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

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28 J. F. VAN IMPE ETAL.-11 -10 -9 4 -7 -6 4 4 -3 -2LolidB)Figure 6. Optimal production P(tf) as a function of BA constant strategy for 277-320 h with SO = 0 mol produces 1978.425 mol penicillin, so thegain using the optimal control is S= 130.7%.3.2.3. 0 these calculations for every Bwithin the given boundaries. The result for the optimal production P(tf) as a function ofparameter B is shown in Figure 6 as an illustration of Conjecture 1. From this plot we concludethat the sequence of optimal controls u*(B,t) is indeed convergent. Note also that forB < the cost has almost reached its limit value.4. PHYSICAL INTERPRETATION OF SINGULAR CONTROLIn Reference 4 the following theorem is proven for the performance index (8): if the specgcrates a, p, and a are functions of substrate concentration Cs only, with continuous derivativesup to second order, then during singular control the substrate concentration remains constantif and only if the product decay constant kh equals zero. This constant value maximizes theratio TI..We have already pointed out that the three specific rates u in (2), a in (3) and p in (4) arefunctions of substrate concentration Cs only. However, owing to the corner point in a, thesekinetics are not continuously differentiable for every value of C,. As a result, the abovetheorem cannot be used to characterize the singular control arc when using the originalHeijnen et al. model.On the other hand, the family of production kinetics (12) is sufficiently smooth to guaranteethe applicability of this theorem. For the original model we prove the following theorem.Theorem IConsider the minimization of performance index (8) subject to the dynamic constraint (6).Suppose that the specific rates are modelled by equations (2)-(4) which are continuouspiecewise smooth functions of the substrate concentration. Then during singular control thesubstrate concentration remains constant if and only if the hydrolysis constant kh = 0 and is
determined by the equationThis constant value maximizes the yield r/a.PENICILLIN G FED-BATCH FERMENTATION 29p = kritProof. In the following a prime denotes derivation with respect to substrate concentrationC,. For every value of the parameter B in the half open interval 10, p d the specific rates (21,(4) and (12) are smooth functions of the substrate concentration Cs only. In this case we knowthat on the singular interval Cs remains constant if and only if &h = 0.‘ Cs satisfiesWe haveBy using equation (4) in the formr’u-u’r=O (18)drdCs-=--dr dpdp dCswe obtain a relation between r’ and u’ which can be written aswithT’ = F&, B)u’R(cl, B) P J[ol. + kritI2 - 4(pcrit - B)plSubstituting (19) in (18) and noting that u’ # 0, we obtainWe now follow a similar line of reasoning (based on Conjecture 1) as used in the determinationof the optimal control for the model involving the original kinetics (3) starting from a modelwith smooth kinetics (12). Thus we consider the limit for B + 0 on both sides of the aboveequation to obtainYx/s[I~-~ritI- (p-kridlQpmU-+m+(As 2 Yp/skrit(p + krit - I P - krit 1)1Yx/sQp,rnax= ( p + brit - 1 p - krit 1)(I p - pcrit I + [I p - krit 1 - (P- pcrit)]2 Yp/skritSolving for p obviously leads toCC = PcritUsing equation (4), the substrate concentration during singular control isperit/ Yx/s + m + Qp,max/ Yp/sCs,siU = KsQs,max - (krit/ Yx/s + m + Qp,max/ Yp/s)= cs.crit
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