## 5 Force Imbalance Levitating Motion Control Test  
After achieving the stable levitation, the step-by-step actuation control method is used to realize the long-distance levitating motion control of a permanent magnet ball. Within the control range of the levitation system, the continuous levitating motion is divided into a series of discrete transition points. The position of each discrete point can be derived from the electromagnetic force equation through the system model. Then, the controller parameters of each transition point are obtained through adjustment. For stability of the controlled object, the continuous position of the permanent magnet ball among the transition points is used to realize the actuation, and the control parameters are quickly updated at each step input signal to ensure the levitating stability of the ball at each transition point. In order to better observe the control effect of the neural network PID controller, the sine signal of \(T=2\pi\) s and square wave signal of \(T=10\) s are used to track and analyze the characteristics of the controller. The simulation results shown in Figure 10 show that the input signal responded well to the control system, and the stable levitation control of the controlled object can be achieved well. The system has a fast response speed and a small tracking error, so it has good dynamic and static performance. For step-by-step actuation control, when the permanent magnet ball directly responds to a large step input signal, due to the non-linear relationship between electromagnetic force on the permanent magnet ball exerted by the actuators and levitation gap, the ball is prone to drop out of control. In addition, the significant overshoot means that the controlled object's dynamic and static characteristics are not stable and accurate. In order to study the effect of different step inputs on the motion performance, the step inputs are set to 1.0 mm, 0.5 mm and 0.2 mm, respectively. The test results are shown in Figure 11 for the descending motion of the permanent magnet ball from the \(-19.40\) mm to the \(-20.40\) mm and the ascending motion from the \(-20.40\) mm to the \(-19.40\) mm. Figure 9: Permanent magnet ball stable levitation effect of the permanent magnet ball. As can be seen from Figure 11, whether the permanent magnet ball is rising or falling, compared with the direct response, the overshoot of the step response is reduced significantly by using the step-by-step levitating motion control method. The smaller the step is input, the smaller the overshoot becomes. The adjustment time of the step-by-step control is a bit longer than the continuous control, but the step response process is more stable and accurate. The reliability of test results of different step inputs is verified by 10 sets of repeatability tests. The repeatability test results of the descending motion are shown in Table 2, and the repeatability test results of the ascending motion are shown in Table 3. In Tables 2 and 3, \(\sigma\)% denotes overshoot, t denotes adjustment time, AVG represents average standard, and \(\sigma\) denotes deviation. Figure 11: Dynamic real-time position response of three step inputs of controlled levitating object. (**a**) Descending motion; (**b**) Ascending motion. Figure 10: Tracking response of neural network PID control. (**a**) Sine wave tracking simulation; (**b**) Square wave tracking simulation. It can be seen from Table 3 that during the descending motion, and compared with the 1.0 mm step input, the overshoot of 0.5 mm step input is reduced by 37.3% and the adjustment time is increased by 34.3%. Compared with the 1.0 mm step input, the overshoot of 0.2 mm step length is reduced by 75.4% and the adjustment time is increased by 37.9%. It can be seen from Table 3 that during the ascending motion, and compared to the 1.0 mm step input, the overshoot of 0.5 mm step input is reduced by 42.7% and the adjustment time is increased by 29.7%.