Observer-based composite adaptive fuzzy echo state network control of uncertain pure-feedback nonlinear systems free from backstepping

Backstepping is a popular control method for uncertain pure-feedback nonlinear systems (PFNSs). However, it usually suffers from the "complexity explosion problem" caused by repeatedly differentiating virtual control inputs and the complex recursive design process. This article focuses on a novel direct control method for uncertain PFNSs with input saturation and unmeasured states without relying on the backstepping framework. The uncertain PFNS is transformed into a canonical system through a coordinate transformation. The tanh function is employed to handle the input saturation, and an observer is established to estimate unmeasurable states. Specially, a fuzzy echo state network, combining fuzzy logic system with echo state network, is created to estimate the ideal controller, and a composite adaptation law that utilizes both tracking error and the approximation of learning error is developed. Without any additional control term, all variables involved are bounded and the tracking error convergence can be guaranteed. The simulation results of two experiments validate the efficacy of the theoretical results.