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State-Space Analysis on The Stability of Limit Cycle Predicted by Harmonic Balance

  • Lee, Byung-Jin (Department of Aerospace Information Engineering, Konkuk University) ;
  • Yun, Suk-Chang (Department of Aerospace Information Engineering, Konkuk University) ;
  • Kim, Chang-Joo (Department of Aerospace Information Engineering, Konkuk University) ;
  • Park, Jung-Keun (Department of Aerospace Information Engineering, Konkuk University) ;
  • Sung, Sang-Kyung (Department of Aerospace Information Engineering, Konkuk University)
  • Received : 2010.08.24
  • Accepted : 2011.06.01
  • Published : 2011.09.01

Abstract

In this paper, a closed-loop system constructed with a linear plant and nonlinearity in the feedback connection is considered to argue against its planar orbital stability. Through a state space approach, a main result that presents a sufficient stability criterion of the limit cycle predicted by solving the harmonic balance equation is given. Preliminarily, the harmonic balance of the nonlinear feedback loop is assumed to have a solution that determines the characteristics of the limit cycle. Using a state-space approach, the nonlinear loop equation is reformulated into a linear perturbed model through the introduction of a residual operator. By considering a series of transformations, such as a modified eigenstructure decomposition, periodic averaging, change of variables, and coordinate transformation, the stability of the limit cycle can be simply tested via a scalar function and matrix. Finally, the stability criterion is addressed by constructing a composite Lyapunov function of the transformed system.

Keywords

References

  1. M. Vidyasagar, Nonlinear systems analysis; 2nd Edition, Englewood Cliffs, New Jersey: Prentice-Hall Inc., 1993.
  2. D. P. Atherton, Nonlinear control Engineering, Van Nostrand Reinhold Company Ltd., London, 1975.
  3. J. M. Hidler and W.Z. Rymer, "Limit cycle behavior in spasticity: Analysis and evaluation," IEEE Transactions on Biomedical engineering, Vol. 47, No. 12, December 2000, pp.1565-1575. https://doi.org/10.1109/10.887937
  4. S. R. Sanders, "On limit cycles and the describing function method in periodically switched circuits," IEEE Transactions on Circuits and Systems, Vol. 40, No. 9, September 1993, pp. 564-572. https://doi.org/10.1109/81.244905
  5. H. Pinheiro, P. Jain, and G. Joos, "Self-sustained oscillating resonant converters operating above the resonant frequency," IEEE Transactions on Power Electronics, Vol. 14, No. 5, September 1999, pp. 803- 809. https://doi.org/10.1109/63.788476
  6. E. Kim, H. Lee and M. Park, " Limit cycle prediction of a fuzzy control system based on describing function method," IEEE Transactions on Fuzzy Systems, Vol. 8, No. 1, February 2000, pp. 11-22. https://doi.org/10.1109/91.824762
  7. A. M. Mohamed and F. P. Emad, "Nonlinear oscillations in magnetic bearing systems," IEEE Transactions on Automatic Control, Vol. 38, No. 8, August 1993, pp. 1242-1245. https://doi.org/10.1109/9.233159
  8. A. I. Mees and A. R. Bergen, "Describing functions revisited," IEEE Transactions on Automatic Control, Vol. 20, No. 4, August 1975, pp. 473-478. https://doi.org/10.1109/TAC.1975.1101025
  9. A. R. Bergen and R. L. Franks, "Justification of describing function method," SIAM Journal of Control, Vol. 9, No. 4, November 1971, pp. 568-589. https://doi.org/10.1137/0309041
  10. F. L. Swern, "Analysis of oscillation in systems with polynomial-type nonlinearities using describing functions," IEEE Transactions on Automatic Control, Vol. 28, No. 1, January 1983, pp. 31-41. https://doi.org/10.1109/TAC.1983.1103144
  11. H. K. Khalil, Nonlinear systems; 2nd edition, Upper Saddle River, New Jersey: Prentice-Hall Inc., 1996.
  12. D. Williamson, "Periodic Motion in nonlinear systems," IEEE Transactions on Automatic Control, Vol. 20, No. 4, August 1975, pp. 479-486. https://doi.org/10.1109/TAC.1975.1101026
  13. J. E. Slotine and W. Li, Applied nonlinear control, Englewood Cliffs, New Jersey: Prentice-Hall Inc., 1991.
  14. Herbert Amann, Ordinary differential equation: An introduction to nonlinear analysis, Walter de Gruyter & Co., 1990.
  15. A. I. Mees and L. O. Chua, "The Hopf bifurcation theorem and its application to nonlinear oscillations in circuits and systems," IEEE Transactions on Circuits and Systems, Vol. 26, No. 4, April 1979, pp. 235-254. https://doi.org/10.1109/TCS.1979.1084636
  16. E. S. Pyatnitskiy and L. B. Rapoport, "Criteria of asymptotic stability of differential inclusions and periodic motions of time-varying nonlinear control systems," IEEE Transactions on Circuits and Systems, Vol. 43, No. 3, March 1996, pp. 219-229. https://doi.org/10.1109/81.486446
  17. Shankar Sastry, Nonlinear systems : Analysis, Stability and Control, Springer-Verlag, 1999.
  18. M. Fiedler and V. Ptak. "On matrices with nonnegative off-diagonal elements and positive principal minors," Czech. Mathematics Journal, Vol. 12, 1962, pp. 382-400.
  19. C. C. Chung and J. Hauser, "Nonlinear H-inf control around periodic orbits," Systems and Control Letters 30, 1997, pp.127-137. https://doi.org/10.1016/S0167-6911(96)00087-4
  20. S. Lee and S. M. Meerkov, "Vibrational feedback control in the problem of absolute stability," IEEE Transactions on Automatic Control, Vol. 36, No. 4, April 1991, pp. 482-484. https://doi.org/10.1109/9.75106
  21. S. Sung, A Feedback Loop Design for MEMS Resonant Accelerometer Using a Describing Function Technique, Ph.D Thesis, Seoul National University, 2003.