# Magnetic Levitation Belt Conveyor Control System Based on Multi-Sensor Fusion
## 2 System Model  
The magnetic levitation ball device mainly consists of electromagnets, photoelectric position sensors, a power supply, amplification and compensation devices, data acquisition cards, control objects (steel balls), and other components. The magnetic levitation ball system is a typical suction-floating levitation system. The basic structure of the system is shown in Figure 2. The electromagnetic force \(F(i,x)\) is generated by passing a certain current through the electromagnet winding. Under the condition that the current in the electromagnet winding is controlled such that the generated electromagnetic force is balanced with the gravity \(mg\) of the steel ball, the steel ball can be suspended in air and in equilibrium. Assuming that the levitating object is not subjected to other disturbing forces, and the force analysis of the object is only subjected to vertical downward gravity \(mg\) and its reverse electromagnetic force \(F(i,x)\), then \(x\) is the distance from the center point of the steel ball to the lower edge of the electromagnet. To simplify the derivation process, the following assumptions are typically made:  
* Neglecting the magnetoresistance in the steel ball and the electromagnet, the magnetoresistance of the magnetic circuit consisting of the electromagnet and the steel ball is concentrated in the suspended air gap. * The leakage flux is neglected, where the flux passes through the air gap of the external pole of the electromagnet. * The magnetic flux is uniformly distributed at the air gap, thus neglecting the edge effects. Figure 2: Basic structure diagram of magnetic levitation ball system. Based on the above assumptions, the mathematical model of the electromagnetic levitation ball system is obtained as follows [24]:  
\[\left\{ \begin{array}{l}m\frac{d^{2}x(t)}{dt^{2}}=F(i,x)+mg\\ F(i,x)=\frac{\mu_{0}\alpha N^{2}}{4}\Big{(}\frac{i}{x}\Big{)}^{2}=K\Big{(} \frac{i}{x}\Big{)}^{2}\\ U(t)=Ri(t)+\frac{\mu_{0}\alpha N^{2}}{2x}\frac{di}{dt}-\frac{\mu_{0}\alpha N^{ 2}i}{2x^{2}}\frac{dx}{dt}\\ F(i_{0},x_{0})+mg=0\end{array} \right. \tag{1}\]  
where \(m\) is the mass of the ball, \(x\) is the levitation distance of the ball, \(i\) is the electromagnet current, \(F(i,x)\) is the electromagnetic suction force, \(g\) is the acceleration due to gravity, \(u_{0}\) is the vacuum permeability, \(A\) is the core area, \(N\) denotes the number of turns in the electromagnet coil, \(U(t)\) represents the voltage across the electromagnet coil, and \(R\) is the resistance of the electromagnet coil. Additionally, \(i_{0}\) and \(x_{0}\) are the electromagnetic current and levitation air gap at the equilibrium point, respectively. It can be observed that this levitation system is a typical nonlinear system, and its nonlinear part must be linearized first. Equation (1) is linearized near the equilibrium operating point (\(i_{0}\),\(x_{0}\)) using Taylor series expansion. Accordingly, the approximate linearized model is obtained as  
\[m\frac{d^{2}x}{dt^{2}}=\frac{2 Ki_{0}}{x_{0}{}^{2}}i-\frac{2{Ki_{0}}^{2}}{x_{0}{ }^{3}}x \tag{2}\]  
The input quantity of the system object is defined as the input voltage of the power amplifier, which is the control voltage \(U_{in}\). Moreover, the output voltage reflected by the output quantity of the system object \(x\) is \(U_{out}\) (sensor output voltage). According to Equation (2), the system state variables are taken to be \(x_{1}=u_{out}\), \(x_{2}=u_{out}\), respectively, and the equation of the state of the system in continuous time is obtained according to the state space theory as  
\[\left\{ \begin{array}{l}\dot{x}=Ax+BU\\ y=Cx\end{array} \right. \tag{3}\]  
where \(A= \begin{bmatrix}0&1\\ \frac{2g}{x_{0}}&0\end{bmatrix} \), \(B= \begin{bmatrix}0\\ -\frac{2g\cdot K_{s}}{i_{0}\cdot K_{a}}\end{bmatrix} \), \(C= \begin{bmatrix}1&0\end{bmatrix} \). The open-loop pole of the system is \(s=\pm\sqrt{2g/x_{0}}\).