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Neural networks have been increasingly employed in Model Predictive Controller (MPC) to control nonlinear dynamic systems. However, MPC still poses a problem that an achievable update rate is insufficient to cope with model uncertainty and external disturbances. In this paper, we present a novel control scheme that can design an optimal tracking controller using the neural network dynamics of the MPC, making it possible to be applied as a plug-and-play extension for any existing model-based feedforward controller. We also describe how our method handles a neural network containing history information, which does not follow a general form of dynamics. The proposed method is evaluated by its performance in classical control benchmarks with external disturbances. We also extend our control framework to be applied in an aggressive autonomous driving task with unknown friction. In all experiments, our method outperformed the compared methods by a large margin. Our controller also showed low control chattering levels, demonstrating that our feedback controller does not interfere with the optimal command of MPC.