Xanthan Gum Fermentation Process Control by Dissolved Oxygen Concentration. Optimal State and Control Profiles

Попчев, Иван; Tsoneva, R. G.

doi:10.1016/s1474-6670(17)58782-0

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“…Here a decomposition method is proposed for solving the discrete time formulation of the problems based on Variational calculus using an augmented Lagrange functional and its decomposition in the time domain. Some partial developments of this method have been given [10,12,13,19] for secondary metabolites and a xanthan gum production. In this paper the method is developed for a general description of batch fermentation processes by means of nonstructured models with slowly varying parameters.…”

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confidence: 99%

“…The extended model is formulated in terms of the process model, Eqs. (10)±(13), and the sensitivity model, Eqs. (48)±(50), and it is described by extended vectors of the states:v e k mkm u k T Y v e P R 2 Y m xY sY p X 51The extended model is the same type of model as the model described by Eqs.…”

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confidence: 99%

“…(15)± (16). This problem is solved by the decomposition method where an augmented Lagrange functional is introduced [13], which is assumed to be in the form of Eq. (43).…”

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

Tsoneva

Patarinska²,

Попчев

1998

Bioprocess Engineering

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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

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confidence: 99%

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confidence: 99%