Vertical vibration suppression of rolling mills based on improved approach rate and nonlinear disturbance observer

Considering the influence of nonlinear stiffness in hydraulic cylinders, a dynamic model of the vertical vibration of the working roll was established. To address the severe oscillation and slow convergence associated with the traditional exponential reaching law (ERL), an improved double power reaching law (IDPRL) was developed. Based on Lyapunov stability theory, the stability, convergence and robustness of the proposed reaching law were theoretically analysed and proven. On this basis, a sliding mode displacement controller (SMDC) employing the IDPRL was designed. To further enhance system robustness and reduce steady-state error, a nonlinear disturbance observer (NDO) was introduced to estimate external disturbances in real time, and the estimated disturbances were incorporated into the control law in a feedforward manner, forming a composite IDPRL-NDO control strategy. Simulation and experimental results demonstrate that, compared with ERL and the fast power reaching law, the proposed IDPRL-NDO strategy effectively alleviates chattering, improves convergence speed by approximately 8% and 11%, and reduces trajectory tracking error by about 75% and 90%, respectively. These results indicate that the proposed method achieves higher tracking accuracy and faster dynamic response, providing an effective and practical control solution for rolling mill vibration suppression.