Computation of multiple limit cycles in nonlinear control systems - a describing function approach

Brito, Alexandro Garro

doi:10.5028/jatm.2011.03017010

Abstract

Limit cycles play an important role in nonlinear systems, provided that many control loops with common nonlinearities like relay, hysteresis, and saturation can present them. Thus, a proper description of this nonlinear phenomenon is highly desirable. A strategy for the linearized analysis is the describing function method, which is a frequency domain approach that allows the limit cycle prediction and stability analysis. Some papers had discussed the method for the simplified analysis; however, they are concentrated in the prediction of only one limit cycle even for systems with multiple conditions. This paper proposes a systematic way of multiple limit cycle determination, as well as the stability analysis of each one. All theoretical/computational issues involved in the approach are also discussed.

Keywords:
Limit cycle; Multimodal optimization; Describing functions

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REFERENCES

Darwen, P., Yao, X., 1996, "Every niching method has its niche: fitness sharing and implicit sharing compared", Lecture Notes in Computer Science, Vol. 1141, pp. 398-407.
Dotson, K.W., Baker, R.L. and Sako, B.H., 2002, "Limit-cycle oscillation induced by nonlinear aerodynamic forces", American Institute of Aeronautics and Astronautics Journal, Vol. 40, No. 11, pp. 2197-2205.
Freedman, H. I., 1990, "A model of predator-prey dynamics as modified by the action of a parasite", Mathematical Biosciences, Vol. 99, No. 2, pp. 143-155.
Gelb, A., Velde, W., 1968, "Multiple-input describing functions and nonlinear system design", McGraw Hill, New York, USA..
Giacomini, H., Neukirch, S., 1997, "Number of limit cycles of the li´enard equation", Physical Review E, Vol. 56, No. 4, pp. 3809-3813.
Gibson, J.E., 1964, "Nonlinear automatic control", McGraw Hill, New York, USA.
Hofbauer, J., So, J.W., 1994, "Multiple limit cycle for three dimensional lotka-volterra equations", Applied Mathmatics Letters, Vol. 7, No. 6, pp. 65-70.
Kienitz, K., 2005, "On the implementation of the eigenvalue method for limit cycle determination in nonlinear systems", Nonlinear Systems, Vol, 45, pp. 25-30.
Leite Filho, W., Bueno, M., 2003, "Analysis of limit-cycle phenomenon caused by actuator's nonlinearity", In Proceedings of the 20th Inter. Conf. Appl. Simul. and Modeling.
Li, J., Tian, Y., Zhang, W. and Miao, S., 2008, "Bifurcation of multiple limit cycles for a rotor-active magnetic bearings system with time-varying stiffness", International Journal of Bifurcation and Chaos, Vol. 18, No. 3, pp. 755-778.
Mendelowitz, L., Verdugo, A. and Rand, R., 2009, "Dynamics of three coupled limit cycle oscillators with application to artificial intelligence", Communications in Nonlinear Science and Numerical Simulation, Vol. 14, No. 1, pp. 270-283.
Nayfeh, A.H.,Mook, D., 1979, "Nonlinear oscillations", Wiley, New York, USA.
Newman, B., 1995, "Dynamics and control of limit cycling motions in boosting rockets", Journal of Guidance, Control and Dynamics, Vol. 18, No. 2, pp. 280-286.
Siljak, D., 1969, "Nonlinear systems - the parameter analysis and design", John Willey & Sons, New York, USA.
Slotine, J.J., Li, W., 1991, "Applied nonlinear control", Prentice Hall, New York, USA.
Somieski, G., 2001, "An eigenvalue method for calculation of stability and limit cycles in nonlinear systems", Nonlinear Dynamics, Vol. 26, pp. 3-22.
Stout, P., Snell, S., 2000, "Multiple on-off valve control for a launch vehicle tank pressurization system", Journal of Guidance, Control and Dynamics, Vol. 23, No. 4, pp. 611-619.
Yu, P., Corless, R., 2009, "Symbolic computation of limit cycles associated with hilbert's 16th problem", Communications in Nonlinear Science and Numerical Simulation, Vol. 14, No. 12, pp. 4041-4056.

Publication Dates

Publication in this collection
Jan-Apr 2011

History

Received
21 May 2010
Accepted
16 Aug 2010

This is an Open Access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

[1] Darwen, P., Yao, X., 1996, "Every niching method has its niche: fitness sharing and implicit sharing compared", Lecture Notes in Computer Science, Vol. 1141, pp. 398-407.

[2] Dotson, K.W., Baker, R.L. and Sako, B.H., 2002, "Limit-cycle oscillation induced by nonlinear aerodynamic forces", American Institute of Aeronautics and Astronautics Journal, Vol. 40, No. 11, pp. 2197-2205.

[3] Freedman, H. I., 1990, "A model of predator-prey dynamics as modified by the action of a parasite", Mathematical Biosciences, Vol. 99, No. 2, pp. 143-155.

[4] Gelb, A., Velde, W., 1968, "Multiple-input describing functions and nonlinear system design", McGraw Hill, New York, USA..

[5] Giacomini, H., Neukirch, S., 1997, "Number of limit cycles of the li´enard equation", Physical Review E, Vol. 56, No. 4, pp. 3809-3813.

[6] Gibson, J.E., 1964, "Nonlinear automatic control", McGraw Hill, New York, USA.

[7] Hofbauer, J., So, J.W., 1994, "Multiple limit cycle for three dimensional lotka-volterra equations", Applied Mathmatics Letters, Vol. 7, No. 6, pp. 65-70.

[8] Kienitz, K., 2005, "On the implementation of the eigenvalue method for limit cycle determination in nonlinear systems", Nonlinear Systems, Vol, 45, pp. 25-30.

[9] Leite Filho, W., Bueno, M., 2003, "Analysis of limit-cycle phenomenon caused by actuator's nonlinearity", In Proceedings of the 20th Inter. Conf. Appl. Simul. and Modeling.

[10] Li, J., Tian, Y., Zhang, W. and Miao, S., 2008, "Bifurcation of multiple limit cycles for a rotor-active magnetic bearings system with time-varying stiffness", International Journal of Bifurcation and Chaos, Vol. 18, No. 3, pp. 755-778.

[11] Mendelowitz, L., Verdugo, A. and Rand, R., 2009, "Dynamics of three coupled limit cycle oscillators with application to artificial intelligence", Communications in Nonlinear Science and Numerical Simulation, Vol. 14, No. 1, pp. 270-283.

[12] Nayfeh, A.H.,Mook, D., 1979, "Nonlinear oscillations", Wiley, New York, USA.

[13] Newman, B., 1995, "Dynamics and control of limit cycling motions in boosting rockets", Journal of Guidance, Control and Dynamics, Vol. 18, No. 2, pp. 280-286.

[14] Siljak, D., 1969, "Nonlinear systems - the parameter analysis and design", John Willey & Sons, New York, USA.

[15] Slotine, J.J., Li, W., 1991, "Applied nonlinear control", Prentice Hall, New York, USA.

[16] Somieski, G., 2001, "An eigenvalue method for calculation of stability and limit cycles in nonlinear systems", Nonlinear Dynamics, Vol. 26, pp. 3-22.

[17] Stout, P., Snell, S., 2000, "Multiple on-off valve control for a launch vehicle tank pressurization system", Journal of Guidance, Control and Dynamics, Vol. 23, No. 4, pp. 611-619.

[18] Yu, P., Corless, R., 2009, "Symbolic computation of limit cycles associated with hilbert's 16th problem", Communications in Nonlinear Science and Numerical Simulation, Vol. 14, No. 12, pp. 4041-4056.

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