A stability criterion for discrete-time fractional-order echo state network and its application

In this paper, combining the theory of numerical solution of fractional-order differential equation, a new model of discrete-time fractional-order echo state network (DFO-ESN) is proposed. In order to ensure that the DFO-ESN can be used for different learning tasks, the stability of DFO-ESN should be guaranteed. Through using an LMI approach, a sufficient stability criterion for DFO-ESN is given. According to the stability criterion, the selection range of reservoir parameters of DFO-ESN can be expanded, such that we can build the DFO-ESN without considering the initial conditions. For the candidate discrete-time Lyapunov function, it is shown that the reservoir states of DFO-ESN are asymptotically stable when time tends to infinity. Finally, two examples demonstrate the feasibility of stability criterion and the learning performance of DFO-ESN.