Composite Learning From Fractional-Order Fuzzy Echo State Network Control Using Historical Data

Considering the infinite memory property of fractional differential equations, and combining the advantages of fuzzy logic systems and reservoir computing of the echo state networks, a fractional-order fuzzy echo state network is proposed for estimating uncertain terms of fractional-order nonlinear systems. In order to improve its learning performance and reduce response oscillations, a modified output weight learning law is constructed by recording historical data over a period of time in a regression matrix in both time and iterative domains, in which a prediction error is defined without using a strict condition, i.e., interval excitation. Furthermore, leveraging Lyapunov stability theory and the frequency distribution model, suitable integral Lyapunov energy functions are formulated based on backstepping framework, proving that the proposed composite learning control scheme can maintain boundedness and stability of all signals within closed-loop system. Finally, the fractional-order Chua-Hartleys chaotic system and the fractional-order single-machine-infinite- bus power system are adopted for simulation to validate the validity of the proposed scheme.