Scientific and Technical Journal


ISSN Print 2221-3937
ISSN Online 2221-3805
In paper, the differential transformation is applied for solving problems of nonlinear optimal
control processes by dynamic objects. An approach based on the multi-step differential transform method
and the Adomian polynomials for approximation of nonlinear terms of differential equations that describe
dynamic process is proposed. It is offered an optimization model of multi-step control process. The model
advantage is the possibility of process simulation of optimal control with piecewise continuous functions,
determination of an optimal control and state trajectories without using numerical integration methods of
differential equations of object dynamics. Herewith, the analytical solution of a problem is allowed, which
essentially reducing amount of calculation.
DOI 10.15276/eltecs.26.102.2017.11
1. Rao, S. (2009). Engineering optimization: theory and practice. John Wiley&Sons Inc, p. 830.
2. Na, T. Y. (1982). Computational methods in engineering boundary value problems. New York: Academic Press, p. 320.
3. Gurman, V. I., Kvokov, V. N., Ukhin, M. Yu. (2008). Approximate optimization methods of control of a flying vehicle [Priblizhennye metody optimizacii upravlenija letatel'nymi apparatami], Automation and remote control, 69:4, pp. 723-732, doi: 10.1134/ S000511790804019X
4. Balashevich, N. V., Gabasov, R., Kalinin, A. I., Kirillova, F. M. (2002). Optimal control of non-linear systems [Optimal'noe upravlenie nelinejnymi sistemami], Computational Mathematics and Math-ematical Physics, 42:7, pp. 931–956.
5. Pukhov, G. E. (1980). Differential transfor-mations of functions and equations [Differencial'nye preobrazovanija funkcij i uravnenij]. Kyiv: Naukova Dunka, p. 419.
6. Zhou, J. K. (1986). Differential transfor-mation and its applications for electrical circuits. Wuhan-China: Huazhong University Press, p. 317.
7. Hatami, M., Ganji, D. D., Sheikholeslami, M. (2016). Differential transform method for me-chanical engineering problems. Elsevier: Science Publishing Co. Inc, p. 422. 8. Baranov, V. L. (1996). Solving nonlinear boundary value problems based on differential trans-
formations [Reshenie nelinejnyh kraevyh zadach na osnove differencial'nyh preobrazovanij], Electronic modeling, 18:4, pp. 58-63.
9. Gusynin A., Gusynin V., Tachinina H. (2016). The use of differential transformations for solving non-linear boundary value problems, Proceedings of NAU, 4(69), pp.45-55, doi: 10.18372/ 2306-1472.70.11422
10. Kanth, A. S. V. R., Aruna, K. (2008). Solution of singular two-point boundary value problems using differential transformation method, Physics Letters A, 372(26), pp. 4671-4673, doi: 10.1016/j.physleta.2008.05.019.
11. Nik, H. S., Effati, S., Yildirim, A. (2013). Solution of linear optimal control systems by differential transform method, Neural. Comput. & Applic., 23:5, pp.1311–1317, doi: 10.1007/s00521-012-1073-4.
12. Avetisyan, A. G., Simonyan, S. O., Kazaryan, D.A. (2009) Solution of linear problem of optimal speed in the field of Pukhov’s differential conversions [Reshenie linejnoj zadachi optimal'nogo bystrodejstvija v oblasti differencial'nyh preobrazovanij Puhova], Bulletin of the Tomsk Polytechnic University, 315:5, pp. 5-13.
13.Uruskiy, O. S., Baranov V. L. (1996). Synthesis of closed laws of terminal control based on differential transformations [Sintez zamknutyh zakonov terminal'nogo upravlenija na osnove differencial'nyh preobrazovanij], Electronic modeling, 3, pp. 3-8.
14. Nazemi, A., Hesam, S., Haghbin, A. (2015). An application of differential transform method for solving nonlinear optimal control prob-
lems, Computational methods for differential equations, 3:3, pp. 200-217.
15. Hwang, I., Li, J., Du, D. (2009). Differential transformation and its application to nonlinear optimal control, Journal of dynamic systems, measurement and control, 131(5):051010-11, doi:10.1115/1.3155013.
16.Gusynin, A. V. (2016). Modified multi-step differential transform method for solving nonlinear ordinary differential equations [Modificirovannyj mnogojetapnyj metod differencial'nyh preobrazovanij dlja reshenija nelinejnyh obyknovennyh differencial'nyh uravnenij], Problems of information technologies, 02(020), pp. 26-34. 17. El-Zahar, E. R. (2015). Applications of
adaptive multi-step differential transform method to singular perturbation problems arising in science and engineering, Appl.Math.Inf.Sci, 9:1, pp.223-232.
18. Nourifar, M., Aftabi Sani, A., Keyhani, A.
(2017). Efficient multi-step differential transform method: Theory and its application to nonlinear os-cillators, Communications in Nonlinear Science and Numerical Simulation, 5, doi: 10.1016/ j.cnsns.2017.05.001.
19. Gusynin, V. P., Gusynin, A. V., Zamirets, O. N. (2016). Solving nonlinear two-point boundary value problems by modified differential transform method [Reshenie nelinejnyh dvuhtochechnyh kraevyh zadach modificirovannym metodom differ-encial'nyh preobrazovanij], Technology of the in-strumentations, 1, pp. 16-21.
20. Al-Eybani, A. M. D. (2015). Adomian de-composition method and differential transform method to solve the heat equations with a power non-linearity, Int. Journal of Engineering Research and Applications, 5:2, pp.94-98.
21. Fatoorehchi, H., Adolghasemi, H. (2013). Improving the differential transform method: A nov-el technique to obtain the differential transforms of nonlinearities by the Adomian polynomials, Applied Mathematical Modeling, 37:8, pp. 6008 - 6017.
22. Pontryagin, L. S., Boltyanskii, V. G., Mishchenko, E. F., Gamkrelidze, R. V. (1983). The mathematical theory of optimal processes [Ma-tematicheskaja teorija optimal'nyh processov]. Mos-cow: Nauka, p. 393.
23. Bellman, R. E. (2003). Dynamic program-ming, Dover Publications p. 384.
24. Gusynin, V., Gusynin, A., Tachinina, H. (2017). Estimate of accuracy of approximate solu-tions of non-linear boundary value problems by the multi-step differential transform method, Proceed-ings of NAU, 1(70), pp. 48-54, doi: 10.18372/2306-1472.70.11422.
25. Ebaid, A. (2012). On a general formula for computing the one-dimensional differential trans-form of nonlinear functions and its application, Pro-ceedings of the American Conference on Applied Mathematics, pp. 92-97.
26. Baranov, V. L. (2000). Differential-Taylor model of optimal control processes [Differencial'no-tejlorovskaja model' optimal'nyh processov uprav-lenija], Electronic modeling, 22:6, pp. 3-17.
27. Gusynin, V. P., Gusyin, A. V., Zamirets Ya. O. (2015). Model of optimization of a multistage process of management aircraft on basis of differen-tial transformations [Model' optimizacii mnogo-jetapnogo processa upravlenija letatel'nym apparat-om na osnove differencial'nyh preobrazovanij], In-formation processing systems, 8(113), pp. 77-82.
Last download:
25 Aug 2018

[ © KarelWintersky ] [ All articles ] [ All authors ]
[ © Odessa National Polytechnic University, 2014-2018. Any use of information from the site is possible only under the condition that the source link! ]