Stability of a linear oscillator with damping force of the fractional-order derivative

It begins with the study of damping representation of a linear vibration system of single degree of freedom (SDOF), from the view point of fractional calculus. By using the idea of stability switch, it shows that the linear term involving the fractional-order derivative of an order between 0 and 2 always acts as a damping force, so that the unique equilibrium is asymptotically stable. Further, based on the idea of stability switch again, the paper proposes a scheme for determining the stable gain region of a linear vibration system under a fractional-order control. It shows that unlike the classical velocity feedback which can adjust the damping force only, a fractional-order feedback can adjust not only the damping force, but also the elastic restoring force, and in addition, a fractional-order PD^α control can either enlarge the stable gain region or narrow the stable gain region. For the dynamic systems described by integer-order derivatives, the asymptotical stability of an equilibrium is guaranteed if all characteristic roots stay in the open left half-plane, while for the systems with fractional-order derivatives, the asymptotical stability of an equilibrium is guaranteed if all characteristic roots stay within a sector in the complex plane. Analysis shows that the proposed method, based on the idea of stability switch, works effectively in the stability analysis of dynamical systems with fractional-order derivatives.