Lie algebra approach in the study of the stability of stochastic linear hybrid systems

Seroka, E.; Socha, L.

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Tytuł artykułu

Lie algebra approach in the study of the stability of stochastic linear hybrid systems

Autorzy

Seroka E. , Socha L.

Treść / Zawartość

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Warianty tytułu

Metody Lie algebr w badaniu stabilności stochastycznych układów hybrydowych

Języki publikacji

Abstrakty

The problem of the stability of a class of stochastic linear hybrid systems with a special structure of matrices and a multiplicative excitation is considered. Sufficient conditions of the exponential p-th mean stability and the almost sure stability for a class of stochastic linear hybrid systems with the Markovian switching are derived. Also, sufficient conditions of the exponential mean-square stability for a class of stochastic linear hybrid systems satisfying the Lie algebra conditions with any switching are found. The obtained results are illustrated by examples and simulations.

W pracy rozważony został problem stabilności klasy liniowych stochastycznych układów hybrydowych ze szczególną strukturą macierzy i multiplikatywnym szumem. Znalezione zostały warunki wystarczające dla eksponencjalnej p-średniej stabilności i eksponencjalnej stabilności prawie na pewno dla klasy stochastycznych liniowych układów hybrydowych z Markowskim przełączaniem. Dodatkowo podane zostały warunki wystarczające dla eksponencjalnej stabilności średnio-kwadratowej dla klasy stochastycznych liniowych układów hybrydowych z dowolnym przełączaniem spełniających warunki Lie algebry. Otrzymane wyniki zilustrowane zostały przykładami i symulacjami.

Słowa kluczowe

hybrid systems nonlinear systems asymptotic stability

Wydawca

Polskie Towarzystwo Mechaniki Teoretycznej i Stosowanej

Czasopismo

Journal of Theoretical and Applied Mechanics

Rocznik

2011

Tom

Vol. 49 nr 1

Strony

31--50

Opis fizyczny

Bibliogr. 23 poz., rys.

Twórcy

autor

Seroka E.

autor

Socha L.

College of Sciences, Cardinal Stefan Wyszyński University in Warsaw, Faculty of Mathematics and Natural Sciences, Warsaw, Poland, ewelina_seroka@o2.pl

Bibliografia

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2. Boukas E.K., 2005, Stochastic Hybrid Systems: Analysis and Design, Birkh¨auser, Boston
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5. Bullo F., Leonard N.E., Lewis A.D., 2000, Controllability and motion algorithms for underactuated Lagrangian systems on Lie Groups, IEEE Trans. Autom. Cont., 45, 8, 1437-1454
6. Dimentberg M.F., Iourtchenko D.V., 2004, Random vibrations with impact: A review, Nonlinear Dynamics, 36, 229-254
7. Gurkan E., Banks S.P., Erkmen I., 2003, Stable controller design for the T-S fuzzy model of a flexible-joint robot arm based on Lie algebra, Proceedings of the 42nd IEEE Conference on Decision and Control, Maul, Hawaii USA, 4714-4722
8. He G., Lu Z., 2006, An input extension control method for a class of secondorder nonholonomic mechanical systems with drift, Proceedings of the 2nd IEEE/ASME International Conference on Mechatronic and Embedded Systems and Applications, 1-6
9. Itkis Y., 1983, Dynamic switching of type-I/type-II structures in tracking servosystems, IEEE Trans. Automat. Control, 28, 531-534
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11. Liberzon D., 2003, Switching in Systems and Control, Birkh¨auser, Boston, Basel, Berlin
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17. Samelson H., 1969, Notes on the Lie algebra, New York, Van Nostrand Reinhold
18. Sussman H.J., Liu W., 1993, Lie bracket extensions and averaging the singlebracket case, In: Nonholonomic Motion Planning, Li Z. and Canny J.F. (Edit.), Boston, 109-148
19. Utkin V.I., 1978, Sliding Modes and their Application in Variable Structures Systems, Mir. Moscow
20. Willems J.L., 1976,Moment stability of linear white noise and coloured noised systems, Stochastic Problems in Dynamics, 67-89
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23. Zhai G., Liu D., Imae J., Koboyashi T., 2006, Lie algebraic stability analysis for switched systems with countinuous-time and discrete-time subsystems, IEEE Trans. Automa. Cont., 53, 152-156

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Bibliografia

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