A new and simple method to construct root locus of general fractional-order systems

Abstract

Recently fractional-order (FO) differential equations are widely used in the areas of modeling and control. They are multivalued in nature hence their stability is defined using Riemann surfaces. The stability analysis of FO linear systems using the technique of Root Locus is the main focus of this paper. Procedure to plot root locus of FO systems in s-plane has been proposed by many authors, which are complicated, and analysis using these methods is also difficult and incomplete. In this paper, we have proposed a simple method of plotting root locus of FO systems. In the proposed method, the FO system is transformed into its integer-order counterpart and then root locus of this transformed system is plotted. It is shown with the help of examples that the root locus of this transformed system (which is obviously very easy to plot) has exactly the same shape and structure as the root locus of the original FO system. So stability of the FO system can be directly deduced and analyzed from the root locus of the transformed IO system. This proposed procedure of developing and analyzing the root locus of FO systems is much easier and straightforward than the existing methods suggested in the literature. This root locus plot is used to comment about the stability of FO system. It also gives the range for the amplifier gain k required to maintain this stability. The reliability of the method is verified with analytical calculations.