Stability analysis of multiple time delayed fractional order systems

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 “fractional-order” 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.