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Practical Direct Collocation Methods for Computational Optimal Control

  • Victor M. Becerra
Chapter
Part of the Springer Optimization and Its Applications book series (SOIA, volume 73)

Abstract

The development of numerical methods for optimal control and, specifically, trajectory optimisation, has been correlated with advances in the fields of space exploration and digital computing. Space exploration presented scientists and engineers with challenging optimal control problems. Specialised numerical methods implemented in software that runs on digital computers provided the means for solving these problems. This chapter gives an introduction to direct collocation methods for computational optimal control. In a direct collocation method, the state is approximated using a set of basis functions, and the dynamics are collocated at a given set of points along the time interval of the problem, resulting in a sparse nonlinear programming problem. This chapter concentrates on local direct collocation methods, which are based on low-order basis functions employed to discretise the state variables over a time segment. This chapter includes sections that discuss important practical issues such as multi-phase problems, sparse nonlinear programming solvers, efficient differentiation, measures of accuracy of the discretisation, mesh refinement, and potential pitfalls. A space relevant example is given related to a four-phase vehicle launch problem.

Keywords

Optimal control Nonlinear programming Collocation methods 

References

  1. 1.
    Becerra, V.: Solving optimal control problems at no cost with PSOPT. Proceedings of IEEE Multi-conference on Systems and Control, Yokohama, Japan, September 7–10 (2010)Google Scholar
  2. 2.
    Benson, D.A.: A Gauss pseudospectral transcription for optimal control. Ph.D. thesis, Department of Aeronautics and Astronautics, MIT, Cambridge, MA (2004)Google Scholar
  3. 3.
    Bertsekas, D.P., Bertsekas, D.P.: Nonlinear Programming, 2nd edn. Athena Scientific, MA (1999)MATHGoogle Scholar
  4. 4.
    Betts, J.T.: Practical Methods for Optimal Control Using Nonlinear Programming. SIAM, Philadelphia (2001)MATHGoogle Scholar
  5. 5.
    Betts, J.T.: Practical Methods for Optimal Control and Estimation Using Nonlinear Programming. SIAM, Philadelphia (2010)MATHCrossRefGoogle Scholar
  6. 6.
    Betts, J.T., Erb, S.O.: Optimal low thrust trajectories to the moon. SIAM J. Appl. Dyn. Syst. 2, 144–170 (2003)Google Scholar
  7. 7.
    Bryson, A., Ho, Y.C.: Applied Optimal Control. Halsted Press, Sydney (1975)Google Scholar
  8. 8.
    Curtis, A.R., Powell, M.J.D., Reid, J.K.: On the estimation of sparse Jacobian matrices. J. Inst. Math. Appl. 13, 117–120 (1974)MATHGoogle Scholar
  9. 9.
    Elnagar, G., Kazemi, M.A., Razzaghi, M.: The pseudospectral legendre method for discretizing optimal control problems. IEEE Trans. Automat. Control 40, 1793–1796 (1995)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Engelsone, A., Campbell, S.: Adjoint estimation using direct transcription multipliers: Compressed trapezoidal method. Optim. Eng. 9, 291–305 (2008)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Gill, P.E., Murray, W., Saunders, M.A.: SNOPT: an SQP algorithm for large-scale constrained optimization. SIAM Rev. 47(1), 99–131 (2001)Google Scholar
  12. 12.
    Hager, W.: Runge–kutta methods in optimal control and the transformed adjoint system. Numer. Math. 87, 247–282 (2000)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Hairer, E., Norsett, S., Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problems. Springer, Berlin (2002)Google Scholar
  14. 14.
    Hartl, R.F., Sethi, S.P., Vickson, R.G.: A survey of the maximum principles for optimal control problems with state constraints. SIAM Rev. 37(2), 181–218 (1995). URL http://www.jstor.org/stable/2132823 Google Scholar
  15. 15.
    Luenberger, D.: Optimization by Vector Space Methods. Wiley, New York (1997)Google Scholar
  16. 16.
    Rao, A., Benson, D., Huntington, G., Francolin, C.: User’s manual for GPOPS version 1.3: A Matlab package for dynamic optimization using the Gauss pseudospectral method (2008)Google Scholar
  17. 17.
    Sethi, S., Thompson, G.: Optimal Control Theory: Applications to Management Science and Economics. Kluwer, Dordecht (2000)MATHGoogle Scholar
  18. 18.
    The Mathworks: Matlab Programming Fundamentals. Natick, MA (2012)Google Scholar
  19. 19.
    The Mathworks: Optimisation Toolbox User’s Guide. Natick, MA (2012)Google Scholar
  20. 20.
    Vanderbei, R.J., Shanno, D.: An interior-point method for nonconvex nonlinear programming. Comput. Optim. Appl. 13, 231–252 (1999)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Wächter, A., Biegler, L.T.: On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming. Math. Program. 106, 25–57 (2006)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.School of Systems EngineeringUniversity of ReadingReadingUK

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