Science China Technological Sciences

, Volume 62, Issue 2, pp 287–297 | Cite as

Stochastic averaging technique for SDOF strongly nonlinear systems with delayed feedback fractional-order PD controller

  • LinCong ChenEmail author
  • Xiao Liang
  • WeiQiu Zhu
  • YaoBing Zhao


A stochastic averaging technique is proposed to study the randomly excited single-degree-of-freedom (SDOF) strongly nonlinear systems with delayed feedback fractional-order proportional-derivative (PD) controller. The delayed feedback fractional-order PD control force is approximated by an equivalent non-delay feedback control force combining with a quasi-linear elastic force and a quasi-linear damping force. The averaged Itô stochastic differential equation for amplitude of the equivalent nonlinear system is derived by the generalized harmonic functions. The analytical stationary probability density function (PDF) is obtained with solving the reduced Fokker-Planck-Kolmogorov (FPK) equation. Two examples of van der Pol oscillator and Rayleigh-Duffing oscillator are studied to illustrate the application and effectiveness of the proposed method. Numerical results display that the proposed method can yield to the high precision, and the time delay could ruin the control effectiveness, but also even amplifies the response of the system more than that of uncontrolled system. Furthermore, the study finds that the parameters of fractional-order α and time delay may cause the stochastic P-bifurcation. It is indicated that the delayed feedback fractional-order PD controller can offer a potentially effective tool for anti-control of stochastic bifurcation


stochastic averaging strongly nonlinear system time delay fractional-order PDα controller 


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Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • LinCong Chen
    • 1
    Email author
  • Xiao Liang
    • 1
  • WeiQiu Zhu
    • 2
  • YaoBing Zhao
    • 1
  1. 1.College of Civil EngineeringHuaqiao UniversityXiamenChina
  2. 2.Departments of Mechanics, State Key Laboratory of Fluid Power Transmission and ControlZhejiang UniversityHangzhouChina

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