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On partial detectability of the nonlinear dynamic systems

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Abstract

Conditions were obtained under which the uniform stability (uniform asymptotic stability) in one part of the variables of the zero equilibrium position of the nonlinear nonstationary system of ordinary differential equations implies the uniform stability (uniform asymptotic stability) of this equilibrium position relative to another, larger part of variables. Conditions were also obtained under which the uniform stability (uniform asymptotic stability) in one part of variables of the “partial” (zero) equilibrium position of the nonlinear nonstationary system of ordinary differential equations implies the uniform stability (uniform asymptotic stability) of this equilibrium position. These conditions complement a number of the well-known results of the theory of partial stability and partial detectability of the nonlinear dynamic systems. Application of the results obtained to the problems of partial stabilization of the nonlinear control systems was considered.

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References

  1. Lyapunov, A.M., Studying a Special Case of the Motion Stability Problem, in Sobr. soch. vol. 2 (Collected Papers), Moscow: Akad. Nauk SSSR, 1956, pp. 272–331.

    Google Scholar 

  2. Rumyantsev, V.V., On Stability of Motion in Part of Variables, Vestn. Mosk. Gos. Univ., Ser. Mat. Mekh. Fiz. Astron. Khim., 1957, no. 4, pp. 9–16.

  3. Rumyantsev, V.V. and Oziraner, A.S., Ustoichivost’ i stabilizatsiya dvizheniya po otnosheniyu k chasti peremennykh (Stability and Stabilization of Motion in Part of Variables), Moscow: Nauka, 1987.

    Google Scholar 

  4. Vorotnikov, V.I., Partial Stability and Control, Boston: Birkhauser, 1998.

    MATH  Google Scholar 

  5. Vorotnikov, V.I. and Rumyantsev, V.V., Ustoichivost’ i upravlenie po chasti koordinat fazovogo vektora dinamicheskikh sister: teoriya, metody i prilozheniya (Stability and Control in Part of Coordinates of the Phase Vector of Dynamic Systems: Theory, Methods, Applications), Moscow: Nauch. mir, 2001.

    Google Scholar 

  6. Vorotnikov, V.I., Partial Stability and Control: State-of-the-art and Outlooks, Avtom. Telemekh., 2005, no. 4, pp. 3–59.

  7. Sontag, E.D. and Wang, Y., Output-to-state Stability and Detectability of Nonlinear Systems, Syst. & Control Lett., 1997, vol. 29, no. 5, pp. 279–290.

    Article  MATH  MathSciNet  Google Scholar 

  8. Sontag, E.D., Mathematical Control Theory: Deterministic Finite Dimensional Systems, New York: Springer, 1998.

    MATH  Google Scholar 

  9. Sontag, E.D., Input to State Stability: Basic Concepts and Results, in Nonlinear Optim. Control Theory, Berlin: Springer, 2007, pp. 163–220.

    Google Scholar 

  10. Shiriaev, A.S. and Fradkov, A.L., Stabilization of Invariant Sets for Nonlinear Non-affine Systems, Automatica, 2000, vol. 36, no. 11, pp. 1709–1715.

    Article  MATH  MathSciNet  Google Scholar 

  11. Shiriaev, A.S., The Notion of V-detectability and Stabilization of Invariant Sets of Nonlinear Systems, Syst. & Control Lett., 2000, vol. 39, no. 5, pp. 327–338.

    Article  MATH  MathSciNet  Google Scholar 

  12. Ingalls, B.P., Sontag, E.D., and Wang, Y., Measurement to Error Stability: A Notion of Partial Detectability for Nonlinear Systems, in Proc. 2002 IEEE Conf. Decision Control, Las Vegas, 2002, pp. 3946–3951.

  13. Peiffer, K. and Rouche, N., Liapounov’s Second Method Applied to Partial Stability, J. Mecanique, 1969, vol. 8, no. 2, pp. 323–334.

    MATH  MathSciNet  Google Scholar 

  14. Khapaev, M.M., Averaging in Stability Theory, Dordrecht: Kluwer, 1993.

    MATH  Google Scholar 

  15. Fradkov, A.L., Miroshnik, I.V., and Nikiforov, V.O., Nonlinear and Adaptive Control of Complex Systems, Dordrecht: Kluwer, 1999.

    MATH  Google Scholar 

  16. Vorotnikov, V.I., Two Classes of Prblems of Partial Stability: On Unification of Notions and Unique Solvability Conditions, Dokl. Ross. Akad. Nauk, 2002, vol. 384, no. 1, pp. 47–51.

    MathSciNet  Google Scholar 

  17. Vorotnikov, V.I., On Stability and Stability in Part of Variables of the Partial Equilibrium Positions of the Nonlinear Dynamic Systems, Dokl. Ross. Akad. Nauk, 2003, vol. 389, no. 3, pp. 332–337.

    MathSciNet  Google Scholar 

  18. Chellaboina, V. and Haddad, V.M., A Unification between Partial Stability and Stability Theory for Time-varying Systems, IEEE Control Syst. Magazine, 2002, vol. 22, no. 6, pp. 66–75. (Erratum: IEEE Control Systems Magazine, 2003, vol. 23, no. 1, pp. 103.)

    Article  MathSciNet  Google Scholar 

  19. Lin, Y., Sontag, E.D., and Wang, Y., A Smooth Converse Lyapunov Theorem for Robust Stability, SIAM J. Control Optim., 1996, vol. 34, no. 1, pp. 124–160.

    Article  MATH  MathSciNet  Google Scholar 

  20. Teel, A. and Praly, L., A Smooth Lyapunov Function from a Class Kl-Estimate Involving Two Positive Semidefinite Functions, ESAIM: Control, Optim. Calculus of Variat., 2000, vol. 5, pp. 313–367.

    Article  MATH  MathSciNet  Google Scholar 

  21. Efimov, D.V., Robastnoe i adaptivnoe upravlenie nelineinymi kolebaniyami (Robust and Adaptive Control of Nonlinear Oscillations), St. Petersburg: Nauka, 2005.

    Google Scholar 

  22. Corduneanu, C., Sur la stabilite partielle, Rev. Roum. Math. Pure Appl., 1964, vol. 9, no. 3, pp. 229–236.

    MATH  MathSciNet  Google Scholar 

  23. Savchenko, A.Ya. and Ignat’ev, A.O., Nekotorye zadachi ustoichivosti neavtonomnykh sister (Some Stability Problems of Nonautonomous Systems), Kiev: Naukova Dumka, 1989.

    Google Scholar 

  24. Hatvani, L., On the Stability of the Solutions of Ordinary Differential Equations with Mechanical Applications, Alkalm. Mat. Lap., 1990/1991, vol. 15, no. 1/2, pp. 1–90.

    MathSciNet  Google Scholar 

  25. Andreev, A.S., Study of Partial Asymptotic Stability, Prikl. Mat. Mekh., 1991, vol. 54, no. 4, pp. 539–547.

    Google Scholar 

  26. Barbashin, E.A. and Tabueva, V.A., Dinamicheskie sistemy s tsilindricheskim fazovym prostranstvom (Dynamic Systems with Cylindrical Phase Space), Moscow: Nauka, 1969.

    Google Scholar 

  27. Halanay, A., Differential Equations: Stability, Oscillations, Time Lags, New York: Academic, 1966.

    MATH  Google Scholar 

  28. Oziraner, A.S., On One Malkin-Masser Theorem, Prikl. Mat. Mekh., 1979, vol. 43, no. 6, pp. 975–979.

    MathSciNet  Google Scholar 

  29. Vorotnikov, V.I., Ustoichivost’ dinamicheskikh sister po otnosheniyu k chasti peremennykh (Stability of Dynamic Systems in Part of Variables), Moscow: Nauka, 1991.

    Google Scholar 

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Authors and Affiliations

  1. Nizhni Tagil Technological Institute, Ural State Technical University, Nizhni Tagil, Russia

    V. I. Vorotnikov & Yu. G. Martyshenko

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  1. V. I. Vorotnikov
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  2. Yu. G. Martyshenko
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Original Russian Text © V.I. Vorotnikov, Yu.G. Martyshenko, 2009, published in Avtomatika i Telemekhanika, 2009, No. 1, pp. 25–38.

This work was supported by the Russian Foundation for Basic Research, project no. 07-01-00483, and the Ministry of Education and Science, project no. 1.2755.07.

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Vorotnikov, V.I., Martyshenko, Y.G. On partial detectability of the nonlinear dynamic systems. Autom Remote Control 70, 20–32 (2009). https://doi.org/10.1134/S0005117909010020

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