## 2 Theoretical Background and Preliminaries
### Nn-Narx Modeling of the MI System  
Nature successfully builds up biological neural systems to perform control and adaptation tasks of living beings effectively in their complicated conditions and dynamically changing environments. To benefit from these assets, ANNs, which resemble biological neural systems, are widely utilized in intelligent system design due to the benefits of the learning and prediction potential of ANN algorithms. Autoregressive neural network models can learn responses of linear and nonlinear dynamical systems from sampled input and output data [21; 24]. Therefore, these models can gain a significant flexibility for multi-loop MRAC-FOPID control structures to be easily employed in real-world applications. NARX modeling can automatically learn response of a wide-range of system models in black box form and this reduces the need for guessing the model structure prior to model identification. The reference model is automatically identified from data that are captured online from the input and output of controlled systems. This point is an important contribution of this study to transform the multi-loop MRAC-FOPID control structures into an intelligent control system and thus increase \begin{table}
\begin{tabular}{l l l} \hline \hline
**Parameter** & **Physical Description** & **Unit** \\ \hline \(m=5.7100\times 10^{-2}\) & mass of ball & [kg] \\ \(g=9.81\) & gravity constant & [\(m/s^{2}\)] \\ \(F_{em}=f(x_{1},x_{3})\) & & [N] \\ \(F_{em}p_{1}=1.7521\times 10^{-2}\) & electromagnetic force & [H] \\ \(F_{em}p_{2}=5.8231\times 10^{-3}\) & electromagnetic force & [m] \\ \(f_{i}(x_{1})\) & & [1/s] \\ \(f_{i}p_{1}=1.4142\times 10^{-4}\) & & [ms] \\ \(f_{i}p_{2}=4.5626\times 10^{-3}\) & & [m] \\ \(c_{i}=2.4300\times 10^{-2}\) & actuator value & [A] \\ \(k_{i}=2.5165\) & & actuator value & [A] \\ \(x_{3 MIN}=3.8840\times 10^{-2}\) & limitation for current & [A] \\ \(u_{MIN}=4.9800\times 10^{-3}\) & limitation for voltage & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physical parameters of the ML system. effectiveness of MRAC-FOPID control in practical nonlinear control applications. For experimental validation, the proposed control structure was employed for the control of a magnetic levitation system. The magnetic levitation control introduces a highly nonlinear, unstable system control problem. The authors observed that the performance of multi-loop MRAC-FOPID control with the MIT rule largely depends on the capability of the reference model to represent the leading closed-loop dynamics of the experimental ML system. In [9], a linear model of the experimental ML system was used to obtain the transfer function of the closed-loop retuning FOPID control system. This linear function was used as the reference model. This modeling approach comes out with two drawbacks for multi-loop MRAC-FOPID control structure which was already mentioned. In the current study, to deal with these drawbacks, NARX modeling was employed to obtain a more accurate reference model. The use of neural networks in system identification allows utilization of machine learning techniques in control, and this enables the multi-loop MRAC-FOPID control structure with NARX model to be more adaptive and intelligent compared to previous configurations. Artificial neural networks can be successfully applied in the modeling of complex nonlinear systems, systems with disturbances and insufficiently known parameters, unpredictable and uncoordinated systems [19; 56]. Figure 4 shows a block diagram of the NARX model. It is composed of time delay blocks (TDL) and conventional artificial neuron models with input and loop weight coefficients (IW and LW, respectively), bias (b), and activation function (f). The vector \(u(t)\) is transmitted to the system as a two-layer input and is realized by passing a time delay block (b1) and in that way a part of Equation (33), is realized.