Abstract
The landing of a passenger aircraft in the presence of windshear is a threat to aviation safety. The present paper is concerned with the abort landing of an aircraft in such a serious situation. Mathematically, the flight maneuver can be described by a minimax optimal control problem. By transforming this minimax problem into an optimal control problem of standard form, a state constraint has to be taken into account which is of order three. Moreover, two additional constraints, a first-order state constraint and a control variable constraint, are imposed upon the model. Since the only control variable appears linearly, the Hamiltonian is not regular. Thus, well-known existence theorems about the occurrence of boundary arcs and boundary points cannot be applied. Numerically, this optimal control problem is solved by means of the multiple shooting method in connection with an appropriate homotopy strategy. The solution obtained here satisfies all the sharp necessary conditions including those depending on the sign of certain multipliers. The trajectory consists of bang-bang and singular subarcs, as well as boundary subarcs induced by the two state constraints. The occurrence of boundary arcs is known to be impossible for regular Hamiltonians and odd-ordered state constraints if the order exceeds two. Additionally, a boundary point also occurs where the third-order state constraint is active. Such a situation is known to be the only possibility for odd-ordered state constraints to be active if the order exceeds two and if the Hamiltonian is regular. Because of the complexity of the optimal control, this single problem combines many of the features that make this kind of optimal control problems extremely hard to solve. Moreover, the problem contains nonsmooth data arising from the approximations of the aerodynamic forces and the distribution of the wind velocity components. Therefore, the paper can serve as some sort of user's guide to solve inequality constrained real-life optimal control problems by multiple shooting.
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References
Bulirsch, R., Montrone, F., andPesch, H. J.,Optimal Control in Abort Landing of a Passenger Aircraft, Proceedings of the 8th IFAC Workshop on Control Applications of Nonlinear Programming and Optimization, Paris, France, 1989.
Miele, A., Wang, T., andMelvin, W. W.,Optimal Abort Landing Trajectories in the Presence of Windshear, Journal of Optimization Theory and Applications, Vol. 55, pp. 165–202, 1987.
Miele, A., Wang, T., andMelvin, W. W.,Quasi-Steady Flight to Quasi-Steady Flight Transition for Abort Landing in a Windshear: Trajectory Optimization and Guidance, Journal of Optimization Theory and Applications, Vol. 58, pp. 165–207, 1988.
Warga, J.,Minimizing Variational Curves Restricted to a Preassigned Set, Transactions on the American Mathematical Society, Vol. 112, pp. 432–455, 1964.
Jacobson, D. H., Lele, M. M., andSpeyer, J. L.,New Necessary Conditions of Optimality for Control Problems with State-Variable Inequality Constraints, Journal of Mathematical Analysis and Application, Vol. 35, pp. 255–284, 1971.
Stoer, J., andBulirsch, R.,Introduction to Numerical Analysis, Springer-Verlag, New York, New York, 1980.
Oberle, H. J.,Numerische Berechnung optimaler Steuerungen von Heizung und Kühlung für ein realistisches Sonnenhausmodell, Habilitationsschrift, Munich University of Technology, Munich, Germany, 1982.
Bulirsch, R.,Die Mehrzielmethode zur numerischen Lösung von nichtlinearen Randwertproblemen und Aufgaben der optimalen Steuerung, Report of the Carl-Cranz Gesellschaft, DLR, Oberpfaffenhofen, Germany, 1971.
Deuflhard, P.,A Modified Newton Method for the Solution of Ill-Conditioned Systems of Nonlinear Equations with Application to Multiple Shooting, Numerische Mathematik, Vol. 22, pp. 289–315, 1974.
Deuflhard, P.,A Relaxation Strategy for the Modified Newton Method, Optimization and Optimal Control, Edited by R. Bulirschet al., Springer-Verlag, Berlin, Germany, pp. 59–73, 1975.
Oberle, H. J., andGrimm, W.,BNDSCO—A Program for the Numerical Solution of Optimal Control Problems, Internal Report No. 515-89/22, Institute for Flight Systems Dynamics, DLR, Oberpfaffenhofen, Germany, 1989.
Bulirsch, R., Montrone, F., andPesch, H. J.,Abort Landing in the Presence of Windshear as a Minimax Optimal Control Problem, Part 2: Multiple Shooting and Homotopy, Journal of Optimization Theory and Applications, Vol. 70, pp. 221–252, 1991.
Montrone, F.,Berechnung der optimalen Steuerung einer Boeing 727 beim Auftreten eines Scherwindes, Diploma Thesis, Department of Mathematics, Munich University of Technology, Munich, Germany, 1989.
Bryson, A. E., andHo, Y. C.,Applied Optimal Control, Ginn and Company, Waltham, Massachusetts, 1969.
McDanell, J. P., andPowers, W. F.,Necessary Conditions for Joining Optimal Singular and Nonsingular Subarcs, SIAM Journal on Control, Vol. 9, pp. 161–173, 1971.
Kelley, H. J., Kopp, R. E., andMoyer, H. G.,Singular Extremals, Topics in Optimization, Edited by G. Leitmann, Academic Press, New York, New York, pp. 63–101, 1967.
Maurer, H.,Optimale Steuerprozesse mit Zustandsbeschränkungen, Habilitationsschrift, University of Würzburg, Würzburg, Germany, 1976.
Bryson, A. E., Denham, W. F., andDreyfus, S. E.,Optimal Programming Problems with Inequality Constraints, I, AIAA Journal, Vol. 1, pp. 2544–2550, 1963.
Oberle, H. J.,Numerical Treatment of Minimax Optimal Control Problems with Application to the Reentry Flight Path Problem, Journal of the Astronautical Sciences, Vol. 36, pp. 159–178, 1988.
Oberle, H. J.,Numerical Solution of Minimax Optimal Control Problems by Multiple Shooting Technique, Report No. 85/3, Department of Mathematics, University of Hamburg, Hamburg, Germany, 1985.
Maurer, H.,On Optimal Control Problems with Bounded State Variables and Control Appearing Linearly, SIAM Journal on Control and Optimization, Vol. 15, pp. 345–362, 1977.
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This paper is dedicated to Professor Hans J. Stetter on the occasion of his 60th birthday.
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Bulirsch, R., Montrone, F. & Pesch, H.J. Abort landing in the presence of windshear as a minimax optimal control problem, part 1: Necessary conditions. J Optim Theory Appl 70, 1–23 (1991). https://doi.org/10.1007/BF00940502
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DOI: https://doi.org/10.1007/BF00940502
Key Words
- Optimal control
- Chebyshev-type optimal control problems
- minimax optimal control problems
- optimal trajectories
- state constraints
- state constraints of third order
- bang-bang controls
- singular controls
- multipoint boundary-value problems
- multiple shooting methods
- flight mechanics
- landing
- abort landing
- windshear problems