Frequency-Domain Characterization of Fractional Variable-Order Systems with Scarpi Derivatives

Abstract

This work presents a frequency-domain analysis framework for fractional variable-order (FVO) linear systems using the Scarpi derivative approach. Unlike classical fractional-order models with fixed differentiation orders, variable-order systems give greater flexibility to more accurately model a more dynamic system whose memory effects change over time or depending on certain operating conditions. The Scarpi derivative is a consistent mathematical way to preserve the necessary properties for analyzing variable-order dynamics and deriving frequency-domain transfer function representations.  A generalized frequency response is created, which represents a linear system modeled using Scarpi (variable-order) operators, then examined the impacts on the magnitude and phase characteristics of the system caused by order variability. Here, methodology generalizes traditional Bode analyses into a variable-order fractional model. Additionally, the methodology illustrates how order variability affects the resonant behavior, bandwidth, stability margins, and robustness of the system. Numerical analysis also illustrated with an example to a complex system with time-varying memory effects. The results of this research indicate that using the Scarpi derivative formulation is an effective means of modeling and characterizing frequency-domain complex systems with nonstationary memory effects. This work has numerous potential applications in control engineering, viscoelastic materials, electrochemical processes, and living systems. The results contribute to the advancement of variable-order fractional calculus by establishing a practical and systematic framework for the frequency analysis of FVO linear systems.