Prescribed performance-barrier Lyapunov function for the adaptive control of unknown pure-feedback systems with full-state constraints

In this paper, an adaptive state-feedback control technique is proposed for a class of unknown pure-feedback systems. A remarkable feature is that not only the problem of full-state constraints and prescribed performance tracking is solved together, but also the design is an approximation-free control scheme for pure-feedback systems with completely unknown nonlinearities. These properties will lead to a difficult task for designing a stable controller. To this end, a novel prescribed performance-barrier Lyapunov function is developed to guarantee that all the state constraints are not violated and the tracking error is preserved within a specified prescribed performance bound at all times, simultaneously. Then, by utilizing the mean value theorem, Nussbaum gain technique, a low-pass filter and a novel bounded estimation approach at each step of back-stepping procedure, a novel adaptive dynamics surface control scheme is developed to remove the difficulties of pure-feedback characteristic, unknown nonlinearities, unknown control direction and explosion of complexity, which can guarantee that the proposed design is universal and low-complexity. Moreover, it is proved that all the signals in the closed-loop system are global uniformly bounded. Two simulation studies are worked out to illustrate the performance of the proposed approach.