A linear single degree-of-freedom (SDOF) oscillator with fractional-order PID controller of acceleration feedback is investigated by the averaging method, and the approximately analytical solution is obtained. Moreover, the numerical solution of the system is obtained by the step-down order method and the power series method progressively. The effects of the parameters in fractional-order PID controller on the dynamical properties are characterized by some new equivalent parameters. The proportional component of fractional-order PID controller is characterized in the form of equivalent mass. The integral component of fractional-order PID controller is denoted in the form of the equivalent linear damping and equivalent mass. The differential component of fractional-order PID controller is denoted in the form of the equivalent linear negative damping and equivalent mass. Those equivalent parameters could distinctly illustrate the effects of the parameters in fractional PID controller on the dynamical response. A comparison between the analytical solution with the numerical results is made, and their satisfactory agreement verifies the correctness of the approximately analytical results. The effects of the parameters in fractional-order PID controller on control performance are further analyzed by some performance parameters of the time response. Finally, the robustness of the fractional-order PID controller based on acceleration feedback is demonstrated through the control of a SDOF quarter vehicle suspension model
