Augmented Lagrange decomposition method for optimal control calculation of batch fermentation processes

Abstract

Decomposition method for solving two optimal control problems and one optimization problem in batch fermentation is proposed. The problems are formulated based on a nonstructured mathematical model with slowly varying parameters and a ®nite cost criterion of maximum end production. Dependence of the model parameters on one physical or chemical parameter, which could easily be used as a control input is introduced analytically in the model equations and three model descriptions are obtained by nonlinear difference equations. Sensitivity functions of state trajectories towards slowly varying coef®cients are introduced to account for model uncertainties. Based on them extended optimal control and optimization problems are formulated.The decomposition method is based on an augmented Lagrange functional, which is decomposed in the time domain by a specially constructed coordinating vector. The solution is obtained in a two level computing structure. The subproblems of the two levels are obtained from the necessary conditions for optimality and then are solved by means of gradient procedures. A computing algorithm is given.Analytical solutions are obtained for production of a xanthan gum. The computing results are given for a model of the Xanthomonas campestris fermentation process. List of symbolsx P R g/kg biomass concentration s P R g/kg substrate concentration p P R g/kg product concentration m P R current variable, denoting state and control variables m 0 Y m xY sY p g/kg initial concentrations2r m Y m xY sY p ± model coef®cients Dt h period of discretization K ± number of steps in the optimization horizon m min Y m max Y m xY sY pY u g/kg; % bound values of variables m m Y f m Y m xY sY p ± scalar continuous and continuously differentiable functions m bi Y i 1Y 2r m Y m xY sY p g/kg sensitivity functions according to the model coef®cients m u Y m xY sY p g/kg sensitivity functions according to the physical or chemical parameter M m Y F m Y m xY sY p ± matrix continuous and continuously differentiable functions J g/kg performance index q m Y m xY sY p ± weight coef®cient in additional performance index L ± Lagrange functional k m P RY m xY sY p ± conjugate variables of Lagrange functional l m P RY m xY sY p ± penalty coef®cients of augmented Lagrange functional q m P RY m xY sY p ± interconnections in time in augmented Lagrange functional e m P RY m xY sY p ± gradients of augmented Lagrange functional according to the conjugate variables d m P RY m xY sY pY u ± gradients of augmented Lagrange functional according to state and control variables a m b 0Y m xY sY pY u ± steps in gradient procedures

Minimal Time Sequential Batch Reactors with Bounded and Impulse Controls for One or More Species

“…Very similar problems (optimizing the production of biomass at a fixed terminal time) have already been tackled with the help of optimal control theory [1,11,10], which has led to computational methods [30,12]. For models with one biological species, a solution of the minimal time problem has been proposed by Moreno in [20] for monotonic as well as nonmonotonic kinetics.…”

Glutaminase Production using Zygosaccharomyces rouxii NRRL‐Y 2547: Effect of Aeration, Agitation Regimes and Feeding Strategies

“…This is accompanied by changes in the mass transfer coefficients due to changes in the fluid characteristics resulting from metabolic products, foaming at high aeration rates and high stirrer speeds, and the presence of anti-foaming agents. Hence, DO values influence all of these model parameters [28]. A constant DO was maintained in the present work, since as pointed out by Large et al [29], it is often difficult to distinguish the effects of changes in the stirrer speed from the concomitant changes in the DO concentration.…”

About minimal time impulsive control of sequential batch reactors with several species

“…Very similar problems (optimizing the production of biomass at a fixed terminal time) have already been tackled with the help of optimal control theory [1], [7], [6] and has led to computational methods [20], [8]. For models with one biological species, a complete solution of the minimal time problem, for monotonic as well as for non-monotonic kinetics, has been proposed by Moreno in [12].…”