American Institute of Mathematical Sciences

Advanced
  • Previous Article
    First-order optimality conditions for convex semi-infinite min-max programming with noncompact sets
  • JIMO Home
  • This Issue
  • Next Article
    The modified cutting angle method for global minimization of increasing positively homogeneous functions over the unit simplex
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 & Management Optimization, 2009, 5 (4) : 835-850. doi: 10.3934/jimo.2009.5.835
[1]

Bangyu Shen, Xiaojing Wang, Chongyang Liu. Nonlinear state-dependent impulsive system in fed-batch culture and its optimal control. Numerical Algebra, Control & Optimization, 2015, 5 (4) : 369-380. doi: 10.3934/naco.2015.5.369
[2]

Jinggui Gao, Xiaoyan Zhao, Jinggang Zhai. Optimal control of microbial fed-batch culture involving multiple feeds. Numerical Algebra, Control & Optimization, 2015, 5 (4) : 339-349. doi: 10.3934/naco.2015.5.339
[3]

Chongyang Liu, Meijia Han. Time-delay optimal control of a fed-batch production involving multiple feeds. Discrete & Continuous Dynamical Systems - S, 2020, 13 (6) : 1697-1709. doi: 10.3934/dcdss.2020099
[4]

M. Alipour, M. A. Vali, A. H. Borzabadi. A hybrid parametrization approach for a class of nonlinear optimal control problems. Numerical Algebra, Control & Optimization, 2019, 9 (4) : 493-506. doi: 10.3934/naco.2019037
[5]

Volker Rehbock, Iztok Livk. Optimal control of a batch crystallization process. Journal of Industrial & Management Optimization, 2007, 3 (3) : 585-596. doi: 10.3934/jimo.2007.3.585
[6]

Ellina Grigorieva, Evgenii Khailov, Andrei Korobeinikov. Parametrization of the attainable set for a nonlinear control model of a biochemical process. Mathematical Biosciences & Engineering, 2013, 10 (4) : 1067-1094. doi: 10.3934/mbe.2013.10.1067
[7]

Bin Li, Xiaolong Guo, Xiaodong Zeng, Songyi Dian, Minhua Guo. An optimal pid tuning method for a single-link manipulator based on the control parametrization technique. Discrete & Continuous Dynamical Systems - S, 2020, 13 (6) : 1813-1823. doi: 10.3934/dcdss.2020107
[8]

Qun Lin, Ryan Loxton, Kok Lay Teo. The control parameterization method for nonlinear optimal control: A survey. Journal of Industrial & Management Optimization, 2014, 10 (1) : 275-309. doi: 10.3934/jimo.2014.10.275
[9]

Sie Long Kek, Mohd Ismail Abd Aziz, Kok Lay Teo, Rohanin Ahmad. An iterative algorithm based on model-reality differences for discrete-time nonlinear stochastic optimal control problems. Numerical Algebra, Control & Optimization, 2013, 3 (1) : 109-125. doi: 10.3934/naco.2013.3.109
[10]

Tao Jiang, Liwei Liu. Analysis of a batch service multi-server polling system with dynamic service control. Journal of Industrial & Management Optimization, 2018, 14 (2) : 743-757. doi: 10.3934/jimo.2017073
[11]

Ying Zhang, Changjun Yu, Yingtao Xu, Yanqin Bai. Minimizing almost smooth control variation in nonlinear optimal control problems. Journal of Industrial & Management Optimization, 2020, 16 (4) : 1663-1683. doi: 10.3934/jimo.2019023
[12]

Dariusz Idczak, Rafał Kamocki. Existence of optimal solutions to lagrange problem for a fractional nonlinear control system with riemann-liouville derivative. Mathematical Control & Related Fields, 2017, 7 (3) : 449-464. doi: 10.3934/mcrf.2017016
[13]

Ellina Grigorieva, Evgenii Khailov. Optimal control of a nonlinear model of economic growth. Conference Publications, 2007, 2007 (Special) : 456-466. doi: 10.3934/proc.2007.2007.456
[14]

Piermarco Cannarsa, Carlo Sinestrari. On a class of nonlinear time optimal control problems. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 285-300. doi: 10.3934/dcds.1995.1.285
[15]

Jitka Machalová, Horymír Netuka. Optimal control of system governed by the Gao beam equation. Conference Publications, 2015, 2015 (special) : 783-792. doi: 10.3934/proc.2015.0783
[16]

Thomas I. Seidman. Optimal control of a diffusion/reaction/switching system. Evolution Equations & Control Theory, 2013, 2 (4) : 723-731. doi: 10.3934/eect.2013.2.723
[17]

Jérome Lohéac, Jean-François Scheid. Time optimal control for a nonholonomic system with state constraint. Mathematical Control & Related Fields, 2013, 3 (2) : 185-208. doi: 10.3934/mcrf.2013.3.185
[18]

Lijuan Wang, Qishu Yan. Optimal control problem for exact synchronization of parabolic system. Mathematical Control & Related Fields, 2019, 9 (3) : 411-424. doi: 10.3934/mcrf.2019019
[19]

Sie Long Kek, Mohd Ismail Abd Aziz, Kok Lay Teo. A gradient algorithm for optimal control problems with model-reality differences. Numerical Algebra, Control & Optimization, 2015, 5 (3) : 251-266. doi: 10.3934/naco.2015.5.251
[20]

M. Predescu, R. Levins, T. Awerbuch-Friedlander. Analysis of a nonlinear system for community intervention in mosquito control. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 605-622. doi: 10.3934/dcdsb.2006.6.605

2018 Impact Factor: 1.025

Tools

Metrics

  • PDF downloads (38)
  • HTML views (0)
  • Cited by (40)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences