Optimal control of switched autonomous systems in microbial fed-batch cultures

Abstract

A switched autonomous system is proposed to formulate the fed-batch culture process by taking the switching instants between the feed and batch processes as control variables. To maximize the concentration of 1,3-propanediol (1,3-PD) at the terminal time, an optimal control model subject to our proposed switched autonomous system and continuous state inequality constraints is then presented. A computational approach is developed to find the optimal switching instants in two aspects. On the one hand, the constraint transcription and smoothing techniques are applied to dealing with the continuous state inequality constraints; on the other hand, the first and the second derivatives of the cost functional, where the transformed state constraints are treated as a penalty function, are derived by the calculus of variations. Numerical results show that, by employing the optimal switching instants, the concentration of 1,3-PD at the terminal time can be increased considerably.

Keywords:

• optimal control
• switched autonomous system
• switching instant
• smoothing technique
• fed-batch culture

2000 AMS Subject Classifications :

• 49J15
• 49K15
• 93C15
• 93C95

Acknowledgements

This work is supported by the National Natural Science Foundation (Nos. 10471014, 10871033 and 70772026).