Optimal control of the penicillin G fed-batch fermentation: An ...

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14 J. F. VAN IMPE ETAL.corresponding parameters are in general time-varying. Secondly, up till now there is a lack ofreliable sensors suited to real-time monitoring of process variables which are needed inadvanced control algorithms. Therefore the earliest attempts at control of a biotechnologicalprocess used no model at all. Successful state trajectories from previous runs which had beenstored in the process computer were tracked using open-loop control. Many industrialfermentations are still operated using this method.During the last two decades, two trends for the design of monitoring and control algorithmsfor fermentation processes have emerged. ' In a jrst approach the difficulties in obtaining anaccurate mathematical process model are ignored. In numerous papers classical methods (e.g.Kalman filtering, optimal control theory, etc.) are applied under the assumption that the modelis perfectly known. Owing to this oversimplification, it is very unlikely that a real-lifeimplementation of such controllers - very often this implementation is already hampered bye.g. monitoring problems - would result in the predicted simulation results. In a secondapproach the aim is to design specific monitoring and control algorithms without the need fora complete knowledge of the process model, using concepts from e.g. adaptive control andnon-linear linearizing control. A comprehensive treatment of these ideas can be found inReference 2 and the references cited therein.We have shown how to combine the best of both trends into one unifying methodology foroptimization of biotechnological processes: optimal adaptive control. 3*4 This is motivated asfollows. Model-based optimal control studies provide a theoretical realizable optimum.However, the real-life implementation will fail in the first place owing to modellinguncertainties. On the other hand, model-independent adaptive controllers can be designed, butthere is a priori no guarantee for at least suboptimality of the results obtained. The gapbetween the two approaches is bridged in two steps. First, heuristic control strategies aredeveloped with nearly optimal performance under all conditions. These suboptimal controllersare based on biochemical knowledge concerning the process and on a careful mathematicalanalysis of the optimal control solution. In a second step, implementation of these profiles inan adaptive model-independent way combines excellent robustness properties with nearlyoptimal performance.As an example, we consider in this paper the development of a heuristic substrate feed ratecontroller for the penicillin G fed-batch fermentation process, based on mathematical andmicrobial insights. There are at least two unstructured models available in the literature thatallow for the optimization of the final penicillin amount with respect to the glucose feedingrate: the model of Heijnen et al. and the model of Bajpai and ReuD. The latter has beenanalysed in References 4 and 7. The analysis in this paper is based on the unstructuredmathematical model proposed by Heijnen et al. ' For the second step of the above approach,i.e. the adaptive implementation, we refer to References 3 and 4.Nowadays penicillin G is an almost common antibiotic; nevertheless the fermentationtechnology and the mathematical description of the production process are still subjects ofinterest. The optimization of product formation during fed-batch fermentation as a part oftotal process control has gained renewed attention. *The category of secondary metabolites includes a large number of extremely valuablecompounds whose mass production has revolutionized public health in modern society. In thispaper the example of penicillin G fed-batch fermentation is only used as a vehicle forpresenting the basic ideas, methodology and results obtained. This fermentation process canbe considered representative for the whole class of processes with secondary metabolite^.^The paper is organized as follows. Section 2 presents the original model of Heijnen et al.and the modifications to make it suitable for standard optimal control, together with the
PENICILLIN G FED-BATCH FERMENTATION 15statement of the complete optimization problem. We also formulate the basic conjecture ofthis paper. In Section 3 we develop for the first time the optimal feed rate profile maximizingthe final amount of product, in verifying the statement by Heijnen et al. that the glucose feedscheme is of crucial importance in obtaining high penicillin yields. Section 4 presents a physicalinterpretation of singular control, based on a mathematical analysis of the optimal controlsolution. In Section 5 we derive a suboptimal strategy based on mathematical and microbialknowledge, that is found to be a useful alternative for the optimal open-loop feed rate profile.It opens perspectives for more reliable, adaptive, model-independent control schemes. Someconclusions are formulated in Section 6.Apart from the derivation of optimal control profiles for this model - which are importanton their own - we believe that the most important contributions of this paper are thefollowing. The realizable gain due to feed profile optimization is in the region of severalhundred per cent. Furthermore, the model is very sensitive towards different feeding policies.These results, together with the fact that the commonly observed biphasic behaviour of thepenicillin fed-batch fermentation has disappeared after optimization, raise some questionsconcerning the validity of this model. In this way this optimal control study can prove to bevery useful for model discrimination purposes: optimization for model discrimination. Formore details see Reference 4. Secondly, the heuristic controllers introduced in Section 5 havean excellent performance in all cases. Moreover, since the control objective (namely to keepthe specific growth rate constant during the production phase) is obtained independently of theexact analytical expressions for the specific rates, these controllers can serve indeed as a basisfor the development of model-independent control algorithms. This is elaborated in detail inReferences 3 and 4: optimal adaptive control.2.1. The original model equations2. THE MODEL OF HEIJNEN ETAL.Heijnen et al. used the following steps in the construction of a simple unstructured modelfor the penicillin G fed-batch fermentation process: (i) definition of relevant compounds in thepenicillin fermentation, (ii) specification of chemical composition and specific enthalpies, (iii)formulation of elemental balances and the enthalpy balance (one of the most interestingfeatures of this approach), (iv) formulation of mass balances for individual compounds, (v)formulation of the weight balance and (vi) selection of the kinetic equations (in this case basedon a literature survey). Their research resulted in the following continuous-time model whichthey believe to be of great possible help in the optimization of the process:dSdX-= -ux+u, - = px, -= dPdt dt dt?rx-khPwithSXPG-=- dGdtCs,inu - 0-OOO8G - O-044rc + 0*068rn + 0*392rs, + 0-032r0 + 0.687rXamount of substrate (glucose) in broth (mol)amount of cell mass in broth (mol dry weight)amount of product (penicillin) in broth (mol)total broth weight (kg)(1)
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