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Biotechnol, Prog. 1994, IO, 606-610
Suboptimal Control of Batch Fermentation Processes Based on Their Classification by the Peculiarity Index Alexandr V. Kazakov*and Sergey L. Ganipol’skiy Moscow State Academy of Food Industry, Moscow, Russia
A new method of suboptimal control for batch and fed-batch fermentation processes is proposed. It combines the properties of program and adaptive control methods. The main idea of the method is to divide the controlled fermentation processes into classes using the integral characteristic of their individual properties-the peculiarity index (PI). It is assumed that any process that belongs to a class is identical to the class’ standard process (CSP). The optimal control programs for the CSP are derived off-line. The value of PI for the current process is determined on-line, and the control program that is used is the optimal one for the corresponding CSP. Industrial scale lysine fermentation processes are taken as an example. It is shown that the efficiency of the method is close to the efficiency of adaptive control, but that it is much simpler and its performance characteristics are better.
Introduction Batch and fed-batch fermentation processes are the most widespread types of processes in biotechnology. The main dificulty in the on-line control of such processes is caused by their stochastic behavior, which is demonstrated by significant deviation in any of a process’ indices and characteristics. For instance, industrial scale lysine fermentation processes have the following coefficients of variation: duration, 24%; activity, 30%;and productivity, 40%. The stochastic behavior of batch fermentation processes prevents the use of the traditional program control, which is widely used to control deterministic batch processes. As for the adaptive control, it is not used in practice (except for baker’s yeast fermentations) because there is not enough information available (monitoring of the broth composition-the concentrations of the biomass, substrate, and metabolites-can be performed with long periods and significant errors) and fast and reliable algorithms for on-line identification of the mathematical model are absent. Some new approaches are now being developed for the “intelligent” control of batch and fed-batch fermentation processes (71,for instance, expert systems (5)or the fuzzyset method (2). In this article, a new, nontraditional approach to batch and fed-batch suboptimal control is proposed that is as simple and reliable as the program control and takes into account the individuality of the process. The main idea of the method is to divide the controlled batch fermentation processes (BFP) into classes using the integral characteristic of their individual properties-the pecularity index (PI). The classification method was described previously (3); it ensures that PI is a n Ndimensional vector, each component of which is the unit normal distribution random variable. Each class unites similar processes that are assumed to be identical to the class’ standard process (CSP). The optimal duration and optimal control program for the CSP are derived off-line. The current process’ PI value and the corresponding class number are determined on-line, and then the duration and control program that are optimal for the CSP are used. This approach is a compromise between program control, which is optimal for the “average” process, and
Figure 1. Division of PI space into six classes for lysine fermentation processes: =, bounds of the class’s areas; -,
coordinates of class’s standard processes.
adaptive control, which is optimal for the each particular process. The first is very simple but has low efficiency ( I I ) , and the second gives the highest efficiency but is very complicated and requires a lot of on-line information to identify the current process.
Theoretical Aspects The Problem Statement. The object of control is the fermentation department of a biochemical plant, which consists of M identical batch bioreactors. Its mean productivity G during period T is described by the following expression: N i=l
where variables with subscript i correspond to the ith BFP: gi, quantity of product; t i , duration; ui, control program; Bi, PI; N , number of BFPs derived during period T , which is determined from the dependence, N
xti= M T i=l
Using the above-mentioned concept of the classification of BFP on 12 classes, expressions 1and 2 could be reduced to the form,
8756-7938/94/3010-0606$04.50/0 0 1994 American Chemical Society and American Institute of Chemical Engineers
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k
k
the eq 8 solution. For example, it may be iterative dynamic programming (8) or Pontryagin's maximum principle (IO). But the most useful approach is to convert this variational problem into a nonlinear programming one by direct parametrization of the control program (6):
and k
N = M T / [ Cyjtjl
(4)
j=1
where variables with subscript j correspond to the j t h class of BFP: y,, probability; e,, PI of CSP; g,, quantity of product; tj, duration; uj, control program. The biotechnological industrial system has a consecutive structure and its productivity is limited by the "bottleneck" stage, which may be fermentation on any downstream stage. To compare the productivity of several stages, it is necessary to use a single scale-the flow rate of the broth, for instance. The flow rate of the fermentation stage, F , is determined by the following expression: k
k
F = [M~yjv,ce,,uj,tj)Y[~yjtjl j=1
(5)
j=l
where t;. is the volume of the broth produced in the j t h CSP. Thus, the standard formulation of the optimal control problem is the following: k
k
where uj(B,tj)is a given function and B is a vector of parameters. Any available software for nonlinear programming may be used for solving this problem (NAG, for example). When the conditional-optimal characteristics are found, the upper-level problem (eqs 9 and lo), which is the constrained optimization one, may be solved. Using the necessary conditions of optimality in the form of the Kuhn-Tucker theorem, the following relationships were derived:
I
t,, if dgj*ldt > c t,, if dgj*ldt < c tj E [G,Gl (12) tj*(c) = root of the equation: dgj*iat - c = o where c is the Lagrange multiplier to be determined. If the conditional-optimal characteristics gj*(tj)are convex, then for any c, vector tZ(c)exists, which obeys the conditions in eq 12. Let us define the following mean indices for the fermentation stage. 1. Duration of the bioreactor's period of calculation: k
which is subject to constraint:
2. Productivity of the bioreactor: k
where tl and t, are the minimal and maximal possible durations of BFP, D is a set of admissible controls, and F"" is the boundary of the flow rate a t the bottleneck stage. Let us decompose the problem (eqs 6 and 7) into two levels. The low-level subproblems: to find the optimal control programs for CSP with given duration:
3. Flow rate of the broth from a bioreactor: b
(15) For every value of 0 < c < c", the corresponding values of these indices exist. The boundary emaxis defined as
where t h is the given duration of BFP. And the upperlevel problem: to find the optimal duration of CSP. k
k
The problem in eqs 9-11 has the following soiu_tion: 1. t(c) is the argmaximum of the function W , if the following condition is met:
7(i0)< P " / M which is subject to constraint: b
(17)
2. i*(c*)is the root of the equation: k
where gj*(ej,u,*,tj),uj*(ej,tj), and V,*(fij,uj*,tj) are solutions of eq 8. We will call gj* and vj* the conditional-optimal caracteristics and uj* the conditional-optimal control program for CSP of the j t h class. Solution of the Formulated Problem. Various methods of the optimal control theory can be applied to
TG*> - P " / M
=0
(18)
The first case indicates that fermentation is the bottleneck stage; the second one indicates that the bottleneck stage is one of the downstream stages.
Example As a n example, the fermentation stage of lysine production was chosen. In one of the plants, the fermen-
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tation stage consists of M = 28 identical stirred-tank bioreactors of 100 m3 volume. Lysine is produced in a fed-batch culture of Brevibacterium flavum, using the molasses' technology (1). Mathematical Model. The mathematical model of the fermentation process was built in the form of a set of first-order differential equations. Each equation describes the material balance of the process for one of the basic components of the broth: biomass, lysine, sugar, auxotrophic substrate (growth factor), and dissolved oxigen. In addition, the fermentor's material balance by the broth is used.
dt
-dR_ - -k,uJ + F$Rf dt
dt
100
(%)
12.1
10.4
2.6
Table 2 class
ki
k3
k io
ki3
1
0.048 0.051 0.061 0.059 0.074 0.071
2.282 2.397 2.773 2.766 3.206 3.252
0.795 1.099 0.926 1.019 0.852 0.885
0.322 0.441 0.304 0.447 0.313 0.469
2 3 4 5 6
(21)
(19)
- S)
where D, is the set of admissible controls, determined by the model equations 19 and 20 and the following constraints:
where X,A, S , and R are concentrations of biomass, lysine, sugar, and growth factor in the broth (g/L), respectively; C is the concentration of dissolved oxygen (% of saturation);F is the flow rate of feed (m3/h);Rf and Sfare concentrations of growth factor and sugar in feed (g/L);Fa is the flow rate of air (m3/h);ux and ua are kinetic functions (h-l); kLa is the volumetric mass-transfer coefficient (kLa = ks k$a/V)(h-l); and k l - k l , are the parameters of the model. The structure of the kinetic functions u, and ua was constructed using the nonlinear stepwise regression method (12):
+
R ux = klo(k,,
k15
To solve this constrained variational problem, the parametrization approach was used. As it has been shown (4), satisfactory results give the following simplest structure of the control function: U(t)=
i
0,
if t
t,
p , i f t, It 5 t, 0, i f t > t,
The parametrized problem takes the form
C
+ R ) (k12 + C) (20)
+
index of sensitivity
- R)
-dS_ - -(k2ux + kQu,+ k,)X + F$3f'
S
parameter ki
Optimal Control Program. The main control action on the lysine fermentation process is the flow rate of feed supply. The problem of choosing the optimal feed supply program has the form
dt
V, = k
Table 1
A +A)
C
13(k,4 S ) (k15 + S ) + C) The following values of the parameters were obtained after identification of the model: k l = 0.068; k2 = 1.15; k3 = 2.62; k4 = 0.682 x k5 = 0.138 x lo4;k6 = 0.421 x lo3;k7 = 60.7; ks = -13.5; ks = 4.75; klo = 0.81; k l l = 1.2; k l z = 54.8; k 1 3 = 0.357; ki4 = 17.4; ki5 = 172; ki6 = 23.6; k17 = 47.9. Independent experimental data was used to verify the model. It was found that the model is adequate according to the Fisher test for all of the directly measured variables-biomass and lysine and sugar concentrations (a = 0.05). The sensitivity of the model's parameters to individual peculiarities of the processes was investigated. The criterion of sensitivity was the index of sensitivity, which is defined for any ki, i = 1-17, as the product of the standard deviation of ki and the derivative of the criterion of identification of the model (eq 19) with respect to k , . To derive these values, the individual models of 143 fermentations were identified. Four indices of sensitivity are presented in Table 1.
As a n example, the problem was solved numerically for typical initial conditions X(0)= 1.0, S(0) == 125, R(0)= 1.3, A(O) = 1.0, C(0) = 100, and V(0) = 56 m3, and the standard values of the other parameters were VY = 75 m3, F* = 1.5 m3/h, S* = 4 g/L, Sf= 220 g L , Rf = 0.95 g/L, and Fa = 5000 m3/h. The following parameters of the feed supply program were derived for duration of the process tf = 55 h: t! = 23.5 h; ti = 40.8 h; Fo = 1.3 m3/h. This program provides the value of the objective function, go = 3704 kg. Classification of the Lysine Fermentation Processes. Classification of fed-batch lysine fermentation processes was done in accordance with the method described in reference 3. The following indices of the completed processes were used to derive a n a posteriori (main) classification of PI: z l , lysine concentration (g/ L); z2, duration of fermentation (h); 23, yield of lysine per sugar (g/g); 2 4 , yield of biomass per sugar (g/g); 25, yield of lysine per biomass (g/g); 26,yield of biomass per growth factor (g/g); and 27, specific productivity (gfih). To build the operative classificator of PI, the parameters k l , k3, klo, and k13 of the mathematical model (eqs 19 and 20) were used, as they are sensitive to the individual peculiarity of the fermentation process. The
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Table 3. Conditional-OptimalCharacteristics and Parameters of Optimal Control Programs of the Six Classes’ Standard Processes tf(h)
class 1 g*(t) VY (m3)
g*(t)
class 2
VY (m3)
g*(t)
VY (m3)
g*W
40.0 45.0 50.0 55.0 60.0 65.0
2.44 2.91 3.40 3.90 4.20 4.30
3.39 3.88 3.86 3.87 3.87 3.87
70.7 75.4 75.0 75.0 75.0 75.0
3.01 3.43 3.46 3.46 3.46 3.46
70.6 75.0 75.0 75.0 75.0 75.0
3.51 3.61 3.62 3.62 3.62 3.79
57.6 62.1 67.3 72.1 74.8 74.9
class 3
calculation using experimental data from 143 industrial processes gave the following expressions of the classificators, each consisting of two components:
class 4
where Dw is the set of admissible controls, defined by the model of the j t h CSP and constraints (eq 22; j = 1-6) and tj and tf are connected by the relationship:
tj = t,
+ to
(27) where to = constant is the mean time of the auxiliary operations (in the example it is assumed that to = 32 h). The problem was solved numerically for the fixed values of tf in the range 45 < tf < 65 h, with a time step of 5 h. The results are presented in Table 3. The conditional-optimal characteristics gj*(tf) were approximated by the expression
+
gj*(tf)= uoj alj{l - exp[-u& - 4011) (28) where aoj, ay, and a3 are parameters (see Table 4). To solve the problem in eqs 9 and 10, the approximated characteristics gj*(tf)were used. For some values of 0 < c < cy, the corresponding values of tJc), j(c), &I, and fit)
72.5 75.0 75.0 75.0 75.0 75.0
3.08 3.22 3.22 3.22 3.22 3.22
73.0 75.0 75.0 75.0 75.0 75.0
aoi
ali
mi
1 2 3 4 5 6
2.398 3.390 3.010 3.590 2.910 3.080
2.983 0.476 0.456 0.031 0.275 0.136
0.043 3.429 0.503 0.236 0.590 2.308
+
where 81 is the ith component of the a posteriori classificator (i = 1, 2) and 8i is the ith component of the operative classificator. The coefficients of canonical correlation are el = 0.791 and e2 = 0.432. Because el > ez, PI space is divided into six parts: three along 81 and two along 8 2 . Each part contains the PI of the fermentation processes that belong to one class. The boundaries of the class’ parts and CSP coordinates were derived in accordance with reference 4. The theoretical probabilities that BFP belongs to the corresponding class are the following: y1 = yz = y5 = y6 = 0.135 and y 3 = y4 = 0.23. Calculation of the Conditional-Optimal Characteristics of the CSP. The next stage of research consists of the development of a mathematical model for each CSP. Such a model has the same structure as the “mean” model (eqs 19 and 20), but differs from it by the values of the sensitive parameters kl, k3, hlo, and k13. The results of the CSP model identification are presented in Table 2. The conditional-optimal characteristics of each CSP are calculated as the solution to eq 8 in the parametrized form, which is analogous to eq 24:
2.93 3.17 3.19 3.19 3.19 3.19
VY (m3)
class
+
+ 8, = 0.001k1 + 0.083k3 + 0.652k10+ O.62Okl3 8, = 0.348k1 + 0.941k3 - o.099klo - o.o07k1,
75.0 75.0 75.0 75.0 75.0 75.0
class 6
VY (m3) g*(t)
Table 4. Parameters of Equation 28
8, = -0.2432, - 0.1462, - 0.2892, - 0.3892, 0.1412, - 0.3762, - 0.0792, 8, = 0.2572, - 0.2652, 0.2262, - 0.0832, 0.2152, - 0.1252, 0.3832, (25)
class 5
VY (m3) g*(t)
Table 5. Parameters of the Fermentation Stage’s Optimal Regimes in Lysine Production t(h) 77 80 82 84 87 90 92 97
tl
(h) t z ( h )
97.0 97.0 97.0 97.0 97.0 97.0 97.0 97.0
73.2 73.9 74.2 74.6 75.1 75.9 77.5 97.0
t3
( h ) t4(h)
76.1 81.0 83.3 85.6 89.0 94.4 97.0 97.0
72.0 76.6 81.4 86.3 93.6 97.0 97.0 97.0
t5
(h) t6(h) t(kgh) 7(m3/h)
74.9 79.1 81.0 83.0 85.9 90.5 97.0 97.0
73.0 74.1 74.6 75.11 75.8 77.0 79.5 97.0
46.4 44.8 44.0 42.9 41.4 40.0 39.2 37.2
0.962 0.932 0.913 0.894 0.865 0.837 0.815 0.770
were calculated using expressions 12-16. The res@@ are tabulated in Table 5. The analysis of functions i(t) and f(t) shows that first is unimodal while the second function is decreasing monotonically. So the conditions 17-18 arg _true.-In the example, the argmaximum of the function i(t) is to = 76.6 h; the corresponding values of the other parameters of the fermentation stage’s regime are the following:
io= 46.5 kgh; f
= 0.98 m3h; t, = 95.6 h; t, = 73 h; t, = 75.2 h; t, = 72 h; t , = 74.1 h; t, = 72.8 h (29) If F a> Mf“, then the optimal regime for the fermentation stage is described by eq 29. In any other case it is determined from eq 18 using Table 5. For-example, assume that Fm” = 24.2-m3/hand M = 28, thenf“ = 0.865 m3/h and Table 5 gives t* = 87 h, t* = 41.4 k g h , tl = 97 h, tz = 75.1 h, t3 = 89.1 h, t4 = 93.6 h, t5 = 85.9 h, and t6 = 75.8 h. Simulation and Testing of the Method. Computer simulation of the proposed method has been performed using experimental data from the same 143 industrial lysine fermentations that were used above. Optimal parameters for each process were calculated as the solution of the problem, which is similar to eq 24 but differs from it only because tf is the parameter to be determined. The mean specific productivity for optimal conditions was calculated as the upper approach of the effectiveness for the adaptive control system. Its value was found to be 48 kg/h. As for the proposed method, in accordance with eq 29 the highest value of its productivity under the same conditions is 46.5 kgh. Thus, control of lysine BFP, based on their PI classification using six classes, provides a calculated efficiency close to that of adaptive control. Experimental testing of the method had been performed a t the same biochemical plant where the initial experimental data was recorded. The mean productivity
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of the bioreactors in the testing fermentations was 38.2 kgh, which is 12.3%more than with ordinary control. The result obtained is only 0.3 of the calculated method's potential but, as was determined for many computeraided control systems in energetics, chemistry, and metallurgy, if the calculated potential of system is more than 10%) it usually can be realized as less than 0.25 (9). Thus, the results obtained may be considered to be satisfactory.
Conclusions The proposed new method of suboptimal control of batch and fed-batch fermentation processes seems to be preferable naw for industrial computer-aided control systems because of its efficiency and simplicity and because of the reliability and universality of its basic applied algorithms. The last feature allows one to build a standard computer-aided control system for industrial fermentation processes that may be adjusted easily to any production.
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(4)Ganipol'skii, S.L. Derivation of the classification and optimal control methods for fed-batch fermentation processes (on lysine biosynthesis example). Ph.D. Dissertation, Inst. Food Technology, Moscow, 1989. (5)Hitzman, 8.;Lubbert, A.; Schugerl, K. An expert system approach for the control of a bioprocess. Biotechnol. Bioeng. 1992,39,33-43. (6)King, R. E.;Aragola, J.; Constantinides, A. Specific optimal control of a batch fermentator. Znt. J . Control 1974,20,869879. (7)Konstantinov, L. V.; Yoshida, T. Knowledge-based control of fermentation processes. Biotechnol. Bioeng. 1992,39,479486. (8) Luus, R. Optimization of Fed-Batch Fermentation by Iterative Dynamic Programming. Biotechnol. Bioeng. 1993,4 1 , 599-602. (9)Minsker, I. N.Comperative analysis problems of functioning computer control systems for technological processes. I n Proceedings of the 8th ZFAC Congress, Kyoto, 1981,p 237. (10)Pontryagin, L. S.; Boltyanskii, V. G.; Gamkrelidze, R. V.; Mishchenco, E. The Mathematical Theory of Optimal Processes; Interscience: 1962. (11)Reuss, M. Modelling and optimization of processes. I n Proceedings of the 8th International Biotechnology Symposium, Paris, 1989;Vol. 1,pp 523-536. (12)Tartakovskii, B. A.; Kazakov, A. V.; Lipovskaya, E. P. Use of the nonlinear stepwise regression method in reconstructing the structure of kinetic relationships for biosynthesis processes. Theor. Found. Chem. Eng. 1990,24,148-154. Accepted May 18, 1994.@ @
Abstract published in Advance ACS Abstracts, June 15, 1994.
