Nussbaum gain adaptive neural control for stochastic pure-feedback nonlinear time-delay systems with full-state constraints

In this paper, the problem concerned with adaptive approximation-based control is discussed for a class of stochastic pure-feedback nonlinear time-delay systems with unknown direction control gains and full-state constraints. In the controller design process, the approximation capability of neural networks is utilized to identify the unknown nonlinearities, the appropriate Lyapunov-Krasovskii functionals are constructed to compensate the unknown time-delay terms, barrier Lyapunov functions (BLFs) are designed to ensure that the state variables are constrained, and the Nussbaum-type gain function is used to solve the difficulties caused by the unknown virtual control gains. Then, based on adaptive backstepping technique and Lyapunov stability theory, a robust control scheme is presented, and the developed controller decreases the number of learning parameters and thus reduces the computational burden. It is shown that the proposed controller can guarantee that all the signals in the closed-loop system are 4-Moment (or 2 Moment) semi-globally uniformly ultimately bounded (SGUUB) and the tracking error converges to a compact set of the origin. Finally, two simulation examples are included to validate the effectiveness of the proposed approach.