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Practical Direct Collocation Methods for Computational Optimal Control

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Modeling and Optimization in Space Engineering

Part of the book series: Springer Optimization and Its Applications ((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.

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Notes

  1. 1.

    1The following web site has a good deal of information about automatic differentiation: http://www.autodiff.org/.

  2. 2.

    2See http://www.psopt.org.

References

  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. 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. Bertsekas, D.P., Bertsekas, D.P.: Nonlinear Programming, 2nd edn. Athena Scientific, MA (1999)

    MATH  Google Scholar 

  4. Betts, J.T.: Practical Methods for Optimal Control Using Nonlinear Programming. SIAM, Philadelphia (2001)

    MATH  Google Scholar 

  5. Betts, J.T.: Practical Methods for Optimal Control and Estimation Using Nonlinear Programming. SIAM, Philadelphia (2010)

    Book  MATH  Google Scholar 

  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. Bryson, A., Ho, Y.C.: Applied Optimal Control. Halsted Press, Sydney (1975)

    Google Scholar 

  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)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  10. Engelsone, A., Campbell, S.: Adjoint estimation using direct transcription multipliers: Compressed trapezoidal method. Optim. Eng. 9, 291–305 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  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. Hager, W.: Runge–kutta methods in optimal control and the transformed adjoint system. Numer. Math. 87, 247–282 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hairer, E., Norsett, S., Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problems. Springer, Berlin (2002)

    Google Scholar 

  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. Luenberger, D.: Optimization by Vector Space Methods. Wiley, New York (1997)

    Google Scholar 

  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. Sethi, S., Thompson, G.: Optimal Control Theory: Applications to Management Science and Economics. Kluwer, Dordecht (2000)

    MATH  Google Scholar 

  18. The Mathworks: Matlab Programming Fundamentals. Natick, MA (2012)

    Google Scholar 

  19. The Mathworks: Optimisation Toolbox User’s Guide. Natick, MA (2012)

    Google Scholar 

  20. Vanderbei, R.J., Shanno, D.: An interior-point method for nonconvex nonlinear programming. Comput. Optim. Appl. 13, 231–252 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. School of Systems Engineering, University of Reading, Reading, RG6 6AY, UK

    Victor M. Becerra

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  1. Victor M. Becerra
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Correspondence to Victor M. Becerra .

Editor information

Editors and Affiliations

  1. Thales Alenia Space SpA, Str. Antica di Collegno 253, Turin, 10146, Italy

    Giorgio Fasano

  2. Stoneybrook Court 114, Halifax, B3M 3J7, Nova Scotia, Canada

    János D. Pintér

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Becerra, V.M. (2012). Practical Direct Collocation Methods for Computational Optimal Control. In: Fasano, G., Pintér, J. (eds) Modeling and Optimization in Space Engineering. Springer Optimization and Its Applications, vol 73. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4469-5_2

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