Direct Method for Stability Analysis of Fractional Delay Systems


  • Mohammad Ali Pakzad Department of Electrical Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran
  • Mohammad Ali Nekui Faculty of Electrical and Computer Engineering, K.N.Toosi University of Technology, Seyed-Khandan, P.O. Box 16315-1355, Tehran, Iran


fractional delay systems, stability windows, Root-Locus.


In this paper, a direct method is presented to analyze the stability of fractional order systems with single and multiple commensurate time delays, against delay uncertainties.. It is shown that this method analytically reveals all possible stability windows exclusively in the parametric space of the time delay. Using the approach presented in this study, first, without using any approximation or substitution, the transcendental characteristic equation is converted to an algebraic one with some specific crossing points. The resulting algebraic equation also enables us to easily determine the delay dependency of the system stability and the sensitivities of crossing roots with respect to time delay. Then, an expression in terms of system parameters and imaginary root of the characteristic equation is derived for computing the delay margin .The number of unstable roots in each interval is calculated with the definition of root tendency on the boundary of each interval. Finally, the concept of stability is expressed as a function of delay. four illustrative examples are presented to confirm the proposed method results.


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