# Development of a Conveyor Cart with Magnetic Levitation Mechanism Based on Multi Control Strategies
## 4 Magnetic Levitation Unit Controller Strategies
### Magnetic Levitation Math Model Analysis  
To build the mathematical model of the magnetic levitation car system, there are some assumptions to be set:  
* Magnetic flux leakage, edge effect and the reluctance between the rail and the electromagnet are ignored [10]  
Figure 5: Result of the suspension force measurement and the force analysis of Maglev unit. (**a**) Force analysis of Maglev unit. (**b**) Result of suspension force measurement.  
Figure 6: The electrical circuit of each magnetic unit control system; dSPACE and MATLAB/SIMULIK installed in the PC; and the eddy sensor located on the top to measure the Y axis displacement.  
* The electromagnet unit is a homogeneous sphere, the magnetic force is concentrated on the center of it [10]
* There is a linear relationship between the output current and input voltage of the power amplifier, and there is no delay [10]  
Constructing a model for the control of a four-unit cart is important. However, when it comes to controlling the four-unit model, factors such as interference and slight deformations make precise control extremely challenging. Consequently, in this paper, for the simplicity of the control system design, we do not consider the four-unit model for control; instead, our focus is on controlling each individual unit.  
To have a good-understanding of the mathematical relationship of the magnetic levitation system, Table 2 shows the parameters named in the nomenclature.  
With the assumptions, the mathematical relationship of the magnetic levitation system can be established through the theoretical derivation. From [PERSON]'s second law:  
\[m\frac{d^{2}x(t)}{dt^{2}}=F(i,x)+mg \tag{1}\]  
where \(m\) is the mass of the float unit, \(g\) is the gravity acceleration, and \(x\) is the air gap of bracket coil and guide rail. \(F(i,x)\) is the electromagnetic force on the magnetic unit, and \(i\) is the instantaneous current of the electromagnet winding. Based on the Kirchhoff laws and Biot-Savart law of the magnetic circuit, the electromagnetic force on the float unit can be deduced as follows [11]:  
\[F(i,x)=-\frac{\mu AN^{2}}{4}(\frac{i}{x})^{2} \tag{2}\]  
where \(K_{c}\) is defined the \(K_{c}=-\mu AN^{2}/4\), which is the constant coefficient related to the magnetic flux of the electromagnetic winding. When the bracket is in equilibrium position, the formula is obtained according to the mechanical balance principle as:  
\[mg+F(i_{0},x_{0})=0 \tag{3}\]  
which \(i_{0},x_{0}\) are the air gap and the current in the coil when the magnetic levitation ball is in balance. Combing Equations (1)-(3) and taking their Laplace transform, the relationship shows Equation (4)  
\[x(s)=\frac{k_{i}}{ms^{2}-k_{s}}i(s) \tag{4}\]  
where \(k_{i}=K_{c}i_{0}{}^{2}/2x_{0}^{2}\) and \(k_{s}=K_{c}i_{0}{}^{2}/2x_{0}^{3}\). In this research, we measure \(x_{0}\) and \(i_{0}\) at the selected equilibrium point, and thus, the \(k_{i}\) and \(k_{s}\) can also be obtained. The specific values are shown in Table 2.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Description** & **Numerical Value** \\ \hline \(x^{*}\) & Target position & 0 mm \\ \(x\) & Gap position & 4 mm \\ \(i\) & Control current & 0.05 A \\ \(\mu_{0}\) & Vacuum permeability & \(4\pi\times 10^{-7}\) (H/m) \\ \(A\) & Magnetic permeability area & \(\pi\times 10^{-4}\) (m\({}^{2}\)) \\ \(N\) & Coil turns & 670 \\ \(m\) & Suspend float & 70 kg \\ \(k_{i}\) & Gain of current and the force & 85 (N/A) \\ \(k_{s}\) & Gain of position and the force & \(1.5\times 10^{5}\) (N/m) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Nomenclature of physical parameters of the magnetic levitation unit.