Cluster Computing

, Volume 20, Issue 4, pp 2891–2903 | Cite as

The numerical method for the optimal supporting position and related optimal control for the catalytic reaction system

  • Qiyu Liu
  • Qunxiong ZhuEmail author
  • Xin Yu
  • Zhiqiang Geng
  • Longjin Lv


This paper considers the numerical approximation for the optimal supporting position and related optimal control of a catalytic reaction system with some control and state constraints, which is governed by a nonlinear partial differential equations with given initial and boundary conditions. By the Galerkin finite element method, the original problem is projected into a semi-discrete optimal control problem governed by a system of ordinary differential equations. Then the control parameterization method is applied to approximate the control and reduce the original system to an optimal parameter selection problem, in which both the position and related control are taken as decision variables to be optimized. This problem can be solved as a nonlinear optimization problem by a particle swarm optimization algorithm. The numerical simulations are given to illustrate the effectiveness of the proposed numerical approximation method.


Optimal supporting position Optimal control Control parameterization method Catalytic reaction system 



This work was partially supported by the National Natural Science Foundation of China (Grant No. 61374096), and the Natural Science Foundation of Zhejiang (Grant No. LY17A010020).


  1. 1.
    Davis, M.E.: Numerical Methods and Modeling for Chemical Engineers. Wiley, New York (1984)Google Scholar
  2. 2.
    Christofides, P.D.: Nonlinear and Robust Control of PDE Systems: Methods and Applications to Transport-Reaction Processes. Birkhauser, Boston (2001)CrossRefzbMATHGoogle Scholar
  3. 3.
    Cao, L., Lu, N.: Experimental study and mathematical description of cationic polymerization reaction in a tubular reactor. J. Beijing Univ. Chem. Technol. 23, 46–52 (1996)Google Scholar
  4. 4.
    Liu, J., Gu, X., Zhang, S.: State and parameter estimation of solid-state polymerization process for PET based on ISR-UKF. CIESC J. 61, 2651–2655 (2010)Google Scholar
  5. 5.
    Kim, H., Miller, D.C., Modekurti, S., Omell, B., Bhattacharyya, D., Zitney, S.E.: Mathematical modeling of a moving bed reactor for post-combustion CO\(_2\) capture. AIChE J. 62, 3899–3914 (2016)CrossRefGoogle Scholar
  6. 6.
    Lao, L., Ellis, M., Christofides, P.D.: Handling state constraints and economics in feedback control of transport-reaction processes. J. Process Control 32, 98–108 (2015)CrossRefGoogle Scholar
  7. 7.
    Aksikas, I., Mohammadi, L., Forbes, J.F., Belhamadia, Y., Dubljevic, S.: Optimal control of an advection-dominated catalytic fixed-bed reactor with catalyst deactivation. J. Process Control 23, 1508–1514 (2013)CrossRefGoogle Scholar
  8. 8.
    Mohammadi, L., Aksikas, I., Forbes, J.F.: Characteristics-based MPC of a fixed bed reactor with catalyst deactivation. IFAC Proc. Vol. 42, 733–737 (2009)CrossRefGoogle Scholar
  9. 9.
    Mohammadi, L., Aksikas, I., Dubljevic, S., Forbes, J.F.: Optimal boundary control of coupled parabolic PDE-ODE systems using infinite-dimensional representation. J. Process Control 33, 102–111 (2015)CrossRefGoogle Scholar
  10. 10.
    Yücel, H., Stoll, M., Benner, P.: A discontinuous Galerkin method for optimal control problems governed by a system of convection–diffusion PDEs with nonlinear reaction terms. Comput. Math. Appl. 70, 2414–2431 (2015)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Wang, Y., Luo, X., Li, S.: “Optimal control method of parabolic partial differential equations and its application to heat transfer model in continuous cast secondary cooling zone,” Adv. Math. Phys. (2015)Google Scholar
  12. 12.
    Ng, J., Dubljevic, S.: Optimal boundary control of a diffusion–convection-reaction PDE model with time-dependent spatial domain: Czochralski crystal growth process. Chem. Eng. Sci. 67(1), 111–119 (2012)CrossRefGoogle Scholar
  13. 13.
    Antoniades, C., Christofides, P.D.: Computation of optimal actuator locations for nonlinear controllers in transport-reaction processes. Comput. Chem. Eng. 24, 577–583 (2000)CrossRefGoogle Scholar
  14. 14.
    Armaou A., Demetriou, M.A.: “Towards optimal actuator placement for dissipative PDE systems in the presence of uncertainty”, 2010, pp. 5662–5667Google Scholar
  15. 15.
    Armaou, A., Demetriou, M.A.: Optimal actuator sensor placement for linear parabolic PDEs using spatial H\(_2\) norm. Chem. Eng. Sci. 22, 7351–7367 (2006)CrossRefGoogle Scholar
  16. 16.
    Antoniades, C., Christofides, P.D.: Integrating nonlinear output feedback control and optimal actuator/sensor placement for transport-reaction processes. Chem. Eng. Sci. 56, 4517–4535 (2001)CrossRefGoogle Scholar
  17. 17.
    Vaidya, U., Rajaram, R., Dasgupta, S.: Actuator and sensor placement in linear advection PDE with building system application. J. Math. Anal. Appl. 394, 213–224 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Privat, Y., Trélat, E., Zuazua, E.: “Optimal location of controllers for the one-dimensional wave equation,” Annales de l’Institut Henri Poincare (C) Non Linear. Analysis 30, 1097–1126 (2013)zbMATHGoogle Scholar
  19. 19.
    Fernández, F.J., Alvarez-Vázquez, L.J., García-Chan, N., Martínez, A., Vázquez-Méndez, M.E.: Optimal location of green zones in metropolitan areas to control the urban heat island. J. Comput. Appl. Math. 289, 412–425 (2015)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Huang, C., Chiang, P.: An inverse study to design the optimal shape and position for delta winglet vortex generators of pin-fin heat sinks. Int. J. Therm. Sci. 109, 374–385 (2016)CrossRefGoogle Scholar
  21. 21.
    Guo, B., Xu, Y., Yang, D.: Optimal actuator location of minimum norm controls for heat equation with general controlled domain. J. Differ. Equ. 261, 3588–3614 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Morris, K.: Linear-quadratic optimal actuator location. IEEE Trans. Autom. Control 56, 113–124 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Hébrard, P., Henrot, A.: A spillover phenomenon in the optimal location of actuators. SIAM J. Control Optim. 44, 349–366 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Guo, B., Yang, D., Zhang, L.: On optimal location of diffusion and related optimal control for null controllable heat equation. J. Math. Anal. Appl. 433, 1333–1349 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Zheng, G., Guo, B., Ali, M.M.: Continuous dependence of optimal control to controlled domain of actuator for heat equation. Syst. Control Lett. 79, 30–38 (2015)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Xin, Y., Zhi-Gang, R., Chao, X.: An approximation for the boundary optimal control problem of a heat equation defined in a variable domain. Chin. Phys. B 23, 040201 (2014)CrossRefGoogle Scholar
  27. 27.
    Li, M., Christofides, P.D.: Optimal control of diffusion-convection-reaction processes using reduced-order models. Comput. Chem. Eng. 32, 2123–2135 (2008)CrossRefGoogle Scholar
  28. 28.
    Lin, Q., Loxton, R., Teo, K.L.: The control parameterization method for nonlinear optimal control: a survey. J. Ind. Manag. Optim. 10, 275–309 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Chen, T., Ren, Z., Xu, C., Loxton, R.: Optimal boundary control for water hammer suppression in fluid transmission pipelines. Comput. Math. Appl. 69, 275–290 (2015)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Chen, T., Xu, C., Lin, Q., Loxton, R., Teo, K.L.: Water hammer mitigation via PDE-constrained optimization. Control Eng. Pract. 45, 54–63 (2015)CrossRefGoogle Scholar
  31. 31.
    Xu, W., Geng, Z., Zhu, Q., Gu, X.: A piecewise linear chaotic map and sequential quadratic programming based robust hybrid particle swarm optimization. Inf. Sci. 218, 85–102 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Liu, W., Yan, N.: Adaptive Finite Element Methods for Optimal Control Governed by PDEs. Science Press, Beijing (2008)Google Scholar
  33. 33.
    Teo, K.L.: A Unified Computational Approach to Optimal Control Problems. Longman Scientific and Technical, London (1991)zbMATHGoogle Scholar
  34. 34.
    Zhou, Q., Luo, J.: Artificial neural network based grid computing of E-government scheduling for emergency management. Comput. Syst. Sci. Eng. 30(5), 327–335 (2015)Google Scholar
  35. 35.
    Zhou, Qingyuan: Research on heterogeneous data integration model of group enterprise based on cluster computing. Clust. Comput. 19(3), 1275–1282 (2016). doi: 10.1007/s10586-016-0580-y CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Qiyu Liu
    • 1
    • 2
  • Qunxiong Zhu
    • 1
    Email author
  • Xin Yu
    • 2
  • Zhiqiang Geng
    • 1
  • Longjin Lv
    • 2
  1. 1.College of Information Science and TechnologyBeijing University of Chemical TechnologyBeijingChina
  2. 2.Ningbo Institute of TechnologyZhejiang UniversityNingboChina

Personalised recommendations