Analysis, Modeling and Stability of Fractional Order Differential Systems 2

8Lyapunov Stability of Commensurate Order Fractional Systems

8.1. Introduction

Stability analysis of a linear integer order differential system is an elementary problem, which is characterized by a popular LMI stability condition [BOY 94]. The true difficulty concerns the stability of nonlinear systems, where different methods have been proposed [KHA 96].

Oddly, the first researchers to propose stability conditions for fractional differential systems directly investigated the nonlinear case, thinking perhaps that the linear case was as elementary as the linear integer order case [LI 09, MOM 04]. However, an LMI condition for linear FDEs has been proposed [SAB 10b] without explicit reference to Lyapunov stability.

In fact, a reasoned approach would consist of first treating the linear case, and then trying to solve all the difficulties induced by the fractional order case. Among all these difficulties is the definition of an appropriate Lyapunov function [TRI 11b]. The choice is not appropriate, as demonstrated in the previous chapter.

Another difficulty, and not the least, concerns the definition of a stability criterion. In the integer order case, an obvious solution is because the energy of a stable system is necessarily decreasing [LYA 07, NAS 68]. This physical principle ...