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

Lur'e systems are feedback interconnection of a linear time-invariant subsystem in the forward path and a memoryless nonlinear one in the feedback path, which have many physical representatives. In this paper, some classical theorems about the L2 input-output stability of integer order Lur'e systems are discussed, and the conditions under which these theorems can be applied in fractional order Lur'e systems with an order between 0 and 1 are investigated. Then, application of circle criterion is compared between Lur'e systems of integer and fractional order using their corresponding Nyquist plots. Furthermore, applying Zames-Falb and generalized Zames-Falb theorems, some classes of stable fractional order Lur'e systems are introduced. Finally, in order to generalize the off-axis circle criterion to fractional order systems, another method is presented to prove one of the theorems used in its overall proof.