## 3 Control Method  
The permanent magnet ball is in equilibrium under the action of dual electromagnets placed vertically. In order to change the displacement of the permanent magnet ball in the vertical position, it is necessary to break the equilibrium of forces acting on the ball. After removing the external magnetic field of the permanent magnet ball and adding the magnetic field instantaneously, the external force and motion of the permanent magnet ball undergo an instantaneous mutation. The process of rebalancing after constant imbalance is called the process of force imbalance. In the process of control, force imbalance is only an instantaneous state of breaking the equilibrium, so it is necessary to accurately control the imbalance and equilibrium of the force. The momentary magnetic imbalance causes the displacement of the permanent magnet ball to change, and the equilibrium is maintained immediately by control. The force control principle that realizes the ball from equilibrium to imbalance and then to equilibrium is called the force imbalance control principle. Neural network covers comprehensive and diverse fields, including computer vision (including image and video fields), voice fields, text-oriented natural language processing, intelligent actuation, smart cities, medical care, and other aspects of life. It is often used to solve nonlinear systems with uncertainties, which is its area of expertise. Because neural network has good adaptability to complex systems with nonlinear and ambiguous models [22; 23], it has been widely used in the field of magnetic levitation technology [24; 25]. Neural network has good tracking performance. PID control with neural network algorithm can change the PID parameters in real time according to the position change of the ball during control, so that the magnetic levitation system always maintains good control ability. The permanent magnet ball obtains a better stable levitation state and levitating motion performance within the levitation range. Neural network PID controller is adopted to actuate a new external magnetic field in the paper, which relies on the actuation principles of force imbalance. For the magnetic levitation actuation and motion control system, the electromagnetic force is exerted by the electromagnetic actuators on the permanent magnet ball. No matter how the position of the permanent magnet ball changes, the upper electromagnetic force always maintains 2 \(mg\) (twice the gravity of the ball), and the lower electromagnetic force always maintains mg. The dynamic models of the electromagnetic actuators 1 and 2 are established, respectively. Laplace transformations are performed on the model, in which the excitation currents \(I_{1}\)(s) and \(I_{2}\)(s) are used as input, whilst the vector combination of  
Figure 3: External magnetic actuation model of dual electromagnetic actuators. levitation gaps [\(\delta_{1}\)\(\delta_{2}\)]T is the output. The magnetic levitation actuation and motion control system of the permanent magnet ball are established as:  
\[ \begin{bmatrix}\delta_{1}\\ \delta_{2}\end{bmatrix} = \begin{bmatrix}(-\frac{k_{11}}{m})/(s^{2}+\frac{k_{k1}}{ m})&0\\ 0&(-\frac{k_{22}}{m})/(s^{2}+\frac{k_{22}}{m})\end{bmatrix}\begin{bmatrix}I_{1} \\ I_{2}\end{bmatrix}  \tag{4}\]  
where \(k_{i1}\) and \(k_{i2}\) (N/A) are, respectively, the current stiffness coefficients of the upper and lower actuators; \(k_{k1}\) and \(k_{k2}\) (N/m) are the displacement stiffness coefficients of the upper and lower actuators, respectively; \(I_{1}\)(s) and \(I_{2}\)(s) are the Laplace transform of the coil winding currents of electromagnetic actuators 1 and 2, respectively; \(\delta_{1}\)(s) and \(\delta_{2}\)(s) are the Laplace transform of levitation gaps between the permanent magnet ball and electromagnetic actuators 1 and 2. Establish the state space model of the actuation system as follows:  
\[ \begin{split}\dot{x}&=\begin{bmatrix}A_{11}&0\\ 0&A_{22}\end{bmatrix}|x+Bi\\ \delta&=Cx\end{split}