Backstepping-sliding-mode-based prescribed-time control for a class of nonlinear systems with disturbances

This article addresses the prescribed-time stability (PTS) problem for a class of nonlinear strict-feedback systems subject to unknown disturbances. Herein, a novel backstepping-sliding-mode (BSM) composite control structure is proposed: a dynamic time-varying sliding-mode surface is integrated into the final step of the backstepping design, effectively suppressing matched disturbances and simplifying the implementation complexity. Specifically speaking, 1) for systems with known nonlinear terms, time-varying gains are directly embedded within the backstepping virtual control laws to achieve PTS; 2) for the case of unknown nonlinear terms, by combining adaptive laws with the BSM framework, the controlled system is guaranteed to converge within the prescribed-time and maintain globally stability thereafter. Moreover, the prescribed time $T_p$ is designer-prespecified and does not depend on initial conditions or controller gains. Finally, two numerical simulation examples validate the convergence performance and robustness of the proposed approaches.