Asymptotic Magnitude Bode Plots of Fractional-Order Transfer Functions

Development of asymptotic magnitude Bode plots for integer-order transfer functions is a well-established topic in the control theory. However, construction of such plots for the fractional-order transfer functions has not received much attention in the existing literature. In the present paper, we investigate in this direction and derive the procedures for sketching asymptotic magnitude Bode plots for some of the popular fractional-order controllers such as $PI^\alpha$, $PI]^\alpha$, $PD^\beta$, $PD]^\beta$, and $PI^\alpha D^\beta$. In addition, we deduce these plots for general fractional commensurate-order transfer functions as well. As applications of this work, we illustrate 1) the analysis of the designed fractional-control loop and 2) the identification of fractional-order transfer function from a given plot.