Inexact Newton matrix-free methods for solving complex biotechnological systems

• Autorzy: Paweł Drąg , Marlena Kwiatkowska
• Języki publikacji: EN

Abstrakty

EN

In the article a 3-dimensional filter method for solving optimal control problems of differential-algebraic equations (DAEs) was presented. Direct multiple shooting method, which is appropriate for the control problems of the multistage DAE systems, leads to the large-scale nonlinear programming problems. In the proposed approach the extended Fletcher's filter with three inputs was used. The filter method promotes global convergence without the need to use a penalty function. The first input of the filter denotes the value of the cost function. The second and third inputs come from two types of equality constraints - consistent initial conditions of the DAE system and continuity constraints on the state trajectories. The new algorithm was tested on the optimal control problem of a fed-batch fermentor for penicillin production. The numerical simulations were executed in MATLAB environment using Wroclaw Center for Networking and Supercomputing.(original abstract)

Słowa kluczowe

EN

Differential equations; MATLAB system; Mathematical analysis

PL

Równania różniczkowe; System MATLAB; Analiza matematyczna

• Czasopismo: Annals of Computer Science and Information Systems
• Rocznik: 2014
• Tom: 2
• Strony: 597--602

Twórcy

autor

Paweł Drąg

• Wrocław University of Technology

autor

Marlena Kwiatkowska

• Wrocław University of Technology

Bibliografia

• An H. -B., Mo Z. -Y., Liu X. -P.. 2007. A choice of forcing terms in inexact Newton method. Journal of Computational and Applied Mathematics. 200:47-60, http://dx.doi.org/10.1016/j.cam.2005.12.030.
• Banga J. R., Balsa-Canto E., Moles C. G., Alonso A. 2005. Dynamic optimization of bioprocesses: Efficient and robust numerical strategies. Journal of Biotechnology. 117:407-419, http://dx.doi.org/10.1016/j.jbiotec.2005.02.013.
• Betts J. T.. 2010. Practical Methods for Optimal Control and Estimation Using Nonlinear Programming, Second Edition. SIAM, Philadelphia, http://dx.doi.org/10.1137/1.9780898718577.
• Biegler L. T. 2010. Nonlinear Programming. Concepts, Algorithms and Applications to Chemical Processes. SIAM, Philadelphia, http://dx.doi.org/10.1137/1.9780898719383.
• Biegler L. T., Campbell S., Mehrmann V. 2012. DAEs, Control, and Optimization. Control and Optimization with Differential-Algebraic Constraints. SIAM, Philadelphia, http://dx.doi.org/10.1137/9781611972252
• Brenan K. E., Campbell S. L., Petzold L. R. 1996. Numerical Solution of Initial- Value Problems in Differential-Algebraic Equations. SIAM, Philadelphia, http://dx.doi.org/10.1137/1.9781611971224
• Brunet R., Guillen-Gosalbez G., Jimenez L.. 2010. Cleaner design of single-product biotechnological facilities through the integration of process simulation, multiobjective optimization, life cycle assessment, and principal compopnent analysis. Ind. Eng. Chem. Res. 51:410-424, http://dx.doi.org/10.1021/ie2011577.
• Dembo R. S., Eisenstat S. C., Steihaug T. 1982. Inexact Newton Methods. SIAM Journal on Numerical Analysis. 19:400-408, http://dx.doi.org/10.1137/0719025.
• Dembo R. S., Steihaug T. 1983. Truncated-Newton algorithm for largescale unconstrained optimization. Mathematical Programming. 26:190-212, http://dx.doi.org/10.1007/BF02592055.
• Diehl M., Bock H. G., Schloder J. P., Findeisen R., Nagy Z., Allgower F. 2002. Real-time optimization and nonlinear model predictive control of processes governed by differential-algebraic equations. Journal of Process Control. 12:577-585, http://dx.doi.org/10.1016/S0959-1524(01)00023-3.
• Drąg P., Styczeń K. 2012. A Two-Step Approach for Optimal Control of Kinetic Batch Reactor with electroneutrality condition. Przegląd Elektrotechniczny. 6:176-180.
• Eisenstat S. C., Walker H. F. 1994. Globally convergent inexact Newton methods. SIAM Journal on Optimization. 4:393-422, http://dx.doi.org/10.1137/0804022.
• Eisenstat S. C., Walker H. F. 1996. Choosing the forcing terms in an inexact Newton method. SIAM Journal on Scientific Computing. 17:16-32, http://dx.doi.org/10.1137/0917003.
• Gear C. W. 1971. The simultaneous numerical solution of differentialalgebraic equations. IEEE Transactions on Circuit Theory. 18:89-95, http://dx.doi.org/10.1109/TCT.1971.1083221.
• Knoll D. A., Keyes D. E. 2004. Jacobian-free Newton-Krylov methods: a survey of approaches and applications. Journal of Computational Physics. 193:357-397, http://dx.doi.org/10.1016/j.jcp.2003.08.010.
• Kwiatkowska M. 2012. Antimicrobial PVC composites. Processing technologies and functional properties of polymer nanomaterials for food packaging : International COST Workshop, Wroclaw, Poland, September 11-12, pp. 40-41.
• Niu D., Jia M., Wang F., He D. 2013. Optimization of nosiheptide fed-batch fermentation process based on hybrid model. Ind. Eng. Chem.Res. 52:3373-3380, http://dx.doi.org/10.1021/ie3022169.
• Nocedal J., Wright S. J. 2006. Numerical Optimization. Second Edition. Springer, New York, http://dx.doi.org/10.1007/978-0-387-40065-5
• Petzold L. 1982. Differential/Algebraic Equations are not ODEs. SIAM Journal on Scientific Computing. 3:367-384, http://dx.doi.org/10.1137/0903023.
• Saad Y., Schultz M. H. 1986. GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7:856-869, http://dx.doi.org/10.1137/0907058.
• Stoner D. L., Poloski A. P., Johnson J. A., Tolle C. R. 2001. Optimization and Control of Dynamic Bioprocesses. Organic Process Research and Development. 5:299-307, http://dx.doi.org/10.1021/op0100091.
• Vassiliadis V. S., Sargent R. W. H., Pantelides C. C. 1994. Solution of a Class of Multistage Dynamic Optimization Problems. 1. Problems without Path Constraints. Ind. Eng. Chem. Res. 33:2111-2122, http://dx.doi.org/10.1021/ie00033a014.
• Vetukuri S. R. R., Biegler L .T., Walther A. 2010. An inexact trust-region algorithm for the optimization of periodic adsorption processes. Ind. Eng. Chem. Res. 49:12004-12013, http://dx.doi.org/10.1021/ie100706c.