An Adaptive Recursive Gain Tracking Control for a Class of Nonaffine Pure-Feedback Nonlinear Systems with Partly Unknown Dynamics

This paper addresses the asymptotic tracking control problem for a class of nonaffine pure-feedback nonlinear systems with partially unknown internal dynamics. Traditional backstepping control is plagued by the notorious complexity explosion arising from repeated differentiation of virtual control signals. Dynamic surface control (DSC) alleviates the computational burden by introducing low-pass filters; nevertheless, it can only guarantee semi-global uniformly ultimately bounded stability instead of exact asymptotic tracking, and its control performance is highly sensitive to filter parameter tuning. Conventional recursive gain control (RGC) eliminates the reliance on filters, yet it merely ensures uniform ultimate boundedness of tracking errors without asymptotic convergence. To overcome these limitations, this paper develops a novel adaptive recursive gain tracking control (ARGTC) scheme. By seamlessly integrating recursive gain construction and online adaptive parameter update laws, the proposed filter-free controller avoids the complexity explosion of backstepping and completely removes the auxiliary low-pass filters indispensable to DSC. Rigorous Lyapunov stability analysis together with Barbálat’s lemma mathematically proves that the closed-loop tracking error converges asymptotically to zero while all closed-loop signals remain uniformly bounded. The designed adaptive update laws obviate the requirement for prior knowledge of the upper bounds of nonlinear uncertainties, thereby reducing design conservatism. Comprehensive comparative numerical simulations are carried out under various operating conditions, including sinusoidal, square-wave and multi-frequency reference tracking, measurement noise disturbances, time-varying parameter perturbations, actuator saturation constraints, multiple initial-condition tests, and filter parameter sensitivity analysis. Quantitative performance metrics and diverse visual results consistently verify that the proposed ARGTC achieves superior transient response, steady-state accuracy, low computational cost, and strong robustness in comparison with backstepping, RGC, DSC, adaptive DSC, approximation-free prescribed performance control approaches and adaptive sliding-mode control (ASMC).