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 d...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 It? 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 RayleighDuffing 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展开更多
基金supported by the National Natural Science Foundation of China(Grant Nos.11672111,11432012,11602089)the Program for New Century Excellent Talents in Fujian Province University,and the Technological Project of Huaqiao University(Grant Nos.ZQN-YX307,ZQNYX505)
文摘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 It? 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 RayleighDuffing 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
