Algorithmization of Maximum Aircraft Range Estimation Using Methods of Nonlinear Programming and Parameter Identification

The article deals with the use of nonlinear programming to solve the problem of maximum aircraft range estimation. The advantage of this approach is that it allows setting the problem in a sufficiently general form, based only on the principal aerodynamic characteristics of the aircraft, for example, without requiring the determination of the guidance law. Due to this, it is possible to obtain the upper limit estimates. The maximum flight range is determined as a solution of optimal control problem. To obtain it, the control signal is expanded in some basis. Estimation of expansion coefficients, performed by parametric identification methods, is the single-criteria multi-parametric optimization problem, which can be solved numerically. In the article, cubic Hermite splines are used as the basis for control approximation. One of their characteristics is that they do not require continuity of the second derivative in their nodes. Therefore, Hermite splines manage to approximate a wider class of signals. This also causes their main drawback — they require larger number of parameters than the classic interpolation cubic splines. The paper considers natural candidate for target functional — maximum flight range. Optimization of control parameters is carried out by the zero order method. One of the more common varieties of population algorithms, particle swarm, is used. Such choice ensures that work with a parameter vector of significant dimension is possible. The article demonstrates that obtained solutions are stable with regard to variations of spline parameters’ values and boundary conditions. It also compares the range estimations obtained using nonlinear programming with another case, when the structure of the guidance law is set explicitly, and only the values of its parameters are subject to optimization. The experiments showed that with a fixed structure of the guidance law the maximum range is slightly lower, probably due to the introduction of additional restrictions. In addition, the paper considers the extension of class of controls to include signals that could be obtained using an artificial neural network. The results show that the application of neural networks in this task does not provide significant advantages compared to cubic splines, although it requires identification of a noticeably larger number of parameters.

1. Byushgens G. S. ed. Aerodynamics, stability and controllability of supersonic aircrafts, Moscow, Nauka, 1998, 816 p. (in Russian).

2. Vasilchenko K. K., Leonov V. A., Pashkovskij I. M., Poplavskij В. K. Aircraft flight tests, Moscow, Mashinostroenie, 1996, 719 p. (in Russian).

3. Hargraves C. R., Paris S. W. Direct trajectory optimization using nonlinear programming techniques, Journal of Guidance, Control, and Dynamics, 1987, vol. 10, pp. 338—342, DOI: 10.2514/3.20223

4. Huang G., Lu Y., Nan Y. А survey of numerical algorithms for trajectory optimization of flight vehicles, Science China. Technological Sciences, 2012, vol. 55, no. 9, pp. 2538—2560, DOI: 10.1007/s11431-012-4946-y

5. Marguet V. Motion planning for multi-agent dynamical systems in a В-spline framework. Ph.D. Thesis, Université Grenoble Alpes, France, 2024.

6. Zav’yalov Ju. S., Kvasov B. I., Miroshnichenko V. L. Methods of spline functions, Moscow, Nauka, 1980, 352 p. (in Russian).

7. Wermel’ V. D. Fundamentals of computational (engineering) geometry, Moscow, Innovacionnoe mashinostroenie, 2021, 352 p. (in Russian).

8. Pupkov K. A., Egupov N. D. eds. Methods of classical and modern theory of automatic control, Moscow, Izd. MGTU im. Baumana, 2004, 656 p. (in Russian).

9. Eberhart R. C., Kennedy J. Particle Swarm Optimization, Proceedings of the IEEE International Conference on Neural Networks, Piscataway, NJ, 1995, pp. 1942—1948, DOI: 10.1109/ICNN.1995.488968

10. Olsson А. E. Particle swarm optimization: theory, technique and applications. Hauppage, USA, Nova Science Publishers, 2011, 305 p.

11. Malyshev V. V. Optimization methods in problems of system analysis and control, Moscow, Publishing house of MAI-PRINT, 2010, 440 p. (in Russian).

12. Chernous’ko F. L., Anan’evskij I. M., Reshmin S. A. Control methods for nonlinear mechanical systems, Moscow, Fizmatlit, 2006, 328 p. (in Russian).

13. Rao А. V. Survey of numerical methods for optimal control, Advances in the Astronautical Sciences, 2010, vol. 135, no. 1, pp. 497—528.

14. Korsun O. N., Stulovskii А. V. Comparison of the direct method and the maximum principle in the problem of the aircraft program control design, Mekhatronika, Avtomatizatsiya, Upravlenie, 2019, no. 6, pp. 367—375 (in Russian), DOI: 10.17587/mau.20.367-375

15. GOST 20058—80. Dynamics of aircraft in the atmosphere. Terms, definitions and symbols, Moscow, Izdatel’stvo standartov, 1981, 54 p. (in Russian).

16. Karpenko А. P. Modern search optimization algorithms. Algorithms inspired by nature, Moscow, Publishing house of MGTU im. Baumana, 2016, 448 p. (in Russian).

17. Mirjalili S., Dong J. S., Lewis А. eds Nature-inspired Optimizers: Theories, Literature Reviews and Applications, Switzerland, AG, Springer Nature, 2020, 239 p.

18. Korsun O. N., Stulo vsky А. V. Recovery of aircraft motion parameters using the optimal control algorithms, Journal of Computer and Systems Sciences International, 2023, vol. 62, no. 1, pp. 61—72, DOI: 10.1134/s1064230723010057

19. Neupokoev F. K. Surface-air missiles firing, Moscow, Voen’izdat, 1991, 344 p. (in Russian).

20. HornikK., Stinchcombe M., White Н. Multilayer feedforward networks are universal approximators, Neural Networks, 1989, vol. 2, no. 5, pp. 359—366, DOI: 10.1016/0893-6080(89)90020-8

21. Haykin S. O. Neural networks: a comprehensive foundation (2nd edition) Upper Saddle River, USA, Prentice Hall, 1998, 842 p.

22. Brunton S. L., Kutz J. N. Data driven science and engineering. Machine learning, dynamical systems, and control, Cambridge, Cambridge University Press, 2019, 472 p.