Nonlinear dynamical systems of fed-batch fermentation and their optimal control

Abstract

In this article, we propose a controlled nonlinear dynamical system with variable switching instants, in which the feeding rate of glycerol is regarded as the control function and the moments between the batch and feeding processes as switching instants, to formulate the fed-batch fermentation of glycerol bioconversion to 1,3-propanediol (1,3-PD). Some important properties of the proposed system and its solution are then discussed. Taking the concentration of 1,3-PD at the terminal time as the cost functional, we establish an optimal control model involving the controlled nonlinear dynamical system and subject to continuous state inequality constraints. The existence of the optimal control is also proved. A computational approach is constructed on the basis of constraint transcription and smoothing approximation techniques. Numerical results show that, by employing the optimal control strategy, the concentration of 1,3-PD at the terminal time can be increased considerably.

Keywords:

• nonlinear dynamical system
• optimal control
• continuous state constraint
• smoothing approximation
• fed-batch fermentation

Acknowledgements

This work is supported by the Shandong Province Natural Science Foundation of China (Nos. ZR2010AQ016 and ZR2011AL003), the Natural Science Foundation for the Youth of China (No. 11001153), the TianYuan Special Funds of the National Natural Science Foundation of China (No. 11126077) and the National Natural Science Foundation of China (No. 10871033).