Stability of Fractional-Delay Systems: A Practical Approach

Abstract

Sometimes, in dealing with fractional-order transfer functions, the exact location of the poles and zeros on the first Riemann sheet is needed. For example, in order to examine the stability of fractional-delay systems, the location of the poles on the first Riemann sheet should be known since the stability is related to the poles that are located on the right half-plane of the first Riemann sheet. The difficulty is due to the fact that most of the practical multi-valued transfer functions consist of large (possibly infinite) number of poles and zeros which makes the problem of determining their location (and consequently, the stability analysis) a challenging task. In this paper, an effective numerical algorithm for determining the location of poles and zeros on the first Riemann sheet is presented. The proposed method is based on the Rouche’s theorem and can be applied to all multi-valued transfer functions defined on a Riemann surface with finite number of Riemann sheets where the origin is a branch point. This covers all practical (finite-dimensional) fractional-order transfer functions and also the so-called fractional-delay systems. An example is presented to confirm the effectiveness of the proposed algorithm.