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Bifurcation control of bounded noise excited Duffing oscillator by a weakly fractional-order \(\varvec{PI}^{\varvec{\lambda }} \varvec{D}^{\varvec{\mu }}\) feedback controller

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Abstract

An innovative bifurcation control method by using a weakly fractional-order \(PI^{\lambda } D^{\mu }\) feedback controller is proposed to eliminate the stochastic jump in the forced response for a bounded noise excited Duffing oscillator. The averaged Itô equations of amplitude modulation and phase difference are derived via stochastic averaging method, from which the reduced Fokker–Planck–Kolmogorov equation is established and solved numerically to obtain the stationary probability density of amplitude. An efficient scheme with high accuracy for simulating the fractional integral and fractional derivative is then explored. By examining the stationary probability density of amplitude of the uncontrolled and controlled systems, the fractional-order \(PI^{\lambda }D^{\mu }\) feedback controller has been demonstrated capable of eliminating the stochastic jump and alleviating the amplitude peak of primary resonance effectively, particularly for the case that the integer-order PID controller fails to perform and even leads to the unstable response. Moreover, analytical results show generally agreement with the results obtained by the proposed simulation scheme.

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References

  1. Colonius, F., Grne, L.: Dynamics, Bifurcations, and Control. Springer, New York (2002)

    Book  Google Scholar 

  2. Chen, G.: Bifurcation Control: Theory and Applications. Springer, New York (2003)

    Book  MATH  Google Scholar 

  3. Chen, G., Moiola, J.L., Wang, H.O.: Bifurcation control: theories, methods and applications. Int. J. Bifurc. Chaos 10, 511–548 (2000)

    MathSciNet  MATH  Google Scholar 

  4. Yabuno, H.: Bifurcation control of parametrically excited Duffing system by a combined linear-plus-nonlinear feedback control. Nonlinear Dyn. 12, 263–274 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ji, J.C.: Local bifurcation control of a forced single-degree-of-freedom nonlinear system: saddle-node bifurcation. Nonlinear Dyn. 25, 369–382 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Basso, M., Evangelisti, A., Genesio, R., Tesi, A.: On bifurcation control in time delay feedback systems. Int. J. Bifurc. Chaos 08, 713–721 (1998)

    Article  MATH  Google Scholar 

  7. Wang, Z.H., Hu, H.Y.: Hopf bifurcation control of delayed systems with weak nonlinearity via delayed state feedback. Int. J. Bifurc. Chaos 15, 1787–1799 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Maccari, A.: Response control for the externally excited van der Pol oscillator with non-local feedback. J. Sound Vib. 331, 987–995 (2000)

    Article  Google Scholar 

  9. Just, W., Fiedler, B., Georgi, M., Flunkert, V., Hövel, P., Schöll, E.: Beyond the odd number limitation: a bifurcation analysis of time-delayed feedback control. Phys. Rev. E 76, 026210 (2007)

    Article  MathSciNet  Google Scholar 

  10. Leung, A.Y.T., Yang, H.X., Zhu, P.: Steady state bifurcation of a periodically excited system under delayed feedback control. Commun. Nonlinear Sci. Numer. Simul. 19, 1142–1155 (2014)

    Article  MathSciNet  Google Scholar 

  11. Tesi, A., Abed, E.H., Genesio, R., Wang, H.O.: Harmonic balance analysis of period-doubling bifurcations with implications for control of nonlinear dynamics. Automatica 32, 1255–1271 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  12. Berns, D.W., Moiola, J.L., Chen, G.R.: Feedback control of limit cycle amplitude from a frequency domain approach. Automatica 34(12), 1567–1573 (1998)

    Article  MATH  Google Scholar 

  13. Leung, A.Y.T., Yang, H.X., Zhu, P.: Bifurcation of a Duffing oscillator having nonlinear fractional derivative feedback. Int. J. Bifurc. Chaos 24, 1450028 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Zhu, W.Q., Wu, Y.J.: Optimal bounded control of harmonically and stochastically excited strongly nonlinear oscillators. Probab. Eng. Mech. 20, 1–9 (2005)

    Article  MathSciNet  Google Scholar 

  15. Zhu, W.Q., Huang, Z.L., Ko, J.M., Ni, Y.Q.: Optimal feedback control of strongly non-linear systems excited by bounded noise. J. Sound Vib. 274, 701–724 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. Balanov, A.G., Janson, N.B., Schöll, E.: Control of noise-induced oscillation by delayed feedback. Phys. D Nonlinear Phenom. 199, 1–12 (2004)

    Article  MATH  Google Scholar 

  17. Feng, C.S., Wu, Y.J., Zhu, W.Q.: Response of Duffing system with delayed feedback control under combined harmonic and real noise excitations. Commun. Nonlinear Sci. Numer. Simul. 14, 2542–2550 (2009)

    Article  Google Scholar 

  18. Feng, C.S., Liu, R.: Response of Duffing system with delayed feedback control under bounded noise excitation. Arch. Appl. Mech. 82(12), 1753–1761 (2012)

    Article  MATH  Google Scholar 

  19. Xu, Y., Ma, S.J., Zhang, H.Q.: Hopf bifurcation control for stochastic dynamical system with nonlinear random feedback method. Nonlinear Dyn. 65, 77–84 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Liu, Z.H., Zhu, W.Q.: Stochastic Hopf bifurcation of quasi-integrable Hamiltonian systems with multi-time-delayed feedback control and wide-band noise excitations. Nonlinear Dyn. 69(3), 935–947 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Arnold, L.: Random Dynamical Systems. Springer, New York (1995)

    Book  MATH  Google Scholar 

  22. Oustaloup, A., Sabatier, J., Lanusse, P.: From fractal robustness to CRONE control. Fract. Calc. Appl. Anal. 2, 1–30 (1999)

    MathSciNet  MATH  Google Scholar 

  23. Podlubny, I.: Fractional-order systems and \(PI^{\lambda }D^{\mu }\)-controllers. IEEE Trans. Autom. Control 44, 208–214 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  24. Agrawal, O.P.: A quadratic numerical scheme for fractional optimal control problems. J. Dyn. Syst. Meas. Control 130, 011010 (2007)

    Article  Google Scholar 

  25. Ladaci, A., Charef, A.: On fractional adaptive control. Nonlinear Dyn. 43, 365–378 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  26. Tavazoei, M.S., Tavakoli-Kakhki, M.: Compensation by fractional-order phase-lead/lag compensators. Control Theory Appl. 8, 319–329 (2014)

    Article  MathSciNet  Google Scholar 

  27. Manuel, F., Silva, J.A.T., Machado, A.M.: Lopes: fractional order control of a hexapod robot. Nonlinear Dyn. 38, 417–433 (2004)

  28. Pop, C.I., Ionescu, C., Keyser, R.E., Dulf, E.H.: Robustness evaluation of fractional order control for varying time delay processes. Signal Image Video Process. 6, 453–461 (2012)

    Article  Google Scholar 

  29. Muresan, C.I., Dulf, E.H., Prodsan, O.: A fractional order controller for seismic mitigation of structures equipped with viscoelastic mass dampers. J. Vib. Control (2014). doi:10.1177/1077546314557553

  30. Manabe, S.: A suggestion of fractional-order controller for flexible spacecraft attitude control. Nonlinear Dyn. 29(1–4), 251–268 (2002)

    Article  MATH  Google Scholar 

  31. Ahn, H.S., Bhambhani, V., Chen, Y.: Fractional-order integral and derivative controller for temperature profile tracking. Sadhana 34, 833–850 (2009)

    Article  MATH  Google Scholar 

  32. Zhuang, D.J., Yu, F., Lin, Y.: Evaluation of a vehicle directional control with a fractional order \(\text{ PD }^{\mu }\) controller. Int. J. Automot. Technol. 9, 679–685 (2008)

    Article  Google Scholar 

  33. Chen, N., Chen, N., Chen, Y.: On fractional control method for four-wheel-steering vehicle. Sci. China Ser. E Technol. Sci. 52(3), 603–609 (2009)

    Article  MATH  Google Scholar 

  34. Shafieezadeh, A., Ryan, K., Chen, Y.: Fractional order filter enhanced LQR for seismic protection of civil structures. J. Comput. Nonlinear Dyn. 3, 021404–021404 (2008)

    Article  Google Scholar 

  35. Zhao, J., Wang, J., Wang, S.: Fractional order control to the electro-hydraulic system in insulator fatigue test device. Mechatronics 23(7), 828–839 (2013)

    Article  Google Scholar 

  36. Zhong, G., Deng, H., Li, J.: Chattering-free variable structure controller design via fractional calculus approach and its application. Nonlinear Dyn. 81(1), 679–694 (2015)

  37. Zhuang, D.J., Yu, F., Lin, Y.: Evaluation of a vehicle directional control with a fractional order PD\(^{\mu }\) controller. Int. J. Automot. Technol. 9(6), 679–685 (2008)

    Article  Google Scholar 

  38. Zhu, W.Q., Lu, M.Q., Wu, Q.T.: Stochastic jump and bifurcation of a Duffing oscillator under narrowband excitation. J. Sound Vib. 165, 285–304 (1993)

    Article  MATH  Google Scholar 

  39. Xu, Z., Chung, Y.K.: Averaging method using generalized harmonic functions for strongly non-linear oscillators. J. Sound Vib. 174, 563–576 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  40. Zhu, W.Q., Huang, Z.L., Suzuki, Y.: Response and stability of strongly non-linear oscillators under wide-band random excitation. Int. J. Non-Linear Mech. 36, 1235–1250 (2001)

  41. Ye, X.Q., Shen, Y.H.: Handbook of Practical Mathematics. Science Press, Beijing (2006). (In Chinese)

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Acknowledgments

This project is supported by the National Science Foundation of Fujian Province under the Grant No.(2014J01014) and the National Natural Science Foundation of China (No: 11302157, 51208217), and Research Award Fund for Outstanding Young Researcher in Higher Education Institutions of Fujian Province. Besides, the first author would like to appreciate the China Scholarship Council for sponsoring the visiting at University of California, Merced under the Grant No.(201408350008).

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Authors and Affiliations

  1. College of Civil Engineering, Huaqiao University, Xiamen, 361021, China

    Lincong Chen, Tianlong Zhao & Jun Zhao

  2. School of Mathematics and Statistics, Xidian University, Xi’an, 710071, China

    Wei Li

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  1. Lincong Chen
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  2. Tianlong Zhao
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  3. Wei Li
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  4. Jun Zhao
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Correspondence to Lincong Chen.

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Chen, L., Zhao, T., Li, W. et al. Bifurcation control of bounded noise excited Duffing oscillator by a weakly fractional-order \(\varvec{PI}^{\varvec{\lambda }} \varvec{D}^{\varvec{\mu }}\) feedback controller. Nonlinear Dyn 83, 529–539 (2016). https://doi.org/10.1007/s11071-015-2345-1

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