# zbMATH — the first resource for mathematics

Stability and stabilization of mechanical systems with switching. (English. Russian original) Zbl 1229.93099
Autom. Remote Control 72, No. 6, 1143-1154 (2011); translation from Avtom. Telemekh. 2011, No. 6, 5-17 (2011).
Summary: Hybrid mechanical systems with switched force fields, whose motions are described by differential second-order equations are considered. We propose two approaches to solving problems of analysis of stability and stabilization of an equilibrium position of the named systems. The first approach is based on the decomposition of an original system of differential equations into two systems of the same dimension but of first order. The second approach is basing on a direct construction of a general Lyapunov function for a mechanical system with switching.

##### MSC:
 93C30 Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems) 70Q05 Control of mechanical systems 93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, $$L^p, l^p$$, etc.) in control theory 93B11 System structure simplification 93C15 Control/observation systems governed by ordinary differential equations
Full Text:
##### References:
 [1] De Carlo, R., Branicky, M., Pettersson, S., and Lennartson, B., Perspectives and Results on the Stability and Stabilisability of Hybrid Systems, Proc. IEEE, 2000, vol. 88, pp. 1069–1082. [2] Shorten, R., Wirth, F., Mason, O., Wulf, K., et al., Stability Criteria for Switched and Hybrid Systems, SIAM Rev., 2007, vol. 49, no. 4, pp. 545–592. · Zbl 1127.93005 [3] Hai Lin and Antsaklis, P.J., Stability and Stabilizability of Switched Linear Systems: A Survey of Recent Results, IEEE Trans. Automat. Control, 2009, vol. 54, no. 2, pp. 308–322. · Zbl 1367.93440 [4] Unsolved Problems in Mathematical Systems and Control Theory, Blondel, V.D. and Megretski, A., Eds., Princeton: Princeton Univ. Press, 2004. [5] Pakshin, P.V. and Pozdyaev, V.V., A Criterion for the Existence of a General Quadratic Lyapunov Function in Linear Second-Order Systems, Izv. Ross. Akad. Nauk, Teor. Sist. Upravlen., 2005, no. 4, pp. 22–27. · Zbl 1126.34345 [6] Kosov, A.A., Vassilyev, S.N., and Zherlov, A.K., Logic-based Controllers for Hybrid Systems, Int. J. Hybrid Syst., 2004, vol. 4, no. 4, pp. 271–299. [7] Marks, R.J., Gravagne, I., Davis, J.M., and DaCunha, J.J., Nonregressivity in Switched Linear Circuits and Mechanical Systems, Math. Comput. Modelling, 2006, vol. 43, pp. 1383–1392. · Zbl 1136.93379 [8] Zubov, V.I., Analiticheskaya dinamika giroskopicheskikh sistem (Analytical Dynamic of Gyroscopic Systems), Leningrad: Sudostroenie, 1970. [9] Merkin, D.R., Vvedenie v teoriyu ustoichivosti dvizheniya (Introfuction into a Theory of Motion Stability), Moscow: Nauka, 1987. [10] Pyatnitskii, E.S., The Decomposition Principle in the Control of mechanical Systems, Dokl. Akad. Nauk SSSR, 1988, vol. 300, no. 2, pp. 300–303. [11] Siljak, D.D., Decentralized Control of Complex Systems, Boston: Academic, 1991. Translated under the title Detsentralizovannoe upravlenie slozhnymi sistemami, Moscow: Mir, 1994. [12] Matrosov, V.M., Metod vektornykh funktsii Lyapunova: analiz dinamicheskikh svoistv nelineinykh sistem (A Method of Lyapunov Functions: Analysis of Dynamical Properties of Nonlinear Systems), Moscow: Fizmatlit, 2001. [13] Chernous’ko, F.L., Anan’evskii, I.M., and Reshmin, S.A., Metody upravleniya nelineinymi mekhanicheskimi sistemami (Methods for Control of Nonlinear Mechanical Systems), Moscow: Fizmatlit, 2006. [14] Zubov, V.I., A Canonical Structure of a Vector Force Field, in Problemy mekhaniki deformiruemogo tverdogo tela (Mechanical Problems of a Deformed Solid Body), Leningrad: Sudostroenie, 1970, pp. 167–170. [15] Chetaev, N.G., Ustoichivost’ dvizheniya. Raboty po analiticheskoi mekhanike (Motion Stability. Papers on Analytical Mechanics), Moscow: Akad. Nauk SSSR, 1962. [16] Krasovskii, N.N., Nekotorye zadachi teorii ustoichivosti dvizheniya (Some Problems of the Theory of Motion Stability), Moscow: Fizmatlit, 1959.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.