## 4 Simulation and Experimental Results  
### Simulation Analysis  
Using MATLAB R2020a to simulate and analyze stability, the parameters of the MSTMP system are listed in Table 3. The mechanical structure parameters provided in Table 3 are sufficient to realize the calculation and solution of the AMB-rotor model. As for more details about the dimensions of the machine parts, considering that this MSTMP is a self-developed product in our laboratory and has certain confidentiality, it is not convenient to provide other parameters unrelated to the modeling and control in this paper. The rotor assembly material constants of the MSTMP are shown in Table 4.  
The AMB-rotor system with a parallel co-frequency ANF results in a change in the closed-loop transfer function. Both the bandwidth factor \(\varepsilon\) and the rotational speed \(\Omega\) adversely affect the stability and even make the system unstable. As shown in Figure 14, as the closed-loop pole trajectory varies in the S-plane as \(\varepsilon\) increases, the ANF has different critical instability of \(\varepsilon\) at different \(\Omega\) (70 Hz, \(\varepsilon\) = 10, 110 Hz, \(\varepsilon\) = 100, and 120 Hz, \(\varepsilon\) = 200). Furthermore, as the root-locus dominant pole trajectory varies in the S-plane as \(\Omega\) increases, the ANF with different \(\varepsilon\) has different critical instability of \(\Omega\) (\(\varepsilon\) = 20, 90 Hz, \(\varepsilon\) = 50, 100 Hz, \(\varepsilon\) = 100, 110 Hz, \(\varepsilon\) = 200, 120 Hz, \(\varepsilon\) = 300, 140 Hz, \(\varepsilon\) = 400, 150 Hz, and \(\varepsilon\) = 500, 160 Hz). In short, dynamic rectification of the bandwidth factor \(\varepsilon\) based on displacement stiffness \(k_{h}\) perturbation can avoid the instability introduced by the inappropriate choice of bandwidth factor \(\varepsilon\) of the ANF. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Materials** & **Young’s Modulus (Pa)** & **Density (kg m\({}^{3}\))** & **Poisson’s Ratio** \\ \hline
1 Cr18 Ni9 Ti & \(1.84\times 10^{11}\) & 7900 & 0.31 \\ Sm2 Co17 & \(1.08\times 10^{11}\) & 8400 & 0.30 \\ GH4169 & \(1.999\times 10^{11}\) & 8240 & 0.30 \\ Silicon steel & \(2.06\times 10^{11}\) & 7650 & 0.30 \\
40 CrNiMoA & \(2.05\times 10^{11}\) & 7850 & 0.30 \\ Aluminum & \(6.19\times 10^{10}\) & 2710 & 0.31 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Rotor Assembly Material Constants of MSTMP. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Symbol** & **Value** \\ \hline Turbo rotor mass & \(m\) & 12.627 kg \\ Equatorial moments of inertia & \(J_{r}\) & 0.0647 kg m\({}^{2}\) \\ Polar moments of inertia & \(J_{z}\) & 0.0496 kg m\({}^{2}\) \\ Distance from O to End-A sensor center & \(a_{s}\) & 0.0363 m \\ Distance from O to End-B sensor center & \(b_{s}\) & 0.0897 m \\ Distance from O to End-A magnetic bearing center & \(a_{m}\) & 0.0088 m \\ Distance from O to End-B magnetic bearing center & \(b_{m}\) & 0.0692 m \\ Current stiffness of End-A magnetic bearing & \(k_{ia}\) & 426 N A\({}^{-1}\) \\ Current stiffness of End-B magnetic bearing & \(k_{ib}\) & 239 N A\({}^{-1}\) \\ Displacement stiffness of End-A magnetic bearing & \(k_{ha}\) & \(8.656\times 10^{5}\) N m\({}^{-1}\) \\ Displacement stiffness of End-B magnetic bearing & \(k_{hb}\) & \(4.869\times 10^{5}\) N m\({}^{-1}\) \\ Protective gap & \(p\) & 0.0002 m \\ Magnetic gap & \(p_{0}\) & 0.0004 m \\ Displacement sensor magnification & \(k_{s}\) & \(8.533\times 10^{6}\) V m\({}^{-1}\) \\ Scale factor & \(K_{p}\) & 3.2 \\ Integration factor & \(T_{i}\) & 2.2 \\ Time constant & \(T_{f}\) & \(2.333\times 10^{-4}\) \\ Incomplete differential factor & \(T_{d}\) & \(3.933\times 10^{-3}\) \\ Power amplifier gain & \(K_{up}\) & \(3.264\times 10^{-4}\) \\ Power amplifier bandwidth & \(W_{up}\) & \(3.081\times 10^{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Structural and Control Parameters of the MSTMP system.