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

Area of Science:

• Control Systems Engineering
• Applied Mathematics
• Dynamical Systems Theory

Background:

• Fractional-order (FO) differential equations are increasingly utilized in advanced modeling and control applications.
• The inherent multivalued nature of FO systems necessitates stability definitions based on Riemann surfaces.
• Existing root locus methods for FO systems are often complex and yield incomplete analyses.

Purpose of the Study:

• To propose a simplified and effective method for plotting and analyzing the root locus of fractional-order linear systems.
• To facilitate a more straightforward assessment of stability for FO systems.
• To determine the range of amplifier gain (k) for maintaining system stability.

Main Methods:

• Transformation of fractional-order systems into equivalent integer-order systems.
• Plotting the root locus of the transformed integer-order system in the s-plane.
• Comparative analysis of root locus plots between original FO systems and their transformed counterparts.

Main Results:

• The proposed method yields root locus plots with identical shape and structure to those of the original FO systems.
• Stability of the FO system can be reliably inferred from the root locus of the transformed integer-order system.
• The method provides clear insights into the gain (k) margins for ensuring stability.

Conclusions:

• The developed root locus plotting technique offers a significantly simpler and more direct approach compared to existing methods for FO systems.
• This simplified analysis enhances the practical application of fractional-order control systems.
• The method's reliability is validated through analytical calculations and comparative studies.