Advertisement

Criterion of Stability of a Linear System with One Harmonic Time-Varying Coefficient Based on a Formalized Filter Hypothesis

  • Anton MandrikEmail author
Conference paper
Part of the Lecture Notes in Networks and Systems book series (LNNS, volume 95)

Abstract

Stability criterion for a linear time-varying (LTV) system with one harmonic time-varying coefficient in feedback is suggested. The found criterion is based on the hypothesis that the linear time-invariant (LTI) part of the system is a low-frequency filter. The criterion is simple and suitable for calculation of stability borders for LTV systems. The suggested criterion is compared with a numerical experiment, Bonjiorno criterion, stationarization method.

Keywords

Stability criterion Linear time-varying system Stability borders Numerical experiment Bonjiorno criterion Stationarization method 

References

  1. 1.
    Taft, V.A.: On the analysis of stability of periodic modes of operation in non-linear automatic control systems. J. Avtomat. i telemek 9 (1959)Google Scholar
  2. 2.
    Hartman, P.: Ordinary Differential Equations. Wiley, New York (1964)zbMATHGoogle Scholar
  3. 3.
    Taft, V.A.: Electrical Circuits with Variable Parameters including Pulsed Control Systems. Pergamon, Oxford (translation from Russian) (1964)Google Scholar
  4. 4.
    Shamma, J.S., Athans, M.: Gain scheduling: potential hazards and possible remedies. IEEE Control Syst. Mag. 12, 101–107 (1992)CrossRefGoogle Scholar
  5. 5.
    Chechurin, S.L., Chechurin, L.S.: Elements of physical oscillation and control theory. In: Proceedings of the IEEE International Conference Physics and Control, St. Petersburg, vol. 2, pp. 589–594 (2003)Google Scholar
  6. 6.
    Bruzelius, F.: Linear Parameter-Varying Systems: An Approach to Gain Scheduling. Chalmers Univ. Technol. (2004)Google Scholar
  7. 7.
    Insperger, T., Stépán, G.: Optimization of digital control with delay by periodic variation of the gain parameters. In: Proceedings of IFAC Workshop on Adaptation and Learning in Control and Signal Processing, and IFAC Workshop on Periodic Control Systems, Yokohama, Japan, pp. 145–150 (2004)Google Scholar
  8. 8.
    Allwright, J.C., Alessandro, A., Wong, H.P.: A note on asymptotic stabilization of linear systems by periodic, piecewise constant, output feedback. Automatica 41, 339–344 (2005)Google Scholar
  9. 9.
    Szyszkowski, W., Stilling, D.S.D.: On damping properties of a frictionless physical pendulum with a moving mass. Int. J. Non Linear Mech. 40(5), 669–681 (2005)ADSCrossRefGoogle Scholar
  10. 10.
    Baños, A., Barreiro, A., Beker, O.: Stability analysis of reset control systems with reset band. IFAC Proc. Vol. 42(17), 180–185 (2009)CrossRefGoogle Scholar
  11. 11.
    Mailybaev, A.A., Seyranian, A.P.: Stabilization of statically unstable systems by parametric excitation. J. Sound Vib. 323, 1016–1031 (2009)ADSCrossRefGoogle Scholar
  12. 12.
    Eissa, M., Kamel, M., El-Sayed, A.T.: Vibration reduction of a nonlinear spring pendulum under multi external and parametric excitations via a longitudinal absorber. Mechanica 46, 325–340 (2011)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Arkhipova, I.M., Luongo, A., Seyranian, A.P.: Vibrational stabilization of the upright statically unstable position of a double pendulum. J. Sound Vib. 331(2), 457–469 (2012)ADSCrossRefGoogle Scholar
  14. 14.
    Abe, A.: Non-linear control technique of a pendulum via cable length manipulation: application of particle swarm optimization to controller design. FME Trans. 41(4), 265–270 (2013)Google Scholar
  15. 15.
    Amer, Y.A., Ahmed, E.E.: Vibration control of a nonlinear dynamical system with time varying stiffness subjected to multi external forces. Int. J. Eng. Appl. Sci. (IJEAS) 5(4), 50–64 (2014)Google Scholar
  16. 16.
    Arkhipova, I.M., Luongo, A.: Stabilization via parametric excitation of multi-dof statically unstable systems. Commun. Nonlinear Sci. Numer. Simul. 19(10), 3913–3926 (2014)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Mandrik, A.V., Chechurin, L.S., Chechurin, S.L.: Frequency analysis of parametrically controlled oscillating systems. In: Proceedings of the 1st IFAC Conference on Modelling, Identification and Control of Nonlinear Systems (MICNON 2015), St. Petersburg, Russia, 24–26 June 2015. IFAC-PapersOnLine, 48(11), 651–655 (2015)Google Scholar
  18. 18.
    Reguera, F., Dotti, F.E., Machado, S.P.: Rotation control of a parametrically excited pendulum by adjusting its length. Mech. Res. Commun. 72, 74–80 (2016)Google Scholar
  19. 19.
     Scapolan, M., Tehrani, M.G., Bonisoli, E.: Energy harvesting using a parametric resonant system due to time-varying damping. Mech. Syst. Signal Process., 1–17 (2016)Google Scholar
  20. 20.
    Mandrik, A.V.: Estimation of stability of oscillations of linear time-varying systems with one time-varying parameter with calculation of influence of higher frequency motions. Autom. Control. Comput. Sci. 51(3), 141–148 (2017)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Peter the Great St. Petersburg Polytechnic UniversitySaint PetersburgRussia

Personalised recommendations