Abstract
In the calculation of periodic oscillations of nonlinear systems –so-called limit cycles – approximative and systematic engineeringmethods of linear system analysis are known. The techniques, working inthe frequency domain, perform a quasi-linearization of the nonlinear system,replacing nonlinearities by amplitude-dependent describing functions.Frequently, the resulting equations for the amplitude and frequency ofpresumed limit cycles are solved directly by a graphical procedure in aNyquist plane or by solving the nonlinear equations or a parameteroptimization problem. In this paper, an indirect numerical approach isdescribed which shows that, for a system of nonlinear differentialequations, the eigenvalues of the quasi-linear system simply indicateall limit cycles and, additionally, yield stability regions for thelinearized case. The method is applicable to systems with multiplenonlinearities which may be static or dynamic. It is demonstrated foran example of aircraft nose gear shimmy dynamics in the presence ofdifferent nonlinearities and the results are compared with those fromsimulation.
Similar content being viewed by others
References
Ackermann, J., Robuste Regelung, Springer-Verlag, Berlin, 1993.
Anderson, M. R., ‘Pilot-induced oscillations involving multiple nonlinearities’, Paper 97-3501, AIAA Atmospheric Flight Mechanics Conference, New Orleans, LA, 1997.
Atherton, D. P., Nonlinear Control Engineering, Van Nostrand Reinhold, London, 1975.
Föllinger, O., Regelungstechnik, Elitera-Verlag, Berlin, 1972.
Föllinger, O., Nichtlineare Regelungen Bd. 1. Grundlagen und Harmonische Balance, 2nd edn., Oldenbourg-Verlag, München, 1978.
Gelb, A. and Van der Velde, W. E., Multiple-Input Describing Functions and Nonlinear System Design, McGraw Hill, New York, 1968.
Graham, D. and McRuer, D., Analysis of Nonlinear Control Systems, Wiley, New York, 1961.
Howe, R. M., Ye, X. A., and Li, B. H., ‘An improved method for simulation of dynamic systems with discontinuous nonlinearities’, Transactions of the Society for Computer Simulation 1, 1984, 33–47.
Katebi, R., ‘Classical nonlinear control. The describing function approach’, in EURACO Network Workshop: Control of Nonlinear Systems. Theory and Application, Portugal, M. Grimble (ed.), The European Robust and Adaptive Control Network, 1996, pp. 1–14.
Lamendola, J. E., ‘Limit cycle PIO analysis with simultaneously acting multiple asymmetric saturation’, Master Thesis, Virginia Polytechnic Institute and State University, Blacksburg, VA, 1998.
Pacejka, H. B., ‘The wheel shimmy phenomenon. A theoretical and experimental investigation with particular reference to the non-linear problem’, Ph.D. Thesis, Delft University, The Netherlands, 1966.
Pillai, V. K. and Nelson, H. D., ‘A new algorithm for limit cycle analysis of nonlinear control systems’, Journal of Dynamic Systems, Measurement and Control 110, 1988, 272–277.
Popov, V. M., ‘Absolute stability of nonlinear systems of automatic control’, Automation and Remote Control 22, 1962, 857–875.
Siljak, D., Nonlinear Systems, Wiley, New York, 1969.
Smith, B. T., Boyle, J. M., Dongarra, J. J., Garbow, B. S., Ikebe, Y., Klema, V. C., and Moler, C. B., Matrix Eigensystem Routines - EISPACK Guide, 2nd edn., Springer-Verlag, Berlin, 1976.
Somieski, G., ‘Stability analysis of a simple aircraft nose landing gear model using different mathematical methods’, Aerospace Science and Technology 8, 1997, 545–555.
Stojic,M. R. and Siljak, D., ‘Sensitivity analysis of self-excited oscillations in nonlinear control systems’, in Proceedings to Sensitivity Methods in Control Theory, Dubrovnik, Yugoslavia, L. Radanovic (ed.), Pergamon Press, Oxford, 1964, pp. 209–219.
Taylor, J. H., ‘General describing function method for systems with many nonlinearities, with application to aircraft performance’, in Proceedings Joint Automatic Control Conference, San Francisco, CA, American Automatic Control Council, 1980, pp. FP9-A1–6.
Unbehauen, H., Regelungstechnik 2, 4th edn., Vieweg-Verlag, Wiesbaden, 1987.
Yeung, K. S., ‘Anwendung von Frequenzkennlinien auf Regelkreise mit zwei Nichtlinearitäten’, Regelungstechnik 12, 1977, 374–381.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Somieski, G. An Eigenvalue Method for Calculation of Stability and Limit Cycles in Nonlinear Systems. Nonlinear Dynamics 26, 3–22 (2001). https://doi.org/10.1023/A:1017384211491
Issue Date:
DOI: https://doi.org/10.1023/A:1017384211491