NONLINEAR CONTROL OF THE DISSOLVED OXYGEN CONCENTRATION FOR OPTIMAL PRODUCTION OF BACILLUS. THURINGIENSIS δ-endotoxins

Transcription

1 NONLINEAR CONTROL OF THE DISSOLVED OXYGEN CONCENTRATION FOR OPTIMAL PRODUCTION OF BACILLUS THURINGIENSIS δ-entoxins Santiago Rómoli a, Adriana N. Amicarelli b, Oscar A. Ortiz a, Jorge R. Vega c, Fernando di Sciascio b, Pablo Aballay a, Gustavo J.E. Scaglia a a Instituto de Ingeniería Química, b Instituto de Automática Universidad Nacional de San Juan, Av. Libertador San Martín Oeste 9, San Juan, J54ARL, Argentina c INTEC, UTN Facultad Regional Santa Fe, Universidad Tecnológica Nacional, Lavaisse 6, Santa Fe, S34EWB, Argentina. sromoli@unsj.edu.ar Abstract. Bacillus thuringiensis is a microorganism that allows the biosynthesis of δ-endotoxins with toxic properties against some insect larvae, being often used for the production of biological insecticides. A key issue for the bioprocess design consists in adequately tracking a pre-specified optimal profile of the dissolved oxygen concentration. To this effect, this paper aims at developing a novel control law based on a nonlinear dynamic inversion method. The closedloop strategy includes an observer based on a Bayesian Regression with Gaussian Process, which is used for on-line estimating the biomass present in the bioreactor. Unlike other approaches, the proposed controller leads to an improved response time with effective disturbance rejection properties, while simultaneously prevents undesired oscillations of the dissolved oxygen concentration. Simulation results based on available experimental data were used to show the effectiveness of the proposal.

2 Keywords: δ-endotoxins Production, Dissolved Oxygen Control, Nonlinear Dynamic Inversion, Batch Process.. Introduction Bioprocesses control strategies are important for reducing production costs and increasing productivity while simultaneously maintaining the product quality. Bacillus thuringiensis (Bt) is an aerobic bacterium used for the production of δ-endotoxins; and the obtained bioinsecticide is of great interest and importance on account of its low environmental impact. Since Bt is an aerobic bacterium, then the dissolved oxygen present in the culture medium is an important production variable. Inadequate levels of oxygen could inhibit or limit the microorganism growth. The control of biotechnological processes that include aerobics microorganisms normally involves the feed of high oxygen flow rates, in order to ensure an excess of the dissolved oxygen concentration (C) with respect to its nominal value (Rocha-Valadez et al., 27). Often a nutrient as dissolved oxygen is a growth-limiting substrate and an agitator is used to minimize spatial gradients of substrate and cell concentrations, thus improving the bioreactor productivity. The agitation speed is chosen to render a proper mixing while avoiding excessive shear forces that might cause cells rupture (Henson, 26). Several articles have studied the effect of the dissolved oxygen in medium for various microorganism cultures. For example, the aeration and agitation rates as well as the type and size of the fermenter and its accessories directly affect the volumetric oxygen transfer coefficient in the production of exopolysaccharides from Enterobacter cloacae WD7 (Bandaiphet and Prasertsan, 26). A continuous-time phenomenological model that describes the oxygen dynamics in an E. coli fed-batch culture has been developed (de Maré and Hagander, 26). In that work, some model parameters have been estimated off-line from empirical formulas while others were assumed to be known. Most research articles on

3 Bt processes suggest that C must be kept below the minimal critical value for the microorganism growth. Data on critical oxygen concentration in Bt fermentations (Moraes et al., 98) indicate high levels of C promote the formation of toxic compounds for Bt. In many instances, the oxygen partial pressure (OPP) is an inhibitor for aerobic cultures if it exceeds a certain threshold level ( bar). Besides, high OPP inhibit microbial growth and product formation (Onken and Liefke, 989). On the other hand, a C lower than the critical value for the microorganisms produces the well-know oxygen limitation effect; and it is apparent that high and low C, as well as oxygen fluctuations, can be detrimental for protein expression (Konz et al., 998). For most microorganisms, the critical C (C*) ranges between. and mg/l. These values are relatively low when compared with the O2 solubility in a diluted aqueous medium at 3 C (around 7.8 mg/l). Therefore, an appropriate O2 flow should continuously be fed into the bioreactor to approximately compensate for its consumption rate. This is a key factor that must be taken into account when designing bioreactors for aerobic microorganism culture. In some fermentations, the C can also be used for estimating the microorganisms concentration in the medium when the O2 consumption is known (di Sciascio and Amicarelli, 28; Amicarelli et al., 24). In particular, the Bt life has two phases: a vegetative phase and a sporulated phase. In batch Bt fermentations, the oxygen concentration is critical during a large part of the microorganisms growth (the vegetative phase), but in addition the oxygen demand decreases during the sporulation phase. Some typical C profiles are available for fed-batch fermentations (Wu et al., 22; Li et al., 29). Also, data on breathing and C for Bt were published (Ribbons, 969). Some experimental results have shown a significant effect of the agitation speed on the Bt production during the stationary phase (Wu et al., 22). For the Bt process, the following has been verified (Ghribi et al., 27): i) a high C increases the cell density while reduces the δ-endotoxins synthesis; and ii) a C

4 profile that maximizes the microorganism productivity is achieved by maintaining the reactor aeration near to 6% - 7% of oxygen saturation during the first 5-7 hours of fermentation, while a 4% of oxygen saturation should be maintained at the sporulation phase, until the end of fermentation and regardless of the carbon source used as substrate (Ghribi et al., 27). All these features allow, in principle, a large-scale production of proteins insecticides with low production costs. A biomass estimator based on Bayesian Regression with Gaussian Process has been proposed in (di Sciascio and Amicarelli, 28) for this bioprocess, and is used integrated in closed loop with the proposed controller in this work. In that work, the time evolution of the biomass concentration was considered to be a stochastic process (i.e., an uncertain dynamic system perturbed by a process noise), where the evolution of the biomass concentration for a particular fermentation can be considered as a realization of the stochastic process. Since the Bt life span exhibits highly different dynamics along its vegetative growth and sporulation phases, then the time evolution of the biomass concentration resembles a non-stationary stochastic process. Bayesian Regression with Gaussian Process is a non-parametric technique based on both experimental data and qualitative knowledge of the bioprocess (di Sciascio and Amicarelli, 28). The Bayesian non-parametric framework is flexible enough to represent a wide variety of bioprocess data. It makes possible interpreting the prior distribution, computing the posterior and full predictive distributions, as well as evaluating the mean predictions and the predictive uncertainties. In practice, biomass, oxygen, and substrate concentrations data can be acquired and recorded at different sampling times, thus conforming a typical missing data problem (Little and Rubin, 24). Then, in a first stage, Bayesian Gaussian Process Regression can be used as an imputation strategy for filling the missing values. In a second stage, Gaussian process regression schema was used for the biomass estimation. The

5 algorithm consists in first choosing a particular form of the covariance function, then computing the posterior distribution of the Bayes rules from the likelihood and prior distributions, and finally integrating the information contained in the observed data. For simplicity the prior mean function was set to be zero, without loss of generality provided that a constant term is included in the covariance function (Williams and Rasmussen, 996; Kuß, 26). In this way, it is possible to analytically express the posterior distribution (see e.g., (Mardia and Marshall, 984)). The hyper-parameters are the parameters of the likelihood and prior distributions. In contrast, hyper-parameters of the covariance function are unknown, and must be determined using the training data. The maximization of the posterior distribution is known as the maximum at posterior (MAP) approach, which is a Bayesian version of the maximum likelihood estimate. The MAP estimate was found by using the mentioned analytical expression of the posterior distribution in a gradient descent, or conjugate-gradient optimization technique, to calculate a local maximum of the posterior distribution. The number of hyper-parameters of the covariance function linearly increases with n, the dimension of the input space. As (Ali et al., 25) said, the trend in the use of observers for chemical process has changed from single-based observer design to hybrid observers design (eg. combination of a reduced-order observer with both the sliding mode and asymptotic/exponential observers), although single-based observers (e.g., the Bayesian Estimators, the asymptotic and exponential observers) will still be applicable in many applications involving chemical process systems. More details about this biomass estimator can be consulted elsewhere (di Sciascio and Amicarelli, 28). In a previous work, the C control in the Bt process has been solved by using a PIDbased controller (Amicarelli et al., 28). However, relatively poor performances were detected in several simulated cases in the change of the C set point. Motivated by this, in

6 the present work, a novel controller based in nonlinear dynamic inversion is designed to track an available optimal C profile (Ghribi et al., 27) of the Bt production process, aiming to improve the performance reached when using previously-developed control laws. The derived control law requires the on-line knowledge of the biomass, and to this effect the biomass estimator based on Bayesian Regression with Gaussian Process (di Sciascio and Amicarelli, 28) is used. The evaluation of the proposed methodology is made on the original Bt process model (Amicarelli et al., 2) integrated with the biomass observer (di Sciascio and Amicarelli, 28). All used fermentation data correspond to experiments that were carried out with glucose-based substrates (Ghribi et al., 27; Amicarelli et al., 2). The paper is organized as follows. Section 2 describes the microorganism and the mathematical phenomenological model for the Bt process. The methodology for the controller design is presented in Section 3. Through several simulated experiments, Section 4 compares the performances reached with the new control law and a classical PID controller. Main conclusions and remarks are summarized in the last section. 2. Modelling 2.. Microorganism and Bioprocess characteristics After the Bt vegetative growth, the beginning of the sporulation phase is induced by a substrate deficiency, called the mean exhaustion point. Normally the sporulation phase is accompanied by the production of crystal proteins known as δ-endotoxins. After sporulation, the process is completed with the cellular wall rupture (cellular lysis), and the subsequent liberation of spores and crystals to the culture medium (Starzak and Bajpai, 99; Aronson, 993; Liu and Tzeng, 2). In a previous work, several Bt fermentations experiments have been used for modeling and validating the bioprocess, and particularly the C dynamics (Atehortúa et al., 27;

7 Amicarelli et al., 2); and such conditions were adopted for the current work. The microorganisms were Bacillus thuringiensis serovar. kurstaki strain 72-45, isolated and stored in the culture collection of the Unidad de Biotecnología y Control Biológico (Colombia) (Amicarelli et al., 2). Growth experiments of the Bt fermentation process were performed in a reactor of 2 liters of nominal volume, with an effective cultivation medium of liters. The ph of the medium was adjusted to 7. with KOH before heat sterilization. Culture conditions at harvest are typified by 9% free spores and δ-endotoxins crystals. The temperature was kept around 3 ºC by using an ON/OFF controller; whereas the ph was automatically controlled between 6.5 and 8.5 through a PID controller. The air flow and the agitation speed rate were set up at 22 l/min and 4 rpm, respectively. Manometric pressure in the reactor was set at 4,368 Pa using a pressure controller. The readings of temperature, ph, and dissolved oxygen were registered by a data acquisition system (using an Advantech PCL card). Biomass quantification was done with the dry weight method, i.e.: dry cell weight (DCW) = (final weight initial weight) / (volume of filtered microbial suspension) (Madigan et al., 997). The glucose concentration was quantified with the reducing sugar method that uses the reagent Dinitrosalicilic Acid (Miller, 959). The reagents concentration used for the ph control was nitric acid (5N) and potassium hydroxide (2N). The foam formation was avoided by aggregating a sterile antifoam solution manually. The Bt δ-endotoxins production is an aerobic operation, i.e., the cells require oxygen as a substrate to achieve cell growth and product formation. The duration of the batch fermentation is limited and depends on the initial conditions of the microorganism culture. All the fermentations were initialized with the same inoculums and different substrate concentration conditions (Atehortúa et al., 27). When the medium is inoculated, the biomass concentration increases at expense of reducing the nutrients. The fermentation

8 concludes when the glucose that limits Bt growth is consumed, or when 9% or more of cellular lysis has occurred. Without considering the latency period (the bioprocess dead time is not considered in this study), the duration of each experiment varies between 4 hours and 8 hours, approximately. The model equations are (Amicarelli et al., 2): Vegetative cells: dx v () t ( t) ks ( t) ke( t) X v ( t) () dt Sporulated cells: dx s () t ks( t) X v( t) (2) dt Total cells: Dissolved Oxygen: dx () t dt ( t) k ( t) X ( t) (3) e v d C () ( 3 ( ) () t dx t K F Csat C t K K2 X ( t) air dt dt (4) Specific growth rate (it was considered a model based on Monod equation (Bailey and Ollis, 986) for each limiting nutrient S and C): St () C () t () t max Ks S( t) Kd C ( t) (5) Substrate Balance: ds( t) ( t) msx v() t dt Yxs / (6) Where: () s s max Gs ( S ( t ) Ps ) max Gs ( S Ps ) e s e initial k t k k (7) () e emax Ge( tpe) max Ge( t Pe) e e e initial k t k k (8)

9 sat: O2 saturation concentration ( concentration in equilibrium with the oxygen partial pressure of the gaseous phase) [g/l] k s max : maximum kinetic constant h k e max : maximum death cell specific rate h Gs : gain constant of the sigmoid equation for spore formation rate Lg Ge : gain constant of the sigmoid equation for death cell specific rate h Ps : position constant of the sigmoid equation for spore formation rate gl Pe : position constant of the sigmoid equation for death cell specific rate h S initial : initial glucose concentration gl t initial : initial fermentation time h For notation, see Table. As stated in the bioprocess modeling, four sets of parameters 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 et al., 27) and (Amicarelli et al., 2) (see Table 2). Maximum glucose concentration in the medium (Smax) was used as the switching criteria among the estimated batch parameter sets. According to experimental results obtained for the δ-endotoxins production of Bt, in some cases one can observe that the C decreases to values lower than the critical one (%) (Atehortúa et al., 27), even when it is assumed that air flow rate is in excess. In Fig., two important facts can be seen: first, the critical influence of the oxygen along a large fraction of the microorganism growth (the vegetative phase), and second, the notorious fall in the demand during the sporulation phase. This situation occurs on most fermentations and the limitations caused by oxygen have been detected under various operating conditions. In the particular case of Fig., the air flow rate was set at 22 l/min, which is assumed to ensure

10 an oxygen excess, or at least to ensure a C higher than the critical C reported for the microorganism (% of the saturation concentration). The C in the culture medium decays along the first fermentation hours (first 5 to 7 hours from a total of 8 hours for completing the process). In some cases, the C can decay even below its critical value. In contrast, the oxygen demand decreases when the sporulation starts (after around the first 8 hours). These results suggest that a C control strategy is necessary for optimizing the Bt process. 3. Controller design One issue of capital importance in bioprocess is the capability of maintain bounded in a desired interval some important variables. More precisely, in some bioprocess is desirable to make the system follow pre-established profiles for the principal variables. This preestablished profiles, usually named optimal, are the best policy to operate a process because ensure lower production costs, higher productivities, better product quality, and/or make the process more environmentally friendly. As it is stated before, in the Bt process, the C was determined to be a critical variable for maximizing the δ endotoxins production. This is due, principally, to the importance of the oxygen concentration in the vegetative phase of the microorganism growth and to the remarkable fall in the dissolved oxygen demand on the sporulation phase. The latest is an essential point because in this process there is not air outlet, so the air should be fed in such a way that an oxygen inhibition could not be probable (Rocha-Valadez et al., 27). Consequently, the air flow is used as the manipulated variable in order to control the dissolved oxygen (Amicarelli et al., 2). Below is presented the methodology based in inverse dynamics for the controller design in order to follow the oxygen profile given by (Ghribi et al., 27), assumed to be an optimal feeding policy (Kamoun et al., 29).

11 3.. Problem definition The principal contribution of this paper is the developing of an original control law able to track a reference profile reported in the literature (Ghribi et al., 27). The controller methodology consist on four steps: in the first one, Eqs. ()-(8) are approximated through the Euler method; then, a state variable named pivotal variable is chosen. After that, by means of nonlinear dynamic inversion, the control action is calculated from the pivotal variable equation; and finally, the controller needs to be correctly tuned through a controller parameter in order to achieve both high performance and stability simultaneously Controller design The control action required to follow the state variable profiles showed in [3] is calculated on the basis of the previously-described process model. To this effect, the discrete version of Eqs. ()-(8) can be written through the Euler approximation as: C, S n n n n n n X v X v T ks ke X v n n n n X S X S T ks X v n n n n n X X T k C, S e X v n n n C, S n S S T m S X v YX S C C T K F C C K K X n n n n n n X X n 3 air sat 2 T (9) Where,

12 n n S C n n s d n, max C S K S K C n ks k s k max s n max exp G exp s initial s s S P G S P s n ke k e k max e n max exp G exp Ge tinitial Pe e t P e () If the bioreactor is operated in a batch mode and with an optimal substrate initial condition the control aim is to follow the oxygen profile so, at the end of the reaction time, the concentration of the sporulated cells concentration (Xs) reached its allowable maximum. Thus, this paper focuses in the control of the dissolved oxygen: from the fifth equation of system (9) the control action (Fair) is obtained. As is said in the previous subsection, the dissolved oxygen concentration is one state variable linked to the others states indirectly. This is because the specific growth rate, µ, is a function of the substrate concentration, S, and of the C. As can be easily see from Eq. (), µ affects all the state variables, thus showing that the C is an adequate variable to adopt as the pivotal one. Consider first, the immediately reachable value of the dissolved oxygen concentration, as proposed here: n n n n ref ref C C K C C () one ( This replacement force the system to evolve from the actual state ( n C ref n C ) to the reference ). Thus, the dissolved oxygen concentration tends gradually to the reference, forcing the whole bioprocess to follow the reference profile. In Eq. (), the controller parameter fulfils < K < making the tracking errors tend to zero when n (this is proved below). Then, the value of the state variable in the next

13 sample time is a function of: the reference profile ( n C ref and n C ref ), the actual state variable ( C n ) and the controller parameter (K). Remark : In Eq. () note that: If K =, the reference trajectory is reached in only one step. Then, the control actions will be very violent, leading to a possible saturation of the actuators. If < K <, the system will slowly reach the reference profile. Replacing Eq. () in the fifth equation of system (9), and solving for the control action: F n air ref C K C C C X X n n n n n n ref n K K2X T T 3 K C C sat n (2) Now, note that, in the current sampling time, X (n+) is unknown. However, it can be replaced for its equation of the third expression of Eq. (9), leading to: F n air n n n n n n n n n ref ref X T k C, S e X v X n K K2X T T n K3 Csat C C K C C C (3) Operating mathematically, the expression of the control action is F n air ref n n n n ref n n n n K k C, S e X v K2X T n K3 Csat C C K C C C (4) Remark 2: The difference between the reference and the real profile is called tracking error, and is given by: n n n e C C (5) ref Theorem : If the system behavior is ruled by Eqs. ()-(8) and the controller is designed by (4), then, e (n), n, when profile tracking problems are considered.

14 Proof. If the system behavior is ruled by Eqs. ()-(8) and the controller is designed by (4), then, e (n), n, when profile tracking problems are considered. In order to demonstrate Theorem, the control action calculated, Eq. (4), is replaced in system (9), leading to n n n n C n n K ref C C ref C X X n K K2X n n T T n 3 n sat K3 Csat C C C T K C C... (6) n n X X n K K2X T Operating mathematically, Then, calculating the tracking error as n n n n ref C C K C C (7) ref n n n ref n n n ref e C C e C C (8) Replacing (8) in (7), Now, note that if < K <, then n n e K e (9) lime n n (2) Hence, Eq. (9) tends to zero when < K < and n, thus proving that the tracking errors tend to zero Monte Carlo Randomized Algorithm In the field of systems and control, probabilistic methods have been found useful especially for problems related to robustness of uncertain systems (Tempo and Ishii, 27). One of these methods, the Monte Carlo Randomized Algorithm, is widely used in many fields such as the diffusing spins, the design of optimum sensors, particle filters, etc

15 (Havangi; Hall and Alexander, 29; Michail et al., 24; Svečko et al., 25). In the control area, Monte Carlo methods allow to estimate an expectation value and they provide effective tools for the analysis of probabilistically robust control schemes. In this subsection, the Monte Carlo method is applied to select an optimal controller parameter. The complete application of this method is explained elsewhere (Tempo and Ishii, 27). In this paper is drafted only one theorem that concern to the control problem addressed. Theorem 2 (Tempo and Ishii, 27): Let ε, δ (, ) where ε is an a priori specified accuracy, and δ, the confidence interval. If log N log (2) then, the empirical maximum satisfies the following inequality with probability greater than -δ, That is, max Prob ˆ J J (22),..., Prob Prob J Jˆ max N (23) where J is the performance function and Ĵ max, the empirical maximum. The theorem establishes that the empirical maximum is an estimate of the true value within an a priori specified accuracy ε with confidence δ if the sample size N satisfies Eq. (2). The algorithm may not produce an approximately correct answer, but the probability of this event is no greater than δ. It is worthy to emphasize that, in Theorem 2, the sample size N is finite and moreover is not dependent on the size of the uncertain set, the structured set of uncertainty matrices, neither the probability density function f, but only on ε and δ.

16 In this paper, it is adopted a confidence value (δ) of., and an accuracy of.7 (ε). Then, from Eq. (2), it is necessary to make, simulations. As was said above, the bioreactor behavior directly depends on the adjustment parameter K. So, this subsection is fundamental for the controller good performance. The method is developed as follows: i) N random values of the parameter is selected; ii) a cost function that evaluates the controller performance is set; and iii) the optimal parameters are determined by means of an optimization procedure. A widely used method consists of defining the cost corresponding to the tracking error, which is calculated as the squared difference between the reference and real profile of the dissolved oxygen (Cheein and Scaglia, 24). The performance function, namely cost function, is equal to the cumulative squared error, which can be numerically approximated as follows: C 2 ref C C T (24) 2 where T is the sample time (within this paper it is adopted as. h, selected by using a Fourier frequency analysis (di Sciascio and Amicarelli, 28)). Although the optimum is not guaranteed, the Monte Carlo Experiment provides an approximate solution based on a large number of trials (N). Hence,, values of each parameter ranging from to were simulated. This parameter range ensures convergence to zero of the tracking error (see Theorem ). The lowest cost is obtained in simulation number 22, where the parameter value was: K Comparison with a PID controller In this Section, the effectiveness of the proposed control law in closed loop with the Bayesian observer (di Sciascio and Amicarelli, 28) is verified through simulation examples. This Section demonstrate, through several examples, that the proposed control

17 law based in inverse dynamics has a better performance than the PID controller, designed in (Amicarelli et al., 28). The principal objective of the control scheme in the Bt δ-endotoxins production is to follow a reference profile of the dissolved oxygen. The foremost challenge of the chosen reference profile is the step at the sixth hour of operation. So as to compare the PID controller with the inverse dynamics based controller, five tests are implemented. In this paper, the proposed controller is similarly tested to (Amicarelli et al., 28) so that the comparison between them is direct. In the first test, the controller performance under normal operating conditions is shown; secondly, a perturbation in the control action is included; in third place, the system is disturbed with a step change; then, a 2% of biomass estimation error and a % model parametric errors are included; and finally, a One-Sided Monte Carlo Randomized Algorithm is applied in order to verify the performance of the proposed controller under a 2% model parameter uncertainty. 4.. Normal operation conditions The controller performance is tested when the process is operated under normal conditions. The model parameters are detailed in Table 2. The optimal controller parameter obtained in the previous section is used. Figure 2a shows the tracking of the reference profile without undesirable oscillations, compared with the PID controller. To better reveal the performance of the control law, the tracking error (see Eq. (5)), is shown in Fig. 2b. It can be observed from the figures that the inverse dynamics based controller is able to accurately maintain the concentration to the set point with reduced rise time compared with the PID controller. This demonstrates the ability of the inverse dynamics controller not only to assess large and small changes in the reference profile but its quick response in adjusting the air flow rate.

18 The effectiveness of the proposed controller is easy to appreciate in Fig. 2. Figure 2a shows the tracking of the reference profile without undesirable oscillations and with improved response time, compared with the PID controller. To better reveal the performance of the control law, the tracking error (Eq. (5)), is shown in Fig. 2b. The control processes that have long dead time, oscillatory output and unstable sub-process, such as biochemical plants, require a higher-level architecture controller than one degree-of-freedom PID controller tuned by one set of three parameters (Maghade and Patre, 22; Zhao et al., 22). Nevertheless, this type of controllers are difficult to design compare with the methodology based in inverse dynamics presented in this paper. Besides this peak represents a 25% lower concentration compared with the reference value, the dissolved oxygen concentration remains in a lower value than the critical one. This means that the microorganism suffer a lack of oxygen for a while, leading to a certain death of the vegetative cells or its sporulation. Since the primary purpose of this process is to maximize the sporulated cells concentration, the undesired peak performed when a PID controller is used, is completely unacceptable. In conclusion it can be said that the PID controller fails to respond as quickly and as accurately as the proposed controller, and the designed controller shows improvement in response time and oscillations Perturbation in the control action So as to demonstrate the controller performance, a random perturbation in the control action is included, as is suggested in (Savran and Kahraman, 24). In this paper, a random perturbation using MATLAB is employed: the control action is affected with a 2% of its value. The function used was random (norm,,.2), which is a random noise normally

19 distributed with zero mean and a standard deviation of.2 (George, 24). The control action is calculated through Eq. (4) and then, the perturbation is added as Eq. (25). n n Fair F (' ',,.2) perturbed air random norm (25) unperturbed The following figures illustrate the previously described results. The control action calculated compared with the feed flow rate in normal operation conditions is shown in Fig. 3a, and the controller performance is evaluated through the tracking error (Fig. 3b). As shown in Fig. 3b, when a Gaussian disturbance is introduced, the tracking error increases. Nevertheless, the tracking error is still low and bounded compared with those obtained by other authors in the literature (Amicarelli et al., 28) Step disturbance in the feed flow rate As (Lübbert and Bay Jørgensen, 2) said, although a rich theory has been developed for the robust control of linear systems, there is a slight knowledge about the robust control of nonlinear systems with constraints, and most bioprocesses are nonlinear. Control systems maintaining stability when the system's dynamics (parameter uncertainty, error in the state variables estimation, fluctuation in the control action) deviates slightly, are said to be robust. Then, one way to ensure the stability, consists in introducing a step-change perturbation in the control action and evaluate the performance of the controller under this perturbation. As proposed by (Tebbani et al., 28), a feed flow rate 4% higher than calculated is adopted. In (Craven et al., 24) the authors discuss the existence of non-modeled disturbances faced during the real-time implementation of a bioprocess. It is not possible predict disturbances in bioprocess due to its inherent complexity and its strongly coupled variables. So, in this paper is introduced a step-change perturbation in the control action throughout the batch time.

20 Figure 4a compares the feed flow rate of the proposed controller normal operation (named Control action in the normal operation) and the calculated control action under the step disturbance (named Control action of the disturbed system). Figure 4b shows that the tracking error obtained with the proposed controller when the system dynamics deviates from its normal condition, remains similar to the normal operation, thus providing a test of robustness Biomass estimation and model parametric errors In general, between several batch fermentations there exist unavoidable differences in process conditions and therefore the task of modeling is normally complicated. The bioprocess always presents strong nonlinearity, disturbance presence from the external environment (foreign microorganism, alterations in the substrate feeding), and even when the process conditions are the same in all fermentations, the microorganisms can behave differently every time. In (Amicarelli et al., 2) it was adopted a first principles model that represents functional (causal) relations between physical variables, succeeding in understanding the bioprocess evolution. The experimental data used in the parameters identification were sufficiently representative of a set of available batch fermentations, and the parameters probability density distributions obtained are consistent with the experimental data record lengths. However, due to the above mentioned, it is necessary to test the designed controller when there are errors in the state variables estimation and in the model parameters. Figure 5a shows an example of the performance of the designed dissolved oxygen controller in closed-loop with a biomass estimator, most specifically shows the controller output for 2% biomass estimation errors and for % model parametric errors compared with the PID controller. In Fig. 5b the tracking error is shown for both cases (with and

21 without estimation and parametric model errors). The inverse dynamics based controller succeeds in following the reference C profile despite the existence of errors in the biomass estimation and in the model parameters Performance of the controller under parameter uncertainty In this subsection, a Monte Carlo simulation is performed to demonstrate the effectiveness of the controller from the statistical viewpoint (Karlgaard and Shen, 23; Hao et al., 24; Michail et al., 24), under parameter uncertainty. In order to prove the robustness of the controller designed, some parameters described in Table 2 are varied in a range of ±2% of its nominal values. Within the Monte Carlo Randomized Algorithms, this is an example of the worst-case problem. In this experiment is adopted a confidence value (δ) of. and an accuracy (ε) of.7. Then, from Eq. (2), it is necessary to make, simulations. Figure 6a shows the C for the, simulations. This figure shows clearly that the performance of the controller is stable, because the state variable tend to the reference profile without undesirable oscillations, although there are a 2% of uncertainly in the bioreactor parameters. Figure 6b depicts the performance function, defined as Eq. (24), for, sets of the system parameters. It is shown that the tracking errors, reflected in the performance function C, remain bounded. Then, it follows that if, simulations with different values randomly chosen of all the parameters are carried out, and the tracking errors remains bounded, there is a 99% probability that the performance of the controller will be proper whatever the parameter values in a range of ±2%. 5. Conclusions

22 A new control law for the tracking of optimal concentration profile in a batch bioreactor has been presented. The proposed method allowed to control a nonlinear system. A One-Sided Monte Carlo Randomized Algorithm is used in order to find the controller parameter that minimizes the tracking error. The controller performance has been compared with a PID controller proposed in (Amicarelli et al., 28). It has been observed that they fail to respond quickly and accurately compared to the controller designed in this paper. This controller show improvement in response time, oscillations and disturbance rejection. Monte Carlo simulation results are provided to demonstrate the effectiveness of the controller in the presence of parameter uncertainty. In general, such methodology can be applied to many nonlinear systems, making it a promising technique for its application to several processes of the biochemical industry. Besides, the present methodology has the advantage of using discrete equations, and therefore a direct implementation in most computer-driven systems is feasible. Results of the implementation indicated the response of the controller was satisfactory in real time which confirmed the applicability of inverse dynamics for controller implementation and the feasibility of the overall design to run a bioreactor process autonomously. Acknowledgments We gratefully acknowledge the Universidad Nacional de San Juan, the National Council of Scientific and Technological Research (CONICET), Argentina, and Universidad Tecnológica Nacional (UTN) by the financial support to carry out this work. References Ali, J. M., Hoang, N. H., Hussain, M. A. and Dochain, D. (25). Review and classification of recent observers applied in chemical process systems. Computers & Chemical Engineering, 76, 27-4.

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24 Karlgaard, C. D. and Shen, H. (23). Robust state estimation using desensitized Divided Difference Filter. ISA Transactions, 52, (5), Konz, J. O., King, J. and Cooney, C. L. (998). Effects of oxygen on recombinant protein expression. Biotechnology progress, 4, (3), Kuß, M. (26). Gaussian Process Models for Robust Regression, Classification, and Reinforcement Learning. Li, X., van der Steen, G., van Dedem, G., van der Wielen, L., van Leeuwen, M., van Gulik, W., Heijnen, J., Ottens, M., Krommenhoek, E. and Gardeniers, J. (29). Application of direct fluid flow oscillations to improve mixing in microbioreactors. AIChE Journal, 55, (), Little, R. J. and Rubin, D. B. (24). Statistical analysis with missing data. John Wiley & Sons. Liu, B. L. and Tzeng, Y. M. (2). Characterization study of the sporulation kinetics of Bacillus thuringiensis. Biotechnology and Bioengineering, 68, (), -7. Lübbert, A. and Bay Jørgensen, S. (2). Bioreactor performance: a more scientific approach for practice. Journal of Biotechnology, 85, (2), Madigan, M. T., Martinko, J. M. and Parker, J. (997). Brock biology of microorganisms. Maghade, D. K. and Patre, B. M. (22). Decentralized PI/PID controllers based on gain and phase margin specifications for TITO processes. ISA Transactions, 5, (4), Mardia, K. V. and Marshall, R. J. (984). Maximum Likelihood Estimation of Models for Residual Covariance in Spatial Regression. Biometrika, 7, (), Michail, K., Zolotas, A. C. and Goodall, R. M. (24). Optimised sensor selection for control and fault tolerance of electromagnetic suspension systems: A robust loop shaping approach. ISA Transactions, 53, (), Miller, G. L. (959). Use of dinitrosalicylic acid reagent for determination of reducing sugar. Analytical chemistry, 3, (3), Moraes, I., Santana, M. and Hokka, C. (98). The influence of oxygen concentration on microbial insecticide production. Adv Biotechnol, Proceedings of 6th International Fermentation Symposium, London, Canada Onken, U. and Liefke, E. (989). Effect of total and partial pressure (oxygen and carbon dioxide) on aerobic microbial processes. Bioprocesses and engineering, Springer, Ribbons, D. W. (969). Automatic assessment of respiration during growth in stirred fermentors. Applied microbiology, 8, (3), Rocha-Valadez, J. A., Albiter, V., Caro, M. A., Serrano-Carreón, L. and Galindo, E. (27). A fermentation system designed to independently evaluate mixing and/or oxygen tension effects in microbial processes: development, application and performance. Bioprocess and Biosystems Engineering, 3, (2), Savran, A. and Kahraman, G. (24). A fuzzy model based adaptive PID controller design for nonlinear and uncertain processes. ISA Transactions, 53, (2), Starzak, M. and Bajpai, R. K. (99). A structured model for vegetative growth and sporulation inbacillus thuringiensis. Applied biochemistry and biotechnology, 28, (), Svečko, J., Malajner, M. and Gleich, D. (25). Distance estimation using RSSI and particle filter. ISA Transactions, 55, (),

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26 Figure captions Fig.. Example of experimental results of dissolved oxygen concentration for some typical batch fermentations from (di Sciascio and Amicarelli, 28). Fig. 2. (a) Comparison of the performance between the proposed controller and the PID controller. (b) Comparison of the tracking errors between the proposed controller and the PID controller. Fig. 3. (a) Control action of the proposed controller with noise compared with the controller without noise. (b) Tracking errors of the proposed controller with noise compared with the PID controller. Fig. 4. (a) Comparison between the system perturbed with a step in all the batch time with the unperturbed system. (b) Comparison of the tracking errors between the proposed controller with a step disturbance and the other two controllers without disturbance. Fig. 5 (a) Dissolved oxygen concentration with a 2% of biomass estimation error and % of model parameters errors of the proposed controller. (b) Tracking errors with a 2% of biomass estimation error and % of model parameters errors of the proposed controller compared with the normal operation of the PID controller. Fig. 6 (a) Dissolved oxygen concentration for, sets of the system parameters, ms, ksmax, Ps, Gs, kemax, Ge, Pe, µmax, YX/S, Ks, Kd, K, K2, K3 and * C of the Monte Carlo Experiment. (b) Performance function C for, sets of the system parameters of the Monte Carlo Experiment.

27 VIII CAIQ25

28 Tables Table. Variables in the model Symbol Description S substrate concentration gl t time h X s sporulated cells concentration gl X v vegetative cells concentration gl X total cell concentration gl C dissolved oxygen concentration gl specific growth rate h max maximum specific growth rate h m s maintenance constant g substrate h g cells k s kinetic constant representing the spore formation h k e death cell specific rate h Y X / S growth yield g cells g substrate K s saturation constant gl

29 Table 2. Set of model parameters for batch and for the intermittent fed-batch culture with total cell retention of Bacillus thuringiensis subsp. Kurstaki Smax < g/l g/l < Smax < 2 g/l 2 g/l < Smax < 32 g/l Smax > 2 g/l µmax h,8,7,65,58 Yx / s gg,7,58,37,5 Ks gl ms g h g,5,5 2,5 3,5 4,5 ks max h Gs gl,5,5,5,5 Ps gl ke max h Ge h Pe h, 5 4, 5 4,7, 5 4,9, 5 6

30 Figure DISSOLVED OXYGEN CONCENTRATION [g/l] 6 x Inverse dynamics Reference PID TRACKING ERROR [g/l] 5 x Inverse dynamics PID TIME [h] TIME [h] Figure 2

31 8 Unperturbed Perturbed 5 x -3 Inverse dynamics PID AIR FLOW RATE [L/h] 8 TRACKING ERROR [g/l] TIME [h] Figure TIME [h] 8 6 Control action of the disturbed system Control action in the normal operation 5 x -3 4 Inverse dynamics with the step disturbance PID 4 AIR FLOW RATE [L/h] TRACKING ERROR [g/l] TIME [h] TIME [h] Figure 4

32 VIII CAIQ x x 5 Inverse dynamics Reference Inverse dynamics PID TRACKING ERROR [g/l] TIME [h] TIME [h] Figure x -3 Reference profile C, PERFORMANCE FUNCTION DISSOLVED OXYGEN CONCENTRATION [g/l] DISSOLVED OXYGEN CONCENTRATION [g/l] TIME [h] 8 TIME [h] Figure