Simple nonsingular control approach to fed-batch fermentation optimization

Determination of the optimal feed rate for fed-batch fermentation is normally a problem in singular control with a state inequality constraint and as such is, in general, difficult to solve, especially for those described by a large number of dynamic mass balance equations. In this article we use a new set of state variables and the culture volume as the control variable. In this way the problem is converted to one of nonsingular control with the magnitude and rate constraints on the manipulated variable and can be numerically solved by a gradient-based technique, thus avoiding the difficulty associated with singular control problems. Examples are given to illustrate the method.     

Cell-mass maximization in fed-batch cultures

Complete solutions are provided for cell-mass maximization for free and fixed final times and constant and variable yields. The optimal feed rate profile is a concatenation of maximum, minimum and singular feed rates. The exact sequence and duration of each feed rate depends primarily on the initial substrate concentration, and degenerate cases arise due to the magnitude constraint on the feed rate and the length of final time t _f. When the final time is free and not in the performance index, it is infinite for constant yield so that any form of feed rate leads to the same amount of cells, while for variable yield the singular feed rate is exponential and maximizes the yield. For fixed final time the singular feed rate for constant yield is exponential and maximizes the specific growth rate by maintaining the substrate concentration constant, while for variable yield, it is semi-exponential and the substrate concentration starts near the maximum specific growth rate and moves toward the maximum yield. A simple sufficient condition for existence of singular feed rate requires an existence of a region bounded by the maxima of specific growth and cellular yield. Otherwise, the optimal feed rate profile is a bang-bang type and the bioreactor operates in batch mode.     

General characteristics of optimal feed rate profiles for various fed-batch fermentation processes

General Characteristics of the optimal feed rate profiles have been deduced for various fed-batch fermentation processes by analyzing singular controls and singular arcs. The optimal control sequences depend on the shapes of the specific growth and product formation rates, mu andpi, and the initial conditions. For fed-batch processes described by four mass balance equations, the most general optimal control sequence consists of a period of maximum feed rate, a period of minimum feed rate (a batch period), a period of singular feed rate (variable and intermediate), and a batch period. Degenerate sequences in which one or more periods are missing can result with a particular set of initial conditions. If the fermentation time is not critical, the singular control maximizes the net yield of product and only when the time is also important, it balances a trade off between the yield of product and the specific growth rate which dictates the fermentation time. With the sequence of optimal control known, the optimal feed rate profile determination is reduced to a problem of determining switching times.     

Optimization of feed rate profile for the monoclonal antibody production

The optimal control algorithm to calculate the optimal feed rate profile of nutrient solution containing two limiting nutrients was proposed. Different from other conventional optimization methods, the proposed algorithm calculated the optimal control profiles for different initial and feed conditions. The singular optimal control algorithm, dynamic programming, and nonsingular transformation algorithm were used for the optimization of simple problems of the 4th order and the performances were compared. With the proposed transformation algorithm, the final MAb concentration increased and the CPU time decreased. For the different initial glucose and glutamine conditions, the optimal control profiles were calculated with the proposed transformation algorithm. As the initial glutamine concentration increased, the final MAb concentration also increased due to the cell viability increase. This was also applied to the different feed compositions. When the glutamine concentration was increased in the feed stream, the final MAb concentration also increased.     

Optimal nonsingular control of fed-batch fermentation

Presented is a new simple method for multidimensional optimization of fed-batch fermentations based on the use of the orthogonal collocation technique. Considered is the problem of determination of optimal programs for fermentor temperature, substrate concentration in feed, feeding profile, and process duration. By reformulation of the state and control variables is obtained a nonsingular form of the optimization problem which has considerable advantage over the singular case since a complicated procedure for determination of switching times for feeding is avoided. The approximation of the state variables by Lagrange polynomials enables simple incorporation of split boundary conditions in the approximation, and the use of orthogonal collocations provides stability for integration of state and costate variables. The interpolation points are selected to obtain highest accuracy for approximation of the objective functional by the Radau-Lobatto formula. The control variables are determined by optimization of the Hamiltonian at the collocation points with the DFP method. Constraints are imposed on state and control variables.The method is applied for a homogeneous model of fermentation with volume, substrate, biomass, and product concentrations as the state variables. Computer study shows considerable simplicity of the method, its high accuracy for low order of approximation, and efficient convergence.     

Feedback optimization of fed-batch fermentation

The problem of feedback optimization of the feed rate for fed-batch fermentation processes is formulated in the framework of singular control theory and switching hypersurfaces. Using four differential balance equations that describe a general class of fedbatch processes and a general objective function to be minimized, it is shown that under certain restrictions the feedback optimization of the feed rate can be realized as a nonlinear function of the state variables, such as the concentrations of cell mass, substrate and product, and the fermentor volume. The restrictions on the initial conditions, the fermentation kinetics and the objective function, that are needed for realization of the feedback optimization, are provided. Fed-batch fermentation models of lysine and alcohol are used to construct switching curves and to illustrate the feedback optimization of the feed flow rates.     

A contribution to optimal control of fed-batch biochemical processes

The problem of looking for high efficient modern control strategies in fermentation technology is very urgent, nowdays. Particular attention should be paid to the processes in fed-batch mode. Both, optimal feedforward and feedback control approaches are suggested. A contribution is considered to have been made in the feedback control where continuous and discrete versions are treated as well. The control laws are carried out by a variation calculus problem and a polynomial pole placement synthesis solution, respectively. All the algorithms result in an optimal substrate feed rate profile. On the basis of recursive least squares identification of the model coefficients an adaptive discrete-time control strategy is proposed. Some satisfying simulation results are dealt with.     

On the optimal control of fed-batch reactors with substrate-inhibited kinetics

The optimal feed rate profiles, for fed-batch fermentation that maximizes the biomass production and accounts for time, are analyzed. The solution can be found only if the final arc of the optimal control is a batch arc, since in this case the final concentrations of substrate and biomass can be determined by ulterior conditions on the mass balance and on the final growth rate of biomass and thus it is possible to solve the resulting time optimal problem by using Green's theorem. This evidences the "turnpike property" of the solution, which tries to spend the maximum time on or at least near the singular arc along which the substrate concentration is maintained constant. The optimality of the final batch arc is related to the time operational cost in the performance index. The sequence of the control depends on the initial conditions for which six different regions, with the respective patterns, have been identified, in case the performance index allows the control sequence to have a final batch.     

Tools for improving feeding strategies in a SBR with several species

This paper analyzes feeding strategies in a sequential batch reactor (SBR) with the objective of reaching a given (low) substrate level as quickly as possible for a given volume of water. Inside the SBR, several species compete for a single substrate, which leads to a minimal time control problem in which the control variable is the feeding rate. Following Gajardo et al. (2008) SIAM J Control Optim 47(6):2827–2856, we allow the control variable to be a bounded measurable function of time combined with possible impulses associated with instantaneous dilutions. For this problem, the extremal trajectories of the singular arc type are characterized as the strategies used to maintain the substrate at a constant level. Since this optimization problem is difficult to solve, this characterization provides a valuable tool for investigating the optimality of various feeding strategies. Our aim is thus to illustrate the use of this tool by proposing potential optimal feeding strategies, which may then be compared with other more intuitive strategies. This aim was accomplished via several numerical experiments in which two specific strategies are compared.     

On-line optimal control for fed-batch culture of baker's yeast production

A method of on-line optimal control for fed-batch culture of bakers yeast production is proposed. The feed rate is taken as the control variable. The specific growth rate of the yeast is the output variable and is determined from the balance equation of oxygen. A moving model is obtained by using the data from the feed rate and the specific growth rate. Based on the moving model, an optimal feed rate for fed-batch culture is then achieved.     

Optimal fed-batch operation of recombinant cells subject to plasmid instability and death

Optimal feed rate strategy is studied for fed-batch culture of recombinant cells with plasmid instability and with different death rates for the plasmid-free cells (PFC) and plasmid-bearing cells (PBC). Most of the fed-batch fermentation is known to have first-order singularity and therefore a single singular arc. However, this study shows that a singular arc with second-order singularity and therefore two distinct singular arcs are possible for a recombinant cell process if PFC and PBC are subjected to death, and their specific growth rates are proportional to each other. Two types of singular arcs are elucidated and analyzed. The optimal policies over the singular arcs are theoretically explored as these findings reveal qualitative information on the singular arc, which is critically important in providing the optimal initial conditions in numerical computation of optimal feed rate profile.     

An optimal operating strategy for fixed-bed bioreactors used in wastewater treatment

Optimization of a fixed-bed bioreactor used in wastewater treatment is addressed. The objective of optimization is to maximize the treatment efficiency of the biofilter by manipulating the feed flow rate while satisfying operational constraints. Numerical results indicate that the optimal input is characterized as being on the boundary of the admissible region. Thus, the characterized optimal solution is implemented using a simple feedback control law, which provides the optimal input profile despite variations in substrate inlet concentration and biomass growth rate.     

Minimal time control of fed-batch bioreactor with product inhibition

This paper is devoted to the minimal time control problem for fed-batch bioreactors, in presence of an inhibitory product, which is released by the biomass proportionally to its growth. We first consider a growth rate with substrate saturation and product inhibition, and we prove that the optimal strategy is fill and wait (bang-bang). We then investigate the case of the Jin growth rate which takes into account substrate and product inhibition. For this type of growth function, we can prove the existence of singular arc paths defining singular strategies. Several configurations are addressed depending on the parameter set. For each case, we provide an optimal feedback control of the problem (of type bang-bang or bang-singular-bang). These results are obtained gathering the initial system into a planar one by using conservation laws. Thanks to Pontryagin maximum principle, Green’s theorem, and properties of the switching function, we obtain the optimal synthesis. A methodology is also proposed in order to implement the optimal feeding strategies.     

Experimental optimization of fed-batch culture for poly-beta-hydroxybutyric acid production

The optimal feed rate profiles of glucose and ammonium hydroxide were calculated using a proposed model, and implemented for the production of poly-beta-hydroxybutyric acid (PHB) by Alcaligenes eutrophus. By implementing these optimal feed rates with a high glucose feed concentration of 700 g/L and an ammonium hydroxide concentration of 7%(w/w), it was possible to achieve a high final cell concentration of 141 g/L and a high PHB concentration of 105 g/L in 40 h of fed-batch operation. The PHB productivity was as high as 2.63 g/(L hr). (c) 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 56: 697-705, 1997.     

On-line optimization of recombinant product in a fed-batch bioreactor

In this paper, an efficient scheme for on-line optimization of a recombinant product in a fed-batch bioreactor is presented. This scheme is based on the parametrization of the system states and the elimination of a subset of the dynamic equations in the mathematical model of the fed-batch bioreactor. The fed-batch bioreactor considered here involves the production of chloramphenicol acetyltransferase (CAT) in a genetically modified E. coli. The optimal inducer and the glucose feed rates are obtained using the proposed optimization approach. This approach is compared with the traditional optimization approach, where all the states and the manipulated variables are parametrized. The approach presented in this paper results in a 5-fold improvement in the computational time for the recombinant product optimization. The optimization technique is employed in an on-line optimization scheme, when parametric drift and a disturbance in the manipulated variable is present. Feedback from the process is introduced through resetting the initial conditions of the model and through an observer for estimating the time varying parameter. The simulation results indicated improvement in the amount of product formed, when the optimal profile is regenerated during the course of the batch.     

On-line state estimation and control in glutamic acid production

An automatic computer control system for glutamic acid production has been developed. A method based on empirical reaction subspace and singular value decomposition was presented for on-line estimation of cell concentration, glutamic acid, and total sugar. The only on-line measured state variable was the oxygen uptake rate. The estimated cell concentration was used as an index for the addition of penicillin. The estimated total sugar concentration was used for system identification. Adaptive control was then applied to manipulate the substrate feeding rate. The total sugar concentration was maintained at a given value during the period of fed-batch culture for glutamic acid production.     

Productivity improvement in xanthan gum fermentation using multiple substrate optimization

A novel and more comprehensive formulation of the optimal control problem that reflects the operational requirements of a typical industrial fermentation has been proposed in this work. This formulation has been applied to a fed-batch bioreactor with three control variables, i.e., feed rates of carbon source, nitrogen source, and an oxygen source, to result in a 148.7% increase in product formation. Xanthan gum production using Xanthomonas campestris has been used as the model system for this optimization study, and the liquid-phase oxygen supply strategy has been used to supply oxygen to the fermentation. The formulated optimization problem has several constraints associated with it due to the nature of the system. A robust stochastic technique, differential evolution, has been used to solve this challenging optimization problem. The infinite dimensional optimization problem has been approximated to a finite dimensional one by control vector parametrization. The state constraints that are path constraints have been addressed by using penalty functions and by integrating them over the total duration to ensure a feasible solution. End point constraints on final working volume of the reactor and on the final residual concentrations of carbon and nitrogen sources have been included in the problem formulation. Further, the toxicity of the oxygen source, H(2)O(2), has been addressed by imposing a constraint on its maximum usable concentration. In addition, the initial volume of the bioreactor contents and feed concentrations have been handled as decision variables, which has enabled a well-grounded choice for their values from the optimization procedure; adhoc values are normally used in the industry. All results obtained by simulation have been validated experimentally with good agreements between experimental and simulated values.     

Run-to-run fed-batch optimization for protein production using recombinant Escherichia coli

A two-phase design approach is introduced to determine the optimal feed rate, fed glucose concentration and fermentation time to maximize protein productivity using recombinant Escherichia coli BL21 (pBAW2) strain. The first phase is applied to determine a primary S-system kinetic model using batch time-series data. Two runs were carried out in the second phase to achieve the maximum protein productivity for the fed-batch fermentation process. The computational results using the S-system kinetic model obtained from the second run are in better agreement with the experiments than those using the kinetic model obtained from batch time-series data. For cross-validation, two extra fed-batch experiments with different feed strategies were carried out for comparison with the optimal fed-batch result. From the experimental results, this approach could improve productivity by at least 3%.     

Bioprocess control from a multivariate process trajectory

A multivariate bioprocess control approach, capable of tracking a pre-set process trajectory correlated to the biomass or product concentration in the bioprocess is described. The trajectory was either a latent variable derived from multivariate statistical process monitoring (MSPC) based on partial least squares (PLS) modeling, or the absolute value of the process variable. In the control algorithm the substrate feed pump rate was calculated from on-line analyzer data. The only parameters needed were the substrate feed concentration and the substrate yield of the growth-limiting substrate. On-line near-infrared spectroscopy data were used to demonstrate the performance of the control algorithm on an Escherichia coli fed-batch cultivation for tryptophan production. The controller showed good ability to track a defined biomass trajectory during varying process dynamics. The robustness of the control was high, despite significant external disturbances on the cultivation and control parameters.     

A parameter optimization approach for the optimal control of large-scale musculoskeletal systems.

This paper describes a computational method for solving optimal control problems involving large-scale, nonlinear, dynamical systems. Central to the approach is the idea that any optimal control problem can be converted into a standard nonlinear programming problem by parameterizing each control history using a set of nodal points, which then become the variables in the resulting parameter optimization problem. A key feature of the method is that it dispenses with the need to solve the two-point, boundary-value problem derived from the necessary conditions of optimal control theory. Gradient-based methods for solving such problems do not always converge due to computational errors introduced by the highly nonlinear characteristics of the costate variables. Instead, by converting the optimal control problem into a parameter optimization problem, any number of well-developed and proven nonlinear programming algorithms can be used to compute the near-optimal control trajectories. The utility of the parameter optimization approach for solving general optimal control problems for human movement is demonstrated by applying it to a detailed optimal control model for maximum-height human jumping. The validity of the near-optimal control solution is established by comparing it to a solution of the two-point, boundary-value problem derived on the basis of a bang-bang optimal control algorithm. Quantitative comparisons between model and experiment further show that the parameter optimization solution reproduces the major features of a maximum-height, countermovement jump (i.e., trajectories of body-segmental displacements, vertical and fore-aft ground reaction forces, displacement, velocity, and acceleration of the whole-body center of mass, pattern of lower-extremity muscular activity, jump height, and total ground contact time).