To address the elastic vibration, difficulty in accurately obtaining model parameters, and input saturation of flexible manipulators under finite sampling frequency conditions, a composite control method combining sliding mode control and autoregressive-model-with-exogenous-input-based LQR is proposed. First, the dynamic model of the flexible manipulator is established using the Lagrange method and the assumed mode method, and is decomposed into a slow subsystem describing rigid-body motion and a fast subsystem describing elastic vibration based on singular perturbation theory. Second, for the slow subsystem, an RBF neural network sliding mode controller with input saturation compensation is designed to achieve precise trajectory tracking and cope with system uncertainties. For the fast subsystem, an autoregressive model with exogenous input(ARX model) is used to identify the relationship between the input torque and the output voltage of the piezoelectric sensor, and an LQR controller is then designed for data-driven vibration suppression. Simulation results show that the proposed composite control method can effectively suppress flexible vibration in both point-to-point control and sinusoidal trajectory tracking tasks, while maintaining good trajectory tracking performance and robustness under finite sampling frequency and input saturation conditions.