In applications of echo state network (ESN), the Wiener-Hopf equation is usually used to learn the ESN’s output
connect weights, but can hardly ensure the stability of the autonomous ESNs. Therefore, The reasons for the loss of the
stability are analyzed firstly, and a sufficient condition of the Lyapunov stability for the autonomous ESNs is proposed
and proved. Then the output connect weight learning problem is translated into an optimization problem with a nonlinear
constraint. Particle swarm optimization algorithm is employed to solve the optimization problem. Finally, the simulation
results show that the method proposed can not only result in high-precision prediction outputs of the ESN, but also ensure its
Lyapunov stability.