PRIMARY RESONANCE OF DUFFING OSCILLATOR WITH FRACTIONAL-ORDER PID CONTROLLER BASED ON VELOCITY FEEDBACK

Compared with the traditional integer-order PID controller, the fractional-order PID(FOPID) controller may present much better control performance due to its two surplus adjustable parameters.The primary resonance of Duffing oscillator with FOPID controller of velocity feedback is investigated by the averaging method, and the approximately analytical solution is obtained.The effects of the parameters in FOPID controller on the dynamical properties are studied and characterized by some equivalent parameters.The proportional component of FOPID controller is characterized in the form of equivalent linear damping.The integral component of FOPID controller is characterized in the form of the equivalent linear damping and equivalent linear stiffness.The differential component of FOPID controller is characterized in the form of the equivalent linear damping and equivalent mass.Those equivalent parameters could distinctly illustrate the effects of the parameters in FOPID controller on the dynamical response.The amplitude-frequency equation for steady-state solution and associated with the stability condition are also presented.A comparison of the analytical solution with the numerical results is made, and their satisfactory agreement verifies the correctness of the approximately analytical results.Finally, the effects on the amplitude-frequency performance of the coefficients and the orders in FOPID controller are analyzed, and the control performances of fractional-order and integer-order PID controller are compared. The results show that the control performance of FOPID controller is better than the traditional integer-order counterpart for controlling the vibration of the primary resonance of Duffing oscillator, when the coefficients of the two controllers are the same.