Geodesic
On the stability of the equilibrium positions of nonlinear mechanical hybrid system
Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, no. 3 (2015), pp. 116-125 Cet article a éte moissonné depuis la source Math-Net.Ru

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Stability of trivial equilibrium position of nonlinear mechanical system with switched potential and dissipative forces is studied. The switched system consists of n subsystems described by Rayleigh equations and some switching law determining at each time instant which subsystem is active. It is assumed that potential and dissipative forces are nonlinear and homogeneous. Asymptotic stability of equilibrium position of subsystem is demonstrated. By the use of Lyapunov multiple function and dwell-time approach, the conditions on switching law are obtained. The fulfilment of these conditions provides asymptotic stability of the equilibrium position of switched system. Also the case of known order of switching between the subsystems is considered. An example is presented to demonstrate effectiveness of the proposed approaches. Furthermore it was shown that the equilibrium position may be unstable if switched law is chosen that doesn’t satisfy derived conditions. Refs 11. Figs 2.
Keywords: switched systems, mechanical systems, asymptotic stability, Lyapunov functions, multiple Lyapunov function. 
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     title = {On the stability of the equilibrium positions of nonlinear mechanical hybrid system},
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P. A. Lakrisenko. On the stability of the equilibrium positions of nonlinear mechanical hybrid system. Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, no. 3 (2015), pp. 116-125. http://geodesic.mathdoc.fr/item/VSPUI_2015_3_a9/

[1] Liberzon D., Morse A. S., “Basic problems in stability and design of switched systems”, Control System Magazine. IEEE, 19:5 (1999), 59–70 | DOI

[2] Decarlo R. A., Branicky M. S., Petterson S., Lennartson B., “Perspectives and results on the stability and stabilizability of hybrid systems”, Proc. of the IEEE, 88:7 (2000), 1069–1082 | DOI

[3] Shorten R. A., Wirth F., Mason O., Wulf K., King C., “Stability criteria for switched and hybrid systems”, SIAM Rev., 49:4 (2007), 545–592 | DOI | MR | Zbl

[4] Aleksandrov A. Iu., Platonov A. V., “On the ultimate boundedness and permanence of solutions for a class of discrete-time switched models of population dynamics”, Vestnik of St. Petersburg State University. Series 10. Applied mathematics. Computer science. Control processes, 2014, no. 1, 5–16 (In Russian)

[5] Aleksandrov A. Iu., Platonov A. V., Chen' Ia., “On the absolute stability of nonlinear switched systems”, Vestnik of St. Petersburg State University. Series 10. Applied mathematics. Computer science. Control processes, 2008, no. 2, 119–133 (In Russian)

[6] Aleksandrov A. Iu., Platonov A. V., “On the stability of hybrid homogeneous systems”, Vestnik of Samarsk. State Tech. University. Series Phys.-math. Sciences, 2010, no. 5(21), 24–32 (In Russian) | DOI

[7] Aleksandrov A. Iu., Kosov A. A., Chen' Ia., “On the stability and stabilisation of switched mechanical systems”, Automatics and telemechanics, 2011, no. 6, 5–17 (In Russian)

[8] Murzinov I. E., “On construction of common Lyapunov function for a family of mechanical systems with one degree of freedom”, Vestnik of St. Petersburg State University. Series 10. Applied mathematics. Computer science. Control processes, 2013, no. 4, 49–57 (In Russian)

[9] Aleksandrov A. Yu., Platonov A. V., Lakrisenko P. A., “Stability analysis of nonlinear mechanical systems with switched force fields”, Proc. of the 21st Mediterranean Conference on Control Automation (Platanias-Chania, Crete, Greece, 2013), 628–633

[10] Aleksandrov A. Iu., Aleksandrova E. B., Ekimov A. V., Smirnov N. V., Collection of Problems on stability theory, “SOLO”, St. Petersburg, 2003, 162 pp. (In Russian)

[11] Zubov V. I., Stability of Motion, Vysshaia Shkola Publ., M., 1973, 272 pp. (In Russian) | MR