Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2018, Article ID 7908378, 11 pages
https://doi.org/10.1155/2018/7908378
Research Article

A High-Precision Single Shooting Method for Solving Hypersensitive Optimal Control Problems

1College of Astronautics, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, China
2Department of Aerospace Engineering and Technology, Politecnico di Milano, 20156 Milano, Italy

Correspondence should be addressed to Binfeng Pan; nc.ude.upwn@gnefnibnap

Received 9 December 2017; Revised 27 February 2018; Accepted 12 March 2018; Published 15 April 2018

Academic Editor: Muhammad N. Akram

Copyright © 2018 Binfeng Pan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. A. V. Rao and K. D. Mease, “Dichotomic basis approach to solving hyper-sensitive optimal control problems,” Automatica, vol. 35, no. 4, pp. 633–642, 1999. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. A. V. Rao, “Riccati dichotomic basis method for solving hypersensitive optimal control problems,” Journal of Guidance, Control, and Dynamics, vol. 26, no. 1, pp. 185–189, 2003. View at Publisher · View at Google Scholar · View at Scopus
  3. Y. Zhao and P. Tsiotras, “Density functions for mesh refinement in numerical optimal control,” Journal of Guidance, Control, and Dynamics, vol. 34, no. 1, pp. 271–277, 2011. View at Publisher · View at Google Scholar · View at Scopus
  4. C. L. Darby, W. W. Hager, and A. V. Rao, “An hp-adaptive pseudospectral method for solving optimal control problems,” Optimal Control Applications and Methods, vol. 32, no. 4, pp. 476–502, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  5. H. Peng, Q. Gao, Z. Wu, and W. Zhong, “Symplectic algorithms with mesh refinement for a hypersensitive optimal control problem,” International Journal of Computer Mathematics, vol. 92, no. 11, pp. 2273–2289, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. E. Aykutlug, U. Topcu, and K. D. Mease, “Manifold-following approximate solution of completely hypersensitive optimal control problems,” Journal of Optimization Theory and Applications, vol. 170, no. 1, pp. 220–242, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. J. T. Betts, Practical Methods for Optimal Control and Estimation Using Nonlinear Programming, Society for Industrial and Applied Mathematics, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  8. M. Gerdts, “Direct shooting method for the numerical solution of higher-index DAE optimal control problems,” Journal of Optimization Theory and Applications, vol. 117, no. 2, pp. 267–294, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  9. D. G. Luenberger and Y. Ye, Linear and Nonlinear Programming, vol. 116 of International Series in Operations Research & Management Science, Springer, New York, NY, USA, 3rd edition, 2008. View at MathSciNet
  10. C. Liu, Z. Gong, and K. L. Teo, “Robust parameter estimation for nonlinear multistage time-delay systems with noisy measurement data,” Applied Mathematical Modelling, vol. 53, pp. 353–368, 2018. View at Publisher · View at Google Scholar · View at MathSciNet
  11. V. Coverstone-Carroll and S. N. Williams, “Optimal low thrust trajectories using differential inclusion concepts,” Journal of the Astronautical Sciences, vol. 42, no. 4, pp. 379–393, 1994. View at Google Scholar · View at Scopus
  12. S. Mehrotra, “On the implementation of a primal-dual interior point method,” SIAM Journal on Optimization, vol. 2, no. 4, pp. 575–601, 1992. View at Publisher · View at Google Scholar · View at MathSciNet
  13. A. L. Herman and B. A. Conway, “Direct optimization using collocation based on high-order Gauss-Lobatto quadrature rules,” Journal of Guidance, Control, and Dynamics, vol. 19, no. 3, pp. 592–599, 1996. View at Publisher · View at Google Scholar · View at Scopus
  14. C. R. Hargraves and S. W. Paris, “Direct trajectory optimization using nonlinear programming and collocation,” Journal of Guidance, Control, and Dynamics, vol. 10, no. 4, pp. 338–342, 1987. View at Publisher · View at Google Scholar · View at Scopus
  15. I. M. Ross and F. Fahroo, “Pseudospectral knotting methods for solving optimal control problems,” Journal of Guidance, Control, and Dynamics, vol. 27, no. 3, pp. 397–405, 2004. View at Publisher · View at Google Scholar · View at Scopus
  16. G. T. Huntington, Advancement and Analysis of a Gauss Pseudospectral Transcription for Optimal Control Problems [Ph.D. thesis], University of Florida, 2007.
  17. F. Topputo and C. Zhang, “Survey of direct transcription for low-thrust space trajectory optimization with applications,” Abstract and Applied Analysis, Article ID 851720, Art. ID 851720, 15 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. Q. Lin, R. Loxton, and K. L. Teo, “The control parameterization method for nonlinear optimal control: a survey,” Journal of Industrial and Management Optimization, vol. 10, no. 1, pp. 275–309, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. R. Loxton, Q. Lin, V. Rehbock, and K. Teo, “Control parameterization for optimal control problems with continuous inequality constraints: new convergence results,” Numerical Algebra, Control and Optimization, vol. 2, no. 3, pp. 571–599, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  20. C. J. Goh and K. L. Teo, “Control parametrization: a unified approach to optimal control problems with general constraints,” Automatica, vol. 24, no. 1, pp. 3–18, 1988. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. M. A. Patterson and A. V. Rao, “GPOPS-II: a MATLAB software for solving multiple-phase optimal control problems using hp-adaptive gaussian quadrature collocation methods and sparse nonlinear programming,” ACM Transactions on Mathematical Software, vol. 41, no. 1, pp. 1–37, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  22. A. E. Bryson and Y. C. Ho, Applied Optimal Control: Optimization, Estimation, And Control, Taylor & Francis, Levittown, PA, USA, 1975. View at MathSciNet
  23. A. V. Rao and K. D. Mease, “Eigenvector approximate dichotomic basis method for solving hyper-sensitive optimal control problems,” Optimal Control Applications and Methods, vol. 21, no. 1, pp. 1–19, 2000. View at Publisher · View at Google Scholar · View at Scopus
  24. A. V. Rao, “Application of a dichotomic basis method to performance optimization of supersonic aircraft,” Journal of Guidance, Control, and Dynamics, vol. 23, no. 3, pp. 570–573, 2000. View at Publisher · View at Google Scholar · View at Scopus
  25. G. Khanna, “High-precision numerical simulations on a CUDA GPU: Kerr black hole tails,” Journal of Scientific Computing, vol. 56, no. 2, pp. 366–380, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  26. Advanpix, “Multiprecision Computing Toolbox for MATLAB,” http://www.advanpix.com.
  27. U. M. Ascher, R. M. Mattheij, and R. D. Russell, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, SIAM, Philadelphia, Pa, USA, 1988. View at Publisher · View at Google Scholar · View at MathSciNet
  28. G. Corliss and Y. F. Chang, “Solving ordinary differential equations using Taylor series,” ACM Transactions on Mathematical Software, vol. 8, no. 2, pp. 114–144, 1982. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  29. R. Barrio, F. Blesa, and M. Lara, “VSVO formulation of the Taylor method for the numerical solution of ODEs,” Computers & Mathematics with Applications, vol. 50, no. 1-2, pp. 93–111, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  30. R. Barrio, “Performance of the Taylor series method for ODEs/DAEs,” Applied Mathematics and Computation, vol. 163, no. 2, pp. 525–545, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  31. P. P. de Groen and M. Hermann, “Bidirectional shooting: a strategy to improve the reliability of shooting methods for ODE,” SIAM Journal on Scientific Computing, vol. 5, no. 2, pp. 360–369, 1984. View at Publisher · View at Google Scholar · View at MathSciNet
  32. U. M. Ascher, R. M. Mattheij, and R. Russell, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, Society for Industrial and Applied Mathematics, 1995. View at Publisher · View at Google Scholar · View at MathSciNet