## 2 Theoretical Background and Preliminaries
### Multi-Loop Mrac
These advantages make ML transportation a suitable candidate for future transportation possibilities [48; 49]. For these reasons, robust control of the ML system introduces a significant control problem that has the potential of producing technological outcomes. Several works have addressed robust control performance of ML systems; for instance, fuzzy logic controllers [50] and FOPID controllers [51; 52; 53; 9; 17] were considered. It is important to stress that disturbance rejection and vibration control are essential problems in control of ML systems. In order to compare among results including experimental and artificial models, the mathematical modeling of ML systems has been studied. Magnetic levitator can be divided into two subsystems, mechanical and electrical [50; 54]. For the electrical subsystem, it is important to single out electromagnetic coil inductance L (H), and its resistance, R (\(\Omega\)). The electrical subsystem can be described with the following well-known differential equation:  
Figure 2:4In order to determine the current in the coil, the resistor \(R_{s}\) is connected in series with the coil. In that way, voltage \(U_{s}\) can be measured across resistor \(R_{s}\), by using A/D convertor for measuring current \(i\). Now, Equation (22) can be rewritten as:  
Figure 2:5  
After applying Laplace transform we obtain:  
Figure 2:6  
On the other side, modelling mechanical subsystem can be performed simply by defining force \(F\), that represents the result of electromagnet activity to the ball:  
Figure 2:7  
where \(m\) is ball mass, \(g\) is gravitation constant, \(K_{f}\) is magnetic force constant which is valid for pair electromagnet and ball, \(i\) is the current which flows through electromagnet, \(x\) is ball distance from magnet. Using second [PERSON]'s law, Equation (25) can be rewritten as:  
Figure 2:8  
The value of electromagnetic coil current in steady state denoted as \(I_{ss}\) can be determined by Equation (26). This current defines a constant of ball position \(X_{ss}\) in a steady state case. If we take into consideration that velocity and acceleration are equal to zero in the state, then Equation (26) is given as:  
Figure 2:9  
Theoretically, regulation of the ball position can be performed by (27). However, external disturbances, uncertainty, system parameter variations, and high nonlinearity degree of the system require a feedback controller installation, which ensures stable control of these mechanical subsystems. In general, adding linearity into the system must be done first, and after that control process begins. In this report, NARX neural network is used for experimental substitution of the real system of magnetic levitation. If the training process is successfully performed, we are going to get values on  
\[J=\frac{1}{2}e_{m}^{2}. \tag{10}\]0 Magnetic Levitation System (MLS) setup in A-Lab. network outputs with minimal errors comparing to the output values of the real system. The nonlinear mathematical model of magnetic levitation system is given by the following equations [55]:  
\[J=\frac{1}{2}e_{m}^{2}. \tag{10}\]1  
\[J=\frac{1}{2}e_{m}^{2}. \tag{10}\]2  
\[J=\frac{1}{2}e_{m}^{2}. \tag{10}\]3  
\[J=\frac{1}{2}e_{m}^{2}. \tag{10}\]4  
\[J=\frac{1}{2}e_{m}^{2}. \tag{10}\]5  
where \(x_{1}\), \(x_{2}\), \(x_{3}\) are ball position, velocity and current respectively. \(u\) is the control signal. The corresponding parameters are given in \[J=\frac{1}{2}e_{m}^{2}. \tag{10}\]6  
Variables \(x_{3 MIN}\) and \(u_{MIN}\) are limitations for current and voltage to avoid break down of device. In this paper, the mathematical output has been compared simultaneously with NN and experimental results during real-time experiments. It is important to have appropriate conversion tables which include the experimental measurement to convert between quantities in order to have accuracy in the simulation.