Stability analysis of multiple time delayed fractional order systems

Abstract

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.

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

“…Pakzad and Nekoui [26] have successfully generalised the direct method [27] for the stability analysis of FDS. A recent paper [28] extended the CTCR method to multiple time‐delayed fractional‐order system while the system also contains fixed time delays and has also been applied to FDSs with coefficient uncertainty and time delay uncertainty in [29]. In this procedure, advanced clustering with frequency sweeping (ACFS) method [5] serves as the first step to the umbrella paradigm called ‘CTCR’ to determine stability switching boundaries.…”

On τ ‐decomposition frequency‐sweeping strategy to rational‐order fractional systems with commensurate delays

“…In [11], a new technique based on the Pade approximation is proposed to determining the stability regions in the time-delay space for any fractional order. Meanwhile, an exact and robust method for the determination of all the possible stability regions in the parametric space has been presented in several literatures [12,13]. Especially, both time-delay space and coefficient space are concerned in [13].…”

“…Meanwhile, an exact and robust method for the determination of all the possible stability regions in the parametric space has been presented in several literatures [12,13]. Especially, both time-delay space and coefficient space are concerned in [13].…”

Direct Method for Stability Analysis of Fractional Delay Systems

“…From the numerical analysis point of view, the effective numerical algorithms have been discussed in [7] and [8] for the evaluation of BIBO stability of fractional order delay systems. In [9] a heavy computation scheme based on the Cauchy's integral has been proposed to test the stability of such systems, and in [10], a technique based on the Lambert W function was used for the same purpose.…”