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Numerical solution to the optimal feedback control of continuous casting process

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Abstract

Using a semi-discrete model that describes the heat transfer of a continuous casting process of steel, this paper is addressed to an optimal control problem of the continuous casting process in the secondary cooling zone with water spray control. The approach is based on the Hamilton–Jacobi–Bellman equation satisfied by the value function. It is shown that the value function is the viscosity solution of the Hamilton–Jacobi–Bellman equation. The optimal feedback control is found numerically by solving the associated Hamilton–Jacobi–Bellman equation through a designed finite difference scheme. The validity of the optimality of the obtained control is experimented numerically through comparisons with different admissible controls. Detailed study of a low-carbon billet caster is presented.

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References

  1. Bardi, M., Capuzzo-Dolcetta, I.: Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations. Systems & Control: Foundations & Applications, Birkhäuser, Boston (1997)

  2. Barron E.N. (1990). Application of viscosity solutions of infinite-dimensional Hamilton– Jacobi–Bellman equations to some problems in distributed optimal control. J. Opti. Theory Appl. 64: 245–268

    Article  Google Scholar 

  3. Brimacombe J.K. (1976). Design of continuous casting machines based on a heat-flow analysis: state-of-the-art review. Can. Metallurg. Q. 15: 163–175

    Google Scholar 

  4. Cannarsa P., Gozzi F. and Soner H.M. (1993). A dynamic programming approach to nonlinear boundary control problems of parabolic type. J. Funct. Anal. 117: 25–61

    Article  Google Scholar 

  5. Carlini E., Falcone M. and Ferretti R. (2004). An efficient algorithm for Hamilton–Jacobi equations in high dimension. Comput. Vis. Sci. 7: 15–29

    Article  Google Scholar 

  6. Cecil T., Qian J., and Osher S. (2004). Numerical methods for high dimensional Hamilton–Jacobi equations using radial basis functions. J. Comput. Phys. 196: 327–347

    Article  Google Scholar 

  7. Crandall, M.G.: Viscosity solutions: a primer. In: Viscosity Solutions and Applications. Lecture Notes in Math, vol. 1660, pp 1–43. Springer, Berlin Heidelberg New York (1997)

  8. Fleming, W.H., Soner, H.M.: Controlled Markov Processes and Viscosity Solutions. Applications of Mathematics; vol. 25, Springer, Berlin Heidelberg New York (1993)

  9. Gozzi F., Sritharan S.S. and Świåch A. (2002). Viscosity solutions of dynamic-programming equations for the optimal control of the two-dimensional Navier-Stokes equations. Arch. Ration. Mech. Anal. 163: 295–327

    Article  Google Scholar 

  10. Guo B.Z., Xia X., Camisani-Calzolari F.R. and Craig I.K. (2002). A semi-discrete approach to modelling and control of the continuous casting process. Steel Res. 71: 220–227

    Google Scholar 

  11. Kocan M. and Soravia P. (1998). A viscosity approach to infinite-dimensional Hamilton–Jacobi equations arising in optimal control with state constraints. SIAM J. Control Opti. 36: 1348–1375

    Article  Google Scholar 

  12. Laitinen E. and Neittaanmäki P. (1988). On numerical simulation of the continuous casting process. J. Eng. Math. 22: 335–354

    Article  Google Scholar 

  13. Li R.H. and Feng G.C. (1980). Numerical Solution of Differential Equations. Higher Education Press, Shanghai (in Chinese).

    Google Scholar 

  14. Li X.J. and Yong J.M. (1995). Optimal Control Theory for Infinite Dimensional Systems. Systems & Control: Foundations & Applications, Birkhäuser Boston, Boston

    Google Scholar 

  15. Miettinen K., Mäkelä M.M. and Männikkö T. (1998). Optimal control of continuous casting by nondifferentiable multiobjective optimization. Comput. Opti. Appl. Int. 11: 177–194

    Article  Google Scholar 

  16. Miettinen K., Mäkelä M.M. and Toivanen J. (2003). Numerical comparison of some penalty-based constraint handling techniques in genetic algorithms. J. Glob. Opti. 27: 427–446

    Article  Google Scholar 

  17. Sontag E.D. (1990). Mathematical Control Theory: Deterministic Finite Dimensional Systems. Springer, Berlin Heidelberg New York

    Google Scholar 

  18. Stoer J. and Rulisch R. (1993). Introduction to Numerical Analysis. Springer, Berlin Heidelberg New York

    Google Scholar 

  19. Bulirsch R. and Stryk O. (1992). Direct and indirect methods for trajectory optimization. Ann. Oper. Res. 37: 357–373

    Article  Google Scholar 

  20. Wang S., Gao F. and Teo K.L. (2000). An upwind finite-difference method for the approximation of viscosity solutions to Hamilton–Jacobi-Bellman equations. IMA J. Math. Control Inform. 17: 167–178

    Article  Google Scholar 

  21. Wang S., Jennings L.S. and Teo K.L. (2003). Numerical solution of Hamilton–Jacobi–Bellman equations by an upwind finite volume method. J. Glob. Opti. 27: 177–192

    Article  Google Scholar 

  22. Zhou X.Y. (1993). Verification theorems within the framework of viscosity solutions. J. Math. Anal. Appl. 177: 208–225

    Article  Google Scholar 

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Authors and Affiliations

  1. Academy of Mathematics and System Sciences, Academia Sinica, Beijing, 100080, People’s Republic of China

    Bao-Zhu Guo

  2. School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag-3, Wits-2050, Johannesburg, South Africa

    Bing Sun

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  1. Bao-Zhu Guo
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  2. Bing Sun
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Correspondence to Bao-Zhu Guo.

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Guo, BZ., Sun, B. Numerical solution to the optimal feedback control of continuous casting process. J Glob Optim 39, 171–195 (2007). https://doi.org/10.1007/s10898-006-9130-0

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  • DOI: https://doi.org/10.1007/s10898-006-9130-0

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