Optimal parametric sensitivity control for the estimation of kinetic parameters in bioreactors

In this paper the well-known problem of optimal input design is considered. In particular, the focus is on input design for the estimation of kinetic parameters in bioreactors. The problem is formulated as follows: given the model structure (f,g), which is assumed to be affine in the input, and the specific parameter of interest theta;(k) find a feedback law that maximizes the sensitivity of the model output to the parameter under different flow conditions in the bioreactor and, possibly, minimize the input or state costs. Analytical solutions to these problems are presented. As an example a bioreactor with a biomass that grows according to the well-known Monod kinetics is considered.     

Nonlinear control of bioreactors with input multiplicities

Control of bioreactors exhibiting two input multiplicities, i.e., steady-state gains having opposite sign, is theoretically analyzed. The nonlinear system is represented by a unity gain linear system cascaded with a nonlinear gain. A conventional PI controller designed for the linear portion of the system followed by the solution of the nonlinear gain equation gives a nonlinear controller. The performance of the nonlinear controller is compared with that of the linear PI controller designed for the overall linear system. The nonlinear PI controller performance is superior to that of the linear PI controller.     

Nonlinear feedforward control of bioreactors with input multiplicities

A steady-state nonlinear feedforward controller (FFC) for measurable disturbances is designed for a continuous bioreactor, which is represented by Hammerstein type nonlinear model wherein the nonlinearity is a polynomial with input multiplicities. The manipulated variable is the feed substrate concentration (S_f) and the disturbance variable is the dilution rate (D). The productivity (Q=DP) is considered as the controlled variable. The desired value of Q=3.73 gives two values of feed substrate concentration. The nonlinearity in the gain is considered for relating output to the manipulated variable and separately for the relation between output to disturbance variable. The FFC is also designed for the overall linearized system. The performance of the FFC is evaluated on the nonlinear differential equation model. The FFC is also designed for the model based on a single nonlinear steady-state equation containing both D and S_f. This nonlinear FFC gives the best performance. The nonlinear FFC is also designed by using only linear gain for the disturbance and nonlinear gain for the manipulated variable. Similarly, nonlinear FFC is also designed by using linear gain for the manipulated variable and the nonlinear gain for the disturbance variable. The performances of these FFC schemes are compared.     

Non-linear control of continuous bioreactors

Control of bioreactors has achieved importance in the recent years. This may be due to the fact that they are difficult to control which may be attributed to its nonlinear dynamic behavior. The model parameters of the bioreactor also vary in an unpredictable manner. The complexity of the biochemical processes inhibits the accurate modeling and also the lack of suitable sensors make the process state difficult to characterize. Considerable emphasis has been placed on the control of fed-batch fermentors because of their prevalence in industries. However, when production of biomass is to be optimized, continuous operation is desirable. Several procedures are available for the nonlinear control of processes, viz., differential geometric approach, internal model control approach, reference synthesis technique, predictive control design, etc., but the major disadvantage of these approaches is the computational time required to perform the prediction optimization. In this study, a nonlinear controller based on a polynomial discrete time model (NARMAX) is evaluated for its performance on a fermentor. It can be shown that a nonlinear self-tuning controller based on NARMAX model can be extended to the control of fermentors. The response is smooth for both load and setpoint changes even when process parameters are assumed to be zero and uncertainty in parameters are present and in the presence of controller constraints. The control action can be made more or less robust by changing the design parameters appropriately. Therefore, nonlinear self-tuning controller is suitable for control of industrial processes.     

Nonlinear control of bioreactors with input multiplicities in dilution rate

Control problems of continuous bioreactors having two input multiplicities in dilution rate on the productivity are analyzed. The nonlinear system is represented by a unity gain linear subsystem cascaded with a nonlinear gain subsystem. A conventional PI controller designed for the linear subsystem followed by the solution of the nonlinear gain equation gives a nonlinear controller. The performance of the nonlinear controller is compared with that of the conventional PI controller and also of the nonlinear controller [1] designed based on the output equation. The present nonlinear PI controller gives a superior performance. A single set of controller settings can be used for both the operating points. Whereas the linear PI controller and the nonlinear controller proposed by Henson and Seborg [1] destabilize the system.     

Dispersal and control of anammox granular sludge at high substrate concentrations

This paper reports about the dispersal and control of anammox granular sludge at high substrate concentrations. The results demonstrate that anammox granular sludge would turn into flocculent sludge when the substrate concentrations exceed the inhibitory threshold concentrations, with an apparent drop in the settling velocity of anammox sludge from 73.73 to 16.49 m/h. Moreover, the sludge was washed out of the reactor and a decrease in the nitrogen removal rate from 23.82 to 16.97 kg N/(m³/day) was observed. The dominant anammox bacteria in the granular and flocculent sludge were Candidatus Kuenenia stuttgartiensis; however, the contents of heme c and extracellular polymeric substances in the flocculent sludge were much lower than in the granular sludge. Furthermore, the chemical composition of extracellular polymeric substances was different. The high nitrite concentrations more than the inhibitory threshold concentrations were regarded as the reason for the observed granular sludge dispersal and deterioration in reactor performance. The apparent dispersed granular sludge and malfunction of reactor performance could be recovered by means of washing out the residual substrate from the reactor and then re-running the reactor from low substrate concentrations.     

Genetic algorithms with filters for optimal control problems in fed-batch bioreactors

When using a genetic algorithm (GA) to solve optimal control problems that can arise in a fed-batch bioreactor, the most obvious direct approach is to rely on a finite dimensional discretization of the optimal control problem into a nonlinear programming problem. Usually only the control function is discretized, and the continuous control function is approximated by a series of piecewise constant functions. Even though the piecewise discretized controls that the GA produces for the optimal control problem may give good performances, the control policies often show very high activity and differ considerably from those obtained using a continuous optimization strategy. The present study introduces a few filters into a real-coded genetic algorithm as additional operators and investigates the smoothing capabilities of the filters employed. It is observed that inclusion of a filter significantly smoothens the optimal control profile and often encourages the convergence of the algorithm. The applicability of the technique is illustrated by solving two previously reported optimal control problems in fed-batch bioreactors that are known to have singular arcs.     

On-line monitoring and control of substrate concentrations in biological processes by flow injection analysis systems

Concentrations of substrates, glucose, and ammionia in biological processes have been on-line monitored by using glucose-flow injection (FIA) and ammonia-FIA systems. Based on the on-line monitored data the concentrations of substrates have been controlled by an on-off controller, a PID controller, and a neural network (NN) based controller. A simulation program has been developed to test the control quality of each controller and to estimate the control parameters. The on-off controller often produced high oscillations at the set point due to its low robustness. The control quality of a PID controller could have been improved by a high analysis frequency and by a short residence time of sample in a FIA system. A NN-based controller with 3 layers has been developed, and a 3(input)-2(hidden)-1(output) network structure has been found to be optimal for the NN-based controller. The performance of the three controllers has been tested in a simulated process as well as in a cultivation process ofSaccharomyces cerevisiae, and the performance has also been compared to simulation results. The NN-based controller with the 3-2-1 network structure was robust and stable against some disturbances, such as a sudden injection of distilled water into a biological process.     

Plots of turnover times versus added substrate concentrations provide only upper bounds to in situ substrate concentrations

In recent years it has become a common practice in the study of microbial activity in aquatic systems to estimate in situ substrate concentrations by plotting substrate turnover times (the substrate concentration divided by microbial uptake rate) against the concentrations of added radioactively labeled substrate. The rationale has been that the uptake rate will remain essentially constant if added concentrations of high specific activity substrates are small compared to in situ concentrations. Unfortunately, a correct mathematical analysis shows that the in situ concentrations estimated by this method are only upper bounds to the true in situ substrate concentrations, regardless of the size of the substrate spikes. The estimated in situ concentration will equal the true in situ concentration only if uptake rate is completely independent of substrate concentration, an unlikely situation in natural systems.     

Simple adaptive pH control in bioreactors using gain-scheduling methods

A simple well-performing adaptive control technique for pH control in fermentations of recombinant protein production processes is described and its design procedure is explained. First, the entire control algorithm was simulated and parameterized. Afterwards it was tested in real cultivation processes. The results show that this simple technique leads to significant reductions in the fluctuations of the pH values in microbial cultures at a minimum of expenditures. The signal-to-noise ratio and thus the information captured by the pH signal were increased by about an order of magnitude. This leads to a substantial improvement in the noise of many other process signals that are used to monitor and control the process. For instance, respiratory off-gas data of CO₂ and its derived carbon dioxide production rate signals from the cultures carry much less noise as compared to those values obtained with conventional pH control. Detailed process analysis revealed that even very small pH jumps of 0.03 values during the fermentation were shown to result in pronounced deflections in CO₂-volume fraction of 8% (peak to peak). The proposed controller, maintaining the pH within the interval of 0.01 around the setpoint, reduces the noise considerably.     

Living with heterogeneities in bioreactors

The presence of spatial gradients in fundamental culture parameters, such as dissolved gases, pH, concentration of substrates, and shear rate, among others, is an important problem that frequently occurs in large-scale bioreactors. This problem is caused by a deficient mixing that results from limitations inherent to traditional scale-up methods and practical constraints during large-scale bioreactor design and operation. When cultured in a heterogeneous environment, cells are continuously exposed to fluctuating conditions as they travel through the various zones of a bioreactor. Such fluctuations can affect cell metabolism, yields, and quality of the products of interest. In this review, the theoretical analyses that predict the existence of environmental gradients in bioreactors and their experimental confirmation are reviewed. The origins of gradients in common culture parameters and their effects on various organisms of biotechnological importance are discussed. In particular, studies based on the scale-down methodology, a convenient tool for assessing the effect of environmental heterogeneities, are surveyed.     

Modelling effects of high product and substrate inhibition on oscillatory behavior in continuous bioreactors

In this study we consider a model for continuous bioreactors which incorporates the effects of high product and substrate inhibition on the kinetics as well as biomass and product yields. We theoretically investigate the possibility of various dynamic behaviors in the bioreactor over different ranges of operating parameters to determine the delineating process conditions which may lead to oscillatory behavior. Application of the singular perturbation technique allows us to derive explicit conditions on the system parameters which specifically ascertain the existence of limit cycles composed of concatenations of catastrophic transitions occurring at different speeds. We discover further that the interactions between the limiting substrate and the growing microorganisms can give rise to high frequency oscillations which can arise during the transients toward the attractor or during the low-frequency cycle. Such a study not only can describe more fully the kinetics in a fermentor but also assist in formulating optimum fermentor operating conditions and in developing control strategy for maintaining optimum productivity.     

Biodegradation of phenol at high initial concentrations in two-phase partitioning batch and fed-batch bioreactors

A two-phase organic-aqueous system was used to degrade phenol in both batch and fed-batch culture. The solvent, which contained the phenol and partitioned it into the aqueous phase, was systematically selected based on volatility, solubility in the aqueous phase, partition coefficient for phenol, biocompatibility, and cost. The two-phase partitioning bioreactor used 500 mL of 2-undecanone loaded with high concentrations of phenol to deliver the xenobiotic to Pseudomonas putida ATCC 11172 in the 1-L aqueous phase, at subinhibitory levels. The initial concentrations of phenol selected for the aqueous phase were predicted using the experimentally determined partition coefficient for this ternary system of 47.6. This system was initially observed to degrade 4 g of phenol in just over 48 h in batch culture. Further loading of the organic phase in subsequent experiments demonstrated that the system was capable of degrading 10 g of phenol to completion in approximately 72 h. The higher levels of phenol in the system caused a modest increase in the duration of the lag phase, but did not lead to complete inhibition or cell death. The use of a fed-batch approach allowed the system to ultimately consume 28 g of phenol in approximately 165 h, without experiencing substrate toxicity. In this system, phenol delivery to the aqueous phase is demand based, and is directly related to the metabolic activity of the cells. This system permits high loading of phenol without the corresponding substrate inhibition commonly seen in conventional bioreactors. (c) 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 55: 155-162, 1997.     

Optimal substrate feeding policy for a fed batch fermentation with substrate and product inhibition kinetics

The optimal substrate feeding policy for the fed batch fermentation which is governed by product and substrate inhibited kinetics is presented. The conjunction point between nonsingular and singular arcs and the feeding policy along the singular arc are derived analytically in terms of the concentrations of substrate and product and the liquid volume. Thus, it is possible to determine the feeding rate by monitoring the state variables (i.e., closed loop control). As a specific example, an optimization study of the fed batch fermentation for ethanol production by Saccharomyces cerevisiae is presented. It is shown that the optimal feeding patterns are heavily dependent upon the initial conditions. The point selectivity provides the guideline for predicting the optimal feeding patterns and explaining the results of rigorous mathematical analysis.     

Event-driven time-optimal control for a class of discontinuous bioreactors

Discontinuous bioreactors may be further optimized for processing inhibitory substrates using a convenient fed-batch mode. To do so the filling rate must be controlled in such a way as to push the reaction rate to its maximum value, by increasing the substrate concentration just up to the point where inhibition begins. However, an exact optimal controller requires measuring several variables (e.g., substrate concentrations in the feed and in the tank) and also good model knowledge (e.g., yield and kinetic parameters), requirements rarely satisfied in real applications. An environmentally important case, that exemplifies all these handicaps, is toxicant wastewater treatment. There the lack of online practical pollutant sensors may allow unforeseen high shock loads to be fed to the bioreactor, causing biomass inhibition that slows down the treatment process and, in extreme cases, even renders the biological process useless. In this work an event-driven time-optimal control (ED-TOC) is proposed to circumvent these limitations. We show how to detect a "there is inhibition" event by using some computable function of the available measurements. This event drives the ED-TOC to stop the filling. Later, by detecting the symmetric event, "there is no inhibition," the ED-TOC may restart the filling. A fill-react cycling then maintains the process safely hovering near its maximum reaction rate, allowing a robust and practically time-optimal operation of the bioreactor. An experimental study case of a wastewater treatment process application is presented. There the dissolved oxygen concentration was used to detect the events needed to drive the controller.     

Oxidation of high ferrous iron concentrations by chemolithotrophic Thiobacillus ferrooxidans in packed bed bioreactors

The bacterial oxidation of high ferrous iron concentrations in batch culture has been studied in a packed bed bioreactor. It has been found that aeration rates from 0.49 to 1.2 VVM did not influence the biofilm oxidation activity during the period of biofilm formation up to 30 g l⁻¹ initial ferrous iron concentration. The contribution of swimming, attached and fixed bacteria to the Fe²⁺ oxidation process has been evaluated. Kinetics data showed that the oxidation rate depends on the aeration rate, when the initial Fe²⁺ concentration exceeded 30 g l⁻¹. The maximum overall Fe²⁺ oxidation rate was 1.8 g l⁻¹ h⁻¹, when the initial ferrous iron concentration was in the range 30 to 45 g l⁻¹.     

Long-term continuous monitoring of dissolved oxygen in cell culture medium for perfused bioreactors using optical oxygen sensors

For long-term growth of mammalian cells in perfused bioreactors, it is essential to monitor the concentration of dissolved oxygen (DO) present in the culture medium to ascertain the health of the cells. An optical oxygen sensor based on dynamic fluorescent quenching was developed for long-term continuous measurement of DO for NASA-designed rotating perfused bioreactors. Tris(4,7-diphenyl-1,10-phenanthroline) ruthenium(II) chloride is employed as the fluorescent dye indicator. A pulsed, blue LED was chosen as the excitation light source. The sensor can be sterilized using an autoclave. The sensors were tested in a perfused rotating bioreactor supporting a BHK-21 (baby hamster kidney) cell culture over one 28-day, one 43-day, and one 180-day cell runs. The sensors were initially calibrated in sterile phosphate-buffered saline (PBS) against a blood-gas analyzer (BGA), and then used continuously during the entire cell culture without recalibration. In the 180-day cell run, two oxygen sensors were employed; one interfaced at the outlet of the bioreactor and the other at the inlet of the bioreactor. The DO concentrations determined by both sensors were compared with those sampled and measured regularly with the BGA reference. The sensor outputs were found to correlate well with the BGA data throughout the experiment using a single calibration, where the DO of the culture medium varied between 25 and 60 mm Hg at the bioreactor outlet and 80-116 mm Hg at the bioreactor inlet. During all 180 days of culture, the precision and the bias were +/-5.1 mm Hg and -3.8 mm Hg at the bioreactor outlet, and +/- 19 mm Hg and -18 mm Hg at inlet. The sensor dynamic range is between 0 and 200 mm Hg and the response time is less than 1 minute. The resolution of the sensor is 0.1 mm Hg at 50 mm Hg, and 0.25 mm Hg at 130 mm Hg.     

Optimal control of epidemics with limited resources

We extend the existing work on the time-optimal control of the basic SIR epidemic model with mass action contact rate. Previous results have focused on minimizing an objective function that is a linear combination of the cost associated with using control and either the outbreak size or the infectious burden. We instead, provide analytic solutions for the control that minimizes the outbreak size (or infectious burden) under the assumption that there are limited control resources. We provide optimal control policies for an isolation only model, a vaccination only model and a combined isolation–vaccination model (or mixed model). The optimal policies described here contain many interesting features especially when compared to previous analyses. For example, under certain circumstances the optimal isolation only policy is not unique. Furthermore the optimal mixed policy is not simply a combination of the optimal isolation only policy and the optimal vaccination only policy. The results presented here also highlight a number of areas that warrant further study and emphasize that time-optimal control of the basic SIR model is still not fully understood.