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Viability Approach to Aircraft Control in Wind Shear Conditions

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Advances in Dynamic and Mean Field Games (ISDG 2016)

Part of the book series: Annals of the International Society of Dynamic Games ((AISDG,volume 15))

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Abstract

This paper addresses the analysis of aircraft control capabilities in the presence of wind shears. The cruise flight phase (flying at the established level with practically constant configuration and speed) is considered. The study utilizes a point-mass aircraft model describing both vertical and lateral motions. As a particular case, a reduced model of lateral motion is derived from the full one. State variables of the models are constrained according to aircraft safety conditions, and differential games where a guiding system, the first player, works against wind disturbances, the second player, are considered. Viability theory is used to find the leadership kernel, the maximal subset of the state constraint where the aircraft trajectories can remain arbitrary long if the first player utilizes an appropriate feedback control, and the second player generates any admissible disturbances. The computations are based on a theoretical background resulting in a grid method developed by the authors. The corresponding software is implemented on a multiprocessor computer system.

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References

  1. Aubin, J.-P.: Viability Theory. Birkhäuser, Boston (1991)

    MATH  Google Scholar 

  2. Aubin, J.-P., Bayen, A. M., Saint-Pierre, P.: Viability Theory: New Directions. Springer-Verlag, Berlin/Heidelberg (2011)

    Book  MATH  Google Scholar 

  3. Bayen, A. M., Mitchell, I. M., Osihi, M. K., Tomlin, C. J.: Aircraft autolander safety analysis through optimal control-based reach set computation. Journal of Guidance, Control, and Dynamics 30 (1), 68–77 (2007)

    Article  Google Scholar 

  4. Botkin, N. D., Hoffmann, K.-H., Mayer, N., Turova, V. L.: Approximation schemes for solving disturbed control problems with non-terminal time and state constraints. Analysis 31, 355–379 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Botkin, N. D., Hoffmann, K.-H., Turova, V. L.: Stable numerical schemes for solving Hamilton–Jacobi–Bellman–Isaacs equations. SIAM J. Sci. Comput. 33 (2), 992–1007 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Botkin, N. D., Turova, V. L.: Dynamic programming approach to aircraft control in a windshear. In: Křivan, V., Zaccour, G. (eds.) Advances in Dynamic Games: Theory, Applications, and Numerical Methods. Annals of the International Society of Dynamic Games, vol. 13, pp. 53–69. Birkhäuser, Boston (2013)

    Chapter  Google Scholar 

  7. Botkin, N. D., Turova, V. L.: Numerical construction of viable sets for autonomous conflict control systems. Mathematics 2, 68–82 (2014)

    Article  MATH  Google Scholar 

  8. Cardaliaguet, P.: A differential game with two players and one target, SIAM J. Control Optim. 34, 1441–1460 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cardaliaguet, P., Quincampoix, M., Saint-Pierre, P.: Set valued numerical analysis for optimal control and differential games. In: Bardi, M., Raghavan, T.E.S., Parthasarathy, T. (eds.) Stochastic and Differential Games: Theory and Numerical Methods. Annals of the International Society of Dynamic Games, vol. 4, pp. 177–274. Birkhäuser, Boston (1999)

    Chapter  Google Scholar 

  10. Chen, Y. H., Pandey, S.: Robust control strategy for take-off performance in a windshear. Optim. Contr. Appl. Met. 10 (1), 65–79 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  11. Falcone, M.: Numerical methods for differential games via PDEs. International Game Theory Review 8(2), 231–272 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cristiani, E., Falcone, M.: Fully-discrete schemes for value function of pursuit- evasion games with state constraints. In: Bernhard, P., Gaitsgory, V., Pourtallier, O. (eds.) Advances in Dynamic Games and Their Applications, Annals of the International Society of Dynamic Games, vol. 10, pp. 177–206. Birkhäuser, Boston (2009)

    Google Scholar 

  13. Fisch, F.: Development of a framework for the solution of high-fidelity trajectory optimization problems and bilevel optimal control problems. Dissertation, Technische Universität München, Institute of Flight System Dynamics (2011)

    Google Scholar 

  14. Krasovskii, N. N., Subbotin, A. I.: Game-Theoretical Control Problems. Springer, New York (1988)

    Book  Google Scholar 

  15. Leitmann, G., Pandey, S.: Aircraft control under conditions of windshear. In: Leondes, C. T. (ed.) Control and Dynamic Systems, vol. 34, part 1, pp. 1–79. Academic Press, New York (1990)

    Google Scholar 

  16. Leitmann, G., Pandey, S.: Aircraft control for flight in an uncertain environment: Takeoff in windshear. J. Optimiz. Theory App. 70 (1), 25–55 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  17. Leitmann, G., Pandey, S., Ryan, E.: Adaptive control of aircraft in windshear. Int. J. Robust Nonlin. 3, 133–153 (1993)

    Article  MATH  Google Scholar 

  18. Miele, A., Wang, T., Melvin, W. W.: Optimal take-off trajectories in the presence of windshear. J. Optimiz. Theory App. 49, 1–45 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  19. Miele, A., Wang, T., Melvin, W. W.: Guidance strategies for near-optimum take-off performance in windshear. J. Optimiz. Theory App. 50 (1), 1–47 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  20. Mitchell, I. M.: A summary of recent progress on efficient parametric approximations of viability and discriminating kernels. EPiC Ser. Comput. Sci. 37, 23–31 (2015)

    Article  Google Scholar 

  21. Patsko, V. S., Botkin, N. D., Kein, V. M., Turova, V. L., Zarkh, M. A.: Control of an aircraft landing in windshear. J. Optimiz. Theory App. 83 (2), 237–267 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  22. Pflüger, D.: Spatially adaptive sparse grids for higher-dimensional problems. Dissertation, Verlag Dr. Hut, München (2010)

    MATH  Google Scholar 

  23. Seube, N., Moitie, R., Leitmann, G.: Aircraft take-off in windshear: A viability approach. Set-Valued Analysis 8, 163–180 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  24. Seube, N., Moitie, R., Leitmann, G.: Viability analysis of an aircraft flight domain for take-off in a windshear. Mathematical and Computer Modelling 36 (6), 633–641 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  25. Zenger, C.: Sparse Grids. In: Hackbusch, W. (ed.) Parallel Algorithms for Partial Differential Equations. Notes on Numerical Fluid Mechanics, vol. 31, pp. 241–251. Vieweg, Braunschweig/Wiesbaden (1991)

    Google Scholar 

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Acknowledgements

This work was supported by the DFG grant TU427/2-1 and HO4190/8-1. Computer resources for this project have been provided by the Gauss Centre for Supercomputing/Leibniz Supercomputing Centre under grant: pr74lu.

Author information

Authors and Affiliations

  1. Department of Mathematics, Technische Universität München, Boltzmannstr. 3, 85748, Garching near Munich, Germany

    Nikolai Botkin & Varvara Turova

  2. Institute of Flight System Dynamics, Technische Universität München, Boltzmannstr. 15, 85748, Garching near Munich, Germany

    Johannes Diepolder, Matthias Bittner & Florian Holzapfel

Authors
  1. Nikolai Botkin
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  2. Johannes Diepolder
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  3. Varvara Turova
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  4. Matthias Bittner
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  5. Florian Holzapfel
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Corresponding author

Correspondence to Nikolai Botkin .

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Editors and Affiliations

  1. Department of Mathematics, Statistics and Computer Science, St. Francis Xavier University, Antigonish, Nova Scotia, Canada

    Joseph Apaloo

  2. Department of Mathematics, University of Padova, Padova, Italy

    Bruno Viscolani

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Botkin, N., Diepolder, J., Turova, V., Bittner, M., Holzapfel, F. (2017). Viability Approach to Aircraft Control in Wind Shear Conditions. In: Apaloo, J., Viscolani, B. (eds) Advances in Dynamic and Mean Field Games. ISDG 2016. Annals of the International Society of Dynamic Games, vol 15. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-70619-1_15

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