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A novel approach to stability analysis of a wide class of irrational linear systems

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Abstract

This paper presents a general methodology for finding stability equivalence regions, of a wide class of linear time-invariant systems with irrational transfer functions, inside a parametric space. The proposed methodology can be applied to distributed-parameter, time-delay and fractional systems. Unlike rational transfer functions which have only a finite number of poles, irrational transfer functions may generally possess an infinite number of poles, branch points and even essential singularities. Due to this, stability of such systems is more difficult to analyze. Two variants of the new methodology are presented. The first one analyzes stability equivalence along a curve in the parametric space, starting from a given parametric point. The second one finds the maximal stability equivalence region in the parametric space around a given parametric point. Both methodologies are based on iterative application of Rouché’s theorem. They are illustrated on several examples, including heat diffusion equation and generalized time-fractional telegrapher’s equation, which exhibit special functions such as \(\sinh \) and \(\cosh \) of \(\sqrt{s}\), the Laplace variable of order 0.5.

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Acknowledgements

M.R. Rapaić is supported by the Ministry of Education, Science and Technological Development of Serbia through project no. 451-03-68/2022-14/200156 “Innovative scientific and artistic research from the FTS (activity) domain.

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Authors and Affiliations

  1. Faculty of Technical Sciences, University of Novi Sad, Trg Dositeja Obradovića 6, Novi Sad, Serbia

    Vukan Turkulov & Milan R. Rapaić

  2. Univ. Bordeaux, CNRS, Bordeaux INP, IMS, UMR 5218, 33400, Talence, France

    Rachid Malti

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  1. Vukan Turkulov
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  2. Milan R. Rapaić
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  3. Rachid Malti
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Correspondence to Vukan Turkulov.

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Turkulov, V., Rapaić, M.R. & Malti, R. A novel approach to stability analysis of a wide class of irrational linear systems. Fract Calc Appl Anal 26, 70–90 (2023). https://doi.org/10.1007/s13540-022-00126-z

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