Preset-Time Sliding Mode Control Based on Prescribed Error Evolution

For a class of second-order nonlinear systems subject to matched disturbances, this paper investigates preset-time establishment and subsequent high-precision maintenance, and proposes a time-varying sliding mode control method based on prescribed error evolution. First, a quintic prescribed error-evolution function satisfying smooth endpoint constraints is constructed to explicitly shape the error establishment process, thereby avoiding excessive transient control demand caused by directly injecting a large initial error into the closed loop. Then, a time-varying sliding variable is constructed around the error-evolution deviation, and a continuous control law is designed by combining a bounded nonlinear mapping with a distance-dependent scheduling gain, achieving coordination between preset-time establishment and input smoothness. The theoretical analysis proves that, under nominal conditions, the closed-loop system can complete the establishment exactly at the preset instant; in the presence of bounded matched disturbances, the system satisfies an explicit small-neighborhood estimate at the preset instant and maintains high precision thereafter. The simulation results show that the proposed method achieves timing-consistent state establishment under different preset establishment scenarios, and significantly reduces the control peak and total input variation, thereby effectively alleviating the risk of actuator saturation.