Abstract
This paper is concerned with the optimal transition and the near-optimum guidance of an aircraft from quasi-steady flight to quasi-steady flight in a windshear. The abort landing problem is considered with reference to flight in a vertical plane. In addition to the horizontal shear, the presence of a downdraft is considered.
It is assumed that a transition from descending flight to ascending flight is desired; that the initial state corresponds to quasi-steady flight with absolute path inclination of −3.0 deg; and that the final path inclination corresponds to quasi-steady steepest climb. Also, it is assumed that, as soon as the shear is detected, the power setting is increased at a constant time rate until maximum power setting is reached; afterward, the power setting is held constant. Hence, the only control is the angle of attack. Inequality constraints are imposed on both the angle of attack and its time derivative.
First, trajectory optimization is considered. The optimal transition problem is formulated as a Chebyshev problem of optimal control: the performance index being minimized is the peak value of the modulus of the difference between the instantaneous altitude and a reference value, assumed constant. By suitable transformations, the Chebyshev problem is converted into a Bolza problem. Then, the Bolza problem is solved employing the dual sequential gradient-restoration algorithm (DSGRA) for optimal control problems.
Two types of optimal trajectories are studied, depending on the conditions desired at the final point. Type 1 is concerned with gamma recovery (recovery of the value of the relative path inclination corresponding to quasi-steady steepest climb). Type 2 is concerned with quasi-steady flight recovery (recovery of the values of the relative path inclination, the relative velocity, and the relative angle of attack corresponding to quasi-steady steepest climb). Both the Type 1 trajectory and the Type 2 trajectory include three branches: descending flight, nearly horizontal flight, and ascending flight. Also, for both the Type 1 trajectory and the Type 2 trajectory, descending flight takes place in the shear portion of the trajectory; horizontal flight takes place partly in the shear portion and partly in the aftershear portion of the trajectory; and ascending flight takes place in the aftershear portion of the trajectory. While the Type 1 trajectory and the Type 2 trajectory are nearly the same in the shear portion, they diverge to a considerable degree in the aftershear portion of the trajectory.
Next, trajectory guidance is considered. Two guidance schemes are developed so as to achieve near-optimum transition from quasi-steady descending flight to quasi-steady ascending flight: acceleration guidance (based on the relative acceleration) and gamma guidance (based on the absolute path inclination).
The guidance schemes for quasi-steady flight recovery in abort landing include two parts in sequence: shear guidance and aftershear guidance. The shear guidance is based on the result that the shear portion of the trajectory depends only mildly on the boundary conditions. Therefore, any of the guidance schemes already developed for Type 1 trajectories can be employed for Type 2 trajectories (descent guidance followed by recovery guidance). The aftershear guidance is based on the result that the aftershear portion of the trajectory depends strongly on the boundary conditions; therefore, the guidance schemes developed for Type 1 trajectories cannot be employed for Type 2 trajectories. For Type 2 trajectories, the aftershear guidance includes level flight guidance followed by ascent guidance. The level flight guidance is designed to achieve almost complete velocity recovery; the ascent guidance is designed to achieve the desired final quasi-steady state.
The numerical results show that the guidance schemes for quasi-steady flight recovery yield a transition from quasi-steady flight to quasi-steady flight which is close to that of the optimal trajectory, allows the aircraft to achieve the final quasi-steady state, and has good stability properties.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Fujita, T. T.,The Downburst, Department of Geophysical Sciences, University of Chicago, Chicago, Illinois, 1985.
Anonymous, N. N.,Aircraft Accident Report: Pan American World Airways, Clipper 759, Boeing 727–235, N4737, New Orleans International Airport, Kenner, Louisiana, July 9, 1982, Report No. NTSB-AAR-8302, National Transportation Safety Board, Washington, DC, 1983.
Anonymous, N. N.,Aircraft Accident Report: Delta Air Lines, Lockheed L-1011-3851, N726DA, Dallas-Fort Worth International Airport, Texas, August 2, 1985, Report No. NTSB-AAR-8605, National Transportation Safety Board, Washington, DC, 1986.
Fujita, T. T.,DFW Microburst, Department of Geophysical Sciences, University of Chicago, Chicago, Illinois, 1986.
Gorney, J. L.,An Analysis of the Delta 191 Windshear Accident, Paper No. AIAA-87-0626, AIAA 25th Aerospace Sciences Meeting, Reno, Nevada, 1987.
Anonymous, N. N.,Flight Path Control in Windshear, Boeing Airliner, pp. 1–12, January–March 1985.
Anonymous, N. N.,Windshear Training Aid, Vols. 1 and 2, Federal Aviation Administration, Washington, DC, 1987.
Bray, R. S.,Aircraft Performance and Control in Downburst Windshear, Paper No. SAE-86-1698, SAE Aerospace Technology Conference and Exposition, Long Beach, California, 1986.
Miele, A., Wang, T., Tzeng, C. Y., andMelvin, W. W.,Transformation Techniques for Minimax Optimal Control Problems and Their Application to Optimal Flight Trajectories in a Windshear: Optimal Abort Landing Trajectories, Paper No. IFAC-87-9221, IFAC 10th World Congress, Munich, Germany, 1987.
Miele, A., Wang, T., Tzeng, C. Y., andMelvin, W. W.,Optimization and Guidance of Abort Landing Trajectories in a Windshear, Paper No. AIAA-87-2341, AIAA Guidance, Navigation, and Control Conference, Monterey, California, 1987.
Miele, A., Wang, T., andMelvin, W. W.,Acceleration, Gamma, and Theta Guidance Schemes for Abort Landing Trajectories in the Presence of Windshear, Rice University, Aero-Astronautics Report No. 223, 1987.
Miele, A., Wang, T., andMelvin, W. W.,Optimization and Acceleration Guidance of Flight Trajectories in a Windshear, Journal of Guidance, Control, and Dynamics, Vol. 10, No. 4, pp. 368–377, 1987.
Miele, A., Wang, T., andMelvin, W. W.,Optimization and Gamma/ Theta Guidance of Flight Trajectories in a Windshear, Paper No. ICAS-86-564, 15th Congress of the International Council of the Aeronautical Sciences, London, England, 1986.
Miele, A., Wang, T., andMelvin, W. W.,Optimal Take-Off Trajectories in the Presence of Windshear, Journal of Optimization Theory and Applications, Vol. 49, No. 1, pp. 1–45, 1986.
Miele, A., Wang, T., andMelvin, W. W.,Guidance Strategies for Near-Optimum Take-Off Performance in a Windshear, Journal of Optimization Theory and Applications, Vol. 50, No. 1, pp. 1–47, 1986.
Miele, A., Wang, T., Melvin, W. W., andBowles, R. L.,Maximum Survival Capability of an Aircraft in a Severe Windshear, Journal of Optimization Theory and Applications, Vol. 53, No. 2, pp. 181–217, 1987.
Miele, A., Wang, T., andMelvin, W. W.,Quasi-Steady Flight to Quasi-Steady Flight Transition in a Windshear: Trajectory Optimization and Guidance, Journal of Optimization Theory and Applications, Vol. 54, No. 2, pp. 203–240, 1987.
Miele, A.,Primal-Dual Sequential Gradient-Restoration Algorithms for Optimal Control Problems and Their Application to Flight in a Windshear, Proceedings of the International Conference on Optimization Techniques and Applications, Edited by K. L. Teo et al., Kent Ridge, Singapore, pp. 53–94, 1987.
Miele, A., Wang, T., Wang, H., andMelvin, W. W.,Optimal Penetration Landing Trajectories in the Presence of Windshear, Rice University, Aero-Astronautics Report No. 216, 1987.
Miele, A., Wang, T., andMelvin, W. W.,Penetration Landing Guidance Trajectories in the Presence of Windshear, Rice University, Aero-Astronautics Report No. 218, 1987.
Frost, W., Chang, H. P., Elmore, K. L., andMcCarthy, J.,Simulated Flight through JAWS Windshear: In-Depth Analysis Results, Paper No. AIAA-84-0276, AIAA 22nd Aerospace Sciences Meeting, Reno, Nevada, 1984.
Ivan, M.,A Ring-Vortex Downburst Model for Flight Simulation, Journal of Aircraft, Vol. 23, No. 3, pp. 232–236, 1986.
Zhu, S., andEtkin, B.,Model of the Wind Field in a Downburst, Journal of Aircraft, Vol. 22, No. 7, pp. 595–601, 1985.
Gonzalez, S., andMiele, A.,Sequential Gradient-Restoration Algorithm for Optimal Control Problems with General Boundary Conditions, Journal of Optimization Theory and Applications, Vol. 26, No. 3, pp. 395–425, 1978.
Miele, A.,Gradient Algorithms for the Optimization of Dynamic Systems, Control and Dynamic Systems, Advances in Theory and Application, Edited by C. T. Leondes, Academic Press, New York, New York, Vol. 16, pp. 1–52, 1980.
Miele, A., andWang, T.,Primal-Dual Properties of Sequential Gradient-Restoration Algorithms for Optimal Control Problems, Part 1, Basic Problem, Integral Methods in Science and Engineering, Edited by F. R. Payne et al., Hemisphere Publishing Corporation, Washington, DC, pp. 577–607, 1986.
Miele, A., andWang, T.,Primal-Dual Properties of Sequential Gradient-Restoration Algorithms for Optimal Control Problems, Part 2, General Problem, Journal of Mathematical Analysis and Applications, Vol. 119, Nos. 1–2, pp. 21–54, 1986.
Author information
Authors and Affiliations
Additional information
This research was supported by NASA Langley Research Center, Grant No. NAG-1-516, by Boeing Commercial Airplane Company, and by Air Line Pilots Association.
The authors are indebted to Dr. R. L. Bowles (NASA-LRC) and Dr. G. R. Hennig (BCAC) for helpful discussions.
Rights and permissions
About this article
Cite this article
Miele, A., Wang, T. & Melvin, W.W. Quasi-steady flight to quasi-steady flight transition for abort landing in a windshear: Trajectory optimization and guidance. J Optim Theory Appl 58, 165–207 (1988). https://doi.org/10.1007/BF00939681
Issue Date:
DOI: https://doi.org/10.1007/BF00939681