Adaptive Fuzzy Echo State Network Control of Fractional-Order Large-Scale Nonlinear Systems With Time-Varying Deferred Constraints

In the traditional constrained control of nonlinear systems, the controller design usually requires that the initial value of the system meets certain strict conditions, and generally only considers static constraints. This article concentrates on the issue of adaptive fuzzy echo state network decentralized control for fractional-order (FO) large-scale nonlinear systems with strong interconnections and time-varying deferred constraints. With the backstepping technique, an FO fuzzy echo state network is constructed to approximate unknown nonlinear functions and interconnected terms in each step, which greatly removes some additional assumptions on unknown functions and provides a higher degree of design freedom and stronger robustness. A shifting function and an error transformation scheme are introduced to handle constraints against the unknown initial tracking condition. Moreover, the constraint conditions are satisfied within a specified time even if they are violated initially by using a time-varying barrier Lyapunov function. Especially, an equivalent definition of the bivariate convex function is given, and an inequality is constructed, which can be used to analyze the stability of FO systems by constructing a bivariate Lyapunov function. According to the FO Lyapunov stability theorem, the proposed adaptive controller can ensure that all the signals involved remain bounded and the tracking error possesses a fast convergence. Finally, an example of the FO single-machine-infinite bus power system illustrates the effectiveness of the proposed control strategy.