Skip to main content
SpringerLink
Log in

Efficient Sequential Quadratic Programming Implementations for Equality-Constrained Discrete-Time Optimal Control

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

Efficient sequential quadratic programming (SQP) implementations are presented for equality-constrained, discrete-time, optimal control problems. The algorithm developed calculates the search direction for the equality-based variant of SQP and is applicable to problems with either fixed or free final time. Problem solutions are obtained by solving iteratively a series of constrained quadratic programs. The number of mathematical operations required for each iteration is proportional to the number of discrete times N. This is contrasted by conventional methods in which this number is proportional to N 3. The algorithm results in quadratic convergence of the iterates under the same conditions as those for SQP and simplifies to an existing dynamic programming approach when there are no constraints and the final time is fixed. A simple test problem and two application problems are presented. The application examples include a satellite dynamics problem and a set of brachistochrone problems involving viscous friction.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. MAYNE, D., A Second-Order Gradient Method for Determining Optimal Trajectories of Nonlinear Discrete-Time Systems, International Journal on Control, Vol. 3,No. 1, pp. 85–95, 1966.

    Google Scholar 

  2. MURRAY, D. M., and YAKOWITZ, S. J., Differential Dynamic Programming and Newton's Methods for Discrete Optimal Control Problems, Journal of Optimization Theory and Applications, Vol. 43,No. 3, pp. 395–414, 1984.

    Google Scholar 

  3. LIAO, L. Z., and SHOEMAKER, C. A., Convergence in Unconstrained Discrete-Time Differential Dynamic Programming, IEEE Transactions on Automatic Control, Vol. 36,No. 6, pp. 692–706, 1991.

    Google Scholar 

  4. MURRAY, D. M., and YAKOWITZ, S. J., The Application of Optimal Control Methodology to Nonlinear Programming Problems, Mathematical Programming, Vol. 21,No. 3, pp. 331–347, 1981.

    Google Scholar 

  5. SEN, S., and YAKOWITZ, S. J., A Quasi-Newton Differential Dynamic Programming Algorithm for Discrete-Time Optimal Control, Automatica, Vol. 23,No. 6, pp. 749–752, 1987.

    Google Scholar 

  6. RAKSHIT, A., and SEN, S., Sequential Rank-One/Rank-Two Updates for Quasi-Newton Differential Dynamic Programming, Optimal Control Applications and Methods, Vol. 11,No. 1, pp. 95–101, 1990.

    Google Scholar 

  7. PANTOJA, J. F., Differential Dynamic Programming and Newton's Method, International Journal on Control, Vol. 47,No. 5, pp. 1539–1553, 1988.

    Google Scholar 

  8. DUNN, J. C., and BERTSEKAS, D. P., Efficient Dynamic Programming Implementations of Newton's Method for Unconstrained Optimal Control Problems, Journal of Optimization Theory and Applications, Vol. 63,No. 1, pp. 23–38, 1989.

    Google Scholar 

  9. GERSHWIN, S. B., and JACOBSON, D. H., A Discrete-Time Differential Dynamic Programming Algorithm with Application to Optimal Orbit Transfer, AIAA Journal, Vol. 8,No. 9, pp. 1616–1626, 1970.

    Google Scholar 

  10. OHNO, K., A New Approach to Differential Dynamic Programming for Discrete-Time Systems, IEEE Transactions on Automatic Control, Vol. 23,No. 1, pp. 37–47, 1978.

    Google Scholar 

  11. OHNO, K., Differential Dynamic Programming and Separable Programs, Journal of Optimization Theory and Applications, Vol. 24,No. 4, pp. 617–637, 1978.

    Google Scholar 

  12. DUNN, J. C., A Projected Newton Method for Minimization Problems with Nonlinear Inequality Constraints, Numerische Mathematik, Vol. 53,No. 4, pp. 377–409, 1988.

    Google Scholar 

  13. GAWANDE, M., and DUNN, J. C., A Projected Newton Method in a Cartesian Product of Balls, Journal of Optimization Theory and Applications, Vol. 59,No. 1, pp. 45–70, 1988.

    Google Scholar 

  14. PANTOJA, J. F., and MAYNE, D. Q., Sequential Quadratic Programming Algorithm for Discrete Optimal Control Problems with Control Inequality Constraints, International Journal on Control, Vol. 53,No. 4, pp. 823–836, 1991.

    Google Scholar 

  15. WRIGHT, S. J., Interior-Point Methods for Optimal Control of Discrete-Time Systems, Journal of Optimization Theory and Applications, Vol. 77,No. 1, pp. 161–187, 1993.

    Google Scholar 

  16. GILL, P. E., MURRAY, W., and WRIGHT, M. H., Practical Optimization, Academic Press, New York, New York, 1981.

    Google Scholar 

  17. GOLUB, G. H., and VAN LOAN, C. F., Matrix Computations, Johns Hopkins University Press, Baltimore, Maryland, 1989.

    Google Scholar 

  18. CHAR, B. W., et al., First Leaves: A Tutorial Introduction to Maple V, Springer Verlag, New York, New York, 1992.

    Google Scholar 

  19. JUNKINS, J. L., and TURNER, J. D., Optimal Continuous Torque Attitude Maneuvers, Journal of Guidance, Control, and Dynamics, Vol. 3,No. 3, pp. 210–217, 1980.

    Google Scholar 

  20. KANE, T. R., LIKINS, P. W., and LEVINSON, D. A., Spacecraft Dynamics, McGraw-Hill, New York, New York, 1983.

    Google Scholar 

  21. GOLDSTEIN, H., Classical Mechanics, Addison-Wesley, Reading, Massachusetts, 1980.

    Google Scholar 

  22. FORRAY, M. J., Variational Calculus in Science and Engineering, McGraw-Hill, New York, New York, 1968.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Structural Dynamics Department, Sandia National Laboratories, Albuquerque, New Mexico

    C. R. Dohrmann (Senior Member of Technical Staff)

  2. Intelligent System Sensors and Controls Department, Sandia National Laboratories, Albuquerque, New Mexico

    R. D. Robinett (Distinguished Member of Technical Staff)

Authors
  1. C. R. Dohrmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. D. Robinett
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dohrmann, C.R., Robinett, R.D. Efficient Sequential Quadratic Programming Implementations for Equality-Constrained Discrete-Time Optimal Control. Journal of Optimization Theory and Applications 95, 323–346 (1997). https://doi.org/10.1023/A:1022635205221

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1022635205221

Search

Navigation