► We introduce a hysteresis functional link artificial neural network (HFLANN) structure to approximate the dynamics of a shape memory alloy actuator. ► HFLANN includes hysteresis operator and functional link artificial neural network. ► A model predictive controller based on the HFLANN is newly derived. ► The effectiveness of the proposed controller is verified through the simulation and experiments. In this paper, a modified Hysteresis Functional Link Artificial Neural Network (HFLANN) is proposed to identify and control a Shape Memory Alloy (SMA) actuator, which has an inherent hysteresis phenomenon. In this structure, a hysteresis operator combined with the Functional Link Artificial Neural Network (FLANN) to employ the hysteresis phenomenon and the dynamic of the SMA actuator. The hysteresis operator is introduced to capture the SMA hysteresis. And the FLANN is employed to approximate the dynamic of the system. In identification problem, the FLANN parameters are trained by Particle Swarm Optimization technique. For control problem, a Model Predictive Controller based HFLANN is derived to control the system. The identification results show that the HFLANN can employ for the SMA dynamic. The simulation and experimental results demonstrated the effectiveness of the proposed algorithm. The SMA hysteresis phenomenon is compensated completely by proposed controller.