Effect of a fractional-order PID controller on the dynamical response of a linear single degree-of-freedom oscillator

The free vibration of a linear single degree-of-freedom (SDOF) oscillator with a fractional-order PID controller of velocity feedback was investigated by the averaging method, and the approximately analytical solution was obtained. The results indicate that the effects of the parameters in the fractional-order PID controller on the dynamical properties are characterized. The proportional component of the fractional-order PID controller is characterized in the form of equivalent linear damping. The integral component of the fractional-order PID controller is characterized in the form of equivalent linear damping and equivalent linear stiffness. The differential component of the fractional-order PID controller is characterized in the form of equivalent linear damping and equivalent linear negative stiffness. Those equivalent parameters could distinctly illustrate the effects of the parameters in the fractional-order PID controller on the dynamical response. A comparison of the analytical solution with numerical results is made, and their agreement verifies the correctness of the approximately analytical results. The effects on system control performance of the coefficients and the orders in the fractional-order PID controller were analyzed by time response performance metrics parameters. Finally, the parameters tuning of the fractional-order PID controller based on velocity feedback was demonstrated through controlling a SDOF quarter vehicle suspension model. © 2016, Editorial Office of Journal of Vibration and Shock. All right reserved.