The microbial production of the bio-fuel ethanol, an alternative, renewable fuel with great potential, has acquired significance in recent times due to declining fossil fuels and the global strive to become energy independent. Zymomonas mobilis is increasingly being promoted as a suitable microorganism for the industrial production of ethanol [4] in bio-reactors. The operation of the bio-reactor in continuous mode, or as a chemostat, is economically efficient from the viewpoint of production and is the most frequent means for large-scale production of bio-chemicals. Before the bio-reactor can be run in continuous mode, the microorganism culture needs to be grown in rich substrate conditions in a batch mode. Upon transition to continuous mode, inherent complexities of the chemical reactions that drive the bio-reactors (owing to the nonlinear dependence of growth rates on substrate concentrations and bio-reactor conditions such as temperatures and pH) result in a host of dynamic characteristics including unstable steady states and oscillations [4]. Furthermore, under continuous operation, the process has to be operated in the presence of constraints which originate from expression of performance criteria (requiring product concentration, for instance, to be above a prescribed value) or from physical limitations on the control actuators (such as valves or pumps). The presence of constraints however, limits the set of initial conditions starting from where closed-loop stability can be achieved and this fact needs to be accounted for in not only determining the appropriate switching time from batch to continuous mode, but in the choice of control design under continuous operation as well.

One control method well suited to handling constraints within an optimal control framework is model predictive control (MPC). In MPC, a control trajectory is computed by solving an optimization problem that minimizes a prescribed performance objective over a finite horizon (predicted using a process model) while satisfying constraints on the input and state variables. To imbue the control design with the desirable features of feedback control, only the first `piece' of the control trajectory is implemented on the system, and the control trajectory is recomputed using the next available state measurement. Several research efforts have focused on the design and stability characterization of model predictive controllers (see, e.g. [5] for an excellent review). In these predictive control approaches, explicit stability constraints are incorporated which require the process state to go to the desired equilibrium point or some neighborhood of the equilibrium point by the end of the horizon. The stability guarantees, however, are based on an assumption of initial feasibility of the optimization problem. More recently, Lyapunov-based predictive controllers were designed that guarantee feasibility from an explicitly characterized set of initial conditions subject to input and state constraints [6] and uncertainty [7] (see [8] for further results and references). The extensive results in model predictive control notwithstanding, application of model predictive control approaches to control of bio-reactors has received limited attention and the existing applications do not explicitly account for the process nonlinearity or provide an explicit characterization of the set of initial conditions starting from where the bio-reactor can be stabilized.

In contrast to continuous operation of bio-reactors, where the control objective is stabilization, the natural control objective during the batch run (preparatory to the operation of a bio-reactor in continuous mode) is achieving desired targets at the end of the batch. When the product is generated using only the batch mode, the desired product properties are utilized to formulate targets at the end of the batch, or in describing optimal trajectory criteria. Several research efforts have focused on developing control designs that accomplish this objective (see e.g., [9]). However, when the process is being run in batch mode only to `prepare' for an eventual transition to continuous mode of operation then it is necessary to account for the stability properties under continuous operation in formulating the operation criteria for the batch mode. Transitioning from batch mode to continuous mode at an inappropriate time, or not accounting for the presence of constraints under continuous operation may result in instability and loss of product under continuous operation. Motivated by these considerations, this work considers the production of ethanol in bio-reactors and addresses the problem of determining the time of transition from batch to continuous operation in a way that ensures stability under continuous mode of operation subject to input constraints. To this end, first a nonlinear model predictive control design is utilized that stabilizes the bio-reactor under continuous mode of operation. The key property of the Lyapunov-based model predictive control design is that it provides guaranteed stability from an explicitly characterized set of initial conditions. Additional constraints are incorporated in the predictive control design that expand on the set of initial conditions that can be stabilized by control designs that only require the value of the Lyapunov function to decay. Then, the explicit characterization of the set of stabilizable initial conditions under continuous mode of operation is used in designing the batch mode of operation. Specifically, a predictive control approach is used in determining the batch duration that results in stabilizable values of the state variables at the end of the batch run. An application of the proposed method is demonstrated on an ethanol production process. A dynamic model of the bio-reactor for producing ethanol, developed in [4], is used to demonstrate the application of the proposed method.

References

[1] P. D. Christofides and N. H. El-Farra, `` Control of Nonlinear and Hybrid Process Systems: Designs for Uncertainty, Constraints and Time-Delays'', Springer-Verlag, Berlin, Germany, 2005.

[2] P. Garhyan, A. Mahecha-Botero, and S. S. E. H. Elnashaie, `` Complex bifurcation/chaotic behavior of acetylcholinesterase and cholineacetyltransferase enzymes system'', App. Math. Model., 9:824-853, 2006.

[3] A. Mahecha-Botero, P. Garhyan, and S. S. E. H. Elnashaie, `` Bifurcation, stabilization, and ethanol productivity enhancement for a membrane fermentor'', Math. & Comp. Model., 41:391-406, 2005.

[4] D. Q. Mayne, J. B. Rawlings, C. V. Rao, and P. O. M. Scokaert, `` Constrained model predictive control: Stability and optimality'', Automatica, 36:789-814, 2000.

[5] P. J. McLellan, A. J. Daugulis, and J. H. Li, `` The incidence of oscillatory behavior in the continuous fermentation of Zymomonas mobilis'', Biotech. Prog., 15:667-680, 1999.

[6] P. Mhaskar., ``Robust model predictive control design for fault-tolerant control of process systems'', Ind. & Eng. Chem. Res., 45:8565-8574, 2006.

[7] P. Mhaskar, N. H. El-Farra, and P. D. Christofides, ``Stabilization of nonlinear systems with state and control constraints using Lyapunov-based predictive control'', Syst. & Contr. Lett., 55:650-659, 2006.

[8] G. Ryhiner, I. J. Dunn, E. Heinzle, and S. Rohani, ``Adaptive online optimal-control of bio-reactors - application to anaerobic degradation'', J. Biotech., 22:89-105, 1992.

[9] D. Shi, P. Mhaskar, N. H. El-Farra, and P. D. Christofides, `` Predictive control of crystal size distribution in protein crystallization'', Nanotechnology, 16:S562-S574, 2005.

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