After the sporulation, the process is completed with the cellular wall rupture. (cellular lysis), and the consequent liberation of spores and crystals to the culture ...
Stable and Robust Dissolved Oxygen Control for the Fermentative Process of Bt δ - Endotoxin Production A . N. Amicarelli
J. M. Toibero
F. A. di Sciascio
Instituto de Automática (INAUT). Universidad Nacional de San Juan. Av. Libertador San Martín 1109 (oeste). J5400ARL. San Juan, Argentina. e-mail:{amicarelli;mtoibero;fernando}@inaut.unsj.edu.ar Abstract−− This paper proposes a robustness analysis for a dissolved oxygen controller based on Lyapunov theory. The control strategy is designed for the Bacillus thuringiensis δ-endotoxins production process. First, the control strategy is presented and second, it is considered with a biomass estimator in closed loop and it is shown a robustness analysis for the errors introduced by both: biomass estimation and modeling errors. Simulation results with real experimental data are included in order to discuss the controller performance. Keywords−− Dissolved Oxygen Control; Lyapunov Theory; Bacillus thuringiensis; Biotechnological Process; Biomass Estimation. 1. INTRODUCTION In designing control algorithms the stability analysis for the control system has fundamental importance. This analysis aggregates a theoretical support to validate the proposed control law. Besides, analysis techniques based on Lyapunov theory result attractive as frequently the systems under study are highly non-linear. From the process control point of view, the biomass concentration is one of the states that characterize the bioprocess; moreover, it is generally the main desired output, and therefore, for control purposes it is necessary to replace the unavailable biomass concentration measurements with reliable and robust online estimations. Bacillus thuringiensis δ-endotoxins production is an aerobic operation, i.e., the cells require oxygen as a substrate to achieve cell growth and product formation [1]. A first principle based model for Bt δ-endotoxins production process was proposed by Rivera [2], a modification to the Rivera model was given by Atehortúa, [3, 4]. Afterwards, Amicarelli [5] improved the model process adding the dissolved oxygen (DO) dynamics due to its importance in the biomass estimation problem and the process control. Several works in the area of biomass estimation can be found in the literature [6, 7, 8, 9, 10, 11] and in the references therein. Many of these works, in aerobics bioprocess, base the biomass estimation on online measurements of dissolved oxygen concentration, substrate or another process variable depending on the particular case. Optimal growth of Bt and product formation is associated with a high absorption rate of oxygen. This assumption could be confirmed by measuring the oxygen
rate for different conditions in batch fermentation of Bt sub. kurstaki. Numerous authors agree that aerobic microorganisms have in general higher oxygen requirements, mainly during the exponential growth phase. Particularly for Bt this requirement decreases during the sporulation phase and in the cellular lysis stage. The decrease in oxygen demand during sporulation phase allows reducing the air supply in the final stage of production, which represents an economy in the process. This work proposes a dissolved oxygen control strategy integrated with a biomass concentration estimator for the biotechnological batch process of δ-endotoxins production of Bt. The controller design is based on Lyapunov Theory and includes a robustness analysis for the errors introduced from the biomass estimator and for the ones due to modeling errors. For estimation purposes, the main variables of this bioprocess are: the concentrations of dissolved oxygen, substrate (basically glucose), and biomass. The paper is organized as follows: Section 2 presents the main characteristics of the Bacillus thuringiensis (Bt) δ-endotoxins production process, and also a phenomenological model of the bioprocess is given. Section 3 is devoted to describing the dissolved oxygen control strategy. Next, in the same Section the robustness analysis is presented. Section 4 gives an analysis of results and discussion and finally, in Section 5 the conclusions are stated. 2. MATERIALS AND METHODS 2.1. Bt δ-endotoxins Production Process Bacillus thuringiensis is one of the microorganisms most frequently studied as toxin producer. Bt is an aerobic spore-former bacterium which, during the sporulation; also produces insecticidal crystal proteins known as δ-endotoxins. It has two stages on its life span: a first stage characterized by its vegetative growth, and a second stage named sporulation phase. When the vegetative growth finalizes, the beginning of the sporulation phase is induced when the mean exhaustion point has been reached. Normally the sporulation is accompanied by the δ-endotoxin synthesis. After the sporulation, the process is completed with the cellular wall rupture (cellular lysis), and the consequent liberation of spores and crystals to the culture medium [12, 13, 14]. The microorganism considered in this work is Bacillus thuringiensis serovar. kurstaki strain 172-0451 isolated in Colombia and stored in the culture collection of Bio-
technology and Biological Control Unit (CIB) [15]. The medium (CIB-1) contained: MnSO4.H2O (0.03 g/L), CaCl2.2H20 (0.041 g/L), KH2PO4 (0.5 g/L), K2HPO4 (0.5 g/L), (NH4)2SO4 (1 g/L), yeast extract (8 g/L), MgSO4.7H2O (4 g/L) and glucose (8 g/L). Growth experiments of the fermentation process with Bacillus thuringiensis were performed in a pilot-scale reactor with a nominal volume of 20 liters. The fermentations were developed with an effective volume of 11 liters of cultivation medium, and they were inoculated to 10% (v/v) with the microorganism Bt culture. The inoculum added consisted of a vegetative phase culture: 5 mL spore suspension with 1 · 107 UFC/mL (stored at –20 °C) was used to inoculate a 500 mL flask containing 100 mL of CIB-1, and incubated with shaking at 250 rpm at 30 °C during 13 h. Fifty milliliters of this culture were aseptically transferred to each one of two 2 L flasks containing 500 mL of CIB-1 and incubated as above for 5 h. The pH medium was adjusted to 7.0 with KOH before its heat sterilization. Culture conditions at harvest are typified by 90% free spores and δendotoxins crystals. The temperature was maintained around 30 ºC by using an ON/OFF control; whereas the pH was fixed between 6.5 and 8.5. Manometric pressure in the reactor was set at 41,368 Pa using a pressure controller. Temperature, pH, dissolved oxygen, and glucose concentration were registered by a data acquisition system using an Advantech® PCL card. Dissolved oxygen was measured by a polarographic oxygen sensor InPro 6000 (Mettler Toledo, Switzerland), and glucose concentration was determined with a rapid off-line measurement method through a glucose analyzer (YSI 2700). The reagents concentration used for the pH control and foam formation were nitric acid (5N), potassium hydroxide (2N) and defoamer (33% v/v). Cell growth was determined as dry cell weight (Dry cell weight (DCW, g /L) = (final weight - initial weight)/(volume of microbial suspension filtered). The foam formation was avoided by manually aggregating a defoamer. 2. 2. Bioprocess Model The following state-space model is the continuous-time version developed by Amicarelli [5]. For notation see Table 1. ⎡ dX v (t ) ⎤ ⎤ ⎢ dt ⎥ ⎡ ( μ (t ) − ks (t ) − ke (t ) ) X v (t ) ⎥ ⎢ ⎥ ⎢ ks (t ) X v (t ) ⎥ ⎢ dX s (t ) ⎥ ⎢ ⎥ ⎢ dt ⎥ ⎢ ⎛ μ (t ) ⎞ ⎥ ⎢ ⎥=⎢ −⎜ + ms ⎟ X v (t ) ⎥ ⎢ dS (t ) ⎥ ⎢ ⎝ Yx / s ⎠ ⎥ ⎢ dt ⎥ ⎢ ⎢ dC (t ) ⎥ ⎢ K3 ⋅ Fair − in ( CDO* − CDO (t ) ) − K1 dX − K 2 ⋅ X (t ) ⎥ ⎥⎦ ⎢ DO ⎥ ⎢⎣ dt ⎢⎣ dt ⎥⎦
(1)
μmax S ( t )
μ (t ) =
(K
s
(2)
+ S (t )) ⎛
⎛
⎞
⎞
⎜ ⎟ ⎜ ⎟ 1 1 ⎟ ⎟ − ks max ⎜ ks ( t ) = ks max ⎜ Gs ( Sinitial − Ps ) ⎟⎟ ⎜⎜ ⎜⎜1 + e Gs ( S (t ) − Ps ) ⎟⎟ ⎝ ⎠ ⎝1 + e ⎠ ⎛ ⎜
ke ( t ) = ke max ⎜ ⎜⎜ ⎝
1+ e
⎞ ⎟ ⎟
1
Ge (t − Pe ) ⎟⎟ ⎠
⎛ ⎜
− ke max ⎜⎜ ⎜ ⎝
1+ e
⎞ ⎟ ⎟
1
Ge (tinitial − Pe ) ⎟⎟
(3) (4)
⎠
Table 1: variables in the bioprocess model Symbol
Description Limiting substrate concentration [ g / L]
S
[ h]
Ts
Sampling time
Xs
Sporulated cells concentration
[ g / L]
Xv
Vegetative cells concentration
[ g / L]
CDO
μ
Dissolved oxygen concentration Specific growth rate [h
μmax
−1
[ g / L]
]
Maximum specific growth rate [h
−1
]
ms
Maintenance constant ⎡⎣ g subs.g cells / h ⎤⎦
ks
Kinetic constant representing the spore formation −1
[h ]
ke
Death cell specific rate [h
YX / S
−1
]
Growth yield ⎡⎣ g cells / g subs.⎤⎦
Ks
Saturation constant [ g / L]
K1 K2
Oxygen consumption constant by growth (dimensionless) −1 Oxygen consumption constant for maintenance [h ]
K3
Ventilation constant [ L
CDO
*
Fair− in
−1
]
O2 saturation concentration (DO concentration in equilibrium with the oxygen partial pressure of the gaseous phase) [ g / L] Air flow that enters to the bioreactor [ L / h]
Four batch cultures with different initial glucose concentration (8, 21, 32 and 40 g/L) were carried out to generate experimental data for model validation and fine parameters tuning. In this context, four parameter sets guarantee a representative covering of an intermittent fed batch culture (IFBC) with total cell retention (TCR) in the operation space according to the work of Atehortúa, [4]. Maximum glucose concentration in the medium ( S max ) was used as the switching criteria among the estimated batch parameter sets. For model parameters see Table 2. Tabla 2: Model Parameters Smax < 10
10 < Smax < 20
20 < Smax < 32
Smax > 32
The following equations define the specific growth speed (Monod equation) μ , the spore formation rate
μmax
0,8
0,7
0,65
0,58
k s and the death cell specific rate ke .
YX / S
0,7
0,58
0,37
0,5
Ks
0,5
2
3
4
0,005
ms
0,005
0,005
0,005
0,5
0,5
0,5
0,5
Gs
1
1
1
1
Ps
1
1
1
1
ke max
0,1
0,1
0,1
0,1
Ge
5
5
5
5
Pe
4
4,7
4,9
6
9,75.10-4
4,502.10-3
3,795.10-3
1,597.10-3
1,589.10-4
0,046.10-3
0,729.10-3
0,561.10-3
4,636.10-4
0,337.10-3
2,114.10-3
1,045.10-3
K2 K3
DO
Fair_in(t)
DO Model
CDO (t )
Fig.1. Control system block diagram
3.2 Robustness Analysis for errors on the biomass estimation. In the previous work [16], the asymptotic stability for this control system on the equilibrium point at the origin ( C DO (t ) = 0 ) has been proved. In other words, if the con-
3. CONTROL STRATEGY 3.1 Nonlinear Controller The fermentations were carried out with substrates based primarily on glucose, for this reason, in this paper the DO concentration is maintained as close as possible to the established benchmark value for this case [1]. A high production of δ-endotoxin is assumed when a dissolved oxygen concentration of 60% of the oxygen saturation concentration in the culture medium is maintained during the first six hours of incubation, and a dissolved oxygen concentration of 40% of the oxygen saturation concentration until the end of the fermentation. The proposed DO controller which allows the process to follow this dissolved oxygen profile throughout the fermentation course was presented in the work of Amicarelli [16]. The controller is based on the complete knowledge of the process model. The dissolved oxygen concentration error is: ref C DO (t ) = CDO (t ) − CDO (t )
C DO (t )
Controller
ks max
K1
ref CDO (t )
trol action (6) is considered, then C DO (t ) → 0 as t → ∞ . In the same mentioned work [16], it is assumed that, there are not parametric modeling errors (i.e. the parameters models K1 , K 2 and K 3 perfectly match with its estimated values) and furthermore, there are not biomass estimation errors X (t ) . The block diagram for this approach can be seen in Fig.1. From (6) and considering Fig. 2, it can be noted that in the control algorithm for CDO in the production of δendotoxin of Bt there are involved variables, which is necessary to estimate, meanly the biomass concentration X (t ) . This way, estimation errors will introduce control errors affecting the control system performance. It is interesting to analyze their effects on the control errors and what happen with the control system stability in this context. The block diagram for this approach can be seen in Fig. 2. ref CDO (t ) C (t ) DO
X (t ) Biomass Estimator
(5)
where C DO (t ) is considered as in Ghribi D. [1], ref
DO Fair_in(t) Controller
CDO (t )
DO Model
CDO (t )
CDO (t )
S (t ) Experimental Data
Fig. 2. System Estimator-Controller Block Diagram
The proposed control law for the airflow command is
The dissolved oxygen dynamics model from (1) is 1 dX (t ) ⎛ ⎞ Fair − in (t ) = + K 2 X (t ) ⎟ ⎜ K L CDO (t ) + K1 K 3 ( C DO* − CDO (t ) ) ⎝ dt ⎠
(6) dC DO (t ) dX (t ) = K 3 ⋅ Fair− in ( C DO * − C DO (t ) ) − K1 − K 2 ⋅ X (t ) . (7) dt dt
where K L is a positive design constant. The control action is specified in liters per hour, and its nominal value is 1320 [ L h −1 ] ). The closed loop equation, replacing (6) in (1), is
dCDO (t ) = K L C DO (t ) dt
(7)
Now, it is possible to write the proposed control law in terms of the estimated biomass concentration and the estimaded model parameters Kˆ 1 , Kˆ 2 and Kˆ 3 as follows Fair− in (t ) =
⎛ dXˆ (t ) ˆ ˆ ⎞ .(8) 1 + K 2 X (t ) ⎟⎟ ⎜ K L C DO (t ) + Kˆ 1 dt Kˆ 3 ( C DO* − C DO (t ) ) ⎜⎝ ⎠
where Xˆ (t ) is the estimated biomass concentration obtained with some estimator [10, 11]. The closed loop equation, replacing (8) in (7) can be rewritten as
(9)
The Lyapunov candidate proposed in [16] with the aim to prove asymptotic stability was V = 12 C 2DO and its derivate over the system trajectories is dCDO (t ) dV ∂V ∂x dC DO (t ) = = CDO = −C DO dt ∂x ∂t dt dt
.
(10)
Now, replacing (9) in (10) ⎡ ⎤ (11) dV dXˆ (t ) dX (t ) = −C DO (t ) ⎢ K '3 K L C DO (t ) + K '3 Kˆ 1 + K '3 Kˆ 2 Xˆ (t ) − K1 − K2 X (t ) ⎥ dt dt dt ⎣ ⎦
where K '3 =
K3 . Kˆ 3
In order to satisfy the asymptotic stability condition, it must be guaranteed that the term − K '3 K L C DO is bigger than
K '3 Kˆ 1
dXˆ (t ) dX (t ) + K '3 Kˆ 2 Xˆ (t ) − K1 − K 2 X (t ) dt dt
.
Next,
it
follows that dX (t ) dXˆ (t ) (12) K '3 K L C DO > K1 + K 2 X (t ) − K '3 Kˆ 2 Xˆ (t ) − K '3 Kˆ 1 dt dt
Denoting the error associated with the estimated biomass as X (t ) = Xˆ (t ) − X (t ) ; and consequently
dX (t ) dXˆ (t ) dX (t ) = − . dt dt dt
A maximum bound for the control error is given by dX (t ) dXˆ (t ) −1 + K '3 Kˆ 1 − K1 + K 2 X (t ) + K '3 Kˆ 2 − K 2 Xˆ (t ) C DO (t ) ≤ ( K '3 K L ) K1 dt dt
(
)
(
)
(13)
Furthermore, considering the triangle inequality, ( K '3 K L )
−1
K1
dX (t ) dXˆ (t ) + K '3 Kˆ 1 − K1 + K 2 X (t ) + K '3 Kˆ 2 − K 2 Xˆ (t ) ≤ dt dt
(
)
(
)
(14)
dX (t ) dXˆ (t ) −1 ⎛ ≤ ( K '3 K L ) ⎜ K1 + K '3 Kˆ 1 − K1 + K 2 X (t ) + K '3 Kˆ 2 − K 2 Xˆ (t ) ⎜ dt dt ⎝
(
)
(
)
⎞ ⎟ ⎟ ⎠
the maximum bound for the control error can be rewritten as dX (t ) dXˆ (t ) −1 ⎛ + K '3 Kˆ 1 − K1 + K 2 X (t ) + K '3 Kˆ 2 − K 2 Xˆ (t ) C DO (t ) ≤ ( K '3 K L ) ⎜ K1 ⎜ dt dt ⎝
(
)
(
)
⎞ (15) ⎟ ⎟ ⎠
Now, considering the control action (8), the C DO is bounded by (15) and the stability condition is maintained in the control system. Now, if there are not biomass estimation errors X (t ) ., i.e. there are only parametric modeling errors, the expression for the bound is given
⎛ dX (t ) C DO (t ) ≤ K L −1 ⎜⎜ K1 + K 2 X (t ) dt ⎝
⎞ ⎟⎟ ⎠
(17)
4. RESULTS AND DISCUSIONS
The duration of the batch fermentation is limited and depends on the initial conditions of the microorganism culture. All fermentations were initialized with the same inoculate and different substrate concentration conditions. When the medium is inoculated, the biomass concentration increases at expense of the nutrients, and the fermentation concludes when the glucose that limits its growth was consumed, or when a value of at least a 90% of cellular lysis is achieved. The duration of each experiment is approximately 18 hours. In previous works [10, 11] the collected data from a set of fermentations (concentrations measurements of dissolved oxygen CDO , substrate S , and biomass X ) were used to design and to propose different biomass estimators for the Bacillus thuringiensis δ-endotoxins production process. In those mentioned works, the biomass concentrations has been completed to have the same size as in dissolved oxygen concentration data set (the experimental measurements were obtained each hour for the biomass and ten times per hour for the dissolved oxygen). This is a missing data problem [17], and Bayesian Gaussian Process regression was utilized as an imputation strategy for filling the missing values [11]. In order to simulate the proposed control scheme (Fig.2), it has been considered a biomass estimator based on Bayesian Regression with Gaussian Process [11], which performance is shown in Fig.3 The biomass concentration obtained by Bayesian Regression of the experimental measurements and the estimated biomass given by the “Bayesian Gaussian Estimator” are considered as real biomass concentration measurements and estimated biomass concentration respectively in the control scheme of Fig. 2. The mentioned estimator provides estimations on the 95% on the confidence level [11]. 15
Biomass Concentration [g/l]
dCDO (t ) K3 ⎛ dXˆ (t ) ˆ ˆ ⎞ dX (t ) = + K 2 X (t ) ⎟⎟ − K1 − K 2 X (t ) . ⎜ K L C DO (t ) + Kˆ 1 dt dt dt Kˆ 3 ⎜⎝ ⎠
10
5
0 0
2
4
6
8
10
12
14
16
18
Time [h]
dX (t ) ⎞ −1 ⎛ C DO (t ) ≤ ( K '3 K L ) ⎜ K '3 Kˆ 1 − K1 + K '3 Kˆ 2 − K 2 X (t ) ⎟ dt ⎝ ⎠
(
)
(
)
(16)
and, conversely, if there are not parametric modeling errors the expression i.e. there are only biomass estimation errors X (t ) , the bound is given
Fig.3. Biomass concentration. The crosses are the biomass experimental measurements, the small circles represent the biomass obtained by Bayesian Regression with Gaussian Process and the solid line is the estimated biomass given by a Bayesian Gaussian Estimator. The gray zone represent the 95% confidence level.
60
55 50 45 40
50
35 0
2
4
6
8
10
12
14
16
18
Time [h] 45
35 0
2
4
6
8
10
12
14
16
18
Time[h]
Fig.4. Dissolved Oxygen. The gray line is the desired dissolved oxygen profile, the solid black line represents the performance of the control system without errors and the dotted line depicts the controller performance when considering estimations errors. -3
1
x 10
0.5
Control Error [g/L]
60
55
40
Fig.7. Dissolved Oxygen. The gray line is the desired dissolved oxygen profile; the solid black line represents the control system performance without parametric model errors and the dotted line is the performance when considering parametric model errors. -3
1
x 10
0.5
0
-0.5
-1
0
-1.5 0
2
4
6
8
10
12
14
16
18
Time[h]
Fig.8. Control Error: for the system considering errors (dotted line) and for the system without considering model parametric errors (solid black line). The gray lines represent the variable bound for the control in this case.
-0.5
-1
-1.5 0
2
4
6
8
10
12
14
16
18
Time [h]
Finally, it has been considered in the proposed control scheme both errors, i.e. estimations and parametric model errors, 65 60
Dissolved Oxygen [%]
Fig.5. Control Error. The solid black line represents the control error for the system without errors and the dotted line is the control error when including estimations errors. The gray lines represent the error bound. 10000 8000
Control Action [L/h]
65
Control Error [g/L]
Dissolved Oxygen [%]
65
Now, it has been considered a parametric model error of 20% and the control system behavior is shown in Fig.7. It can be noted that the system shows an adequate performance also for the important parametric model errors present in the system. Next, Fig. 8 shows the control errors for this case and the established bound for parametric model errors.
Dissolved Oxygen [%]
Now, from (8) it can be seen that is necessary to estimate the biomass concentration derivative. In order to obtain this signal, X and Xˆ have been filtered using a conventional alpha-beta filter [18]. Figure 4 shows the performance of the dissolved oxygen controller in closed-loop with the biomass estimator, i.e. the controller output when considering biomass estimation errors. In Fig.5 the control error is shown for both cases: with and without estimation errors. Moreover, in this same figure it can be observed the maximum bound for this situation according to (17).
6000
55 50 45
4000
40
2000
35 0
2
4
6
8
10
12
14
16
18
Time [h] 0 -2000
0
2
4
6
8
10
12
14
16
18
Time [h]
Fig.6. Control Action, considering biomass estimation errors (dotted line) and without biomass estimation errors (solid black line)
Fig.9. Dissolved Oxygen: desired profile (gray line); controlled output without errors (solid black line) and controlled output when considering both parametric model and estimation errors (dotted line).
REFERENCES
Through Adequate Control of Aeration. Enzyme and Microbial Technology Vol. 40, pp. 614-622, 2007. [2] Rivera, D., Margaritis, A. and De Lasa, H. A sporulation kinetic model for batch growth of B. thuringiensis. Canadian Journal of Chemical Engineering, 77, 903–910, 1999. [3] Atehortúa, P., Álvarez, H. and Orduz, S. Comments on: A sporulation kinetic model for batch growth of B. thuringiensis. The Canadian Journal of Chemical Engineering, 84(3), 2006. [4] Atehortúa, P., Alvarez, H. and Orduz, S. Modeling of growth and sporulation of Bacillus thuringiensis in an intermittent fed batchculturewith total cell retention. Bioprocess and Biosystems Engineering, 30, 447–456, 2007. [5] Amicarelli, A., di Sciascio, F. and Álvarez, H. Including dissolved oxygen dynamics to the Bacillus thuringiensis endotoxins production process. The Canadian Journal of Chemical Engineering, submitted for publication, 2009. [6] Bastin, G., and Dochain, D. On-line Estimation and Adaptive Control of Bioreactors. Elsevier, Amsterdam, 1990. [7] Dochain, D. State and Parameter Estimation in Chemical and Biochemical Processes: A Tutorial Journal of Process Control, V.13, 801-818, 2003. [8] Leal Ascencio, R. Artificial Neural Networks as a Biomass Virtual Sensor for a Batch Process. Proc. of the 2001 IEEE Int. Symposium on Intelligent Control Sep.5, Mexico City, Mexico, 2001. [9] Li, B. Artificial Neural Network Based Software Sensor for Biomass during Microorganism Cultivation, Ph.D. thesis, South China University of Technology, 2003. [10] Amicarelli, A., di Sciascio, F., Álvarez, H. and Ortiz, O. (In Spanish) Estimación de Biomasa en un proceso Batch: Aplicación a la producción de _ endotoxinas de Bt. In XXII Interamerican congress of chemical engineering, 2006. [11] di Sciascio, F. and Amicarelli, A. N. Biomass. Estimation in Batch Biotechnological Processes by Bayesian Gaussian Process Regression, Computers and Chemical Engineering Vol. 32, pp. 3264 – 3273, 2008. [12] Starzak, M. and Bajpai, R. A structured model for vegetative growth and sporulation in Bacillus thuringiensis. Applied Biochemistry and Biotechnology, 28/29,699–718, 1991. [13] Aronson, A. I. The two faces of Bacillus thuringiensis: insecticidal proteins and post exponential survival. Molecular Microbiology, 7, 489–496, 1993. [14] Liu, B. L. and Tzeng, Y. M. Caracterization study of the sporulation kinetics of Bacillus thuringiensis. Biotechnology and Bioengineering, 68, 1–17, 2000. [15] Vallejo, F., González, A., Posada, A., Restrepo, A. and Orduz, S. Production of Bacillus thuringiensis subsp. Medellín by batch and fed-batch culture. Biotechnology. Techniques, 13, 279–281, 1999. [16] Amicarelli A., Toibero J. M., Quintero O., di Sciascio F., Carelli R. Estrategias De Control De Oxígeno Disuelto Aplicadas A La Fermentación Batch De Bt. XIII Congreso Latinoamericano de Control Automático. VI Congreso Venezolano de Automatización y Control. 25 al 28 de Noviembre de 2008 - Mérida, Venezuela. [17] Little R. and Rubin D. Statistical Analysis with Missing Data, New York, Wiley, 2002.
[1] Ghribi, D., Zouari, N., Trabelsi, H. and Jaoua, S. Improvement of Bacillus thuringiensis δ-endotoxin Production by Overcome of Carbon Catabolite Repression
[18] Kalata, P., The tracking index: A generalized parameter for α - β and α – β - γ target trackers, IEEE Transactions on Aerospace and Electronic Systems, vol. 20, Nº 2, pp. 174–182, 1994.
1
x 10
-3
Control Error [g/L]
0.5
0
-0.5
-1
-1.5 0
2
4
6
8
10
12
14
16
18
Time [h]
Fig.10. Control Error. The solid black line represents the control error for the system without errors and the dotted line is the control error with the presence of parametric model errors and estimation errors. The gray lines represent the control error variable bound.
Equations (15), (16) and (17) give a bound for the estimation errors of for the parametric modeling errors depending on the case. Note that, considering the control action (8), the C DO is bounded by (15) in the most general case preserving the control system stability condition. Note that, in spite of this prove of asymptotic stability for this control system, certain restrictions should be considered for the controller implementation: i) the air flow has a maximum value physically plausible (given by the valve used). ii) it is only possible to perform a DO control by increasing the DO concentration, but a decrease on the concentration value can be expected only due to the microorganisms consumption, to the addition of antifoam agents (or pH control agents), but not as a direct consequence of this control scheme 5. CONCLUSIONS
This paper considered a robustness analysis for a dissolved oxygen controller based on Lyapunov theory. The presented control strategy was designed for the Bacillus thuringiensis δ-endotoxins production process, and was integrated with a biomass estimator in closed loop. The robustness analysis for errors introduced from the biomass estimation and modeling errors was investigated. This work provides three expressions to bound estimation errors, parametric modeling errors and for the case when both errors are present. Considering the proposed control action it is demonstrated that the control error is bounded preserving the stability condition for the control system. Acknowledgments This work was supported by the Universidad Nacional de San Juan and CONICET (Consejo Nacional de Investigaciones Científicas y Técnicas).
