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Optimal take-off trajectories in the presence of windshear

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Abstract

This paper is concerned with optimal flight trajectories in the presence of windshear. With particular reference to take-off, eight fundamental optimization problems [Problems (P1)–(P8)] are formulated under the assumptions that the power setting is held at the maximum value and that the airplane is controlled through the angle of attack.

Problems (P1)–(P3) are least-square problems of the Bolza type. Problems (P4)–(P8) are minimax problems of the Chebyshev type, which can be converted into Bolza problems through suitable transformations. These problems are solved employing the dual sequential gradient-restoration algorithm (DSGRA) for optimal control problems.

Numerical results are obtained for a large number of combinations of performance indexes, boundary conditions, windshear models, and windshear intensities. However, for the sake of brevity, the presentation of this paper is restricted to Problem (P6), minimax ∣Δh∣, and Problem (P7), minimax ∣Δγ∣. Inequality constraints are imposed on the angle of attack and the time derivative of the angle of attack.

The following conclusions are reached: (i) optimal trajectories are considerably superior to constant-angle-of-attack trajectories; (ii) optimal trajectories achieve minimum velocity at about the time when the windshear ends; (iii) optimal trajectories can be found which transfer an aircraft from a quasi-steady condition to a quasi-steady condition through a windshear; (iv) as the boundary conditions are relaxed, a higher final altitude can be achieved, albeit at the expense of a considerable velocity loss; (v) among the optimal trajectories investigated, those solving Problem (P7) are to be preferred, because the altitude distribution exhibits a monotonic behavior; in addition, for boundary conditions BC2 and BC3, the peak angle of attack is below the maximum permissible value; (vi) moderate windshears and relatively severe windshears are survivable employing an optimized flight strategy; however, extremely severe windshears are not survivable, even employing an optimized flight strategy; and (vii) the sequential gradient-restoration algorithm (SGRA), employed in its dual form (DSGRA), has proven to be a powerful algorithm for solving the problem of the optimal flight trajectories in a windshear.

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Author information

Authors and Affiliations

  1. Aero-Astronautics Group, Rice University, Houston, Texas

    A. Miele (Professor of Astronautics and Mathematical Sciences) & T. Wang (Senior Research Associate)

  2. Delta Airlines, Atlanta, Georgia

    W. W. Melvin (Captain)

  3. Airworthiness and Performance Committee, Air Line Pilots Association (ALPA), Washington, DC

    W. W. Melvin (Captain)

Authors
  1. A. Miele
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  2. T. Wang
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  3. W. W. Melvin
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Additional information

Portions of this paper were presented at the AIAA Atmospheric Flight Mechanics Conference, Snowmass, Colorado, August 19–21, 1985. The authors are indebted to Boeing Commercial Aircraft Company, Seattle, Washington and to Pratt and Whitney Aircraft, East Hartford, Connecticut for supplying some of the technical data pertaining to this study.

This research was supported by NASA-Langley Research Center, Grant No. NAG-1-516. The authors are indebted to Dr. R. L. Bowles, NASA-Langley Research Center, Hampton, Virginia, for helpful discussions.

This paper is based in part on Refs. 1–5.

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Miele, A., Wang, T. & Melvin, W.W. Optimal take-off trajectories in the presence of windshear. J Optim Theory Appl 49, 1–45 (1986). https://doi.org/10.1007/BF00939246

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