Minimum-Phase Realization of Complex-Order Fractional Derivatives and Integrals

This study addresses the challenge of preserving minimum-phase behavior in fractional-order systems (FOS) of complex orders, which are vital for control and signal processing applications involving memory effects and phase asymmetry. Traditional approximation techniques often compromise phase integrity, leading to instability or poor invertibility. Building upon El-Khazali’s first-order rational approximation, we propose a frequency-domain strategy to ensure minimum-phase realization using a Routh-Hurwitz-based safe-band verification. The method identifies analytically valid regions in the (α, β) parameter space where all approximated transfer functions maintain minimum-phase behavior across desired frequency bands. Simulation results confirm the accuracy of the method and reveal key design trends via test function maps and frequency responses. An interactive MATLAB App was developed to support real-time visualization and parameter exploration. The proposed framework provides a low-order, stable, and analytically tractable solution for minimum-phase approximation of complex-order Laplace operators.