On Leonov’s method for computing the linearization of the transverse dynamics and analysis of Zhukovsky stability

Authors

  • Anton S. Shiriaev Norwegian University of Science&Technology, NO-7491, Trondheim, Norway
  • Ramil R. Khusainov Innopolis University, Universitetskaya ul., 1, Innopolis, 420500, Russian Federation
  • Shamil N. Mamedov Innopolis University, Universitetskaya ul., 1, Innopolis, 420500, Russian Federation
  • Sergey V. Gusev St. Petersburg State University, Universitetskaya nab., 7–9, St. Petersburg, 199034, Russian Federation
  • Nikolay V. Kuznetsov St. Petersburg State University, Universitetskaya nab., 7–9, St. Petersburg, 199034, Russian Federation; Institute for Problems in Mechanical Engineering of the Russian Academy of Sciences, Bolshoy pr. V.O., 61, St. Petersburg, 199178, Russian Federation; University of Jyv¨askyl¨a, P.O. Box 35, FI-40014, Jyv¨askyl¨a, Finland https://orcid.org/0000-0002-6474-9657

DOI:

https://doi.org/10.21638/11701/spbu01.2019.402

Abstract

The paper is focused on a comprehensive discussion of one of results of Prof. G. A. Leonov aimed at analysis of Zhukovsky stability of a solution of a nonlinear autonomous system by linearization. The main contribution is in deriving a linear system that approximates dynamics of the original nonlinear systems transverse to the vector-flow on a nominal behavior. As illustrated, such a linear comparison system becomes instrumental in analysis and re-design of classical feedback controllers developed earlier for stabilization of motions of nonlinear mechanical systems.

Keywords:

moving Poincar´e section, Zhukovsky stability, transverse linearization

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References

Литература

Leonov G.A. Generalization of the Andronov-Vitt theorem // Regular and Chaotic Dynamics. 2006. Vol. 11, No. 2. С. 281–289. https://doi.org/10.1070/RD2006v011n02ABEH000351

Kuznetsov N.V., Leonov G.A. Strange attractors and classical stability theory: stability, instability, Lyapunov exponents and chaos. // In: Handbook of Applications of Chaos Theory / Eds. Ch.H. Skiadas, Ch. Skiadas. Chapman and Hall/CRC, 2016. P. 105–134. https://doi.org/10.1201/b20232-7

Leonov G.A., Kuznetsov N.V. Time-varying linearization and the Perron effects // International Journal of Bifurcation and Chaos. 2007. Vol. 17, No. 4. P. 1079–1107. https://doi.org/10.1142/S0218127407017732

Shiriaev A.S., Freidovich L.B., Gusev S.V. Transverse linearization for controlled mechanical systems with several passive degrees of freedom // IEEE Transactions on Automatic Control. 2010. Vol. 55, No. 4. P. 893–906. https://doi.org/10.1109/TAC.2010.2042000

Fradkov A.L. Swinging control of nonlinear oscillations // International Journal of Control. 1996. Vol. 64, No. 6. P. 1189–1202. https://doi.org/10.1080/00207179608921682

Spong M.W., Corke P., Lozano R. Nonlinear control of reaction wheel pendulum // Automatica. 2001. Vol. 37. P. 1845–1851. https://doi.org/10.1016/S0005-1098(01)00145-5

Spong M.W., Block D.J., ˚Astr¨om K.J. The mechatronic control kit for education and research // Proceedings of the IEEE Conference on Control Applications. 2001. P. 105–110.

Shiriaev A.S., Fradkov A.L. Stabilization of invariant sets for nonlinear systems with applications to control of oscillations // International Journal of Robust and Nonlinear Control. 2001. P. 215–240. https://doi.org/10.1002/rnc.568

Формальский А.М. Физико-технический практикум “Управление маятником при помощи маховика”. М.: Изд-во Московского ун-та, 2003.

Srinivas K.N., Behera L. Swing-up control strategies for a reaction wheel pendulum // International Journal of Systems Science. 2008. Vol. 39, No. 12. P. 1165–1177. https://doi.org/10.1080/00207720802095137

Freidovich L., La Hera P., Mettin U., Robertsson A., Shiriaev A. S., Johansson R. Stable periodic motions of inertia wheel pendulum via virtual holonomic constraints // Asian Journal of Control. 2009. Vol. 11, No. 5. P. 548–556. https://doi.org/10.1002/asjc.135


References

Leonov G.A., “Generalization of the Andronov-Vitt theorem”, Regular and Chaotic Dynamics 11(2), 281–289 (2006). https://doi.org/10.1070/RD2006v011n02ABEH000351

Kuznetsov N.V., Leonov G.A., Strange attractors and classical stability theory: stability, instability, Lyapunov exponents and chaos, in Handbook of Applications of Chaos Theory, 105–134 (Eds. Ch. H. Skiadas, Ch. Skiadas, Chapman and Hall/CRC, 2016). https://doi.org/10.1201/b20232-7

Leonov G.A., Kuznetsov N.V., “Time-varying linearization and the Perron effects”, International Journal of Bifurcation and Chaos 17(4), 1079–1107 (2007). https://doi.org/10.1142/S0218127407017732

Shiriaev A. S., Freidovich L.B., Gusev S.V., “Transverse linearization for controlled mechanical systems with several passive degrees of freedom”, IEEE Transactions on Automatic Control 55(4), 893–906 (2010). https://doi.org/10.1109/TAC.2010.2042000

Fradkov A.L., “Swinging control of nonlinear oscillations”, International Journal of Control 64(6), 1189–1202 (1996). https://doi.org/10.1080/00207179608921682

Spong M.W., Corke P., Lozano R., “Nonlinear control of reaction wheel pendulum”, Automatica 37, 1845–1851 (2001). https://doi.org/10.1016/S0005-1098(01)00145-5

Spong M.W., Block D.J., ˚Astr¨om K.J., “The mechatronic control kit for education and research”, Proceedings of the IEEE Conference on Control Applications, 105–110 (2001).

Shiriaev A.S., Fradkov A.L., “Stabilization of invariant sets for nonlinear systems with applications to control of oscillations”, International Journal of Robust and Nonlinear Control, 215–240 (2001). https://doi.org/10.1002/rnc.568

Formal’ski A.M., Controlling an inverted pendulum by an inertia wheel (Moscow Univ. Press, Moscow, 2003). (In Russian)

Srinivas K.N., Behera L., “Swing-up control strategies for a reaction wheel pendulum”, International Journal of Systems Science 39(12), 1165–1177 (2008). https://doi.org/10.1080/00207720802095137

Freidovich L., La Hera P., Mettin U., Robertsson A., Shiriaev A.S., Johansson R., “Stable periodic motions of inertia wheel pendulum via virtual holonomic constraints”, Asian Journal of Control 11(5), 548–556 (2009). https://doi.org/10.1002/asjc.135

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Published

2019-11-28

How to Cite

Shiriaev, A. S., Khusainov, R. R., Mamedov, S. N., Gusev, S. V., & Kuznetsov, N. V. (2019). On Leonov’s method for computing the linearization of the transverse dynamics and analysis of Zhukovsky stability. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 6(4), 544–554. https://doi.org/10.21638/11701/spbu01.2019.402

Issue

Vol. 6 No. 4 (2019)

Section

In memoriam of G. A. Leonov

License

Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.