Exact method for the stability analysis of time delayed linear-time invariant fractional-order systems

In this study, a practical analytical procedure is introduced for determining the stability robustness map of a general class of linear-time invariant fractional-order systems with single and multiple commensurate delays of retarded type, against delay uncertainties. The complexity arises due to the exponential type transcendental terms and fractional order in their characteristic equation (CE). It is shown that this procedure analytically reveals all possible stability regions exclusively in the parametric space of the time delay. Using the presented method in this study, first, the authors will eliminate the transcendental terms of exponential type from the CE and then, they can determine all the locations where roots pass through the imaginary axis. By definition of root tendency on the boundary of each interval, the number of unstable roots in each region is calculated. Finally, the concept of stability is expressed in the intervals of delay values. The effectiveness of the proposed method results is illustrated via six numerical examples and to gain a better understanding of the problem, the root-locus curve of these systems has been depicted as a function of delay parameter changes.