Preservation of Stability and Synchronization of a Class of Fractional‐Order Systems

Armando Fabián Lugo-Peñaloza,J.j Flores-Godoy ,Guillermo Fernández‐Anaya

doi:10.1155/2012/928930

Armando Fabián Lugo-Peñaloza, J.j Flores-Godoy + Show 1 more

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https://doi.org/10.1155/2012/928930

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Abstract

We present sufficient conditions for the preservation of stability of fractional‐order systems, and then we use this result to preserve the synchronization, in a master‐slave scheme, of fractional‐order systems. The systems treated herein are autonomous fractional differential linear and nonlinear systems with commensurate orders lying between 0 and 2, where the nonlinear ones can be described as a linear part plus a nonlinear part. These results are based on stability properties for equilibria of fractional‐order autonomous systems and some similar properties for the preservation of stability in integer order systems. Some simulation examples are presented only to show the effectiveness of the analytic result.

Highlights

The applications of fractional calculus to science and engineering have been growing in the last few years 1 ; this is due in part to the properties of these operators
There are many different works on the synchronization of fractional autonomous systems that can be described as a linear plus a nonlinear part 11, in such works several schemes are proposed to ensure that the error dynamics satisfies the conditions from the celebrated theorem for autonomous commensurate differential systems with fractional order between 0 and 1 by ; this means that the error dynamics must hold a linear relation in order to achieve the synchronization
There is some other interesting theme, the preservation of stability and synchronization, which is the main issue in this work. This problem can be stated as follows: if we have an original autonomous nonlinear system that can be described as a linear plus a nonlinear part whose origin is stable, we want to investigate some kinds of modifications that can occur to the fractional order, the linear part, and the nonlinear part in such a way that the origin of the modified system is stable

Summary

Introduction

The applications of fractional calculus to science and engineering have been growing in the last few years 1 ; this is due in part to the properties of these operators. There is some other interesting theme, the preservation of stability and synchronization, which is the main issue in this work This problem can be stated as follows: if we have an original autonomous nonlinear system that can be described as a linear plus a nonlinear part whose origin is stable, we want to investigate some kinds of modifications that can occur to the fractional order, the linear part, and the nonlinear part in such a way that the origin of the modified system is stable. This subject is important because such modifications can be interpreted as perturbations on the system. In 19 , the authors have reached conditions for the preservation of stability for integer-order systems in the presence of nonlinear modifications to the Jacobian matrix; such modifications can be applied on the characteristic polynomial or in form of a nonlinear polynomial matrix evaluation

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