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October  2009, 5(4): 835-850. doi: 10.3934/jimo.2009.5.835

Modelling and optimal control for nonlinear multistage dynamical system of microbial fed-batch culture

Chongyang Liu 1, , Zhaohua Gong 2, , Enmin Feng 3, and  Hongchao Yin 4,

1. 

Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, Liaoning, China
2. 

School of Mathematics and Information Science, Shandong Institute of Business and Technology, Yantai 264005, Shandong, China
3. 

Department of Applied Mathematics, Dalian University of Technology, Dalian, Liaoning, 116024, P.R.
4. 

School of Energy and Power Engineering, Dalian University of Technology, Dalian 116024, Liaoning, China

Received  July 2008 Revised  May 2009 Published  August 2009

In this paper, we propose a new controlled multistage system to formulate the fed-batch culture process of glycerol bio-dissimilation to 1,3-propanediol (1,3-PD) by regarding the feeding rate of glycerol as a control function. Compared with the previous systems, this system doesn't take the feeding process as an impulsive form, but a time-continuous process, which is much closer to the actual culture process. Some properties of the above dynamical system are then proved. To maximize the concentration of 1,3-PD at the terminal time, we develop an optimal control model subject to our proposed controlled multistage system and continuous state inequality constraints. The existence of optimal control is proved by bounded variation theory. Through the discretization of the control space, the control function is approximated by piecewise constant functions. In this way, the optimal control model is approximated by a sequence of parameter optimization problems. The convergence analysis of this approximation is also investigated. Furthermore, a global optimization algorithm is constructed on the basis of the above descretization concept and an improved Particle Swarm Optimization (PSO) algorithm. Numerical results show that, by employing the optimal control policy, the concentration of 1,3-PD at the terminal time can be increased considerably.
Keywords: Control parametrization, Nonlinear multistage system, Optimal control, Fed-batch culture, Improved PSO algorithm..
Mathematics Subject Classification: Primary: 49J15, 49K15; Secondary: 93C1.
Citation: Chongyang Liu, Zhaohua Gong, Enmin Feng, Hongchao Yin. Modelling and optimal control for nonlinear multistage dynamical system of microbial fed-batch culture. Journal of Industrial and Management Optimization, 2009, 5 (4) : 835-850. doi: 10.3934/jimo.2009.5.835
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