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

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Abstract

In this paper, an improved stable digital rational approximation of the fractional-order operator \(s^\alpha ,\alpha \in \,R\) 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 \(s^\alpha \). 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 \(s^\alpha \), and highlights the major disadvantage of the latter approach that leads to undesirable rational models with complex conjugate poles and zeros.

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Acknowledgments

The authors would like to thank the anonymous reviewers for their useful comments and suggestions to improve the quality of the paper.

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Authors and Affiliations

  1. LRES Laboratory, Electrical Engineering Department, Skikda University, BP. 26, Route d’El-Hadaîek, 21000, Skikda, Algeria

    Fouzia Leulmi & Youcef Ferdi

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  1. Fouzia Leulmi
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  2. Youcef Ferdi
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Correspondence to Fouzia Leulmi.

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Leulmi, F., Ferdi, Y. Improved Digital Rational Approximation of the Operator \(S^{\alpha }\) Using Second-Order s-to-z Transform and Signal Modeling. Circuits Syst Signal Process 34, 1869–1891 (2015). https://doi.org/10.1007/s00034-014-9928-9

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  • DOI: https://doi.org/10.1007/s00034-014-9928-9

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