## 4 Force Imbalance Stable Levitation Test  
The overshoot \(\sigma\)% is the primary parameter of the active magnetic levitation actuation system. The actual meaning of the overshoot in the paper is the magnitude of position change of the permanent magnet ball in the vertical direction. The overshoot characterizes the feasibility of the permanent magnet ball's levitating motion and the stability during the motion. A lower overshoot can enable the controlled ball to obtain more precise position control in practical applications and reduces the possibility of damage to the inspected surface by the controlled object. Therefore, the optimization of the overshoot is the significance of the active magnetic levitation actuation and the premise of the system stability. ANSYS Maxwell finite element analysis conditions are closer to the test environment. According to the simulation results of \(F\) on the permanent magnet ball, the coefficients \(a\), \(b\), \(c\) and \(d\) of (2) are fitted by the least square method. The multi-parameter fitting diagram is shown in Figure 5. According to the fitting result, the coefficient values of (2) are \(a_{1}=a_{2}=111300.000\), \(b_{1}=b_{2}=-301.000\), \(c_{1}=c_{2}=8.691\), and \(d_{1}=d_{2}=4.199\). The equation of electromagnetic force \(F\) is exerted on the permanent magnet ball by the electromagnetic actuator, i.e.,Through calculation, the parameter is determined as \(k_{\text{i1}}=0.14371\), \(k_{\text{S1}}=-16.6152\), \(k_{\text{i2}}=-0.027568\), \(k_{\text{S2}}=-4.197625\) in (4). According to the actual values of current stiffness coefficients and displacement stiffness coefficients, the coefficient matrices in (5) are:  
\[A_{11}=\left[ \begin{array}{cc}0&63.69\\ 32&0\end{array} \right]A_{22}=\left[ \begin{array}{cc}0&16.1\\ 32&0\end{array} \right]B=\left[ \begin{array}{cc}1&0\\ 0&0\\ 0&0.25\\ 0&0\end{array} \right]C=\left[ \begin{array}{cc}0&-0.551&0&0\\ 0&0&0.4228\end{array} \right] \tag{12}\]  
The neural network PID controller and model-based controller are built in the simulation environment, and the simulation results of overshoot and response time are compared. For the simulation conditions the same mathematical model is used, and the same initial parameters are used for different control methods. The parameter values are \(k_{\text{p}}=-12.6\), \(k_{\text{i}}=-9\) and \(k_{\text{d}}=-0.84\). The error variable is used as the discrete input signal of the neural network PID controller system equation, and the output of the system equation is used as the control variable. For the simulation results of Figure 6, the overshoot of step response output of the neural network PID controller of the magnetic levitation actuation and motion control system is \(4.9\%\), and the overshoot of the model-based controller is \(7.5\%\). Magnetic levitating system with the neuron network PID controller has a better stability. The system with dual actuators has a small overshoot and short adjustment time, so the system has good dynamic and static characteristics. The proposed controller parameters are adjusted more stably, indicating that the mechanical structure of the system is more reasonable, the control method is more applicable, and the tracking error of the control system is smaller. The neural network PID controller with dual electromagnetic actuators can realize the stable levitation of the permanent magnet ball. The steady-state error of output signals is less than \(0.1\%\), which meets system performance requirements. Figure 5: External magnetic field simulation and data fitting of the magnetic levitation actuation and motion control system. (**a**) ANSYS Maxell magnetic field simulation of dual electromagnetic actuators; (**b**) Data fitting of electromagnetic force.