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On Asymptotic Stability of Homogeneous Singular Systems with Switching

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Abstract

This paper considers the stability of singular switched systems with homogeneous functions in the right-hand sides of the corresponding equations. Stability conditions of the complete system with an arbitrary switching mode are established through its decomposition into the isolated subsystems of fast and slow motions. A stabilizing feedback control design method based on measurements of the fast variables only is suggested.

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References

  1. Tikhonov, A.N., On Dependence of Solutions of Differential Equations on Small Parameter, Mat. Sb., 1948, vol. 22 (64), no. 2, pp. 193–204.

    MathSciNet  Google Scholar 

  2. Zhang, Y., Naidu, D.S., Cai, C.X., and Zou, Y., Singular Perturbations and Time Scales in Control Theories and Applications: An Overview 2002–2012, Int. J. Inform. Syst. Sci., 2014, vol. 9, no. 1, pp. 1–36.

    Google Scholar 

  3. Klimushev, A.I. and Krasovskii, N.N., Uniform Asymptotic Stability of Systems of Differential Equations with Small Parameter at Derivatives, Prikl. Mat. Mekh., 1961, vol. 25, no. 4, pp. 680–690.

    Google Scholar 

  4. Yang, C., Zhang, Q., and Zhou, L., Stability Analysis and Design for Nonlinear Singular Systems, Lecture Notes in Control and Information Sciences, Berlin: Springer-Verlag, 2013.

    Google Scholar 

  5. Lobry, C. and Sari, T., Singular Perturbation Methods in Control Theory, in Control lineaire appl. Travaux en cours, 2005, vol. 64, Hermann, Paris, pp. 151–177.

  6. Shorten, R., Wirth, F., Mason, O., Wulf, K., and King, C., Stability Criteria for Switched and Hybrid Systems, SIAM Rev., 2007, vol. 49, no. 4, pp. 545–592.

    Article  MathSciNet  MATH  Google Scholar 

  7. Shpilevaya, O.Ya. and Kotov, K.Yu., Switched Systems: Stability and Design (Review), Avtometriya, 2008, vol. 44, no. 5, pp. 71–87.

    Google Scholar 

  8. Hai, L. 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.

    Article  MathSciNet  MATH  Google Scholar 

  9. Unsolved Problems in Mathematical Systems and Control Theory, Blondel, V.D. and Megretski, A., Eds., Oxford: Princeton Univ. Press, 2004.

  10. Liberzon, D., Switching in Systems and Control, Boston: Birkhauser, 2003.

    Book  MATH  Google Scholar 

  11. Pakshin, P.V. and Pozdyayev, V.V., An Existence Criterion of a Common Lyapunov Function for a Set of Linear Second-Order Systems, Izv. Ross. Akad. Nauk, Teor. Sist. Uprvalen., 2005, no. 4, pp. 22–27.

    Google Scholar 

  12. Vassilyev, S.N. and Kosov, A.A., Analysis of Hybrid Systems Dynamics Using the Common Lyapunov Functions and Multiple Homomorphisms, Autom. Remote Control, 2011, vol. 72, no. 6, pp. 1163–1183.

    Article  MathSciNet  MATH  Google Scholar 

  13. Aleksandrov, A.Yu., Kosov, A.A., and Chen, Y., Stability and Stabilization of Mechanical Systems with Switching, Autom. Remote Control, 2011, vol. 72, no. 6, pp. 1143–1154.

    Article  MathSciNet  MATH  Google Scholar 

  14. Aleksandrov, A.Yu., Kosov, A.A., and Platonov, A.V., On the Asymptotic Stability of Switched Homogeneous Systems, Syst. Control Lett., 2012, vol. 61, no. 1, pp. 127–133.

    Article  MathSciNet  MATH  Google Scholar 

  15. Merkin, D.R., Vvedenie v teoriyu ustoichivosti dvizheniya (Introduction to the Theory of Motion Stability), Moscow: Nauka, 1987.

    MATH  Google Scholar 

  16. Barbashin, E.A., Funktsii Lyapunova (Lyapunov Functions), Moscow: Nauka, 1970.

    MATH  Google Scholar 

  17. Aleksandrov, A.Yu., Ustoichivost’ dvizhenii neavtonomnykh dinamicheskikh sistem (Stability of Motions of Nonautonomous Dynamic Systems), St. Petersburg: S.-Peterburg. Gos. Univ., 2004.

    Google Scholar 

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Correspondence to A. A. Kosov or M. V. Kozlov.

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Russian Text © A.A. Kosov, M.V. Kozlov, 2019, published in Avtomatika i Telemekhanika, 2019, No. 3, pp. 45–54.

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Kosov, A.A., Kozlov, M.V. On Asymptotic Stability of Homogeneous Singular Systems with Switching. Autom Remote Control 80, 429–436 (2019). https://doi.org/10.1134/S0005117919030032

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  • DOI: https://doi.org/10.1134/S0005117919030032

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