ARTICLE Software Sensor Design Considering Oscillating Conditions as Present in Industrial Scale Fed-Batch Cultivations V. Lyubenova,1 S. Junne,2 M. Ignatova,1 P. Neubauer2 1
Institute of System Engineering and Robotics, Bulgarian Academy of Sciences, Acad. G. Bontchev str. Bl. 2, PO. Box 79, Sofia, Bulgaria; telepohone: þ359-2870-9179; fax: þ359 2 8703 361; e-mail:
[email protected] 2 Chair of Bioprocess Engineering, Department of Biotechnology, Technische Universita¨t Berlin, Berlin, Germany
Introduction ABSTRACT: Investigations of inhomogeneous dynamics in industrial-scale bioreactors can be realized in laboratory simulators. Such studies will be improved by on line observation of the growth of microorganisms and their product synthesis at oscillating substrate availability which represents the conditions in industrial-scale fed-batch cultivations. A method for the kinetic monitoring of such processes, supported by on line measurements accessible in industrial practice, is proposed. It consists of a software sensor (SS) system composed of a cascade structure. Process kinetics are simulated in models with a structure including time-varying yield coefficients. SSs for measured variable kinetics have classical structures. The SS design of unmeasured variables is realized after a linear transformation using a logarithmic function. For these software sensors, a tuning procedure is proposed, at which an arbitrary choice of one tuning parameter value that guarantees stability of the monitoring system allows the calculation of the optimal values of six parameters. The effectiveness of the proposed monitoring approach is demonstrated with experimental data from a glucose-limited fed-batch process of Bacillus subtilis in a laboratory two-compartment scale down reactor which tries to mimic the conditions present in industrial scale nutrientlimited fed-batch cultivations. Biotechnol. Bioeng. 2013;110: 1945–1955. ß 2013 Wiley Periodicals, Inc. KEYWORDS: software sensor design; industrial scale fedbatch processes; scale down reactor; inhomogeneities; tuning procedure
Lyubenova and Junne contributed equally to this work. Correspondence to: V. Lyubenova Contract grant sponsor: DFG Contract grant number: NE 1360/3-1; NE 1360/2-1 Contract grant sponsor: Bulgarian National Fund ‘‘Scientific Researches’’ Contract grant number: DTK—02/27/17.12.2009 Received 23 June 2012; Revision received 12 January 2013; Accepted 5 February 2013 Accepted manuscript online 21 February 2013; Article first published online 26 March 2013 in Wiley Online Library (http://onlinelibrary.wiley.com/doi/10.1002/bit.24870/abstract) DOI 10.1002/bit.24870
ß 2013 Wiley Periodicals, Inc.
It is described, that an unwanted substrate gradient in industrial-scale bioreactors appears near the feeding zone, especially if substrate limited fed-batch procedures are applied (Bylund et al., 1998; Enfors et al., 2001). While the concentration of the high-concentrated feed solution close to the inlet is high, leading to substrate excess, other regions of the reactor are characterized by substrate depletion. Due to the movement of cells in the turbulent fluid flow of a bioreactor, they are continuously exposed to alternating environmental conditions between substrate excess and starvation (Neubauer et al., 2005). Similar observations can be made for the distribution of oxygen, compounds added for pH regulation and others (Neubauer and Junne, 2010). Such inhomogeneous processes show a high batch to batch variability compared to well mixed processes (i) due to the continuous change of the composition of the liquid phase and the hydrodynamic conditions (volume, viscosity, density) during a cultivation, and (ii) the strong dependence of the biological system from the local environment and its fast reaction and adaptation which is in the order of milliseconds for metabolic fluxes and in the order of seconds to minutes for adaptation at the level of gene expression. A current obstacle in studying inhomogeneities in industrial-scale bioreactors is the lack of on line measurements of main process variables under consideration of an oscillating environment and its impacts for growth and product synthesis. The effectiveness of investigations in the scale down reactors would be improved if relevant parameters could be estimated on line under oscillating conditions. The challenge is to monitor a multitude of parameters which influence the biological system on one side with a limited number of available on line sensors on the other side. Therefore it is widely accepted that the only way to properly control a large scale bioprocess is to include feed-back control strategies, which allow the process to be guided with a steady adaptation.
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A widely used approach for process monitoring is to apply a software sensor (SS) to track unmeasured variables (Assis and Filho, 2000; Bastin and Dochain, 1990). A model-based SS is an algorithm built from a dynamic model of a process which uses certain on line measured parameters in order to estimate either variables which cannot be measured on line, or to estimate unknown parameters. Therefore the operational model has to be (i) as accurate as possible to mimic the main characteristics and dynamics of the process, and (ii) simple enough for monitoring and control design. This is not a trivial task due to nonlinear parameter dependence in instationary processes and due to the lack of reproducibility of experiments. The basis of the adaptive observers/estimators design is a general dynamic model as proposed by Bastin and Dochain (1990). These software sensors are considered as classical ones and have been widely applied in the past simultaneously with other approaches for nonlinear systems, such as extended Kalman and Luenberger filters (Soons et al., 2006; Veloso et al., 2009), neural-network based observers (Acun˜a et al., 1998; Georgieva and Feyo de Azevedo, 2009), the high-gain approach (Selis¸teanu et al., 2012), multirate observers (Lyubenova et al., 2011), and others. Many industrial applications for SS are reported, mainly in the area of refinery processes, the pulp and paper industry and for biotechnological processes for biomass estimation and fed-batch process control (Fortuna et al., 2007). For example, the biomass and product concentration in fed-batch processes with yeast was validated with industrial fermentation data with a multiway projection to latent structures (MPLS) method (Thaysen and Jørgensen, 2004). The glucose concentration in industrial fed-batch cultivations for antibiotic production was estimated with a fuzzy neural network model basically using the oxygen uptake rate and airflow as input variables (Imanishi et al., 2002). Recently, a new approach for monitoring very complex biotechnological processes has been proposed (Ignatova and Lyubenova, 2011; Lyubenova and Ignatova, 2011). This approach is based on reduced operational models which present process kinetics with single terms including time-varying yield coefficients that comprise unmodeled dynamics. In this article, a new stepwise SS for kinetic monitoring of aerobic fed-batch bioprocesses at industrial-scale bioreactors is proposed. The monitoring scheme is based on a SS with a cascade structure that makes use of a novel formulation of the process kinetics (Ignatova and Lyubenova, 2011). A model for such bioprocesses is derived simply considering measurements which are commonly available in industrial practice and includes key process parameters only. A software sensor system (SSS) of the carbon source consumption rate and three time-varying yield coefficients is obtained in two steps using on line measurements of biomass concentration, specific oxygen consumption and carbon dioxide production rates. The SSS is tuned and applied using
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data from fed-batch cultivations with Bacillus subtilis performed in a scale down bioreactor (Junne et al., 2011). Hence, the methodology is developed for process optimization in the technical scale under industrial-scale relevant conditions.
Methods Process Model Usually, bioprocesses are simulated based on the biochemical network of reactions considering the corresponding stoichiometry. For process model derivation, a choice of key variables has to be made from the expert’s point of view. The model including the dynamics of these variables is assumed to be appropriate for the derivation of monitoring and control algorithms. For the investigated class of processes, on line measured or real-time estimated parameters are the biomass concentration, the specific oxygen consumption rate, and the respiratory quotient. The glucose concentration is not measured on line, but its dynamics are considered. The following reduced model is applied: dCX F ¼ RX ðtÞ CX V dt
(1)
dCS F F ¼ RS ðtÞ þ CS0 CS V V dt
(2)
dCO2 F ¼ qRO2 ðtÞX CO2 þ kL aðCO 2 CO2 Þ V dt dRQ ¼ RRQ ðtÞ dt
(3)
(4)
where CX, CS, CO2 , and RQ are biomass, glucose, dissolved oxygen concentrations, and the respiratory quotient, respectively; F ¼ (dV/dt) is the substrate feed rate, V is the culture volume, CS0 is the substrate concentration in the feed, kLa is the oxygen transfer coefficient, and CO 2 is the oxygen saturation concentration. The specific dissolved oxygen consumption is included in order to increases the number of measured parameters in the SS design. Within the novel approach, the rates of biomass growth (RX), glucose consumption (RS), specific dissolved oxygen consumption ðqRO2 Þ, and respiratory quotient dynamics (RRQ) are considered as nonlinear functions of the unknown time-varying yield coefficients YXS(t), YO2 S ðtÞ, YRQS(t), and the glucose consumption rate, RS(t), as follows: RX ðtÞ ¼ RS ðtÞYXS ðtÞ
(5)
qRO2 ðtÞ ¼ RS ðtÞYO2 S ðtÞ
(6)
RRQ ðtÞ ¼ RS ðtÞYRQS ðtÞ
(7)
Equations (4) and (7), are derived assuming that the changes of the respiratory quotient can be expressed as: dRQ ¼ RS ðtÞð1 YXS ðtÞÞkYEth=pyr ðtÞ dt
(8)
where Ypyr=Eth is the yield coefficient for ethanol production at the pyruvic acid branch point. Since the respiratory quotient RQ characterizes the degree of fermentative growth, it is postulated that its changes are proportional to the split of carbon towards ethanol synthesis at the pyruvic acid branch point under substrate limited conditions. Hence, the yield YRQS(t) will represent the term of Equation (8) including both ‘‘true’’ yield coefficients, YXS ðtÞ and YEth=pyr ðtÞ, as follows: YRQS ðtÞ ¼ ð1 YXS ÞkYEth=pyr ðtÞ
(9)
Software Sensor System (SSS) The structure of the SSS of the four parameters, YXS, YO2 S , YRQS, and RS is provided in Figure 1. As a first step, the estimation of the biomass growth rate (RX) and the respiratory quotient dynamics (RRQ), is realized by classical software sensors SS1 and SS2, respectively. Their outputs are used together with the estimation of the specific dissolved oxygen consumption (qRO2 ), as input information of software sensor 3 (SS3). First Step of the Stepwise SS Design In the first step, the following algorithms for the estimation of the rates RX and RRQ, are proposed: dC^X F ¼ R^X ðtÞ CXm þ C1X ðCXm C^X Þ V dt
Figure 1.
(10a)
dR^X ¼ C2X ðCXm C^X Þ dt
^ dRQ ^ ¼ R^RQ ðtÞ þ C1RQ ðRQm RQÞ dt
dR^RQ ¼ C2RQ ðRQm ^ RQÞ dt
(10b)
(11a)
(11b)
where C1X, C2X, C1RQ, C2RQ are design parameters, and CXm and RQm are measured values of the variables CX and RQ. The estimation algorithms of Equations (10) and (11) have the structure of observer-based estimators. The stability analysis of second order observer-based estimators similar to Equations (10) and (11) has been well described by Bastin and Dochain (1990) and will not be further commented on in this article.
Second Step of Stepwise SS Design—Estimator of RS(t), YXS(t), YO2 S ðtÞ, YRQS(t) The second step of SS design includes the application of an algorithm for the estimation of the four parameters RS(t), YXS(t), YO2 S ðtÞ, and YRQS(t) on the basis of qRO2 m derived from on line measurements, while RX and RRQ values are obtained with Equations (10) and (11) in the first step. For the SS3 structure derivation, a linear transformation of nonlinear models (5)–(7) is applied using logarithmic linearization. In this way, the SS3 design is realized according to the linear control theory.
Scheme of the software sensor system approach.
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The following auxiliary parameters are introduced: wRs ¼ lnðRs Þ wYs ¼ lnðYXS Þ
The following fifth order estimator for the parameters wYs, wYO2 , wYRQ, and wRs is derived: (12)
(23a)
(23b)
(13)
wYO2 ¼ lnðYO2 S Þ
(14)
^ YO2 dw dqO2 ^ Rs w ^ YO2 Þ ¼ p^ þ C2 ðln qO2 w dt qO2 dt
wRQ ¼ lnðYRQS Þ
(15)
^ YRQ dw dRRQ ^ Rs w ^ YRQ Þ (23c) ¼ p^ þ C3 ðln RRQ w dt RRQ dt
With the differentiation of Equations (5)–(7) and (12)– (15), the following dynamic equations for RX, qRO2 , and RRQ can be derived: dRX dRS dYXS dwRs dwYs ¼ þ ¼ þ RX dt RS dt YXS dt dt dt
(16)
dqO2 dRS dYO2 S dwRs dwYO2 ¼ þ ¼ þ qO2 dt RS dt YO2 S dt dt dt
(17)
dRRQ dRS dYRQS dwRS dwRQ ¼ þ ¼ þ RRQ dt RS dt YRQS dt dt dt
(18)
Based on Equations (16)–(18), the dynamics of the auxiliary parameters wYs, wYO2 , and wYRQ are expressed as functions of RX, qRO2 , and RRQ, its time-derivatives and the wRs time-derivative as follows: dwYs dRX dwRs ¼ dt RX dt dt dwYO2 dqO2 dwRs ¼ dt qO2 dt dt dwYRQ dRRQ dwRs ¼ dt RRQ dt dt
(19a)
(19b)
^ Rs dw ^ Rs w ^ Ys Þ ¼ p^ þ C4 ðln RX w dt ^ Rs w ^ YO2 Þ þ C40 ðln qO2 w dp^ ^ Rs w ^ Ys Þ ¼ C5 ðln RX w dt
(23d)
(23e)
where C1, C2, C3, C4, C40 , and C5 are SS design parameters. The values of RX and RRQ, its time-derivatives at the right side of Equations (23a) and (23c) are obtained by Equations (10) and (11). Time-derivatives of qRO2 in (23b) are calculated by numerical differentiation. The structure of the software sensor (23) is based on the dynamic equations (16)–(18). In Equations (23a)–(23c), the timederivative of wRs is replaced with the auxiliary parameter p: ^_ Rs ¼ ^p. The estimation of parameter p is renewed by w Equation (23e), which is turn-driven by the deviation ^ Rs w ^ Ys Þ. C5 ðln RX w The estimations of the parameters RS, YXS, YO2 S , and YRQS are calculated using the estimations of parameters wYs, wYO2 , wYRQ, and wRs and a recalculation by Equations (12)–(15): ^ Ys Þ Y^XS ¼ expðw
(24a)
^ YO2 Þ Y^O2 S ¼ expðw
(24b)
^ YRQ Þ Y^RQS ¼ expðw
(24c)
^ Rs Þ R^S ¼ expðw
(24d)
(19c)
Considering the relationships (5)–(7) and (12)–(15), natural logarithms of rates RX, qRO2 , and RRQ are considered as on line measurements in a transformed space. They are linear functions of auxiliary parameters:
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^ Ys dw dRX ^ Rs w ^ Ys Þ ¼ p^ þ C1 ðln RX w dt RX dt
The proposed structure of SS3 includes sufficient design parameters to guarantee the observability of process parameters with available on line measurements.
ln RX ¼ lnðYXS Þ þ lnðRS Þ ¼ wYs þ wRs
(20)
Tuning of the Stepwise SS of RS(t), YXS(t), YO2 S ðtÞ, YRQS(t)
ln qRO2 ¼ lnðYO2 S Þ þ lnðRS Þ ¼ wYO2 þ wRs
(21)
ln RRQ ¼ lnðYRQS Þ þ lnðRS Þ ¼ wYRQ þ wRs
(22)
The tuning is realized by stability analysis of system (23a–e). The dynamics of the estimation errors vector T ~ Ys w ~ YO2 w ~ YRQ w ~ Rs ~p is presented in equax ¼ w tion system (35) (see Appendix). The characteristic
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polynomial of matrix A in (35) is derived as follows: l5 þ l4 ðC1 þC2 þC3 þC4 þ C40 Þþ l3 ðC2 C3 þC1 ðC2 þC3 Þþ ðC1 þ C2 þ C3 ÞðC4 þ C40 Þ C4 C40 Þþ l2 ðC1 C2 C3 þC2 C3 ðC4 þ C40 Þ þ C1 ðC2 þ C3 ÞðC4 þ C40 Þ þ lðC1 C2 C3 ðC4 þ C40 Þ C2 C40 C5 Þ C2 C3 C40 C5 . The following equation system (equalities and inequalities) has to be solved to realize tuning in a general case assuming that the system has five different Eigenvalues l1–l5:
2l0 þ 2l00 þ l5 ¼ C1 C2 C3 < 0
(26a)
l02 þ l002 þ 4l0 l00 þ 2l5 ðl0 þ l00 Þ ¼ C2 C3 þ C1 ðC2 þ C3 Þ > 0
(26b)
2ðl0 þ l00 Þl0 l00 þ l5 ðl02 þ l002 Þ þ 4l5 l0 l00
l 1 þ l2 þ l3 þ l4 þ l5 ¼ C1 C2 C3 C4 C40 < 0
Applying the simplifications Z1, Z2, and Z3, the system (25) is reduced as follows:
(25a)
¼ C1 C2 C3 < 0
(26c)
l02 l002 þ 2l5 l0 l00 ðl0 þ l00 Þ ¼ C2 C40 C5 > 0
(26d)
l3 l4 þ l1 l2 þ ðl1 þ l2 Þðl3 þ l4 Þ l02 l002 l5 ¼ C2 C3 C40 C5 < 0
þ l5 ðl1 þ l2 þ l3 þ l4 Þ ¼ C2 C3 þ C1 ðC2 þ C3 Þ þ ðC1 þ C2 þ C3 Þ ðC4 þ
C40 Þ
C4
Based on the relations (26d) and (26e), the following relation between the design parameter C3 and the Eigenvalues is then introduced:
C40
>0
(26e)
(25b) C3 ¼
l3 l4 ðl1 þ l2 Þ þ l1 l2 ðl3 þ l4 Þ þ l3 l4 l5 þ l1 l2 l5
¼ C1 C2 C3 C2 C3 ðC4 þ C40 Þ C1 ðC2 þ C3 Þ
2l0 þ 2l00 þ l5 ¼ A < 0
ðC4 þ C40 Þ (25c)
l1 l2 l3 l4 l5 ¼ C2 C3 C40 C5 < 0
(25d)
(25e)
According to the stability conditions, the Eigenvalues have to be negative. Hence, inequalities are introduced. As described in the literature (Birkhoff and Mac Lane, 1996), the general quantitative polynomials cannot be solved algebraically (in terms of finite numbers of additions, subtractions, multiplications, divisions, and root extractions). For this reason, the tuning procedure is performed by some simplifications (Z1–Z3) with respect to the number of different Eigenvalues of matrix A and to the number of estimator (23) tuning parameters functionally related to these Eigenvalues. Z1 : C4 ¼ C40 > 0; Z2 : l1 ¼ l2 ¼ l12 ¼ l0 ; Z3 : l3 ¼ l4 ¼ l34 ¼ l00 where l0 and l00 are negative numbers.
(28a)
l02 þ l002 þ 4l0 l00 þ 2l5 ðl0 þ l00 Þ ¼ B > 0
(28b)
2ðl0 þ l00 Þl0 l00 þ l5 ðl02 þ l002 Þ þ 4l5 l0 l00 ¼ C < 0 (28c)
l1 l2 l3 l4 þ l3 l4 l5 ðl1 þ l2 Þ þ l1 l2 l5 ðl3 þ l4 Þ ¼ C1 C2 C3 ðC4 þ C40 Þ C2 C40 C5 > 0
(27)
The following auxiliary parameters A, B, and C are introduced:
þ ðl1 þ l2 Þðl3 þ l4 Þl5
0 l l þ l5 ðl0 þ l00 Þ 0 00
Substituting the terms C2C3 and (C2 þ C3) from (26b) with their equivalents from Equations (26c) and (26a), respectively, the following third order polynomial of parameter C1 is derived as follows: C13 þ AC12 þ BC1 þ C ¼ 0
(29)
One of the roots of (29) is denoted as: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q pffiffiffiffi 3 q pffiffiffiffi A 3 C1 ¼ þ Q þ Q 2 2 3 3 A AB q ¼ 2 þC 3 3
Q¼
a3 3
þ
(30)
q2 2
A2 þB 3 The design parameter C2 is calculated with Equation (26a): a¼
C2 ¼ C1 C3 A
Lyubenova et al.: Software Sensor for Industrial Scale Fed-Batch Biotechnology and Bioengineering
(31)
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Values of the parameters C3 and C1 are obtained applying Equations (27) and (30), respectively. The value of parameter C40 is calculated from Equation (26e) by: C40 ¼
l02 l002 l5 C2 C3 C5
(32)
To satisfy the stability condition (25e), the value of C5 can be negative. The parameter C4 receives its values from Z1. In general, tuning of the SS3 includes the calculation of the design parameters C1, C2, C3, C4, and C40 using Equations (30), (31), (27), Z1, and (32), respectively. Three Eigenvalues l0 , l00 , and l5 have to be chosen. In this article, an optimal tuning is proposed based on the minimization of the square of sum of estimation errors: 2 2 sumðe2 Þ ¼ R~X þ q~R2O2 þ R~RQ
(33)
where R~X ¼ R^X R^S Y^XS q~RO2 ¼ qRO2
m
R^S Y^O2 S
R~RQ ¼ R^RQ R^S Y^RQS
(34a) (34b) (34c)
^ X and R ^ RQ are estimations of RX and RRQ obtained from R ^ S ; Y^ XS ; Y^ O S , and estimators (10) and (11), qRO2 m , and R 2 ^ Y RQS are estimates of RS(t), YXS(t), YO2 S ðtÞ, YRQS(t). They are obtained from estimator (23) by applying relationship (24). Considering that l0 ¼ l00 ¼ l1, the tuning is reduced to the determination of the two Eigenvalues l1 and l5. If one of them is fixed arbitrarily, the value of the other one can be obtained by solving the optimization problem defined by Equation (33). In order to solve it, an evolutionary algorithm approach in Matlab1 Version 7.0.1 (Mathworks, Inc., Natick, MA) was applied. The tuning approach will be demonstrated on the basis of experimental data.
from two glucose limited fed-batch cultivations of a nonsporulating B. subtilis mutant realized in a two compartment reactor as a scale-down process simulator (Junne et al., 2011; Neubauer and Junne, 2010) (Fig. 2). Data from one experiment are used for the demonstration of theoretical results. Data from a second fermentation are applied for tuning validation. Experiments are realized at a constant feed rate. In this way, different substrate per cell density ratios (that is amount of substrate provided per cell) are examined. The calculation of the specific oxygen uptake rate (qRO2 m ) and of the carbon dioxide production rate (qRCO2 m ) was performed on the basis of a gas balance of the input and output flow using O2 and CO2 measurements in the off-gas. Specific oxygen uptake rate (qRO2 m ), cell dry weight, and respiratory quotient (RQ ¼qRCO2 m =qRO2 m ) are regarded as on line measurements for the proposed monitoring scheme, although the biomass concentration CX was determined off line with dry cell mass determination after washing and overnight drying of the cells. Two sets of experimental data are shown in Figure 3. Both experiments were performed under the same conditions, but as it can be seen they varied due to a lack of absolute reproducibility under the oscillating conditions applied. In order to demonstrate adaptive performance of the proposed monitoring scheme shown in Figure 1, the tuning of the software sensors is realized using the data of one experiment while the data of the other are used for validation. For this purpose, dynamic simulations of the system were conducted within the Matlab computing environment. In Figures 4 and 5, the estimation results realized with SS1 and SS2, are shown. The optimal values of tuning parameters of observer-based estimators (10) and (1111) are
Results Case Study at a Nutrient-Limited Fed-Batch Process in a Scale Down Reactor As described in the methods section, the estimation of the biomass growth rate and the respiratory quotient dynamics are realized by second order software sensors (SS1 and SS2). Their outputs simultaneously and the specific oxygen consumption rate (qRO2 m ) derived from on line measurements are used as input information for the second step that includes the derivation of a fifth order software sensor (SS3) with four time-varying kinetic parameters estimated simultaneously. The obtained cascade SS is applied to data
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Figure 2. Scheme of the laboratory scale down reactor system where a stirred tank reactor (homogeneous conditions) is combined to a plug flow reactor where the feed is introduced (inhomogeneous conditions) in a nutrient-limited fed-batch cultivation.
Figure 3. Experimental data of two fed-batch cultivations of a non-sporulating Bacillus subtilis mutant realized in a two-compartment reactor at the feeding phase: cell dry weight (a), glucose concentration (b), specific oxygen consumption rate (c), and respiratory quotient (d) (Junne et al., 2011).
Figure 4. SS1 results from data of the feeding phase from two independent fermentations: (a) comparison between measured (solid line) and estimated (dashed line) cell dry weight, (b) estimates of the volumetric growth rate based on cell dry weight measurements.
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Figure 5.
SS2 results of two experiments during the feeding phase: (a) comparison between measured (solid line) and estimated (dashed line) respiratory quotient, (b) estimates of the rate of changes of the respiratory quotient.
chosen as balance between the convergence time of the solution and the robustness of the observer to disturbances. The following values are calculated: C1X ¼ 10, C2X ¼ 25, C1RQ ¼ 2, C2RQ ¼ 1. The values of C1X and C2X correspond to Eigenvalues l1,2 ¼ 5, while the values of C1RQ and C2RQ correspond to Eigenvalues l1,2 ¼ 1. In Figure 4a, almost full coincidence between the measured (solid lines) and estimated values (dashed lines) of CX is observed. In Figure 5a, the estimates of RQ with software sensor SS2 follow the dynamics of the RQ measurement with a good accuracy. As it can be seen in Figure 5a, the SS2 operates as filter of RQ measurements. These results prove that SS1 and SS2 provide accurate estimations of RX and RRQ (Figs. 4b and 5b) for both experiments. Hence, the outputs of estimators (10) and (11) could be used as on line inputs for the estimator (23). The advantages of the proposed tuning procedure are demonstrated in Figures 6 and 7 where the estimation results ^ S ; Y^ XS ; Y^ O S , and Y^ RQS of SS3 are shown. The estimates R 2 are presented with lines. In order to show the accuracy of received estimates, reference values have to be derived. Laboratory measurements of glucose concentration are used for this purpose. The obtained discrete estimates of the glucose consumption rate RS ref for both experiments are shown in Figure 7b and d. The reference values of YXS, YO2 S , and YRQS are calculated using the relationships (5)–(7), and
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RS ref . The corresponding reference values are shown in Figures 6, 7a and c. The results in Figures 6 and 7 are obtained using the parameter values shown in Table I as tuning No. 1. These values are received applying the tuning procedure of SS3 with data of one experiment. The SS3 performance is demonstrated in Figures 6a and b and 7a and b. In Figures 6c and d and 7c and d verification results are calculated using data from the second experiment. The obtained estimations for the data from experiment 1 and the validation with data from experiment 2 converge to the corresponding reference values. These results confirm the adaptive character of the proposed software sensor. In order to demonstrate user-friendly performance of the proposed tuning procedure, different versions of tuning have been tested. The results are presented as tuning Nos. 2– 5. In Table I, tuning Nos. 2 and 3 are performed with 10% variations of the original l5 value. This leads to an optimal value of the other one (l1) and to the determination of design parameters C1, C2, C3, C4, and C40 on the basis of Equations (30), (31), (27), Z1, and (32), respectively. As it can be seen in the last column of Table I, the decrease of the absolute value of l5 leads to an increased error of estimation calculated using Equation (33). An opposite effect is observed on the estimation error increasing l5. In tuning Nos. 4 and 5, the influence of the design parameter C5 is
Figure 6. SS3 results of two experiments during the feeding phase: (a) and (b): comparison between measured (solid line) and estimated (dashed line) YXS and YO2 S for experiment 1; (c) and (d): validation of SS3 tuning using the data of the repeated cultivation experiment (experiment 2).
Figure 7. SS3 results of two experiments during the feeding phase: (a) and (b): comparison between measured (solid line) and estimated (dashed line), YRQS and, RS for experiment 1; (c) and (d): validation of SS3 tuning using the data of the repeated cultivation experiment (experiment 2).
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Table I. No. 1 2 3 4 5
Performance investigation of SS3 under deviations of the tuning parameters. C5
l1
l5
C1
C2
C3
C4
C40 0
Error
1 1 1 1.1 0.9
76.5 68.86 84.54 76.37 76.7
500 450 550 500 500
512.7 461.34 564.13 512.44 512.6
274.9 246.9 303.38 273.45 274.46
18.4 16.54 20.33 18.3 18.39
3.382e6 2.454e6 4.537e6 3.036e6 3.744e6
3.382e6 2.454e6 4.5374e6 3.0360e6 3.7439e6
0.448 0.662 0.347 0.445 0.449
investigated. As observed, 10% changes of its value do not reflect on the estimation error comparing with tuning No. 1. Hence, a value of C5 ¼ 1 is acceptable for the proposed tuning procedure.
Conclusion The obtained theoretical results are applied for monitoring of a nutrient-limited fed-batch process in a scale down bioreactor. In this reactor concept, cells are subjected to oscillating environmental conditions. The presented methodology allows the integration of scale-dependent effects into a software design approach. As it has been shown in the scale down experiments with B. subtilis, the substrate uptake rate is decreased under oscillatory conditions. The accumulation of intracellular pyruvic acid concentration leads to a different distribution of carbon in the metabolic pathways that follow (Junne et al., 2011). The analysis of NAD(P)H and NAD(P)þ revealed a change of cellular reduction state under oscillatory conditions. Hence, many indicators show that B. subtilis cells differs in their physiologic conditions when opposed to oscillating conditions, although particular mechanisms are unclear yet and currently in the scope of further investigations. However, the impact of oscillations on the cell needs to be considered in process development and control. The application of a software sensor design as described in this study focusing on the substrate uptake rate and corresponding yield coefficients supports this consideration. One drawback could be the lack of appropriate on line monitoring of the biomass concentration (mostly performed with optical density measurements) and influences of lysed cell material. In order to account for this problem, suggested SS relying on the oxygen consumption and exhaust gas analysis can be applied for estimating the biomass concentration (Cazzador and Lubenova, 1995; Lubenova, 1996; Lubenova et al., 1993). The proposed tuning procedure is adaptive to the degree of oscillations which can be set in the experimental approach in the scale down reactor tailored to the industrial application. In comparison to other state estimation methods (e.g., extended Kalman Filters), the presented methodology has the advantage, that it is based on the methodology that the dynamics of each process variable is considered as a sum of two main parts: process kinetics and transport dynamics. Here the process kinetics is summarized in a term including
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Biotechnology and Bioengineering, Vol. 110, No. 7, July, 2013
reaction rates and time-varying yield coefficients. In this way, the model structure keeps small (in comparison to unstructured models). The idea for linear transformation of non-linear models allows the design of SS applying linear control theory. The measurements obtained by the new SSS improve the process monitoring and allow studying process dynamics. This provides reasons to extend theoretical investigations and their applications in future. Especially the integration of physiological parameters that are possible to measure on line will broaden the spectrum of applicability of the presented approach, while a strong correlation exists between yield coefficient dynamics and the physiological cell status. This work was supported by DFG grants NE 1360/3-1 and NE 1360/2-1, and partially by the Bulgarian National Fund ‘‘Scientific Researches’’ under contract No. DTK—02/27/17.12.2009.
Appendix Stability analysis of SS (23) Consider the dynamics of T ~ Ys w ~ YO2 w ~ YRQ w ~ Rs ~ p estimation error vector x ¼ w _x ¼ Ax þ u; 2 C1 0 6 0 C2 6 6 0 0 A¼6 6 6 4 C4 C40 C5 0 0 u ¼ 0 0 dp
C50
0
C1
0
C2
C3 C 004
C2 0 C4 C4
C 005
C5
1
3
1 7 7 7 1 7 7; 7 1 5 0
(35)
dt
l1, l2, l3, l4, and l5 are the Eigenvalues of matrix A, related to C1–C5, C40 as follows:
1. l1 þ l2 þ l3 þ l4 þ l5 ¼ C1 C2 C3 C4 C40 < 0; 2. l3l4 þ l1l2 þ (l1 þ l2)(l3 þ l4) þ l5(l1 þ l2 þ l3 þ l4) ¼ C2C3 þ C1(C2 þ C3) þ (C1 þ C2 þ C3)(C4 þ C40 ) C4 C40 > 0; 3. l3l4(l1 þ l2) þ l1l2(l3 þ l4) þ l3l4l5 þ l1l2l5 þ (l1 þ l2)(l3 þ l4)l5 ¼ C1C2C3 C2C3(C4 þ C40 ) C1(C2 þ C3)(C4 þ C40 ) < 0;
4. l1l2l3l4 þ l3l4l5(l1 þ l2) þ l1l2l5(l3 þ l4) ¼ C1C2C3(C4 þ C40 ) C2C40 C5 > 0; 5. l1l2l3l4l5¼ C2C3C40 C5 < 0. Assumptions: A1. The design parameters C1–C5, C40 are chosen such that matrix A has negative real Eigenvalues as follows: l1 ¼ l2 ¼ l1;2 < 0;
l3 ¼ l4 ¼ l3;4 < 0;
l5 ¼ l < 0
A2. The input u of the system (24) is bounded jðdu=dtÞj < M1 M1 —upper bound. Hence, since A is a stable matrix (see Assumption A1) and the input u of the system (24) is bounded (Assumption A2), the state x is also bounded (Bastin and Dochain, 1990) and the boundedness of estimation vector x as well. References Acun˜a G, Latrille E, Be´al C, Corrieu G. 1998. Static and dynamic neural network models for estimating biomass concentration during thermophilic lactic acid bacteria batch cultures. J Ferment Bioeng 85(6):615– 622. Assis A, Filho RM. 2000. A new approach for designing model-based indirect sensors. Comp Chem Eng 24(2–7):1099–1103. Bastin G, Dochain D. 1990. On-line estimation and adaptive control of bioreactors. Amsterdam, Oxford, New York, Tokyo: Elsevier. 378 p. Birkhoff G, Mac Lane S. 1996. A survey of modern algebra, 5th edition. New York: Macmillan. p 418–421. Bylund F, Collet E, Enfors SO, Larsson G. 1998. Substrate gradient formation in the large-scale bioreactor lowers cell yield and increases byproduct formation. Bioprocess Eng 18:171–180. Cazzador L, Lubenova VN. 1995. Nonlinear estimation of specific growth rate for aerobic fermentation processes. Biotechnol Bioeng 47:626–632. Enfors SO, Jahic M, Rozkov A, Xu B, Hecker M, Ju¨rgen B, Kru¨ger E, ´ Beirne D, Noisommit-Rizzi N, Reuss M, Schweder T, Hamer G, O Boone L, Hewitt C, McFarlane C, Nienow A, Kovacs T, Tra¨gardh C, Fuchs L, Revstedt J, Friberg PC, Hjertager B, Blomsten G, Skogman H, Hjort S, Hoeks F, Lin HY, Neubauer P, van der Lans R, Luyben K, Vrabel P, Manelius A. 2001. Physiological responses to mixing in large scale bioreactors. J Biotechnol 85:185. Fortuna L, Graziani S, Rizzo A, Xibilia MG. 2007. Soft Sensors for Monitoring and Control of Industrial Processes. In: Fortuna L, Graziani S, Rizzo A,
(Eds.) Advances in Industrial Control. Springer-Verlag GmbH: Heidelberg. 272 p. Georgieva P, Feyo de Azevedo S. 2009. Studies in computational intelligence, Vol. 218. Springer: Berlin, Heidelberg. p 99–125. Ignatova MN, Lyubenova VN. 2011. Control of biotechnological processes new formalization of kinetics: Theoretical aspects and applications. Saarbru¨cken: LAP LAMBERT Academic Publishing, GmbH & Co. 120 p. Imanishi T, Hanai T, Aoyagi I, Uemura J, Araki K, Yoshimoto H, Harima T, Honda H, Kobayashi T. 2002. Software sensing for glucose concentration in industrial antibiotic fedbatch culture using fuzzy neural network. Biotechnol Bioprocess Eng 7:275–280. Junne S, Klingner A, Kabisch J, Schweder T, Neubauer P. 2011. A twocompartment bioreactor system made of commercial parts for bioprocess scale-down studies: Impact of oscillations on Bacillus subtilis fed-batch cultivations. Biotech J 6(8):1009–1017. Lubenova VN. 1996. On-line estimation of biomass concentration and non stationary parameters for aerobic bioprocesses. J Biotechnol 46:197– 207. Lubenova VN, Ignatova MN, Tzonkov S. 1993. On-Line estimation of specific growth rate for a class aerobic batch processes. Chem Biochem Eng 7(2):101–106. Lyubenova VN, Ignatova MN. 2011. Cascade software sensors for monitoring of activated sludge waste water treatment processes. Compt Rend Acad Bulg Sci 64(3):395–404. Lyubenova VN, Ignatova MN, Salonen K, Kiviharju K, Eerika¨inen T. 2011. Control of a-amylase production by Bacillus subtilis. Bioprocess Biosyst Eng 34(3):367–374. Neubauer P, Junne S. 2010. Scale-down simulators for metabolic analysis of large-scale bioprocesses. Curr Opin Biotech 21:114–121. Neubauer P, Ha¨ggstro¨m L, Enfors SO. 2005. Influence of substrate oscillations on acetate formation and growth yield in Escherichia coli glucoselimited fed-batch cultivations. Biotech Bioeng 47:139–146. Selis¸teanu D, Petre E, Roman M, Sendrescu D. 2012. Estimation of kinetic rates in a baker’s yeast fed-batch bioprocess by using non-linear observers. Control Theory Appl, IET 6(2):243–253. Soons ZITA, Voogt JA, van Straten G, van Boxtel AJB. (2006). Constant specific growth rate in fed-batch cultivation of Bordetella pertussis using adaptive control. J Biotechnol 125:252–268. Thaysen M, Jørgensen SB. 2004. Application of software sensors for monitoring and prediction in fermentation processes. Proceedings 7th ADCHEM Symposium: 429–434. Veloso A, Rocha I, Ferreira EC. 2009. Monitoring of fed-batch E. coli fermentations with software sensors. Bioprocess Biosys Eng 32(3):381– 388.
Lyubenova et al.: Software Sensor for Industrial Scale Fed-Batch Biotechnology and Bioengineering
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