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Stability Analysis of Fractional Order Systems Described in Lur'e Structure

Mousavi, Shima Sadat | 2012

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 44130 (05)
  4. University: Sharif University of Technology
  5. Department: Electrical Engineering
  6. Advisor(s): Tavazoei, Mohammad Saleh
  7. Abstract:
  8. Most of the real-world systems can be modeled as closed loop systems which are made up of a LTI subsystem in the forward path and a memoryless nonlinear subsystem in the feedback path. Such systems are called “Lur’e systems”, the name of the first who introduced these systems. Because of the generality and the wide applicability of this type of systems, studying the stability of them has been notable and interesting to many scientists for a long time. Meanwhile, in recent decades, many scientists in physics and engineering sciences have been interested in applications of the fractional order systems and as a result focused on studying the features of the fractional order systems. An important reason of this interest is the more genuine description of many actual systems made by this kind of systems.In this thesis, input-output stability of kind L2 for the systems described in the Lur’e structure is discussed. For this reason, considering the common theorems about the stability of the Lur’e systems of integer order, validity of them about fractional order systems is studied. Accordingly, at first, some classic theorems about the stability of Lur’e systems e.g. Circle Theorem are discussed. As a result, if the subsystem of fractional order has an impulse response with bounded and norms, the Circle Theorem can be applicable to determine the stability of the system. Then, application of Circle Theorem is compared between Lur’e systems of integer and fractional order using their respective Nyquist diagrams. The other theorem discussed in this thesis is Popov Theorem. Furthermore, the conditions underwhich the Zames-Falb and Generalized Zames-Falb Theorems which use multipliers and integral quadratic constraints respectively, can be generalized to fractional order Lur’e systems are studied. By means of these theorems, a series of stable fractional order systems are introduced. Finally, while generalizing the off-axis Circle Theorem to fractional order systems, another method is represented to prove one of the theorems used in its overall proof
  9. Keywords:
  10. Fractional Order System ; Lur'e Systems ; Absolute Stability ; Bounded-Input Bounded-Output Stability

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