Neural network based semi-empirical approach to the modeling of longitudinal motion and identification of aerodynamic characteristics for maneuverable aircraft

Mathematica modeling, numerical technique and program complexes


Egorchev M. V.*, Tiumentsev Y. V.**

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia



The simulation problem for longitudinal motion of a maneuverable aircraft is considered including identification of its aerodynamic characteristics, such as the coefficients of aerodynamic axial and normal forces, as well as the pitch moment coefficient. This problem is solved in the class of modular semi-empirical dynamic models that combine the possibilities of theoretical and neural network modeling.

This approach differs significantly from the traditionally accepted method for solving problems of this class [6-8], based on the use of the linearized model of the disturbed motion of the aircraft and using the representation of the dependencies for the aerodynamic forces and moments acting on the aircraft in the form of their expansion into a Taylor series, leaving in it, as a rule, only members not higher than the first order.

Accordingly, the solution of the identification problem with this approach is reduced to reconstructing from the experimental data the dependences describing the coefficients of the Taylor expansion, in which the derivatives of the dimensionless coefficients of the aerodynamic forces and moments with respect to the various parameters of the motion of the aircraft are determining.

In contrast, the semi-empirical approach realizes the reconstruction of the relations for the force coefficients and moments   as some whole non-linear dependencies on the corresponding arguments, without resorting to their series expansion and to linearization, i.e. the functions themselves, represented in the ANN-form, are evaluated, and not the coefficients of their expansion in a series. Each of these dependencies is implemented as a separate ANN-module, built into a semi-empirical ANN-model. Derivatives etc. if necessary, can be found using the results obtained during formation of the ANN-modules for the coefficients of forces and moments within the semi-empirical ANN-model.

A mathematical model of the longitudinal motion of a maneuverable aircraft is derived, which is used as a basis in the formation of the corresponding semi-empirical ANN-model, as well as for the generation of a training set. An algorithm for such a generation is proposed, which provides a fairly uniform coverage of the possible values of state variables and controls for the maneuverable aircraft by training examples. Next, a semi-empirical ANN-model of the longitudinal controlled motion of the aircraft is formed, including the ANN-modules realizing the functional dependences for the coefficients . In the process of learning the obtained ANN-model, the identification problem for these coefficients is solved. The corresponding results of computational experiments characterizing the accuracy of the formed ANN-model as a whole, as well as the accuracy of the solution of the problem of identification of aerodynamic coefficients are given.


nonlinear dynamical system, aircraft longitudinal motion, aerodynamic model identification, semi-empirical model, neural network learning


  1. Egorchev M.V., Tiumentsev Yu.V. Vestnik Moskovskogo aviatsionnogo instituta, 2014, vol. 21, no. 4, pp. 13–24.

  2. Egorchev M.V., Tiumentsev Yu.V. Learning of semi-empirical neural network model of aircraft three-axis rotational motion, Optical Memory and Neural Networks (Information Optics), 2015, vol. 24, no. 3, pp. 210–217.

  3. Egorchev M.V., Tyumentsev Yu.V. Sbornik nauchnykh trudov XV Vserossiiskoi nauchno-tekhnicheskoi konferentsii «Neiroinformatika-2013», Moscow, 2013, vol. 2, pp. 22–31.

  4. Egorchev M.V., Kozlov D.S., Tyumentsev Yu.V., Chernyshev A.V. Vestnik informatsionnykh i komp’yuternykh tekhnologii, 2013, no. 9, pp. 3–10.

  5. Egorchev M.V., Tyumentsev Yu.V. Sbornik nauchnykh trudov XVI Vserossiiskoi nauchno-tekhnicheskoi konferentsii «Neiroinformatika-2014», Moscow, 2014, vol. 2, pp. 263–272.

  6. Berestov L.M., Poplavskii B.K., Miroshnichenko L.Ya. Chastotnye metody identifikatsii letatel’nykh apparatov (Frequency-domain methods for aircraft identification), Moscow, Mashinostroenie, 1985, 184 p.

  7. Klein V., Morelli E.A. Aircraft system identification: Theory and practice, Reston, VA: AIAA, Inc., 2006, 498 p.

  8. Tischler M.B., Remple R.K. Aircraft and rotorcraft system identification: Engineering methods with flight-test examples, Reston, VA: AIAA, Inc., 2006, 558 p.

  9. Vereshchikov D.V., Kuznetsov A.D. Trudy MAI, 2016, no. 89, available at:

  10. Niewald P.W., Parker S.L. Flight-test techniques employed to successfully verify F/A-18E in-flight lift and drag, Journal of Aircraft, 2000, vol. 37, no. 2, pp. 194–200.

  11. Mulder J.A., van Sliedregt J.M. Estimation of drag and thrust of jet-propelled aircraft by non-steady flight-test maneuvers, Delft Univ. of Technology, Memorandum M-255, Dec. 1976, 32 p.

  12. Cybenko G. Approximation by superposition of a sigmoidal function, Mathematics of Control, Signals and Systems, 1989, vol. 2, pp. 303–314.

  13. Hornik K., Stinchcombe M., White H. Multilayer feedforward networks are universal approximators, Neural Networks, 1989, vol. 2, pp. 359–366.

  14. Gorban A.N. Sibirskiy zhurnal vychislitel’noy matematiki, 1998, vol. 1, no. 1, pp. 11–24.

  15. Haykin S. Neural networks: A comprehensive foundation: 2nd Edition, Prentice Hall, 2006, 823 p.

  16. Bochkarev A.F. Aeromekhanika samoleta: Dinamika poleta (Aeromechanics of airplane: Flight dynamics), Moscow, Mashinostroenie, 1985, 360 p.

  17. Nguyen L.T., Ogburn M.E., Gilbert W.P., Kibler K.S., Brown P.W., Deal P.L. Simulator study of stall/post-stall characteristics of a fighter airplane with relaxed longitudinal static stability, NASA TP-1538, Dec. 1979, 223 p.

  18. Muja M., Lowe D. G. Scalable Nearest Neighbor Algorithms for High Dimensional Data, IEEE Transactions on Pattern Analysis and Machine Intelligence, 2014, vol. 36, no. 11, pp. 2227–2240.

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