Stability Investigation and Stabilization of Nonlinear Switched Mechanical Systems via Decomposition

A nonlinear hybrid mechanical system is studied. It is assumed that on the system homogeneous switched positional forces, linear gyroscopic forces and homogeneous dissipative ones act. Conditions are determined under which the trivial equilibrium position of the considered system is asymptotically stable for any admissible switching law. It is well known that to prove the asymptotic stability uniform with respect to swishing law, it is sufficient to construct a common Lyapunov function for the family of subsystems corresponding to the hybrid system. However, till now there are no general constructive methods for finding of common Lyapunov functions, even for families composed of linear subsystems. The problem is especially difficult for mechanical systems with switched force fields, since such systems are described by differential equations of the second order. This results in the appearance of certain special properties of motions and essentially complicates the analysis of system dynamics. In particular, well known results obtained for switched systems of general form might be inefficient or even inapplicable for mechanical systems. In the present paper, a new approach to the stability analysis is proposed. The approach is based on the decomposition of the original system consisting of differential equations of the second order into two isolated first order subsystems of the same dimension. It should be noted that one of the isolated subsystem is switched, whereas another one is non-switched. Thus, the decomposition method permits us to reduce the problem of a common Lyapunov function constructing for the original family of nonlinear systems of dimension with a special structure to that for an auxiliary family of subsystems of dimension which, generally, do not possess a special structure. The problem of stabilization of the equilibrium position of a system for any switching mode scaling the potential with the aid of small forces of radial correction is considered. For a model of the magnetic bearing of a rotor with nonlinear switched circular forces, the stabilizing feedback control law is constructed by the use of linear gyroscopic and nonlinear dissipative forces.

гибридные механические системы, асимптотическая устойчивость, управление, функции Ляпунова, стабилизация, декомпозиция, закон переключения, hybrid mechanical systems, asymptotic stability, control, Lyapunov functions, stabilization, decomposition, switching law

1. Shorten R., Wirth F., Mason O., Wulf K., King C. Stability Criteria for Switched and Hybrid Systems // SIAM Rev. 2007. V. 49. N. 4. P. 545-592.

2. Hai Lin, Antsaklis P. J. Stability and Stabilizability of Switched Linear Systems: a Survey of Recent Results // IEEE Trans. Automat. Contr. 2009. V. 54, N. 2. P. 308-322.

3. Liberzon D. Switching in Systems and Control. Boston, MA: Birkhauser, 2003. 233 p.

4. DeCarlo R., Branicky M., Pettersson S., Lennartson B. Perspectives and Results on the Stability and Stabilisability of Hybrid Systems // Proc. IEEE. 2000. V. 88. P. 1069-1082.

5. Васильев С. Н., Косов А. А. Анализ динамики гибридных систем с помощью общих функций Ляпунова и множественных гомоморфизмов // Автоматика и телемеханика. 2011. № 6. С. 27-47.

6. Vassilyev S. N., Kosov A. A., Malikov A. I. Stability Analysis of Nonlinear Switched Systems via Reduction Method // Preprints of the 18th IF AC World Congress. Milano, Italy. Aug. 28 - Sept. 2, 2011. P. 5718-5723.

7. Александров А. Ю., Косов А. А. Об устойчивости и стабилизации положений равновесия нелинейных неавтономных механических систем // Изв. РАН. Теория и системы управления. 2009. № 4. С. 13-23.

8. Александров А. Ю., Косов А. А. Об устойчивости и стабилизации нелинейных нестационарных механических систем // Прикл. математика и механика. 2010. Т. 74, № 5. С. 774-788.

9. Александров А. Ю., Косов А. А., Чэнь Я. Об устойчивости и стабилизации механических систем с переключениями // Автоматика и телемеханика. 2011. № 6. С. 5-17.

10. Матросов В. М. Метод векторных функций Ляпунова: анализ динамических свойств нелинейных систем. М.: Физматлит, 2001. 384 с.

11. Пятницкий Е. С. Принцип декомпозиции в управлении динамическими системами // Докл. АН СССР. 1988. Т. 300. № 2. С. 300-303.

12. Черноусько Ф. Л., Ананьевский И. М., Решмин С. А. Методы управления нелинейными механическими системами. М.: Физматлит, 2006. 328 с.

13. Зубов В. И. Аналитическая динамика гироскопических систем. Л.: Судостроение, 1970. 320 с.

14. Меркин Д. Р. Гироскопические системы. М.: Наука, 1974. 344 с.

15. Меркин Д. Р. Введение в теорию устойчивости движения. М.: Наука, 1987. 304 с.

16. Зотеев В. Е. Параметрическая идентификация диссипативных механических систем на основе разностных уравнений. М.: Машиностроение, 2009. 344 с.

17. Пановко Г. Я. Основы прикладной теории колебаний и удара. Л.: Машиностроение, 1976. 326 с.

18. Gendelman O. V., Lamarque C. H. Dynamics of linear oscillator coupled to strongly nonlinear attachment with multiple states of equilibrium // Chaos, Solitons and Fractals. 2005. V. 24. P. 501-509.

19. Gourdon E., Lamarque C. H. Energy pumping for a larger span of energy // J. of Sound and Vibration. 2005. V. 285. P. 711-720.

20. Александров А. Ю., Косов А. А., Платонов А. В., Фадеев С. С. Об устойчивости и стабилизации механических систем с переключениями силовых полей // Мехатроника, автоматизация, управление. 2013. № 12. С. 9-16.

21. Aleksandrov A. Yu., Chen Y., Kosov A. A., Zhang L. Stability of Hybrid Mechanical Systems with Switching Linear Force Fields // Nonlinear Dynamics and Systems Theory. 2011. V. 11, N. 1. P. 53-64.

22. Зубов В. И. Математические методы исследования систем автоматического регулирования. Л.: Судостроение, 1959. 324 с.

23. Aleksandrov A. Yu., Kosov A. A., Platonov A. V. On the Asymptotic Stability of Switched Homogeneous Systems // Systems & Control Letters. 2012. V. 61, N. 1. P. 127-133.

24. Александров А. Ю., Жабко А. П., Косов А. А. Анализ устойчивости и стабилизация нелинейных систем на основе декомпозиции // Сиб. мат. журнал. 2015 (принята к печати).

25. Post R. F. Stability Issues in Ambient-Temperature Passive Magnetic Bearing Systems. February 17, 2000. Lawrence Livermore National Laboratory. Technical Information Department's Digital Library. http://www.llnl.gov/tid/Library.html

26. Метелицын И. И. К вопросу о гироскопической стабилизации // Докл. АН СССР. 1952. Т. 86. № 1. С. 31-34.