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A new methodology, based on advanced clustering with frequency sweeping (ACFS), is presented for the stability analysis of fractional-order systems with multiple time delays against delay uncertainties. The problem is known to be notoriously complex, primarily because the systems are infinite dimensional due to delays. Multiplicity of the delays in this study complicates the analysis even further. And "fractionalorder" feature of the systems makes the problem much more challenging compared to integer order systems. ACFS does not impose any restrictions in the number of delays, and it can directly extract the 2-D cross sections of the stability views in any two delay domain. We show that this procedure analytically reveals all possible stability regions exclusively in the space of the delays. As an added strength, it does not require the delay-free system under consideration to be stable. The main contribution of this document is that we demonstrate for the first time that the stability maps of a fractional-order system with multiple time delays can be obtained efficiently. Two illustrative examples are presented to confirm the proposed method results.

The aim of this study is to offer a new analytical method for the stability testing of neutral type linear time-invariant (LTI) time-delayed fractional-order systems with commensurate orders and multiple commensurate delays. It is evident from the literature that the stability assessment of this class of dynamics remains unsolved yet and this is the first attempt to take up this challenging problem. The method starts with the determination of all possible purely imaginary characteristic roots for any positive time delay. To achieve this, the Rekasius transformation is used for the transcendental terms in the characteristic equation. An explicit analytical expression in terms of the system parameters which reveals the stability regions (pockets) in the domain of time delay has been presented. The number of unstable roots in each delay interval is calculated with the definition of root tendency (RT) on the boundary of each interval. Two example case studies are also provided, which are not possible to analyze using any other methodology known to the authors.

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