Stability Analysis for a Class of Fractional-Order System with Commensurate Order

Aiming at a class of linear time-invariant fractional-order system (LTI-FOS) with commensurate order, which is similar to traditional second order system, the range of damping ratio is derived for the closed-loop stability of this special kind of FOS based on sufficient and necessary conditions. The stable parameter regions is wider than that of traditional second order system. The existing fractional Nyquist stability criterion ad logarithmic frequency stability criterion can not be directly used while fractional order integral unit exists in open-loop transfer function. A method of drawing auxiliary arc on the Nyquist diagram and auxiliary line on the Bode diagram is given. Then the stability and relative stability analysis for this special kind of FOS is evaluated using this method. Finally a conclusion under the condition of the same angular frequency and damping ratio is drawn: the bigger of the fractional order, the smaller of relative stability.