0
Research Papers

Alternative Criterion for Investigation of Pitchfork Bifurcations of Limit Cycle in Relay Feedback Systems

[+] Author and Article Information
Huidong Xu

State Key Laboratory of Advanced Design
and Manufacture for Vehicle Body,
Hunan University,
Changsha, Hunan 410082, China;
Key Laboratory of Advanced Design and
Simulation Techniques for Special Equipment,
Ministry of Education,
Hunan University,
Changsha, Hunan 410082, China;
College of Mechanical and Vehicle Engineering,
Hunan University,
Changsha, Hunan 410082, China

Guilin Wen

e-mail: glwen@hnu.edu.cn
State Key Laboratory of Advanced Design
and Manufacture for Vehicle Body,
Hunan University,
Changsha, Hunan 410082, China;
Key Laboratory of Advanced Design and
Simulation Techniques for Special Equipment,
Ministry of Education,
Hunan University,
Changsha, Hunan 410082, China;
College of Mechanical and Vehicle Engineering,
Hunan University,
Changsha, Hunan 410082, China

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received March 8, 2013; final manuscript received October 16, 2013; published online February 13, 2014. Assoc. Editor: Eric A. Butcher.

J. Comput. Nonlinear Dynam 9(3), 031004 (Feb 13, 2014) (7 pages) Paper No: CND-13-1061; doi: 10.1115/1.4025744 History: Received March 08, 2013; Revised October 16, 2013

Relay feedback systems are strongly nonlinear due to their switching properties. Some nonlinear properties of relay feedback systems have been verified to be preferable to modern control engineering, whereas others might drive the system to be more complex or even unpredictable. An alternative criterion is proposed to investigate the pitchfork bifurcations of the limit cycle of relay feedback systems in this paper. The proposed critical criterion is explicitly formulated by the coefficients of the characteristic polynomial equation instead of the eigenvalues of the Jacobian matrix. It is more convenient and efficient for detecting the existence of this type of bifurcation than the classical critical criterion. Numerical simulations show the pitchfork bifurcation behaviors in relay feedback systems and demonstrate that the proposed criterion is a general and exact analytic method for determining pitchfork bifurcations in maps.

FIGURES IN THIS ARTICLE
<>
Copyright © 2014 by ASME
Your Session has timed out. Please sign back in to continue.

References

Tsypkin, Y. Z., 1984, Relay Control Systems, Cambridge University Press, Cambridge, UK.
Schuck, O. H., 1959, “Honeywell's History and Philosophy in the Adaptive Control Field,” Proceedings of the Self Adaptive Flight Control Symposium, P. C.Gregory, ed., Wright- Patterson AFB, Ohio.
Åström, K. J., and Hägglund, T., 1995, PID Controllers: Theory, Design and Tuning, second ed., Instrument Society of America, Research Triangle Park, NC.
Yu, C. C., 1999, Autotuning of PID Controllers: A Relay Feedback Approach, Springer-Verlag, Berlin.
Wang, Q. G., Lee, T. H., and Lin, C., 2003, Relay Feedback: Analysis, Identification and Control, Springer-Verlag, London.
Palmor, Z. J., Halevi, Y., and Efrati, T., 1995, “A General and Exact Method for Determining Limit Cycles in Decentralized Relay Systems,” Automatica, 31, pp. 1333–1339. [CrossRef]
Varigonda, S., and Georgiou, T. T., 2001, “Dynamics of Relay Relaxation Oscillators,” IEEE Trans. Autom. Control, 46, pp. 65–77. [CrossRef]
Lin, C., Wang, Q. G., Lee, T. H., and Lam, J., 2002, “Local Stability of Limit Cycles for Time-Delay Relay-Feedback Systems,” IEEE Trans. Circuits Syst., I: Fundam. Theory Appl., 49, pp. 1870–1875. [CrossRef]
Goncalves, J. M., Megretski, A., and Dahleh, M. A., 2001, “Global Stability of Relay Feedback Systems,” IEEE Trans. Autom. Control, 45, pp. 550–562. [CrossRef]
Kowalczyk, P., and di Bernardo, M., 2001, “On a Novel Class of Bifurcations in Hybrid Dynamical Systems—The Case of Relay Feedback Systems,” Proceedings of the Hybrid Systems: Computation and Control, Springer-Verlag, Berlin.
Wen, G. L., Wang, Q. G., and Lee, T. H., 2007, “Quasi-Period Oscillations of Relay Feedback Systems,” Chaos, Solitons Fractals, 34, pp. 405–411. [CrossRef]
Amrani, D., and Atherthon, D. P., 1989, “Designing Autonomous Relay Systems With Chaotic Motion,” Proceedings 28th IEEE Conference on Decision and Control, Tampa, FL, pp. 932–936.
Cook, P. A., 1985, “Simple Feedback Systems With Chaotic Behaviour,” Syst. Control Lett., 6, pp. 223–227. [CrossRef]
Genesio, R., and Tesi, A., 1990, “Chaos Prediction in a Third-Order Relay System,” Dipartimento di Sistemi ed Informatica, University of Florence, Italy, Internal Report No. RT 29/90.
Kuznetsov, Y. A., 1998, Elements of Applied Bifurcation Theory, second ed., Springer-Verlag, New York.
Perko, L., 2001, Differential Equations and Dynamical Systems (Texts in Applied Mathematics), third ed., Springer-Verlag, New York.
Guckenheimer, J., and Holmes, P., 1986, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, New York.
Champanerkar, J., and Blackmore, D., 2007, “Pitchfork Bifurcations of Invariant Manifolds,” Topol. Appl, 154, pp. 1650–1663. [CrossRef]
Xu, K. D., 1995, “Stochastic Pitchfork Bifurcation: Numerical Simulations and Symbolic Calculations Using MAPLE,” Math. Comput. Simul., 38, pp. 199–209. [CrossRef]
Varela, S., Masoller, C., and Sicardi, A. C., 2000, “Numerical Simulations of the Effect of Noise on a Delayed Pitchfork Bifurcation,” Physica A, 283, pp. 228–232. [CrossRef]
Lasalle, J. P., 1986, The Stability and Control of Discrete Processes, Springer, Berlin.
Brown, B. M., 1965, “On the Distribution of the Zeros of a Polynomial,” Q. J. Math., 16, pp. 241–256. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

The evolvement of the trajectory of systems (1) and the Poincaré map

Grahic Jump Location
Fig. 2

Bifurcation diagram of the fixed points of map (9): (a) Δx1 = 0.00005, (b) Δx1 = -0.00005, and (c) Δx1 = ±0.00005

Grahic Jump Location
Fig. 3

Symmetric periodic motion of systems (1) before the pitchfork bifurcation (v = -1.14, Δx1 = ±0.00005)

Grahic Jump Location
Fig. 4

The asymmetric fixed points bifurcating from the symmetric fixed point (a) v = -1.13, Δx1 = 0.00005; (b) v = -1.13, Δx1 = -0.00005; and (c) v = -1.13, Δx1 = ±0.00005

Grahic Jump Location
Fig. 5

Asymmetric stable periodic motions of systems (1) after the pitchfork bifurcation: (a1) v = -1.13, Δx1 = 0.00005; (b1) v = -1.13, Δx1 = 0.00005; (a2) v = -1.13, Δx1 = -0.00005; (b2) v = -1.13, Δx1 = -0.00005; (a3) v = -1.13, Δx1 = ±0.00005; and (b3) v = -1.13, Δx1 = ±0.00005

Grahic Jump Location
Fig. 6

Chaos motion of systems (1) after the pitchfork bifurcation: v = -1.07

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In