## 4 Closed-Loop Stability Analysis
(2) Levitation with disturbance under uncertain time delay. The simulation results for the PID controller without disturbance (air-gap response, air-gap changing velocity, and control current) are illustrated in Figure 4. The control current has fluctuations in Figure 4, which is related to the excessive gain. However, a small gain will cause the system to be unstable. The essential reason is that the maglev system is a highly nonlinear system, and the PID is a controller based on linearization theory, which is easy to fail when applied to a nonlinear maglev system. The simulation results for the SMC controller without disturbance are included in Figures 5 and 6 depicts the simulation results for the proposed adaptive robust levitation controller without disturbance. Figures 4-6 show that the performance of the PID controller deteriorated with 29.4% overshoot, 0.248 mm steady-state error, and 2.1 s settling time and the SMC controller with 0.248 mm steady-state error and 1.05 s settling time. The performance of the proposed adaptive robust levitation controller was excellent without overshoot and steady-state error. \begin{table}
\begin{tabular}{c c} \hline \hline
**Physical Quantity** & **Value** \\ \hline Levitation mass (\(k_{S}\)) & 100 \\ Coil resistance (\(\Omega\)) & 0.420 \\ Coil inductance (\(mtH\)) & 177.8 \\ Number of coil turns \(N_{m}\) & 340 \\ Magnetic-pole width (\(mm\)) & 28 \\ Magnetic-pole length (\(mm\)) & 700 \\ Air permeability \(\mu_{0}/(H\cdot m^{-1})\) & \(4\pi\times 10^{-7}\) \\ Target levitation air gap (\(mm\)) & 8.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter values for the magnetic-suspension system. Figure 4: Simulation results for the PID controller without disturbance (dashed line: the reference airgap). Figure 5: Simulation results for the sliding mode controller (SMC) without disturbance (dashed line: the reference airgap). The simulation results for the PID controller with external disturbances (the air-gap response, the air-gap changing velocity, and the control current) are shown in Figure 7. The simulation results for the SMC controller with disturbances are included in Figures 8 and 9 and depict the simulation results for the proposed adaptive levitation controller with external disturbances. Figure 7 shows that levitation failure appeared with the PID controller. The air gap of the system with the SMC controller in Figure 8 fluctuated above 3.5% and the current response had chattering phenomenon. The proposed adaptive levitation controller could still effectively levitate the train. After stabilization, the air gap of the system fluctuated within 1%. The control performance indexes can be summarized in Table 2 as below. Figure 6: Simulation results for the proposed robust levitation controller without disturbance (red dashed line: the reference airgap). Figure 7: Simulation results for the PID controller with external disturbance. (**a**) Nonlinear external disturbance. (**b**) Airgap response. Figure 8: Simulation results for the PID controller with external disturbance. Figure 9: Simulation results for the proposed robust levitation controller with external disturbance. The simulation results demonstrated the effectiveness of the presented robust levitation controller to deal with uncertain time delay and external disturbances while staying asymptotically stable.