“…The integration of ML and MPC has experienced consistent growth in recent years, resulting in various categories of applications [45]- [47], e.g., offline modeling utilizes measurement data to create ML models for MPC [2], [48], [49]; online learning adjusts MPC model coefficients in real-time [26], [50], [51]; ML in imitation of MPC replicates MPC behavior in realtime, with successful applications in various industries [52]- [54] and improvements of computational efficiency [52], [55], [56]; ML in control structure of MPC involves ML as an add-on or embedded controller [57]- [59]; finally, MPC can also work as a safe learning controller in learning algorithms to address control constraints [60], [61]. Complex nonlinear interactions that may be difficult for conventional mathematical and statistical models to capture, particularly in complex systems, can be captured by ML-based models such as nonlinear autoregressive models with exogenous inputs (NARX) [62], [63], feed-forward neural networks (FNNs) [64], deep neural networks (DNNs) [65], and recurrent neural networks (RNNs) [66], can be used as process models, have the potential to effectively represent complicated physical systems, have demonstrated the ability to successfully simulate dynamic processes inside the MPC framework, giving precise approximations and quicker convergence in MPC. In this regard, it would be beneficial to place a stronger emphasis on MPC controllers utilizing NARX models, as this approach aligns with the methodology outlined in our paper.…”