unknown input reconstruction has received considerable attention lately. Among the existing algorithms, second-order sliding mode observers have proven to be very effective. Here, we use the estimation algorithm presented by De Battista, Picó, Garelli, and Navarro (2012) to estimate the net growth rate and the glucose consumption rate using the measurements of the cell and glucose concentrations. It has been shown (De Battista et al., 2012) that this observer is equivalent to the super-twisting algorithm, hence the employed algorithm exhibits finite time convergence to the time-varying unknown signal (particularly important to preserve the closed loop stability and performance) and robustness. For the development of the sliding mode observer used to estimate the growth rate from the measurement of the biomass concentration, it is assumed (De Battista et al., 2012) that the

process may be cast as x_ ¼ f ðx; tÞ þ μðtÞx

ð28Þ

with x being the biomass concentration (xð0Þ 4 0) and μðtÞ being the growth rate, whose derivative accepts an absolute bound, i.e., jμ_ j o ρ. Hence, no structure of the kinetics μðtÞ is imposed or required. Then, the proposed observer for (28) is   f ðx; tÞ þ ρz2 þ2ρβjσ j1=2 signðσ Þ z1 z_ 1 ¼ x z_ 2 ¼ α signðσ Þ

σ ¼ ρ  1 lnðx=z1 Þ μ^ ¼ ρz2

ð29Þ

where z1 ðz1 ð0Þ 4 0Þ is the estimated biomass and z2 is an image of μ^ , the estimated growth rate; σ is an image of the observer error and α, β are observer parameters. Here, the process and observer structures (28) and (29) are used twice to estimate on one hand the net growth rate and on the other hand the consumption rate of glucose. Recalling that ξ1 and ξ2 respectively represent biomass and glucose concentrations, then two sliding mode observers (SMO1 and SMO2) can be designed (Sbarciog, Coutinho, & Vande Wouwer, 2013c) assuming the structure of the output dynamics of the form

ξ_ 1 ¼ f 1 ðξ1 ; tÞ þ μðtÞξ1

ð30Þ

ξ_ 2 ¼ f 2 ðξ2 ; tÞ þ φðtÞξ2

ð31Þ

to estimate in finite time the signals μðtÞ and φðtÞ from the measurements of the cell and glucose concentrations:   f ðξ ; tÞ z_ 1 ¼ 1 1 þ ρ1 z2 þ 2ρ1 β1 jσ 1 j1=2 signðσ 1 Þ z1

ξ1

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M. Sbarciog et al. / Control Engineering Practice 32 (2014) 123–135

z_ 2 ¼ α1 signðσ 1 Þ

σ1 ¼ ρ

1 1

lnðξ1 =z1 Þ

μ^ ¼ ρ1 z2 z_ 3 ¼



_ jo ρ2 , z1 (z1 ð0Þ 4 0) and z3 ðz3 ð0Þ 4 0Þ are with jμ_ j o ρ1 and jφ respectively the estimates of the cell and glucose concentrations. Note that a straightforward match exists between the dynamics (30) and the biomass equation (12), thus the estimated signal μðtÞ readily represents the net growth rate ðμðtÞ ¼ μ1 ðξÞ  μ2 ðξÞÞ:

ð32Þ

f 2 ðξ2 ; tÞ

ξ2

 þ ρ2 z4 þ 2ρ2 β2 jσ 2 j1=2 signðσ 2 Þ z3

f 1 ðξ1 ; tÞ ¼  Db  ξ1 ;

μðtÞ ¼ k1  rðξÞ=ξ1

For the equivalence of the dynamics (31) to the glucose equation (13)

z_ 4 ¼ α2 signðσ 2 Þ

f 2 ðξ2 ; tÞ ¼ D  ðξin2  ξ2 Þ;

σ 2 ¼ ρ2 1 lnðξ2 =z3 Þ φ^ ¼ ρ2 z4

φðtÞ ¼ γ ðtÞ  ξ1 =ξ2

where γ ðtÞ ¼ k2  rðξÞ=ξ1 ¼ aμ1 ðξÞ  μ3 ðξÞ is the net glucose consumption rate. Hence the estimated signal φðtÞ does not represent

ð33Þ

−1

Bleed rate: Db (d ); Dilution rate: D (d−1)

1.8 D D

1.6

b

1.4 1.2 1 0.8 0.6 0.4 0.2 0

0

5

10

15

20

25

30

35

40

Time (d)

18 measured real state estimated

7000

Glucose concentration: ξ2 (mmol/L)

Cells concentration: ξ1 (106 cells/L)

8000

6000 5000 4000 3000 2000 1000

measured real state estimated

17 16 15 14 13 12 11 10

0

5

10

15

20

25

30

35

9

40

0

5

10

15

Time (d)

1

25

30

35

40

25

30

35

40

0.01 calculated estimated

0.8

calculated estimated

0.009 0.008 Net consumption rate:− γ

Net growth rate: μ

20 Time (d)

0.6

0.4

0.2

0.007 0.006 0.005 0.004 0.003 0.002

0

0.001 −0.2

0

5

10

15

20 Time (d)

25

30

35

40

0

0

5

10

15

20 Time (d)

Fig. 4. Open loop estimates of cells and glucose concentrations, and net growth and glucose consumption rates: (a) process inputs; (b) measured, simulated, and estimated cells concentration; (c) measured, simulated, and estimated glucose concentration; (d) simulated and estimated net growth rate; (e) simulated and estimated net consumption rate.

M. Sbarciog et al. / Control Engineering Practice 32 (2014) 123–135

Thus, using the estimates μ^ ðtÞ and γ^ ðtÞ provided by the observers (32), (33), and (34) the approximation of the linearizing feedback controller (14) and (15) becomes

the glucose consumption rate; however this can be computed from the estimated signal as

γ ðtÞ ¼ φðtÞ  ξ2 =ξ1

129

ð34Þ

1

Db ¼

since measurements/estimates of ξ1 and ξ2 are available. Fig. 4 shows the estimates provided by the two observers for some arbitrary process inputs, when the process outputs are affected by measurement noise. In animal cell cultures the glucose concentration is more prone to measurement noise than the cells concentration, which can be measured quite accurately. The observer parameters have been set to α1 ¼ 1, β 1 ¼ 2, ρ1 ¼ 0:75, α2 ¼ 1:75, β2 ¼ 2:5, ρ2 ¼ 1, which ensures fast convergence and accurate estimates.

D¼

ξ1

ðμ^ ðtÞξ1  λ1 ðξ 1  ξ1 ÞÞ

ð35Þ

1 ð  γ^ ðtÞξ1 þ λ2 ðξ 2  ξ2 ÞÞ ðξin2  ξ2 Þ

ð36Þ

where λ1, λ2 are the controller parameters (to be tuned) and ξ 1 , ξ 2 are the controller inputs. Under the assumption that the estimation errors k1  rðξÞ=ξ1  μ^ ðtÞ and k2  rðξÞ=ξ1  γ^ ðtÞ are negligible, the linear decoupled dynamics are obtained as the model of the

7000

14 13.5

Glucose concentration: ξ2 (mmol/L)

6

Cells concentration: ξ1 (10 cells/L)

6500 6000 5500 5000 4500 4000 3500 3000

13 12.5 12 11.5 11

2500 2000

0

5

10

15

20

25

10.5

30

0

5

10

Time (d)

20

25

30

20

25

30

4000

12

3000

11

0

5

10

15

20

2

1.5

1

0.5

0

10 30

25

b

b

13

−1 Bleed rate: D (d ), Dilution rate: D (d )

5000

D D

2.5

−1

14

ξ

6000

2

15

Input feedback controller

Input feedback controller

ξ

1

7000

2000

15 Time (d)

0

5

10

Time (d)

15 Time (d)

K

T

p1 p2

0.6

0.4

0.2

0

−0.2

T

i2

0.2 Integral times: Ti1, Ti2

0.8 Proportional gains: K p1, Kp2

i1

K

0.15

0.1

0.05

0

0

5

10

15 Time (d)

20

25

30

−0.05

0

5

10

15

20

25

30

Time (d)

Fig. 5. Closed loop response for nominal values of the system parameters – kinetics-independent controller: (a) measured cell concentration; (b) measured glucose concentration; (c) inputs of the linearizing feedback controller; (d) process inputs; (e) proportional gains PI controllers; (f) integral times PI controllers.

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time representation of the PI controllers is given by

inner loop:

ξ_ 1 ¼ λ1 ðξ 1  ξ1 Þ

ð37Þ

ξ_ 2 ¼ λ2 ðξ 2  ξ2 Þ

ð38Þ

ξ j ðkÞ ¼ ξ j ðk  1Þ þ K pj ðej ðkÞ  ej ðk  1ÞÞ þ

Independent of the implemented linearizing control law (either (17), (18) or (35), (36)), classical PI controllers are used in the outer loop to ensure the tracking of the setpoint changes and to reject the disturbances acting on the process. The incremental discrete

ξm j ðkÞ ¼

5500

ð39Þ

T s ej ðkÞ

Bj ðq  1 Þ 1 ξ ðkÞ þ υðkÞ; Aj ðq  1 Þ j 1q1

ð40Þ

j ¼ 1; 2

14

5000

Glucose concentration: ξ (mmol/L) 2

Cells concentration: ξ1 (106 cells/L)

T ij

where K pj and T ij respectively represent the proportional gains ref and the integral time constants; ej ðtÞ ¼ ξj ðtÞ  ξj ðtÞ with j¼1,2 are the control errors; Ts represents the sampling period and k is the discrete time index ðt ¼ kT s Þ. Conventionally, the parameters K pj and T ij are time-invariant. However, to cope with the nonlinear nature of the process (inexact feedback linearization) we adapt the controller parameters periodically. To this end, we use a discrete time representation of the inner loop ((23), (24) or (37), (38)):

3.3. Receding horizon optimization based tuning of PI controllers

4500 4000 3500 3000 2500 2000

K pj

0

5

10

15

20

25

13.5

13

12.5

12

11.5

11

30

0

5

10

Time (d)

15

20

25

30

20

25

30

Time (d)

12

3000

2000

11

0

5

10

15

20

10 30

25

D D

2

b

1.5

1

b

4000

ξ

13

Input feedback controller

Input feedback controller

ξ

2

1

5000

−1 Bleed rate: D (d ),Dilution rate: D (d −1)

14

0.5

0

0

5

10

15 Time (d)

Time (d)

0.9

0.2 K

T

p1

0.8

i1

0.15

K

T

i2

0.7 0.1 Integral times: Ti1, Ti2

Proportional gains: K p1, Kp2

p2

0.6 0.5 0.4 0.3 0.2

0.05 0 −0.05 −0.1

0.1 −0.15

0 −0.1

0

5

10

15 Time (d)

20

25

30

−0.2

0

5

10

15

20

25

30

Time (d)

Fig. 6. Closed loop response for a 20% increase on μmax – kinetics-based controller: (a) measured cells concentration; (b) measured glucose concentration; (c) inputs of the linearizing feedback controller; (d) process inputs; (e) proportional gains PI controllers; (f) integral times PI controllers.

M. Sbarciog et al. / Control Engineering Practice 32 (2014) 123–135 m 1 where ξm Þ and Bj ðq  1 Þ are 1 and ξ2 are the model outputs; Aj ðq polynomials in the shift operator q found by discretizing the dynamics (23), (24) or (37), (38) with a sampling period Ts; υðtÞ is the uncorrelated noise with zero mean value. Using (39) and (40) iteratively, predictions of the process outputs over a horizon m Np (ξj ðk þ lÞ, l ¼ 1…N p ) are computed. Then the parameters of the

PI controllers are found by minimizing Np

J i ¼ ∑ ðξi ðk þ lÞ  ξi ðk þ lÞÞ2 þ αi ðΔξ i ðk þ l  1ÞÞ2 m

ð41Þ

with Δξ i ðkÞ ¼ ξ i ðkÞ  ξ i ðk  1Þ, i¼ 1,2. In principle the tuning of the PI controllers may be done at every sampling instant. However,

16

5000

Glucose concentration: ξ2 (mmol/L)

measured real state estimated

4000

1

Cells concentration: ξ (106 cells/L)

ref

l¼1

6000

3000

2000

1000

0

131

0

5

10

15

20

25

15

14

13

12

11

10

30

measured real state estimated

0

5

10

Time (d)

15

20

25

30

Time (d)

Glutamine concentration: ξ3 (mmol/L)

4 real state estimated

3.5 3 2.5 2 1.5 1 0.5 0

0

5

10

15

20

25

30

Time (d) Fig. 7. Estimation of cell, glucose, and glutamine concentrations provided by the EKF: (a) measured, simulated, and estimated cells concentration; (b) measured, simulated, and estimated glucose concentration; (c) simulated and estimated glutamine concentration.

−3

1.2

12 calculated estimated

1

x 10

calculated estimated

11 Net consumption rate: − γ

Net growth rate: μ

10 0.8 0.6 0.4 0.2

9 8 7 6 5 4

0 3 −0.2

0

5

10

15 Time (d)

20

25

30

2

0

5

10

15

20

25

30

Time (d)

Fig. 8. Estimation of the net growth and consumption rates provided by the SMO1 and SMO2: (a) simulated and estimated net growth rate; (b) simulated and estimated net consumption rate.

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this leads to unnecessary computational effort. Therefore, the tuning is triggered periodically (every Na samples, with Na r N p ) as long as the control errors are higher than some thresholds. Note that the physical control inputs Db and D may saturate at their respective lower and upper bounds (0 rDb rD, 0 r D r Dmax ), which implies also the saturation of the feedback linearizing controller inputs. Hence, an anti-windup mechanism is included in the control law (39), which consists of the augmentation of the control error at time instant k  1 with the difference existing between the a admissible control ξ j ðk 1Þ and the computed control ξ j ðk  1Þ: , a

enj ðk  1Þ ¼ ej ðk  1Þ þ ðξ j ðk  1Þ  ξ j ðk  1ÞÞ

K pj þ

K pj T ij

ξ j ðkÞ ¼ ξ

T ij

! Ts

ð42Þ

Glucose concentration: ξ2 (mmol/L)

15

4000

3000

2000

1000

0

5

10

15

20

25

14.5 14 13.5 13 12.5 12 11.5 11 0

30

5

10

Time (d)

16

4000 13 3000 12 2000 11

1000

15

20

−1

14

10

25

30

b

1 0.8 0.6 0.4 0.2 0 0

10 30

25

D D

1.2

b

5000

Input feedback controller ξ

ξ

2

1

15

5

20

1.4 −1 Bleed rate: D (d ), Dilution rate: D(d )

6000

0

15 Time (d)

7000

Input feedback controller

T s ej ðkÞ  K pj enj ðk  1Þ

Figs. 5 and 6 present simulation results of the closed loop employing the two control approaches. For the kinetics-based controller, the gains have been tuned as mentioned in Section 3 (λ1 ¼ 11:8319, λ2 ¼ 6:9534), while for the kinetics-independent controller, their tuning is very simple and does not require any sophisticated

5000

0

!

4.1. Performance evaluation of the closed loop system

1

Cells concentration: ξ (106 cells/L)

K pj þ

K pj

4. Simulation results and discussion

6000

0

a j ðk  1Þ þ

5

10

15

20

25

30

Time (d)

Time (d)

0.6

0.1 K

T

p1

i1

K

0.5

T

0.08

i2

0.4

0.06 i1

Integral times: T , Ti2

p1

Proportional gains: K , K

p2

p2

0.3 0.2 0.1 0 −0.1

0.04 0.02 0 −0.02

0

5

10

15 Time (d)

20

25

30

−0.04 0

5

10

15

20

25

30

Time (d)

Fig. 9. Closed loop response for the kinetics mismatch case – kinetics-dependent controller: (a) measured cell concentration; (b) measured glucose concentration; (c) inputs of the linearizing feedback controller; (d) process inputs; (e) proportional gains PI controllers; (f) integral times PI controllers.

M. Sbarciog et al. / Control Engineering Practice 32 (2014) 123–135

glutamine concentrations provided by the EKF. The outputs of the kinetics-independent controller are computed based on the estimates of cell and glucose concentrations, net growth rate and glucose consumption rate provided by the two SMO observers. The simulation results presented here include the controlled outputs ξ1, ξ2 and their respective setpoints; the inputs of the inner loop controller computed by the two PI controllers represented with continuous line and the inputs which are admissible (to comply with the physical constraints on the flow rates) represented with dashed line; the process inputs calculated by the linearizing feedback controller: the dilution and the bleed rates; the parameters of the PI controllers. Fig. 5 illustrates the system closed loop response for several setpoint changes, in the case of nominal values of the system

mathematical procedure. Here, the gains have been selected as λ1 ¼ λ2 ¼ 5, based on some knowledge of the process dynamics. A sampling period of 0.05 d is used. For the tuning algorithm of the PI controllers the prediction horizon Np has been set to 20 samples and the penalty coefficients α1 and α2 in the cost functions (41) have been chosen to be equal to 10. The PI parameters are adapted every N a ¼ 10 samples as long as the control errors are higher than the imposed thresholds (i.e., je1 ðtÞj 4 50, je2 ðtÞj 4 0:5), otherwise no adaptation occurs. The optimization problems are solved using the Nelder–Mead algorithm. Constraints on the dilution and bleed rates 1 are imposed: 0 r D r 3:75 d , 0 r Db r D. All the investigations are carried out in the presence of measurement noise. The outputs of the kinetics-based controller are computed based on the estimates of cell, glucose and

Glucose concentration: ξ (mmol/L) 2

15

5000

4000

1

Cells concentration: ξ (106 cells/L)

6000

3000

2000

1000

0

0

5

10

15

20

25

14.5 14 13.5 13 12.5 12 11.5 11

30

0

5

10

Time (d)

5000

15

4000

14

3000

13

2000

12

1000

11

2

15

20

Input feedback controller ξ

Input feedback controller

ξ

10

25

30

20

25

30

1.5 Bleed rate: Db (d ),Dilution rate: D (d −1)

16

5

20

b

1

0.5

0

10 30

25

D D

−1

6000 1

16

0

15 Time (d)

7000

0

133

0

5

10

Time (d)

15 Time (d)

1.8

0.14 K

T

p1

1.6

i1

K

i2

i2

1.4 Integral times: T , T

1.2

0.1

i1

p1

Proportional gains: K , K p2

T

0.12

p2

1 0.8 0.6

0.08 0.06 0.04

0.4 0.02

0.2 0

0

5

10

15 Time (d)

20

25

30

0

0

5

10

15

20

25

30

Time (d)

Fig. 10. Closed loop response for the kinetics mismatch case – kinetics-independent controller: (a) measured cell concentration; (b) measured glucose concentration; (c) inputs of the linearizing feedback controller; (d) process inputs; (e) proportional gains PI controllers; (f) integral times PI controllers.

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parameters, for the kinetics-independent controller. Similar results (not shown here) are obtained also with the kinetics-based controller. Fast tracking of the imposed reference signals is achieved: in closed loop a settling time of 2–3 d is achieved, while open loop operation leads to settling times in the range 5–10 d or even more. Hence, the proposed control structure increases the loop speed at least two times. This property is preserved also in the case of parametric uncertainty such as illustrated in Fig. 6 for the kinetics-based controller, where an increase of 20% in the magnitude of the maximum specific growth of the process has been considered. 4.2. Robustness with respect to kinetics structure mismatch Not only the parameters of the reaction kinetics may not be accurately known, but also the assumed structure may differ substantially from the real one. In this section we investigate the influence of the kinetics structure mismatch on the performance of the control loop. To this end we assume that the real process has the kinetics r 1 ðξÞ ¼ μmax 

ξ2 ξ3 KI Lac KI Amm    ξ K Glc þ ξ2 K Gln þ ξ3 KI Lac þ ξ4 KI Amm þ ξ5 1

¼ μ1 ðξÞ  ξ1

ð43Þ

kdmax ξ 1 þ ðkdAmm =ξ5 Þn 1 ¼ μ2 ðξÞ  ξ1

ð44Þ

r 2 ðξÞ ¼

which may characterize cell cultures (Jang & Barford, 2000), while the model preserves the kinetics given by (8), (9). The additional parameters appearing in the kinetics (43) and (44) have been selected as KI Lac ¼ 90 mmol=L, KI Amm ¼ 15 mmol=L, kdAmm ¼ 4:5 mmol=L, n¼2, while the others have been preserved as indicated in Table 1. The parameters of the linearizing feedback controllers (for both kineticsbased and kinetics-independent approaches) and of the tuning algorithm have been kept constant as given in Section 4.1, thus no retuning of the controllers has been performed. Figs. 7 and 8 show respectively the estimations provided by the EKF and by the SMO1 and SMO2. As expected, accurate estimations of the net growth rate and consumption rate are obtained since the sliding mode observers do not use a structure of the kinetics. On the other hand, the estimation errors when the EKF is employed are quite large, especially for the glucose and glutamine concentrations. Nevertheless, they do not affect the quality of the control, and both approaches produce comparable results in terms of control efficiency (Figs. 9 and 10), in spite of the significant difference between the kinetics of the process and those of the assumed model. 4.3. Discussion As shown by the closed loop simulation results, the proposed framework is an effective tool in controlling perfusion cell cultures. Both approaches, either kinetics-based or kinetics-independent, are robust with respect to parametric and kinetic uncertainties and with respect to measurement noise. While this feature is a direct consequence of using sliding mode observers to estimate the rates as unknown inputs in the kinetics-independent approach, in the kinetics-based approach the robustness is ensured by the structure and the design of the control loop. A comparison of the two approaches aiming at establishing the superiority of one over the other is pointless, as they have been provided as solutions for two distinct situations which may occur in practice. Each approach inherits the advantages and disadvantages of the estimation algorithm it employs: in the kinetics-based approach the estimation errors depend on the accuracy of the model kinetics, however the noise is almost entirely filtered out, which leads to a smoother

control law; in the kinetics-independent approach a fast converging estimation will be more influenced by the noise, which generates larger fluctuations in the control law.

5. Conclusions In this paper a simple multivariable output-feedback control strategy has been presented for the regulation of cell and glucose concentrations in perfusion cell cultures. It assumes a cascade form, where a linearizing feedback controller is used in the inner loop to provide decoupled linear dynamics and the two adaptive PI controllers are used in the outer loops for performance and disturbance rejection. It is supposed that only measurements of the cell and glucose concentrations are available, all the other process variables needed for the implementation of the control law are estimated. In the field of cell cultures control the proposed cascade strategy brings several innovative aspects: (i) it provides a solution for the multivariable control, which allows the simultaneous consideration of the productivity and economics in the system operation; (ii) it can be applied to a general class of perfusion systems; (iii) it has two approaches, a kinetics-based one for the case when an approximate system model is available and a kinetics-independent one which allows an effective control without requiring the laborious work for a model identification. Moreover, it is simple to implement and provides great performance and robustness properties.

Acknowledgments The authors gratefully acknowledge the support of the FEDER program 2007–2013, in the framework of Hainaut-Biomed, and the research project OCPAM. This paper presents research results of the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme, initiated by the Belgian State, Science Policy Office. The authors also gratefully acknowledge the support of FNRS and CNPq in the framework of a bilateral research project led by Michel Gevers (UCL, Belgium) and Carlos E. de Souza (LNCC, Brazil). The scientific responsibility rests with the author(s). References Åström, K., & Wittenmark, B. (1995). Adaptive control. Boston: Addison-Wesley. Banik, G., & Heath, C. (1995). Partial and total cell retention in a filtration base homogeneous perfusion reactor. Biotechnology Progress, 11, 584–588. Coutinho, D., & Vande Wouwer, A. (2013). A robust non-linear feedback control strategy for a class of bioprocesses. IET Control Theory & Applications, 7(6), 829–841. Dalm, M., Cuijten, S., van Grunsven, W., Tramper, J., & Martens, D. (2004). Effect of feed and bleed rate on hybridoma cells in an acoustic perfusion bioreactor: Part 1. Cell density, viability and cell-cycle distribution. Biotechnology and Bioengineering, 88, 547–557. De Battista, H., Picó, J., Garelli, F., & Navarro, J. (2012). Reaction rate reconstruction from biomass concentration measurement in bioreactors using modified second-order sliding mode algorithms. Bioprocess and Biosystems Engineering., 35, 1615–1625. de Tremblay, M., Perrier, M., Chavarie, C., & Archambault, J. (1992). Optimization of fed-batch culture of hybridoma cells using dynamic programming: Single and multi feed cases. Bioprocess Engineering, 7, 229–234. Deschênes, J.-S., Desbiens, A., Perrier, M., & Kamen, A. (2006a). Multivariable nonlinear control of biomass and metabolite concentrations in a high-celldensity perfusion bioreactor. Industrial & Engineering Chemistry Research, 45, 8985–8997. Deschênes, J.-S., Desbiens, A., Perrier, M., & Kamen, A. (2006b). Use of cell bleed in a high cell density perfusion culture and multivariable control of biomass and metabolite concentrations. Asia-Pacific Journal of Chemical Engineering, 1, 82–91. Dewasme, L., Coutinho, D., & Vande Wouwer, A. (2011). Adaptive and robust linearizing control strategies for fed-batch cultures of microorganisms

M. Sbarciog et al. / Control Engineering Practice 32 (2014) 123–135

exhibiting overflow metabolism. In: J.-A. Cetto, J.-L. Ferrier, & J. Filipe (Eds.), Informatics in control, automation and robotics. Lecture notes in electrical engineering, Vol. 89 (pp. 283–305). Berlin, Heidelberg: Springer-Verlag. Dowd, J., Kwok, K., & Piret, J. (2001a). Glucose-based optimization of CHO-cell perfusion cultures. Biotechnology and Bioengineering, 75, 252–256. Dowd, J., Kwok, K., & Piret, J. (2001b). Predictive modeling and loose-loop control for perfusion bioreactors. Biochemical Engineering Journal, 9, 1–9. Farza, M., Nadri, M., & Hammouri, H. (2000). Nonlinear observation of specific growth rate in aerobic fermentation. Bioprocess and Biosystems Engineering, 23 (4), 359–366. Friedman, L., Moreno, J., & Iriarte, R. (2011). Sliding modes after the first decade of the 21st century. Berlin, Heidelberg: Springer-Verlag. Gnoth, S., Jenzsch, M., Simutis, R., & Lübbert, A. (2008). Control of cultivation processes for recombinant protein production: A review. Bioprocess and Biosystems Engineering, 31, 21–39. Henson, M., & Seborg, D. (1997). Nonlinear process control. Upper Saddle River: Prentice Hall, NJ. Henson, M. A. (2006). Biochemical reactor modeling and control. IEEE Control Systems Magazine, 26(4), 54–62. Jain, E., & Kumar, A. (2008). Upstream processes in antibody production: Evaluation of critical parameters. Biotechnology Advances, 26, 46–72. Jang, J., & Barford, J. (2000). An unstructured kinetic model of macromolecular metabolism in batch and fed-batch cultures of hybridoma cells producing monoclonal antibody. Biochemical Engineering Journal, 4, 153–168. Jazwinski, A. (1970). Stochastic processes and filtering theory. New York: Academic Press. Komolpis, K., Udomchokmongkol, C., Phutong, S., & Palaga, T. (2010). Comparative production of a monoclonal antibody specific for enrofloxacin in a stirred-tank bioreactor. Journal of Industrial and Engineering Chemistry, 16, 567–571. Leith, D.-J., & Leithead, W.-E. (2000). Survey of gain-scheduling analysis and design. International Journal of Control, 73, 1001–1025.

135

Mjalli, F. S., & Al-Asheh, S. (2005). Neural-networks-based feedback linearization versus model predictive control of continuous alcoholic fermentation process. Chemical Engineering and Technology, 28(10), 1191–1200. Ozturk, S., Thrift, J., Blackie, J., & Naveh, D. (1997). Real-time monitoring and control of glucose and lactate concentrations in a mammalian cell perfusion reactor. Biotechnology and Bioengineering, 53, 372–378. Sbarciog, M., Coutinho, D., & Vande Wouwer, A. (2013a). An adaptive cascade structure for the estimation and control of perfusion animal cell cultures. In Preprints of the 12th IFAC symposium on computer applications in biotechnology (pp. 284–289), Mumbai, India. Sbarciog, M., Coutinho, D., & Vande Wouwer, A. (2013b). A cascade MPC-feedback linearizing strategy for the multivariable control of animal cell cultures. In Preprints of the 9th IFAC symposium on nonlinear control systems (pp. 247–252), Toulouse, France. Sbarciog, M., Coutinho, D., & Vande Wouwer, A. (2013c). Kinetics independent multivariable robust control of animal cell cultures. In Proceedings of ICSTCC (pp. 479–484), Sinaia, Romania. Sbarciog, M., Saraiva, I., & Vande Wouwer, A. (2013). Accelerating animal cell growth in perfusion mode by multivariable control: Simulation studies. Bioprocess and Biosystems Engineering, 36, 517–530. Seborg, D., Edgar, T. F., & Mellichamp, D. A. (1989). Process dynamics and control. New York: John Wiley & Sons. Smets, I. Y., Claes, J. E., November, E. J., Bastin, G. P., & Van Impe, J. F. (2004). Optimal adaptive control of (bio)chemical reactors: Past, present and future. Journal of Process Control, 14, 785–805. Xu, Y., Sun, J., Mathew, G., Jeevarajan, A., & Anderson, M. (2004). Continuous glucose monitoring and control in a rotating wall perfused bioreactor. Biotechnology and Bioengineering, 87, 473–477. Zhu, G.-Y., Zamamiri, A., Henson, M. A., & Hjortso, M. A. (2000). Model predictive control of continuous yeast bioreactors using cell population balance models. Chemical Engineering Science, 55, 6155–6167.