Stability Analysis of a Fractional-Order Linear System Described by the Caputo–Fabrizio Derivative

Hong Li,Jun Cheng,Hou-Biao Li,Shou-Ming Zhong

doi:10.3390/math7020200

Abstract

In this paper, stability analysis of a fractional-order linear system described by the Caputo–Fabrizio (CF) derivative is studied. In order to solve the problem, character equation of the system is defined at first by using the Laplace transform. Then, some simple necessary and sufficient stability conditions and sufficient stability conditions are given which will be the basis of doing research of a fractional-order system with a CF derivative. In addition, the difference of stability domain between two linear systems described by two different fractional derivatives is also studied. Our results permit researchers to check the stability by judging the locations in the complex plane of the dynamic matrix eigenvalues of the state space.

Highlights

IntroductionResearch into fractional-order systems has become a hot subject because of many advantages of fractional derivatives
It permits us to check the stability by judging the locations in the complex plane of the dynamic matrix eigenvalues of the state space
A new definition of the fractional derivative without a singular kernel has been proposed by Caputo and Fabrizio in 2015 [15]. This new fractional derivative is less affected by past, compared with the classical Caputo fractional derivative which shows a slow stabilization

Summary

Introduction

Research into fractional-order systems has become a hot subject because of many advantages of fractional derivatives. Lyapunov approach [12,13] are used to investigate the stability of fractional order linear time invariant systems. A new definition of the fractional derivative without a singular kernel has been proposed by Caputo and Fabrizio in 2015 [15]. This new fractional derivative is less affected by past, compared with the classical Caputo fractional derivative which shows a slow stabilization. The characteristic equation in this paper can be extended to all linear fractional-order systems described by the CF derivative. Theorems in this paper can be used to analyse local stability of nonlinear system described by the CF derivative

Problem Formulation and Preliminaries

Main Results

Numerical Examples

Conclusions