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Oscillation conditions of nonlinear systems with static feedback

  • Adaptive and Robust Systems
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Abstract

Conditions are proposed of the availability of the oscillation property in the sense of Yakubovich in a system with a nonlinear nominal portion enclosed by static nonlinear output feedback.A technique is worked out to calculate analytical estimates for the amplitude of oscillations in the system.The relation between estimates for the amplitude of oscillations and the index of excitability of the system with respect to the input is established.Examples are given of the computer modeling for systems of the 2nd and the 3rd order,including Van der Pol and Lorentz systems conforming the applicability of the suggested solutions.

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REFERENCES

  1. Fradkov, A.L., Miroshnik, I.V., and Nikiforov, V.O., Nelineinoe i adaptivnoe upravlenie slozhnymi dinamicheskimi sistemami (Nonlinear and Adaptive Control of Complex Dynamic Systems), St. Petersburg: Nauka, 2000.

    Google Scholar 

  2. Jiang, Z.-P., Teel, A., and Praly, L., Small-Gain Theorem for ISS Systems and Applications, Math. Control Signal Syst., 1994, no. 7, pp. 95–120.

  3. Sepulchre, R., Jankovic, M., and Kokotivic, P.V., Constructive Nonlinear Control, New York: Springer-Verlag, 1996.

    Google Scholar 

  4. Leonov, G.A., Burkin, I.M., and Shepelyavyi, A.I., Chastotnye metody v teorii kolebanii (Frequency Methods in the Theory of Oscillations), St. Petersburg: Nauka, 1992.

    Google Scholar 

  5. Martinez, S., Cortes, J., and Bullo, F., Analysis and Design of Oscillatory Control Systems, IEEE Trans. Automat. Control, 2003, vol. 48, no. 7, pp. 1164–1177.

    Google Scholar 

  6. Fradkov, A.L., Feedback Resonance in Nonlinear Oscillators, Proc. 5th Europ. Control Conf., Karlsruhe, 1999.

  7. Fradkov, A.L., Kiberneticheskaya fizika (Cybernetic Physics), St. Petersburg: Nauka, 2003.

    Google Scholar 

  8. Yakubovich, V.A., Frequency Conditions of Self-Oscillations in Nonlinear Systems with One Stationary Nonlinearity, Sib. Mat. Zh., 1973, vol. 14, no. 12, pp. 1100–1129.

    Google Scholar 

  9. Yakubovich, V.A., Frequency Conditions of Oscillations in Nonlinear Controllable Systems with One Univalent or Hysteretic Nonlinearity, Avtom. Telemekh., 1975, no. 12, pp. 51–64.

  10. Hill, D. and Moylan, P., Dissipative Dynamics Systems. Basic Input-Output and State Properties, J. Franklin Inst. 1980, vol, 309, pp. 327–357.

    Google Scholar 

  11. Polushin, I.G., Fradkov, A.L., and Hill, D.D., Passivity and Passification of Nonlinear Systems (Review), Avtom. Telemekh., 2000, no. 3, pp. 3–37.

  12. Angeli, D., Sontag, E.D., and Wang, Y., A Characterization of Integral Input to State Stability, Syst. Control Lett., 1999, vol. 38, pp. 209–217.

    Google Scholar 

  13. Sontag, E.D. and Wang, Y., Output-to-State Stability and Detectability of Nonlinear Systems, Syst. Control Lett., 1997, vol. 29, pp. 279–290.

    Google Scholar 

  14. Sontag, E.D., Smooth Stabilization Implies Coprime Factorization, IEEE Trans. Automat. Control, 1989, vol. 34, pp. 435–443.

    Google Scholar 

  15. Sontag, E.D. and Wang, Y., Various Results Concerning Set Input-to-State Stability, Proc. IEEE CDC 95, IEEE Publications, 1995, pp. 1330–1335.

  16. Angeli, D., Input-to-State Stability of PD-Controlled Robotic Systems, Automatica, 1999, vol. 35, pp. 1285–1290.

    Google Scholar 

  17. Sontag, E.D. and Wang, Y., Notions of Input to Output Stability, Syst. Control Lett., 1999, vol. 38, pp. 235–248.

    Google Scholar 

  18. Nemytskiy, V.V., Oscillations in Multidimention Dynamical Systems, Proc. Int. Sympos. on Nonlinear Oscillations, 1963.

  19. Sontag, E.D., Asymptotic Amplitudes and Cauchy Gains. A Small Gain Principle and an Application to Inhibitory Biological Feedback, Syst. Control Lett., 2000, vol. 47, pp. 167–179.

    Google Scholar 

  20. Fradkov, A.L. and Pogromsky, A. Yu., Introduction to Oscillations and Chaos, Singapore: World Scientific, 1998.

    Google Scholar 

  21. Hayachi, C., Nonlinear Oscillations in Physical Systems, New York: McGraw-Hill, 1964.

    Google Scholar 

  22. LaSalle, J. and Lefschetz, S., Stability by Liapunov’s Direct Method, New York: Academic, 1961. Translated under the title Issledovanie ustoichivosti pryamym metodom Lyapunova, Moscow: Mir, 1964.

    Google Scholar 

  23. Arcak, M. and Teel, A., Input-to-State Stability for a Class of Lurie Systems, Automatica, 2002, vol. 38, pp. 1945–1949.

    Google Scholar 

  24. Fomin, V.N., Fradkov, A.L., and Yakybovich, V.A., Adaptivnoe upravlenie dinamicheskimi sistemami (Adaptive Control of Dynamic Systems), Moscow: Nauka, 1981.

    Google Scholar 

  25. Sontag, E.D., and Wang, Y., New Characterization of the Input to State Stability Property, IEEE Trans. Automat. Control, 1996, vol. 41, pp. 1283–1294.

    Google Scholar 

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Authors and Affiliations

  1. Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, St.Petersburg, Russia

    D. V. Efimov & A. L. Fradkov

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  1. D. V. Efimov
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  2. A. L. Fradkov
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Additional information

Translated from Avtomatika i Telemekhanika, No.2, 2005, pp.92–107.

Original Russian Text Copyright © 2005 by Efimov, Fradkov.

This work was supported by the Russian Foundation for Basic Research,project no.02-01-00765,the Fund of assistance of the native science,and Program no.19 of the Presidium of the Russian Academy of Sciences,project no.1.4.

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Efimov, D.V., Fradkov, A.L. Oscillation conditions of nonlinear systems with static feedback. Autom Remote Control 66, 249–264 (2005). https://doi.org/10.1007/s10513-005-0048-7

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  • DOI: https://doi.org/10.1007/s10513-005-0048-7

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