Skip to main content
SpringerLink
Log in

Control of an aircraft landing in windshear

  • Contributed Papers
  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

The problem of the feedback control of an aircraft landing in the presence of windshear is considered. The landing process is investigated up to the time when the runway threshold is reached. It is assumed that the bounds on the wind velocity deviations from some nominal values are known, while information about the windshear location and wind velocity distribution in the windshear zone is absent. The methods of differential game theory are employed for the control synthesis.

The complete system of aircraft dynamic equations is linearized with respect to the nominal motion. The resulting linear system is decomposed into subsystems describing the vertical (longitudinal) motion and lateral motion. For each subsystem, an, auxiliary antagonistic differential game with fixed terminal time and convex payoff function depending on two components of the state vector is formulated. For the longitudinal motion, these components are the vertical deviation of the aircraft from the glide path and its time derivative; for the lateral motion, these components are the lateral deviation and its time derivative. The first player (pilot) chooses the control variables so as to minimize the payoff function; the interest of the second player (nature) in choosing the wind disturbance is just opposite.

The linear differential games are solved on a digital computer with the help of corresponding numerical methods. In particular, the optimal (minimax) strategy is obtained for the first player. The optimal control is specified by means of switch surfaces having a simple structure. The minimax control designed via the auxiliary differential game problems is employed in connection with the complete nonlinear system of dynamical equations.

The aircraft flight through the wind downburst zone is simulated, and three different downburst models are used. The aircraft trajectories obtained via the minimax control are essentially better than those obtained by traditional autopilot methods.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Abbreviations

b :

mean aerodynamic chord, m

c x ,c y ,c z :

aerodynamic force coefficients, body-axes system

g :

acceleration of gravity, m sec−2

I x ,I y ,I z ,I xy :

inertia moments, kg m2

l :

wing span, m

m :

aircraft mass, kg

m x ,m y ,m z :

aerodynamic moment coefficients, body-axes system

M x ,M y ,M z :

aerodynamic moments, N m

P :

thrust force, N

q :

dynamic pressure, kg m−1 sec−2

S :

reference surface, m2

V :

absolute velocity, m sec−1

\(\hat V\) :

relative velocity, m sec−1

W :

wind velocity, m sec−1

V xg ,V yg ,V zg :

absolute velocity components, m sec−1

\(\hat V_{xg} ,\hat V_{yg} ,\hat V_{zg} \) :

relative velocity components, m sec−1

W xg ,W yg ,W zg :

wind velocity components, m sec−1

x g ,y g ,y g :

coordinates of the aircraft center of mass, m, ground-fixed system

α:

angle of attack, deg

β:

sideslip angle, deg

γ:

bank angle, deg

δ a :

aileron deflection, deg

δ e :

elevator deflection, deg

δ r :

rudder deflection, deg

δ as :

aileron setting (control), deg

δ es :

elevator setting (control), deg

δ ps :

engine control lever setting, deg

References

  1. Krasovskii, N. N.,Game Problems about Contact of Motions, Nauka, Moscow, Russia, 1970 (in Russian).

    Google Scholar 

  2. Krasovskii, N. N., andSubbotin, A. I.,Game-Theoretical Control Problems, Springer Verlag, New York, New York, 1988.

    Google Scholar 

  3. Kein, V. M., Parikov, A. N., andSmurov, M. Iu.,On Means of Optimal Control by the Extremal Aiming Method, Journal of Applied Mathematics and Mechanics, Vol. 44, No. 3, pp. 306–310, 1980.

    Article  Google Scholar 

  4. Titovskii, I. N.,Game Theoretical Approach to the Synthesis Problem of Aircraft Control in Landing, Uchenye Zapiski TsAGI, Vol. 12, No. 1, pp. 85–92, 1981 (in Russian).

    Google Scholar 

  5. Botkin, N. D., Kein, V. M., andPatsko, V. S.,The Model Problem of Controlling the Lateral Motion of an Aircraft during Landing, Journal of Applied Mathematics and Mechanics, Vol. 48, No. 4, pp. 395–400, 1984.

    Article  Google Scholar 

  6. Korneev, V. A., Melikyan, A. A., andTitovskii, I. N.,Stabilization of Aircraft Glide Path in Wind Disturbances in the Minimax Formulation Izvestia Akademii Nauk SSSR, Tekhnicheskaya Kibernetika, No. 3, pp. 132–139, 1985 (in Russian).

    Google Scholar 

  7. 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.

    Article  MathSciNet  Google Scholar 

  8. Miele, A., Wang, T., Tzeng, C. Y., andMelvin, W. W.,Optimal Abort Landing Trajectories in the Presence of Windshear, Journal of Optimization Theory and Applications, Vol. 55, No. 2, pp. 165–202, 1987.

    Article  MathSciNet  Google Scholar 

  9. Miele, A., Wang, T., Wang, H., andMelvin, W. W.,Optimal Penetration Landing Trajectories in the Presence of Windshear, Journal of Optimization Theory and Applications, Vol. 57, No. 1, pp. 1–40, 1988.

    Article  MathSciNet  Google Scholar 

  10. Chen, Y. H., andPandey, S.,Robust Control Strategy for Take-Off Performance in a Windshear, Optimal Control Applications and Methods, Vol. 10, No. 1, pp. 65–79, 1989.

    Google Scholar 

  11. Leitmann, G., andPandey, S.,Aircraft Control for Flight in an Uncertain Environment: Take-Off in Windshear, Journal of Optimization Theory and Applications, Vol. 70, No. 1, pp. 25–55, 1991.

    Article  MathSciNet  Google Scholar 

  12. Bulirsch, R., Montrone, F., andPesch, H. J.,Abort Landing in the Presence of Windshear as a Minimax Optimal Control Problem, Part 1: Necesary Conditions, Journal of Optimization Theory and Applications, Vol. 70, No. 1, pp. 1–23, 1991.

    Article  Google Scholar 

  13. 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, No. 2, pp. 223–254, 1991.

    Article  Google Scholar 

  14. Ivan, M.,A Ring-Vortex Downburst Model for Real-Time Flight Simulation of Severe Windshears, AIAA Flight Simulation Technology Conference, St. Louis, Missouri, pp. 57–61, 1985.

  15. Zhu, S., andEtkin, B.,Model of Wind Field in a Downburst, Journal of Aircraft, Vol. 22, No. 7, pp. 595–601, 1985.

    Google Scholar 

  16. Ostolsavskii, I. V., andStrazheva, I. V.,Flight Dynamics: Trajectories of Aircraft, Mashinostroenie, Moscow, Russia,. 1969 (in Russian).

    Google Scholar 

  17. Aleksandrov, A. D. andFedorov, S. M., EditorsSystems of Digital Aircraft Control, Mashinostroenie, Moscow, Russia, 1983 (in Russian).

    Google Scholar 

  18. Fedorov, S. M., Editor,Automatic Control of Airplanes and Helicopters, Transport, Moscow, Russia, 1977 (in Russian).

    Google Scholar 

  19. Isaacs, R.,Differential Games, John Wiley and Sons, New York, New York, 1965.

    Google Scholar 

  20. Pontryagin, L. S.,Linear Differential Games, Part 2, Soviet Mathematics Doklady, Vol. 8, No. 4, pp. 910–912, 1967.

    Google Scholar 

  21. Pshenichnii, B. N., andSagaidak, M. I.,Differential Games of Prescribed Duration, Kibernetika, Vol. 6, No. 2, pp. 72–83, 1970, (in Russian).

    Google Scholar 

  22. Kein, V. M.,Optimization of Control Systems with Minimax Criterion, Nauka, Moscow, Russia, 1985 (in Russian).

    Google Scholar 

  23. Botkin, N. D., andPatsko, V. S.,Positional Control in a Linear Differential Game, Engineering Cybernetics, Vol. 21, No. 4, pp. 69–76, 1983.

    Google Scholar 

  24. Botkin, N. D., Kein, V. M., Krasov, A. I., andPatsko, V. S.,Control of Aircraft Lateral, Motion during Landing in the Presence of Wind Disturbances, Report No. 81104592/0283007880, VNTI Center and Civil Aviation Academy, Leningrad, Russia, 1983.

    Google Scholar 

  25. Subbotin, A. I., andPatsko, V. S., Editors,Algorithms and Programs of Solution of Linear Differential Games, Ural Scientific Center, Academy of Sciences of the USSR, Sverdlovsk, Russia, 1984 (in Russian).

    Google Scholar 

  26. Botkin, N. D., Kein, V. M., Patsko, V. S., andTurova, V. L.,Aircraft Landing Control in the Presence of Windshear, Problems of Control and Information Theory, Vol. 18, No. 4, pp. 223–235, 1989.

    MathSciNet  Google Scholar 

  27. Botkin, N. D., Patsko, V. S., andTurova, V. L.,Development of the Algorithms for Constructing Extremal Wind Disturbances, Report No. 188003467/02880054701, VNTI Center and Institute of Mathematics and Mechanics, Sverdlovsk, Russia, 1987 (in Russian).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics and Mechanics, Russian Academy of Sciences, Ekaterinburg, Russia

    V. S. Patsko (Senior Research Associate), N. D. Botkin (Senior Research Associate), V. L. Turova (Research Associate) & M. A. Zarkh (Research Associate)

  2. Civil Afviration Academy, St. Petersburg, Russia

    V. M. Kein (Professor)

Authors
  1. V. S. Patsko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. D. Botkin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. V. M. Kein
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. V. L. Turova
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. M. A. Zarkh
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by N. V. Banichuk

Rights and permissions

Reprints and permissions

About this article

Cite this article

Patsko, V.S., Botkin, N.D., Kein, V.M. et al. Control of an aircraft landing in windshear. J Optim Theory Appl 83, 237–267 (1994). https://doi.org/10.1007/BF02190056

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02190056

Key Words

Search

Navigation