Conditions for a Class of Rational Approximants of Fractional Differentiators/Integrators to Enjoy the Interlacing Property

Fractional-order controllers are based on fractional differentiators and integrators, which extend classical integral or derivative actions to improve performance and robustness of feedback loops. The fractional (irrational) operators are usually realized by continued fractions leading to rational approximations. Stable and minimum-phase transfer functions, with interlaced distribution of real zeros and poles, are also requested as suitable approximations. This paper proves that two classes of continued-fractions-based approximations have negative zeros and poles enjoying the interlacing property, for any fractional order v, with |v| < 1, and for any degree, m, of the approximating transfer function.