Improved Digital Rational Approximation of the Operator $$S^{\alpha }$$ S α Using Second-Order s-to-z Transform and Signal Modeling

Abstract

In this paper, an improved stable digital rational approximation of the fractional-order operator $$s^\alpha ,\alpha \in \,R$$s?,??R is developed. First, a novel efficient second-order digital differentiator is derived from the transfer function of the digital integrator proposed by Tseng. Then, the fractional power of the new s-to-z transform is expanded using power series expansion (PSE)-signal-modeling technique to obtain stable rational approximation of $$s^\alpha $$s?. Simulation results show that the proposed rational approximation has better frequency characteristics in almost the whole frequency range than that of existing first-order s-to-z transforms based approximations for different values of the fractional-order $$\alpha $$?. This paper also shows the benefit of using PSE-signal-modeling approach with first- or second-order mapping functions over PSE-truncation approach that is used in recent works for rational approximation of the operator $$s^\alpha $$s?, and highlights the major disadvantage of the latter approach that leads to undesirable rational models with complex conjugate poles and zeros.