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

TL;DR

This paper investigates the stability of fractional order Lur'e systems, extending classical integer order stability theorems to fractional orders between 0 and 1, and introduces new criteria and methods for their analysis.

Contribution

It extends classical stability theorems to fractional order Lur'e systems, compares stability criteria between integer and fractional cases, and proposes new methods for stability proof.

Findings

Classical theorems can be applied to fractional order systems under certain conditions.

Comparison of circle criterion application between integer and fractional order systems.

Introduction of new stability classes and methods for fractional order Lur'e systems.

Abstract

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.