Abstract
Singular perturbation theory has played an important role in aircraft performance analysis and in the development of feedback guidance laws for aerospace vehicles. It is expected that this area of mathematics will serve as an important modeling and analysis tool in the development of guidance and flight control algorithms for future air and space transportation systems as well. This paper summarizes the theoretical concepts which are important for applications of this theory to problems in flight mechanics. In particular, emphasis will be given to problem formulation and solution approaches that are useful in applying singular perturbation theory for deriving nonlinear guidance algorithms for aerospace vehicles. The intent is to give an account of some of the progress that has been made and to provide a guide to some of the applications that have been treated successfully.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Kaiser, F., Der Steigflug mit Strahlflugzeugen Teilbericht 1: Bahngeschwindigkeit besten Steigens, Versuchs-Bericht No. 26202144, Messerschmitt A. G., Lechfeld, Germany, 1944.
Merritt, S. R., Cliff, E. M., and Kelley, H. J., Energy-Modelled Climb and Climb-Dash: The Kaiser Technique, Automatica, Vol. 21, pp. 319–321, 1985.
Rutowski, E. S., Energy Approach to the General Aircraft Performance Problem, Journal of the Aeronautical Sciences, Vol. 21, pp. 187–195, 1954.
Bryson, A. E., Desai, M. N., and Hoffman, W. C., The Energy-State Approximation in Performance Optimization of Supersonic Aircraft, Journal of Aircraft, Vol. 6, pp. 481–487, 1969.
Hendrick, J. K., and Bryson, A. E., Three-Dimensional, Minimum-Time Turns for a Supersonic Aircraft, Journal of Aircraft, Vol. 9, pp. 115–121, 1972.
Kokotovic, P. V., and Sannuti, P., Singular Perturbation Method for Reducing the Model Order in Optimal Control Design, IEEE Transactions on Automatic Control, Vol. 13, pp. 377–384, 1968.
Kokotovic, P. V., and Yackel, R. A., Singular Perturbation of Linear Regulators: Basic Theorems, IEEE Transactions on Automatic Control, Vol. 17, pp. 29–37, 1972.
O’Malley, R. E., Jr., Singular Perturbation of the Time-Invariant Linear State Regulator Problem, Journal of Differential Equations, Vol. 12, pp. 117–128, 1972.
Kokotovic, P. V., O’Malley, R. E., Jr., and Sannuti, P., Singular Perturbations and Order Reduction in Control Theory: An Overview, Automatica, Vol. 12, pp. 123–132, 1976.
Saksena, V. R., O’Reilly, J., and Kokotovic, P. V., Singular Pertubations and Time-Scale Methods in Control Theory: Survey 1976–1983, Automatica, Vol. 20, pp. 273–293, 1984.
Kelley, H. J., and Edelbaum, T. N., Energy Climbs, Energy Turns, and Asymptotic Expansions, Journal of Aircraft, Vol. 7, pp. 93–95, 1970.
Kelley, H. J., Boundary Layer Approximations to Powered-Flight Altitude Transients, Journal of Spacecraft and Rockets, Vol. 7, p. 879, 1970.
Kelley, H. J., Aircraft Maneuver Optimization by Reduced-Order Approximation, Control and Dynamic Systems, Edited by C. T. Leondes, Academic Press, New York, New York, Vol. 10, pp. 131–178, 1973.
Ashley, H., Multiple Scaling in Flight Vehicle Dynamic Analysis: A Preliminary Look, AIAA Guidance, Control, and Dynamics Conference, Huntsville, Alabama, pp. 1–9, 1967.
Wasow, W., Asymptotic Expansions for Ordinary Differential Equations, Interscience, New York, New York, 1965.
Kokotovic, P. V., Khalil, H. K., and O’Reilly, J., Singular Perturbation Methods in Control: Analysis and Design, Academic Press, New York, New York, 1986.
Nayfeh, A. H., Perturbation Methods, John Wiley and Sons, New York, New York, 1973.
O’Malley, R. E., Jr., Introduction to Singular Perturbations, Academic Press, New York, New York, 1974.
Tihonov, A. N., Systems of Differential Equations Containing a Small Parameter Multiplying the Derivative, Matematika Periodiceskii Sbornik, Vol. 31, pp. 575–586, 1952.
Levin, J. J., and Levinson, N., Singular Perturbations of Nonlinear Systems of Differential Equations and an Associated Boundary-Layer Equation, Journal of Rational Mechanics and Analysis, Vol. 3, pp. 274–280, 1954.
Vasileva, A. B., Asymptotic Behavior of Solutions to Certain Problems Involving Nonlinear Differential Equations Containing a Small Parameter Multiplying the Highest Derivatives, Soviet Mathematical Surveys, Vol. 18, pp. 13–81, 1963.
Ardema, M. D., Solution of the Minimum Time-to-Climb Problem by Matched Asymptotic Expansions, AIAA Journal, Vol. 14, pp. 843–850, 1976.
Shy, Y. Y., Matched Asymptotic Solutions for Optimum Lift-Controlled Atmospheric Entry, AIAA Journal, Vol. 9, pp. 2229–2238, 1971.
Mease, K. D., and McCreary, F. A., Atmospheric Guidance Law for Planar Skip Trajectories, AIAA Atmospheric Flight Mechanics Conference, Snowmass, Colorado, pp. 409–415, 1985.
Naidu, D. S., Three-Dimensional Atmospheric Entry Problem Using the Method of Matched Asymptotic Expansions, IEEE Transactions on Aerospace and Electronic Systems, Vol. 25, pp. 660–667, 1989.
Calise, A. J. and Melamed, N., Optimal Guidance of Aeroassisted Transfer Vehicles Based on Matched Asymptotic Expansions, AIAA Guidance, Navigation, and Control Conference, New Orleans, Lousiana, pp. 1048–1058, 1991.
Bryson, A. E., Jr., and Ho, Y. C., Applied Optimal Control, John Wiley and Sons, New York, New York, 1975.
Kelley, H. J., Singular Perturbations for a Mayer Variational Problem, AIAA Journal, Vol. 8, pp. 1177–1178, 1970.
Ardema, M. D., Linearization of the Boundary-Layer Equations of the Minimum Time-to-Climb Problem, Journal of Guidance, Control and Dynamics, Vol. 2, pp. 434–436, 1979.
Calise, A. J., Extended Energy Management Methods for Flight Performance Optimization, AIAA Journal, Vol. 15, pp. 314–321, 1977.
Calise, A. J., and Moerder, D. D., Singular Perturbation Techniques for Real-Time Aircraft Trajectory Control, NASA CR 3597, 1982.
Calise, A. J., A New Boundary-Layer Matching Procedure for Singularly Perturbed Systems, IEEE Transactions on Automatic Control, Vol. 23, pp. 434–438, 1978.
Calise, A. J., Optimization of Aircraft Altitude and Flight-Path Angle Dynamics, Journal of Guidance, Control, and Dynamics, Vol. 7, pp. 123–125, 1984.
Ardema, M. D., Nonlinear Singularly Perturbed Optimal Control Problems with Singular Arcs, Automatica, Vol. 16, pp. 99–104, 1980.
Calise, A. J., and Corban, J. E., Optimal Control of Two-Time-Scale Systems with State-Variable Inequality Constraints, Journal of Guidance, Control, and Dynamics, Vol. 15, pp. 468–476, 1992.
Ardema, M. D., and Rajan, N., Separation of Time Scales in Aircraft Trajectory Optimization, Journal of Guidance, Control, and Dynamics, Vol. 8, pp. 275–278, 1985.
Shinar, J., On Applications of Singular Perturbation Techniques in Nonlinear Optimal Control, Automatica, Vol. 19, pp. 203–211, 1983.
Calise, A. J., Markopoulos, N., and Corban, J. E., Nondimensional Forms for Singular Perturbation Analyses of Aircraft Energy Climbs, AIAA Guidance, Navigation, and Control Conference, New Orleans, Louisiana, pp. 704–709, 1991.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1994 Springer Science+Business Media New York
About this chapter
Cite this chapter
Calise, A.J. (1994). Singular Perturbations in Flight Mechanics. In: Miele, A., Salvetti, A. (eds) Applied Mathematics in Aerospace Science and Engineering. Mathematical Concepts and Methods in Science and Engineering, vol 44. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-9259-1_7
Download citation
DOI: https://doi.org/10.1007/978-1-4757-9259-1_7
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4757-9261-4
Online ISBN: 978-1-4757-9259-1
eBook Packages: Springer Book Archive