Abstract
In the paper the optimal control of highly nonlinear differential-algebraic systems (DAEs) is discussed. The direct shooting method is seen as an efficient tool for control of the complex real-life technological processes, where dynamics and conservation laws are presented. To stabilize the optimization algorithm, the multiple shooting method was proposed. The multiple shooting approach introduces new decision variables and constraints to the problem, but it can preserve the stability of the process, the continuity of the differential state trajectories and enables parallel computation of the mathematical model. The conditions for the frequency of shots, to establish the well-conditioned optimization problem, are considered. The proposed method was tested on the mathematical model of the fed-batch fermentor for penicillin production process, which is a highly nonlinear multistage differential-algebraic system. The numerical simulations were executed in MATLAB environment using Wroclaw Center for Networking and Supercomputing.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
E. Balsa-Canto, V.S. Vassiliadis, J.R. Banga, Dynamic optimization of single and multi-stage systems using a hybrid stochastic-deterministic method. Ind. Eng. Chem. Res 44, 1514–1523 (2005)
J.R. Banga, E. Balsa-Canto, C.G. Moles, A.A. Alonso, Dynamic optimization of bioprocesses: efficient and robust numerical strategies. J. Biotechnol. 117, 407–419 (2005)
J.T. Betts, Practical Methods for Optimal Control and Estimation Using Nonlinear Programming, 2nd edn. (SIAM, Philadelphia, 2010)
L.T. Biegler, Nonlinear Programming, Concepts, Algorithms and Applications to Chemical Processes (SIAM, Philadelphia, 2010)
L.T. Biegler, S. Campbell, V. Mehrmann, DAEs, Control, and Optimization, in Control and Optimization with Differential-Algebraic Constraints, ed. by L.T. Biegler, S. Campbell, V. Mehrmann (SIAM, Philadelphia, 2012)
L.T. Biegler, I.E. Grossmann, Retrospective on optimization. Comput. Chem. Eng. 28, 1169–1192 (2004)
K.E. Brenan, S.L. Campbell, L.R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations (SIAM, Philadelphia, 1996)
M. Cannon, Efficient nonlinear model predictive control algorithms. Annu. Rev. Control 28, 229–237 (2004)
M. Caracotsios, W.E. Stewart, Sensitivity analysis of initial value problems with mixed ODEs and algebraic equations. Comput. Chem. Eng. 9, 359–365 (1985)
E.F. Carrasco, J.R. Banga, Dynamic optimization of batch reactors using adaptive stochastic algorithms. Ind. Eng. Chem. Res. 36, 2252–2261 (1997)
M. Diehl, H.G. Bock, J.P. Schlöder, R. Findeisen, Z. Nagy, F. Allgöwer, Real-time optimization and nonlinear model predictive control of processes governed by differential-algebraic equations. J. Process Control 12, 577–585 (2002)
P. Dra̧g, K. Styczeń, A two-step approach for optimal control of kinetic batch reactor with electroneutrality condition. Przegla̧d Elektrotechniczny 6(88), 176–180 (2012)
A. Flores-Tlacuahuac, L.T. Biegler, E. Saldivar-Guerra, Dynamic optimization of hips open-loop unstable polymerization reactors. Ind. Eng. Chem. Res. 44, 2659–2674 (2005)
A. Flores-Tlacuahuac, S.T. Moreno, L.T. Biegler, Global optimization of highly nonlinear dynamic systems. Ind. Eng. Chem. Res. 47, 2643–2655 (2008)
I.E. Grossmann, L.T. Biegler, Part II. Future perspective on optimization. Comput. Chem. Eng. 28, 1193–1218 (2004)
A. Hartwich, K. Stockmann, C. Terboven, S. Feuerriegel, W. Marquardt, Parallel sensitivity analysis for efficient large-scale dynamic optimization. Optim. Eng. 12, 489–508 (2011)
D.B. Leineweber, I. Bauer, H.G. Bock, J.P. Schlöder, An efficient multiple shooting based reduced SQP strategy for large scale dynamic process optimization. Part 1: Theoretical aspects. Comput. Chem. Eng. 27, 157–166 (2003)
J. Nocedal, S.J. Wright, Numerical Optimization, 2nd edn. (Springer, New York, 2006)
K. Styczeń, P. Dra̧g. A modified multipoint shooting feasible-SQP method for optimal control of DAE systems. Proceedings of the Federated Conference on Computer Science and Information Systems, Szczecin, Poland, pp. 477–484, 18–21 September 2011
V.S. Vassiliadis, R.W.H. Sargent, C.C. Pantelides, Solution of a class of multistage dynamic optimization problems. 1. Problems without path constraints. Ind. Eng. Chem. Res. 33, 2111–2122 (1994)
V.S. Vassiliadis, R.W.H. Sargent, C.C. Pantelides, Solution of a Class of Multistage Dynamic Optimization Problems. 2. Problems with Path Constraints. Ind. Eng. Chem. Res. 33, 2123–2122 (1994)
Acknowledgments
The project was supported by the grant of National Science Centre Poland DEC-2012/07/B/ST7/01216.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Dra̧g, P., Styczeń, K. (2016). Direct Shooting Method for Optimal Control of the Highly Nonlinear Differential-Algebraic Systems. In: Fidanova, S. (eds) Recent Advances in Computational Optimization. Studies in Computational Intelligence, vol 610. Springer, Cham. https://doi.org/10.1007/978-3-319-21133-6_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-21133-6_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-21132-9
Online ISBN: 978-3-319-21133-6
eBook Packages: EngineeringEngineering (R0)