This paper studies the adaptive asymptotic tracking problem for a class of unknown nonlinear systems in pure-feedback form. Different from the traditional literatures which only tackle the bounded tracking problem for pure-feedback systems, this paper investigates the asymptotic tracking problem by developing a novel controller design method. Moreover, the differentiable assumption on nonaffine functions is canceled, and only a mild semi-bounded assumption is required as the controllability condition. By utilizing Lyapunov theorem, it is proved that all the variables of the resulting closed-loop system are semiglobally uniformly ultimately bounded, and the output tracking error can converge to zero asymptotically by choosing design parameters appropriately. Finally, a simulation result is presented to verify the effectiveness of the proposed control scheme.

Introduction

In the last several decades, adaptive control techniques have been found to be powerful for controlling the trianglestructural nonlinear systems in terms of either pure-feedback or strict-feedback [1]–[۱۷]. Specifically, pure-feedback systems do not have the explicit control input, which makes the control design very difficult and draws much interest in the control community for a long time [8]–[۱۷]. In [10], to solve the prescribed performance tracking control problem, a low-complexity control scheme is designed for a class of unknown pure-feedback systems. In [11], a predefinedtracking-constrained-based adaptive control scheme is developed for a class of switched stochastic nonlinear systems in the pure-feedback form with dead zone output. By employ the mean value theorem to convert the nonaffine function into an affine form, all these studies referred above have presented a unified and general framework for pure-feedback nonlinear control system design. However, there are still a number of issues should have been further studied, such as, the mean value theorem requires the nonaffine function must be differentiable with respect to the control variables or input. In the hope to overcome these problems, in [12], a pioneering modeling method is presented under the mild assumptions. Instead of utilizing mean value theorem and implicit function theorem, this control method does not require that the nonaffine functions must be differentiable. Subsequently, the controllability conditions are relaxed to semi-bounded and discontinuity in [13] and [14], respectively. In [15], the further research is devoted to a class of more general MIMO pure-feedback nonlinear systems with periodic disturbances.