Lyapunov stability of fractional-order nonlinear systems: A distributed-order approach

This paper discusses the stability issues of fractional-order nonlinear scalar systems by using the distributed-order operators and the order sensitivity method. A positivity check method is proposed by the use of initialized fractional calculus. By doing so, the fractional-order system is converted to a corresponding distributed-order one, and a group of Lyapunov function candidates of the distributed-order system are derived from the Volterra integral equations. Particularly, it is proved that the stability conditions of fractional-order and integer-order nonlinear systems are consistent with each other, which is the main contribution of this paper, and it also provides a way to the stability analysis of distributed-order nonlinear systems. Several examples are illustrated to validate the above conclusions.