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Multiple-Subarc Gradient-Restoration Algorithm, Part 1: Algorithm Structure

Author

Listed:
  • A. Miele

    (Rice University)

  • T. Wang

    (Rice University)

Abstract

Rapid progresses in information and computer technology allow the development of more advanced optimal control algorithms dealing with real-world problems. In this paper, which is Part 1 of a two-part sequence, a multiple-subarc gradient-restoration algorithm (MSGRA) is developed. We note that the original version of the sequential gradient-restoration algorithm (SGRA) was developed by Miele et al. in single-subarc form (SSGRA) during the years 1968–86; it has been applied successfully to solve a large number of optimal control problems of atmospheric and space flight. MSGRA is an extension of SSGRA, the single-subarc gradient-restoration algorithm. The primary reason for MSGRA is to enhance the robustness of gradient-restoration algorithms and also to enlarge the field of applications. Indeed, MSGRA can be applied to optimal control problems involving multiple subsystems as well as discontinuities in the state and control variables at the interface between contiguous subsystems. Two features of MSGRA are increased automation and efficiency. The automation of MSGRA is enhanced via time normalization: the actual time domain is mapped into a normalized time domain such that the normalized time length of each subarc is 1. The efficiency of MSGRA is enhanced by using the method of particular solutions to solve the multipoint boundary-value problems associated with the gradient phase and the restoration phase of the algorithm. In a companion paper [Part 2 (Ref. 2)], MSGRA is applied to compute the optimal trajectory for a multistage launch vehicle design, specifically, a rocket-powered spacecraft ascending from the Earth surface to a low Earth orbit (LEO). Single-stage, double-stage, and triple-stage configurations are considered and compared.

Suggested Citation

  • A. Miele & T. Wang, 2003. "Multiple-Subarc Gradient-Restoration Algorithm, Part 1: Algorithm Structure," Journal of Optimization Theory and Applications, Springer, vol. 116(1), pages 1-17, January.
    • Handle: RePEc:spr:joptap:v:116:y:2003:i:1:d:10.1023_a:1022114117273
    DOI: 10.1023/A:1022114117273
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    References listed on IDEAS

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    1. A. Miele & T. Wang, 2003. "Multiple-Subarc Gradient-Restoration Algorithm, Part 2: Application to a Multistage Launch Vehicle Design," Journal of Optimization Theory and Applications, Springer, vol. 116(1), pages 19-39, January.
    2. A. Miele & S. Mancuso, 1998. "Optimal Ascent Trajectories and Feasibility of Next-Generation Orbital Spacecraft," Journal of Optimization Theory and Applications, Springer, vol. 97(3), pages 519-550, June.
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    Citations

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    Cited by:

    1. N. Yokoyama & S. Suzuki & T. Tsuchiya, 2008. "Convergence Acceleration of Direct Trajectory Optimization Using Novel Hessian Calculation Methods," Journal of Optimization Theory and Applications, Springer, vol. 136(3), pages 297-319, March.
    2. A. Miele & T. Wang, 2006. "Optimal Trajectories and Guidance Schemes for Ship Collision Avoidance," Journal of Optimization Theory and Applications, Springer, vol. 129(1), pages 1-21, April.
    3. A. Miele & T. Wang, 2005. "Maximin Approach to the Ship Collision Avoidance Problem via Multiple-Subarc Sequential Gradient-Restoration Algorithm," Journal of Optimization Theory and Applications, Springer, vol. 124(1), pages 29-53, January.
    4. Mauro Pontani & Bruce Conway, 2014. "Optimal Low-Thrust Orbital Maneuvers via Indirect Swarming Method," Journal of Optimization Theory and Applications, Springer, vol. 162(1), pages 272-292, July.
    5. A. Miele & M. Ciarcià, 2008. "Optimal Starting Conditions for the Rendezvous Maneuver, Part 1: Optimal Control Approach," Journal of Optimization Theory and Applications, Springer, vol. 137(3), pages 593-624, June.
    6. A. Miele & T. Wang & J. A. Mathwig & M. Ciarcià, 2010. "Collision Avoidance for an Aircraft in Abort Landing: Trajectory Optimization and Guidance," Journal of Optimization Theory and Applications, Springer, vol. 146(2), pages 233-254, August.
    7. A. Miele & M. Ciarcià & M. W. Weeks, 2007. "Guidance Trajectories for Spacecraft Rendezvous," Journal of Optimization Theory and Applications, Springer, vol. 132(3), pages 377-400, March.
    8. A. Miele & M. W. Weeks & M. Ciarcià, 2007. "Optimal Trajectories for Spacecraft Rendezvous," Journal of Optimization Theory and Applications, Springer, vol. 132(3), pages 353-376, March.

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    1. Mauro Pontani & Bruce Conway, 2014. "Optimal Low-Thrust Orbital Maneuvers via Indirect Swarming Method," Journal of Optimization Theory and Applications, Springer, vol. 162(1), pages 272-292, July.
    2. A. Miele & M. W. Weeks & M. Ciarcià, 2007. "Optimal Trajectories for Spacecraft Rendezvous," Journal of Optimization Theory and Applications, Springer, vol. 132(3), pages 353-376, March.
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    5. A. Miele & T. Wang, 2005. "Maximin Approach to the Ship Collision Avoidance Problem via Multiple-Subarc Sequential Gradient-Restoration Algorithm," Journal of Optimization Theory and Applications, Springer, vol. 124(1), pages 29-53, January.
    6. N. Yokoyama & S. Suzuki & T. Tsuchiya, 2008. "Convergence Acceleration of Direct Trajectory Optimization Using Novel Hessian Calculation Methods," Journal of Optimization Theory and Applications, Springer, vol. 136(3), pages 297-319, March.
    7. A. Miele & T. Wang & C. S. Chao & J. B. Dabney, 1999. "Optimal Control of a Ship for Course Change and Sidestep Maneuvers," Journal of Optimization Theory and Applications, Springer, vol. 103(2), pages 259-282, November.
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    10. A. Miele & M. Ciarcià & M. W. Weeks, 2007. "Guidance Trajectories for Spacecraft Rendezvous," Journal of Optimization Theory and Applications, Springer, vol. 132(3), pages 377-400, March.

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