### Analytical construction of relay dynamic system point mappings taking into account delays

**Published:**10.12.2019

**Authors:** Simonyants R.P., Bulavkin V.N.

**Published in issue: **#12(96)/2019

**DOI: **10.18698/2308-6033-2019-11-1944

**Category:** Aviation and Rocket-Space Engineering | **Chapter:** Aircraft Dynamics, Ballistics, Motion Control

The article considers solving the problem of analytical construction of Poincare maps for finding simple and complex limit cycles in a relay dynamic system involving constant perturbation and delay. Application of Neymark’s theory of multivariate point transformations allows reducing the problem under consideration to the search for a multifold fixed point thus overcoming the difficulty of finding complex periodic motions. The choice of switching lines as arcs without contact taking into account delays significantly simplified the task of analytical construction of point mapping. The results of analytical constructions are confirmed by numerical simulation of movements. The results obtained can find practical application in developing reactive systems for controlling the orientation and stabilization of the spacecraft. Compared with the previously known solution, a more complete result is obtained, which is of particular importance in the study of systems with high efficiency of executive bodies.

**References**

**[1]**Raushenbakh B.V., Tokar E.N. Upravlenie orientatsiey kosmicheskikh apparatov [Spacecraft orientation control]. Moscow, Nauka Publ., 1974. pp. 269–270.

**[2]**Simonyants R.P. Vestnik MGTU im. N.E. Baumana. Ser. Priborostroyeniye — Herald of the Bauman Moscow State Technical University. Series: Instrument Engineering, 2016, no. 3, pp. 88–101. DOI: 10.18698/0236-3933-2016-3-88-101

**[3]**Gaushus E.V. Issledovanie dinamicheskikh sistem metodom tochechnykh preobrazovaniy [Study of dynamic systems by the method of point transformations]. Moscow, Nauka Publ., 1976, 368 p.

**[4]**Neymark Y.I. Metod tochechnykh otobrazheniy v teorii nelineynykh kolebaniy [Point mapping method in the theory of nonlinear oscillations]. Moscow, Nauka Publ., 1972, 472 p.

**[5]**Sieber J. Nonlinearity, 2006, no. 19 (11), pp. 2489–2527.

**[6]**Landry M., Campbell S.A., Morris K., Aguilar C. Dynamics of an inverted pendulum with delayed feedback control. SIAM J. Applied Dynamical Systems, 2005, vol. 4, no. 2, pp. 333–351. DOI: 10.1137/030600461

**[7]**Somov E.I., Butyrin S.A., Somov S.E. Izvestiya Samarskogo nauchnogo tsentra RAN — Izvestia of Samara Scientific Center of the Russian Academy of Sciences, 2014, vol. 16, no. 6, pp. 156–164.

**[8]**Somov S.E. Izvestiya Samarskogo nauchnogo tsentra RAN — Izvestia of Samara Scientific Center of the Russian Academy of Sciences, 2010, vol. 12, no. 4, pp. 227–232.

**[9]**Norbury J., Wilson R.E. Journal of Computational and Applied Mathematics, 2000, vol. 125, no. 1-2, pp. 201–215. DOI: 10.1016/S0377-0427(00)00469-6

**[10]**Feofilov S.V., Kozyr A.V. Izvestiya Tulskogo gosudarstvennogo universiteta. Tekhnicheskie nauki — Izvestiya Tula State University. Technical Sciences, 2018, no. 6, pp. 135–147.

**[11]**Kowalczyk P. The dynamics and event-collision bifurcations in switched control systems with delayed switching. Journal of Physics D: Applied Physics, submitted in July 2019. Available at: http://prac.im.pwr.edu.pl/~kowalczykp/ kowalczyk17_Artc6.pdf (accessed September 18, 2019).

**[12]**Iwasaki T. SICE Journal of Control, Measurement, and System Integration, 2018, vol. 11, no. 1, pp. 002–013.

**[13]**Tang S., Tang B., Wang A., Xiao Y. Nonlinear Dynamics 2015, no. 81 (3), pp. 1575–1596.

**[14]**Yoon Y.E., Johnson E.N. Determination of Limit Cycle Oscillation Frequency in Linear Systems with Relay Feedback. Proceedings of the AIAA Guidance, Navigation, and Control Conference, January 8—12, 2018. United States, Florida, Kissimmee, AIAA Publ., 2018, pp. 178–196.

**[15]**Krasnoshchechenko V.I. Matematicheskoe modelirovanie i chislennye menody — Mathematical Modeling and Computational Methods, 2015, no. 2 (6), pp. 87–104.