Stability and resonance conditions of second-order fractional systems

Abstract

The interest of studying fractional systems of second order in electrical and mechanical engineering is first illustrated in this paper. Then, the stability and resonance conditions are established for such systems in terms of a pseudo-damping factor and a fractional differentiation order. It is shown that a second-order fractional system might have a resonance amplitude either greater or less than one. Moreover, three abaci are given allowing the pseudo-damping factor and the differentiation order to be determined for, respectively, a desired normalized gain at resonance, a desired phase at resonance, and a desired normalized resonant frequency. Furthermore, it is shown numerically that the system root locus presents a discontinuity when the fractional differentiation order is an integral number.

Asymptotic behaviour of solutions to non-commensurate fractional-order planar systems

“…Lemma 3.1. Let 0 < α 1 < α 2 ≤ 1 and a, b, c ∈ R. Then, the following statements hold for the function Q defined in (12).…”

“…In addition to the algorithmic approach as in [16], a number of analytic approaches have been used to investigate the zeros of characteristic polynomials of systems of fractional order systems. In [12], the stability and resonance conditions are established for fractional systems of second order in terms of a pseudo-damping factor and a fractional differentiation order. The method in [12] has been successfully extended in [26] for a wide class of second kind non-commensurate elementary systems.…”

“…In [12], the stability and resonance conditions are established for fractional systems of second order in terms of a pseudo-damping factor and a fractional differentiation order. The method in [12] has been successfully extended in [26] for a wide class of second kind non-commensurate elementary systems. By the substitution method, a variation of constants formula and the properties of the Mittag-Leffler function in the stable domain, in [10], the authors have shown the asymptotic stability for fractional order systems with (block) triangular coefficient matrices.…”

Dynamic robust stabilization of fractional-order linear systems with nonlinear uncertain parameters: an LMI approach

“…Therefore, fractional-order differential equations can more completely and precisely describe systems having responses with long memory transients than the ordinary integer-order differential equations Tavazoei, 2014, 2017). Accordingly, stability and stabilization of fractional-order systems is an important and challenging problem since many physical and real-world processes are modeled with fractional-order state equations (Badri and Sojoodi, 2018;Ivanova et al, 2018;Lu and Chen, 2009;Ma et al, 2014). Unfortunately, uncertainties arising from neglected dynamics, uncertain physical parameters, parametric variations in time, and many other sources are inevitable in real physical system.…”

Analytical study of resonance regions for second kind commensurate fractional systems

“…In this case, viewed the complexity of the algebraic methods as Routh and Jury criteria [11], stability has been essentially investigated using graphical methods as in [30,20,28]. However, very recently, [5,12] establish numerically the stability and resonance limits of uncommensurate systems.…”