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Stability Analysis of Fractional Delay Systems in the Frequency Domain

Mesbahi, Afshin | 2014

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 45704 (05)
  4. University: Sharif University of Technology
  5. Department: Electrical Engineering
  6. Advisor(s): Haeri, Mohammad
  7. Abstract:
  8. An algebraic method is presented for assessing the bounded input bounded output (BIBO) stability of a class of fractional delay systems with commensurate orders and multiple commensurate delays of retarded type. In the proposed method, first, by mapping the principal sheet of the Riemann surface and a pseudo-delay transformation, an auxiliary polynomial is generated. Then, this auxiliary polynomial is employed to find roots of the characteristic equation on the imaginary axis. The properties of the root path close to these roots are used to identify intervals of delay values, in which the system is stable. Moreover, a delay-independent BIBO-stability criterion is provided. Based on the proposed method, a fractional order PID controller is designed to stabilize the fractional delay systems with commensurate orders and multiple commensurate delays, while the time delays in the system may belong to several distinct intervals. The presented results are extended to fractional-order time-delay differential systems of retarded type with two independent time delays. Furthermore, a simple inequality constraint is established to obtain pure imaginary poles of the scalar systems. The proposed method is extended to analyze the stability of fractional delay systems of neutral type such that the extended method handles the existing stability differences between retarded and neutral types.
    Stability analysis of time-delay systems is a difficult task, especially when the system dimension is high. This complexity can be reduced when the system decomposes into separated or cascade sub-systems with lower orders. To do so, necessary and sufficient conditions are provided to determine whether or not the system matrices are simultaneously similar to a block triangular or diagonal form. Moreover, the similarity transformation matrix is calculated based on eigenbases of the system matrices. In order to perform such decomposition, one needs to determine a linear transformation matrix. Furthermore, the given conditions are adapted to a simple but effective condition to derive all possible scalar sub-systems for a given linear system.
  9. Keywords:
  10. Stability ; Stability Analysis ; Fractional Order System ; Time Delay ; Block Triangular Matrix ; Fractional Delay Systems ; Block Diagonal Matrix

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