Biotechnol. frog. 1002, 8, 118-125
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Estimation of Fermentation Parameters Using Partial Data James Gomes and Ani1 S. Menawat* Department of Chemical Engineering, Tulane University, New Orleans, Louisiana 70118
Frequently state variables in fermentation processes cannot be measured for the lack of sensors or interference. For example, in complex medium fermentations it is nearly impossible to obtain reliable estimates of the cell concentration. Absence of such data makes it difficult to model the fermentation process accurately. In such situations a mathematical technique called external differential representation (EDR) can be used to identify the system model by estimating the parameters in the absence of part of the data. The objective of this method is to rewrite the system equations in terms of the higher order differential equations in the input and output variables. This reduces the number of state variables needed to estimate the parameters. Additionally, relationships between the measured variables and the measured outputs and known inputs can also be developed which can be used t o predict the unmeasured states. The reconstructibility of the system with the selected output is central t o this analysis. This method is demonstrated on two systems-gibberellin and gramicidin S fermentations. The external differential representation in each case is used to identify the system parameters using only partial data, and the accuracy in predicting the unmeasured variables is checked with the complete data.
Introduction It is often not possible to measure all the states of a process, especially in biological systems. This restricts our ability to model these systems properly. Generally, an approximate form or geometrical shape of the model can be written from prior experiences or from first principles, but the parameters cannot be identified for lack of complete data. When sufficient information can be gained about the system by the measurement of all or part of the outputs so that its path to the initial conditions can be retraced, the system is said to be reconstructible with that set of outputs. In other words, retracing the path would provide us with sufficient information to identify the system parameters. This paper presents how a system model can be rewritten in terms of the outputs and inputs only and how it can be used for parameter estimation. In this approach, it is not necessaryto linearize the state space of the system. The transformed system is completely reconstructible and can be used to estimate parameters, predict unknown states, and design control strategies. Reconstructibility is a system property associated with the information contained in the output measurements. All outputs do not carry the same amount of information and the information content of one output or a set of outputs can be higher than others. Finding an optimum set of outputs to reconstruct the system is important in fermentation processes. It is not possible to measure the cell mass concentration in complex medium fermentations due to the presence of dissolved solids. Probes used to monitor the pH, dissolved oxygen concentration (DO), and glucose concentration foul and drift during the course of the fermentation. Noise in these processescan seriously affect the accuracy of the readings. Also, probes for online measurement of desired products (e.g., an antibiotic) are usually not available. Most of these problems can be solved if a method can be devised by which some measurements can be used to estimate the unknown state
* To whom correspondence should be addressed. 8756-7938/92/3008-0118$03.00/0
variables. Mathematical techniques used to design such observers that can predict the state of the system depend on a model of the system. Of course, the quality of the prediction depends on the accuracy of the model. Fermentation processes are macroscopicmanifestations of complex metabolic reactions that occur within a cell. Almost all models available in the literature revolve around the empirical Monod equation, which exhibits the typical hyperbolic shape for growth rate. Hence, all such models possess nonlinear properties associated with the hyperbolic geometry, in the evolution of the cell and substrate concentrations. Several generalized growth models have been proposed (Savageau, 1980; Turner and Pruitt, 1978; Turner et al., 1976), which show different nonlinear properties under various limiting conditions. This inherent nonlinearity of microbial models is a consequence of the many biochemical reactions occurring within a cell. Several structured modeling approaches have also appeared in literature. Among these, the single-cell model (Shuler and Domach, 19831, the cybernetic model (Kompala et al., 1984) and various metabolic network models (Delgado and Liao, 1991; Palsson and Lightfoot, 1984; Palsson et al., 1984, 1985; Schauer and Heinrich, 1983; Schlosser and Bailey, 1990) are noteworthy. Models are growing in complexity to explain the experimental observations in sufficient detail. These models have strong nonlinearities and their application in control strategies has been a topic of rigorous discussion. A void, however, exists in the literature about parameter identification in the absence of complete data. The usual approach in handling a nonlinear problem is to linearize the model equations and solve it as a linear problem. However, linearization by truncation of higher order terms in series expansions is only locally valid around the operating point. For example, simulations of the Taylor series linearization of the Monod equation show large deviations from the true curve. Methods of global linearization, which are based on differential geometry, are more appropriate in cases of severe nonlinearities. Several such techniques have appeared in the literature,
0 1992 American Chemical Society and American Institute of Chemical Engineers
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each with its own set of restrictions on the system equations. A rigorous development of these techniques requires extensive analysis (Desoer and Wang, 1980;Hermann and Krener, 1977;Hunt et d.,1983;van der Schaft, 1982). Krener and Isidori (1983) proposed linearization by output injection of a single output to globally linearize unforced nonlinear systems. This method was extended to multiple outputs in forced and unforced systems (Krener and Respondek, 1985). The system is divided arbitrarily into forced and unforced segments that are treated independently in the construction of the observer. Nonlinear reconstructibilityis central to this analysis. To avoid the arbitrary division of the system into forced and unforced segments as required by Krener's approach, Keller (1987) proposed the generalized observer canonical form (GOCF) method. This technique is applicable to nonlinear systems that are linear in the input variables. As opposed to Krener's method, the GOCF depends on the first n derivatives of the inputs. This method is also concerned with the reconstruction of the system. On the basis of this reconstruction, an observer can be designed to estimate the unknown state variables. Similarly, these techniques have been applied in designing control strategies for biochemical systems. Kantor (1986) has used the technique of feedback linearization proposed by Hunt et al. (1983) for the control of achemostat. This approach redefines the input variable such that the transformed system appears linear. However, the restrictive condition of involutivity is required for this method of linearization. To relax these restrictions and make these techniques more accessible, Kravaris (Kravaris and Soroush, 1990; Kravaris, 1988; Kravaris and Chung, 1987) combined the techniques of Gilbert and Ha (1984) and Hirschorn (1979) to globally linearize systems in an input-output sense. If there is no input-output invariance and the system is of finite relative order, this method can be used to obtain a linear input-output relation. This linear transformation has been used to design controls for several systems. It, however, required an external filter to reconstruct the unmeasured states of the system. All the above techniques assume complete knowledge of the model parameters. Methods developed for global linearization of nonlinear systems apply only to a restricted class of models (Hunt et al., 1983; Krener and Respondek, 1985). Any fermentation model that incorporates the Monod structure in the cell, substrate, and product dynamics falls outside this category. An alternate approach would be to design nonlinear observers for the system based on the observability (reconstructibility) analysis. One such method which expresses the state variables of the system in terms of the measured outputs and the inputs of the system is called the external differential representation (EDR) (van der Schaft, 1989,1987). The state variables are expressed as higher order differentialequations of the measured outputs and the inputs. No attempt is made to linearize the state space of the system. The resulting system contains complete information and thus can be used for identification of parameters, for reconstruction of the unmeasured states, and to design control strategies. Application of this method to batch cultures and single-stage chemos t a b is presented here. Kinetic parameters of the models are determined using only partial experimental data with the external differential representation. The unmeasured state variables are also predicted and compared with the actual experimental data.
External Differential Representation Consider a general smooth nonlinear system in the state space form given by
YE
y = h(x)
YC RP
(1)
where x is the state variable, u is the input, and y is the measurement (observation). The EDR converts this system into a set of higher-order differential equations in the terms of the inputs and outputs: (2) R ~ ( U ,zi, ...,u ~ -y ,~3,, ...,y"-') = o i Ep where Ui and yj denote the j t h time derivative of the input u and output function y, respectively. Such a transformation is possible if and only if the system is completely reconstructible by the selected output(@to ensure that the output(s) tracks the dynamics of the system completely. Complete reconstructibility is possible if the system description satisfies the observability rank condition (van der Schaft, 1987,1982). Observability is a global concept and several parallel relationships (Hermann and Krener, 1977) may be defined depending on the premise of the problem. We are concerned with local weak observability for it can be tested algebraically. The algebra is developed by successive Lie derivatives of the observation in the direction of the system evolution. The Lie derivative is a multidimensional generalization of the directional derivative. I f f is a C" vector field on Rn and h is a Cm function on an, then the Lie derivative of h with respect to f is defined as
...
Lf(h) = ( d h f ) = -fl dh + + -fn ah (3) ax1 ax* L&) is also a Cmvector field on 8"; thus the higher order Lie derivatives can be defined inductively by LP(h) = h L:(h)
= L,[L;-'(h)l
k = 1,2,3, ... (4)
= (dL:-'(h),f)
Definition1: A nonlinear system is said to be observable in some neighborhood Uof xg, the point of interest, if the one-formsLP-l(dh(x)),k = 1, n, are linearly independent for all x in U. This may be conveniently tested by constructing the matrix A of the one-forms as follows:
...
r
-
(5)
If the rank of this matrix is n,then the system is observable under h(x). This is called the observability rank condition. The nonlinear system and the observation equations (eq 1) can be rewritten in the implicit form
Fi(x, k, u) = ki- fi(x,u)= 0 Fi(x,y ) = yi-n- hi-,(x) = 0
i = 1, ...,n
i = n + 1,...,n + p ( 6 )
The objective of EDR is to eliminate the state variable x and its derivatives in eq 6, in terms of the inputs, outputs, and their derivatives. This is possible, for the system is completely reconstructible. The EDR is obtained by successively differentiating the output with the system equations. The state variablesx can now be reconstructed in terms of the outputs, inputs, and their derivatives. The
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Table I. Comparison of Gibberellin Model Parameters cell as substrate as
parameters
Monod
observation
observation
0.0263 1.681 0.199 9.833 X 5.0X
0.0249 3.159 0.175 4.390 X 5.7 x 10-2
~~
0.0278 10.59 0.205 5.903 X 4.9 x 10-2
Pcm
K, Y m
1/N
e2
rigorous mathematical treatment of the external representation algorithm and the proof of the intermediate steps have been presented by van der Schaft (1987). Each step of the algorithm produces the next higher derivative of the observed variable. Derivatives of nonobserved variables are eliminated in terms of the state variable x , the input u,the observed output y (equivalently h(x)),and the derivatives of u and y. These equations are then rearranged into three segments: (i) the untransformed state equations, (ii) the transformed state equations, and (iii) the observation equations. Also, at each step the constant system rank assumption is checked. The constant system rank assumption with a given number of outputs is related to the observability rank of the system (van der Schaft, 1987). The algorithm terminates when the rank of two consecutive transformed equation sets are equal. If the system is completely observable with the outputs, then the final equations are expressed only in terms of the outputs and inputs. In the more general situation, where complete observability may not be present, the final version comprises the unobservable,the observable, and the external differential segments, described by the following equations:
x 2 = +(u,...,u ~ * y, - ~..., , yk*-') Ri(U,
dim x 2 = n - ii (7)
u, ...,uk'-l, y, 4, ...,yk*) = 0
There are two possible observations-the cell mass concentration or the substrate concentration. Since the system satisfies definition 1 for either measurement, it is completely reconstructible. Therefore, we have two independent options. First consider the measurement of the substrate concentration. The EDR is given by
+
where p ( S ) = /.tmS/(S Km)is the specific growth rate. The EDR is defined only in terms of the substrate concentration. So it is clear from eq 9 that by measuring only substrate concentration the parameters of the model can be identified. In conjunction with the EDR the equation for predicting the cell mass concentration can be obtained as
Similarly, when only the cell mass concentration is measured the EDR becomes
This equation is used to estimate the kinetic parameters of the system. The equation used to predict the unmeasured substrate concentration is 77
k
i E &J
Here k* denotes the iteration at which the algorithm terminates. The state space is divided into the unobservable part x1 and the reconstructed part x2. The first equation represents the dynamics of the unobservable states. The second equation represents the observable space expressed in the inputs, outputs, and their derivatives. It is clear that the external differential representation cannot reconstruct the dynamics of xl.
Applications in Fermentation The external differential representations for two fermentation processes are presented: first, gibberellin fermentation in a batch reactor (Borrow et al., 1961) modeled by the Monod equation, and second, gramicidin S production modeled by Blanch and Rogers (1971). These two processes have been selected because sufficient experimental data is available for appreciating the predictive qualities of the EDR. A nonlinear observability analysis is performed to identify the observable states of the model. Next, the observation is used to derive the EDR. In the case of the Monod model, which is the simpler of the two, only the final result is presented. A detailed derivation of the EDR is shown for the Blanch and Rogers model. The Monod Model. The Monod model for batch fermentations with constant yield and maintenance coefficient is given as
The gibberellin fermentation data was first fitted assuming that both the cell mass and substrate data were available. The values for the kinetic parameters, namely, pm, K m , Y, and m, were obtained using the Gill-Murray nonlinear least squares optimization algorithm (Scales, 1985). These parameters were then used to simulate the process and the resulting profiles are shown in Figure la. The usual overestimation in the cell mass concentration values and underestimation of the substrate concentration is noticed. Similarly,the EDR equations were used to estimate the parameters of the model when only one of the states is measured. Equations 9 and 11 are the EDR representations for the substrate and cell mass concentration measurements, respectively. The results are compared in Table I. The values for the maximum specific growth rate (pm)and the yield coefficient (Y) are approximately equal. However, the Kmand m values show discrepancies. This is more pronounced in the case of the substrate concentration as observation. It may seem that there is a loss in parameter identifiability because m and Yare present in eq 9 as the product mY only. However, the initial condition on $ is not known a priori and must be computed from the initial cell mass and substrate concentrations. Consequently, the substrate dynamics as written in eq 8, becomes an integral part of the estimation routine. To obtain a new set of trial parameters, this equation must be satisfied at every iteration. This computation at the initial condition makes m and Y identifiable. However,there is a loss in sensitivity of these
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200
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n o
--5
--5
.6 -
.Q
P
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-
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I
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L?
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0
50
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(h)
Figure 1. (a, upper left panel) Monod equation profiles for gibberellin batch fermentation data (Borrowet al., 1961). Symbols: 0, experimental values for cell mass concentration; 0, experimental values for substrate concentration; -, curve fit for the Monod equation. (b, upper right panel) Prediction of the substrate concentration with the EDR when cell mass concentration is measured. Symbols: 0 ,experimental values for cell mass concentration; 0,experimental values for substrate concentration; +, predicted values of the substrate concentration; -, EDR curve fit for cell mass concentration measurements. (c, lower left panel) Prediction of the cell mass concentration with the EDR when substrate concentration is measured. Symbols: 0 , experimental values for cell mass concentration; 0,experimental values for substrate concentration; U, predicted values of the cell mass concentration; -, EDR curve fit for substrate concentration measurements. (d, lower right panel) Comparison of parameters obtained with the Monod equation, cell mass concentration as measurement, and substrate concentration as measurement. Symbols: 0 ,experimental values for cell mass concentration; 0, experimental values for substrate concentration; -, profiles for the Monod equation parameters; - - - -, profiles for cell mass concentration as measurement; - - -, profiles for substrate concentration as measurement.
two parameters as is evident in the numerical values obtained (cf. Table I). Nevertheless, the value obtained for the yield coefficient Y is of the correct order of magnitude. Good judgement about the quality of parameters estimated can only be made along with their sensitivity profiles (Gomes and Menawat, 1991a,b; Menawat and Balachander, 1991). The fermentation model (eq 8)can often predict negative substrate concentrations because of ita structural deficiency. Since the substrate concentration values below zero are not admissible (Figure lb), they were not used in the parameter estimation. Hence the full strength of the
optimization routine is not realized. This results in the discrepancy observed in the cell mass prediction. The prediction of substrate concentration is far better because eq 11 does not suffer from the mY interaction problem of eq 9. In the case of cell mass concentration as the observation (Figure IC),the predicted values of the substrate concentration are within a 10% error of the experimental data. A comparison of the three seta of parameters obtained is presented in Figure Id. The substrate concentration profiles are essentially identical. Similarly, the cell concentration profiles are the same when this parameter is observed. The prediction from substrate
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measurement is skewed downward, which is due to the structural deficiency in the model as discussed above. The quality of data fitting reflects that upon comparing the least squares sum of the errors. Model for Gramicidin S Production. Blanch and Rogers (1971) presented a chemostat model for the production of gramicidin S as follows:
C=*kP-CD S+B
P = F - - S- + B 3 =y
A k P
S+8
AP-PD
+ (S, - S)D
+ p(S)P + AP + PD = 0
F ~ ( xk,, U ) S + r p ( S ) k P- (S, - S)D = 0
g/L
79 g l L ir, h-’
k,mg of dry weight/mg 1/N ez
k
Blanch and Rogers
EDR
1.190 0.750 5.100 0.905 3.790 2.16 X 10-2
0.798 0.204 4.271 1.145 4.911 5.72 X 10-4
(14)
F5(x,y ) = y2 - s = 0
p’(S)P yp’(S)kP + D 0 -1 (17) which now imposes two restrictions on the system for it to be of full rank:
The former restriction is trivially satisfied since the terms inside the brackets are always greater than zero. The latter restriction is same as that obtained from the observability analysis, i.e., Smust be a nonzero quantity. Hence, outside this restriction where S # 0, the rank of the matrix s2 = 3. With pz = 52 - SI = 1> 0 we move to the next step of the algorithm. Here
CrS =-
S+8
The order of the system n = 3 and the number of observations p = 2. The rank condition = sk (3) axj i=n,+l, ...,n+p;i=l,...,n
around (9,a, 9 ) (15)
where k denotes the number of the iteration, SI = 2, so that p1 = S I - SO = 2 - 0 = 2. For the first step nk = n the implicit equations (14) are replaced and renamed with the following set of equations:
y))
i=nk+l ,...,nk+pk;i=l,...,n
= Pk
(19)
is trivially satisfied and the system may now be represented as
+
F1(x,u, u, y , y , y) = ji, - ?rp(S)P+ [ P + p(S)P + AP PDID + p’(S)PS + [ p ( S ) + A + DIP + P b = 0
F ~ ( xU ,,y , 9 ) =
where
1
+r+DI rAS)k
rank ( %akj (x,
F 4 ( x , y )= y , - P = O
rank
%I
r
F,(X,k, U) = C - p(S)kP + CD = 0
P - fC
parameters a,h-*
For the second step nk = nk-l-Pk-l= 1. The rank condition now yields the matrix
If the dilution rate D , which is the input to the system, is set to zero, the model represents a batch fermentation. This form is more suitable for testing for reconstructibility since it is independent of the inputs to the system. The system is reconstructible with C or S only when the substrate concentration is nonzero, i.e., S > 0. However, there is no such restriction with the product concentration P as measurement. Since antibiotic fermentations are typically conducted in complex medium where the cell concentration is not readily possible, we choose the pair (S,P) as observation for deriving the EDR of the process. We shall follow the steps to demonstrate the EDR analysis in detail. The model equations, along with the observation equations, are rewritten in the implicit form:
F,(x, k, U )
Table 11. Comparison of Gramicidin S Model Parameters
+ r p ( S ) k P - (Sf-S)D = 0
(20)
F,(x, y ) = y1- P = 0
F4(x,y ) = y2 - s = 0 The algorithm terminates in the next step. By solving these four equations, the EDR for the model is obtained as
P = ap(S)P- [ P + p(S)P + AP + PDID - p’(S)PS[p(S)
+ A + DIP - PB
F1(x,k, U ) = C - p(S)kP + CD = 0 C = [ P + p(S)P + AP + PDI-k
A
F&,
U,y ,
9 ) = 9, + r p ( S ) k P- (Sf-S ) D = 0 (16) F 4 ( x , y )= y , - P = 0
The first two equations are used to estimate the kinetic parameters of the system using only the measurements of the substrate and the product concentration. The cell mass concentration is then predicted using the third equation. The EDR may be applied to batch fermentations by setting the dilution rate D = b = 0.
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5
5
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18
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The (hr)
o n
1
0
0 0
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16
Time (hr)
Figure 2. (a, upper left panel) Prediction of cell mass concentration with EDR in gramicidin S batch fermentation. Symbols: 0, experimental values for cell mass concentration; 0, experimental values for substrate concentration; W, predicted values of the cell mass concentration; -, EDR fit of the substrate concentration; - - -, Blanch-Rogers model profile for the substrate concentration; - _ -, -Blanch-Rogers model profile for the cell mass concentration. (b, upper right panel) Comparison of EDR parameters with Blanch-Rogers model parameters. Symbols: 0,experimental values for product concentration; 0,experimental values for substrate concentration data; -, EDR fit of the cell mass and substrate concentrations; - - - -, Blanch-Rogers model profile for the cell mass and substrate concentrations. (c,bottom panel) Prediction of cell mass concentration in gramicidinS single-stagechemwtat fermentation usingparameters obtained in the batch fermentation case. Symbols: 0,experimentalvaluesfor cellmassconcentrations; 0,experimental values for substrate concentration; W, predicted values of the cell mass concentration; -, EDR fit of the substrate concentration. The EDR for the Blanch and Rogers model for batch fermentation can be easily obtained by setting the input D (dilution rate) and its derivative to zero. In this case it is assumed that only the substrate concentration (S) and the product concentration (P) can be measured. The resulting equations obtained are
These equations are used to estimate the kinetic parameters of the system, namely, a,8, y, r, and k . The cell
mass concentration is then predicted with
c = [P + p(S)P+ rP1-k'Ir
(23)
The Gill-Murray nonlinear least squares optimization routine was used to estimate the kinetic parameters of the system. This method treats the optimization as a large residual problem. The Gill-Murray algorithm is more efficient than other general-purpose minimization algorithms for ita ability to switch automatically between quadratic Newton's and the linear steepest descent algorithms as needed (Scales, 1985). The kinetic parameters obtained using this optimization routine are presented in Table 11. The cell mass concentration is
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predicted using these kinetic parameters and is shown in Figure 2a. The overprediction at the maximum cell mass concentration is about 10%. Considering the fact that this method provides cell mass concentration values that cannot be measured, these predicted values are quite acceptable. The parameters obtained using the optimization algorithm match closely with values reported by Blanch and Rogers. Some difference in the value for /3 is observed. This discrepancy may also be attributed to the model's insensitivity of the Monod constant 0. A good basis for comparison of the parameters is the average error per data point (1/N Ce2). Since the average error for the Blanch and Rogers parameter values was not reported, it was obtained by simulation. It is clear that the parameter values obtained by the EDR have a smaller error. It may also be noticed in Figure 2b that the EDR profiles for the substrate and product concentrations are closer to the experimental values than the corresponding Blanch-Rogers profiles. However, the cell concentration is overpredicted by the EDR. This is quite reasonable from the perspective of the optimization algorithm. In the BlanchRogers parameter set, the cell mass data is also present, hence it distributes the significance of data equally among all the measured states, whereas a better overall data fit is obtained in the EDR, where emphasis is placed on the substrate and product concentration data points in the absence of cell mass data. This results in the observed overprediction. This is clear from the comparison of Blanch-Rogers model parameters with the EDR parameters as shown in Figure 2b. Hence, it can be concluded that the observed phenomenon is due to the structure of the model and it is the best that can be expected from this model. The parameters obtained by fitting the batch data are used to simulate the single-stage chemostat fermentation shown in Figure 2c. As in the batch fermentation case, there is an overprediction in the cell mass concentration. The simulated profile for the substrate concentration does not follow the initial rise in the substrate concentration. Similarly, the simulated cell mass concentration profile does not follow the lag present in the data. Once again it is due to the structural limitations of the model since it is not designed to represent transient conditions. All models having the pure Monod structure in their specific growth rate and specific substrate consumption rate [i.e., F ( S )= pmS/(Km S ) ] suffer this fate. A probable reason may be that the empirical Monod equation implicitly assumes a steady state in the intermediates.
+
measurements a self-defeating effort. Also, biosensors to measure a product of interest may not be available. However,it should be noted that the system identification is only as good as the model. The mathematics does not take into consideration any possible physical inconsistencies present in the model. The only assumption made in this technique is that of constant rank. Hence, in a sense this method is exact. As a direct consequence of the EDR, nonlinar state observers can be obtained. These equations can be used to predict the unknown states as shown here for the cell concentration.
Notation
c D dh F k
K, Lf m
n P
P R S
S
Sf U
U X
X Y
Y Y
cell mass concentration, g/L dilution rate, h-l gradient of the observation function h implicit functions turnover constant, mg of dry weight/mg of gramicidin S Monod constant, g/L Lie derivative in the direction of the function f maintenance coefficient [substrate concentration (g/L)/cell mass concentration (g/L)I order of the system number of observations product concentration, g/L higher order differential equation rank of matrix at intermediate stages substrate concentration, g/L feed concentration, g/L input of the system manifold on which the input u is defined state variable of the system manifold on which the state variable x is defined output of the system (observation) manifold on which the output y is defined yield coefficient [cell mass concentration (g/L)/ substrate concentration (g/L)I
Greek Letters a specific growth rate (Blanch& Rogers model),h-' B constant, g/L yield coefficient (Blanch & Rogers model) [subY strate (g/L)/cell mass (g/L)I specific growth rate (Monod kinetics), h-l I.L if rate of generation of mature cells, h-l
Conclusions
Literature Cited
Most models of fermentation processes can be expressed in the form of the system given by eq 1. If the system is reconstructible in the measured values of the outputs and inputs, then it is possible to monitor the dynamics of all the state variables. Otherwise, only those state variables which are observable with the measured values can be monitored using the EDR. In its final form, the EDR contains fewer equations in the state variables. If the system is completely reconstructible with the chosen outputs, then the EDR is expressed only in terms of these measurements. The external differential representation is a powerful tool for parameter estimation. This method reduces the data required for identifying the system completely. This is extremely useful in systems where part of the data cannot be measured. For example, in complex medium fermentations the presence of dissolved solids can make cell mass
Blanch, H. W.; Rogers, P. L. Production of Gramicidin S Batch and Continuous Culture. Biotechnol. Bioeng. 1971,13,843864.
Borrow, A,; Jeffreys, E. G.; Kessell, R. H. J.;Lloyd, E. C.; Lloyd, P. B.; Dixon, I. S. The Metabolism of Gibberella fujikuroi in Stirred Culture. Can. J. Microbiol. 1961, 7, 227-276. Delgado, J. P.; Liao, J. C. IdentifyingRate-ControllingEnzymes in Metabolic Pathways without Kinetic Parameters. Biotechnol. B o g . 1991, 7, 15-20. Desoer, C. A.; Wang, Y. Foundations of Feedback Theory for Nonlinear Dynamical Systems. IEEE Trans. Circuits Syst. 1980, CAS-27, 104-123. Gilbert, E.; Ha, I. J. An Approach to Nonlinear Feedback Control with Applications to Robotics. IEEE Trans. Syst. Man. Cybern. 1984, SMC-14, 879-884. Gomes, J.; Menawat, A. S. Effect of Inputs on Parameter Estimation and Prediction of Unmeasured States. AIChE J . 1991a, submitted for publication.
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Gomes, J.; Menawat, A. S. Observability, Relative Order and External Differential Representation. Int. J. Control 1991b, submitted for publication. Hermann, R.; Krener, A. J. Nonlinear Controllability and Observability. IEEE Trans. Autom. Control 1977, AC-22,728740.
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