## 4 Simulation and Experimental Results
### Experimental Results  
To validate the effectiveness of the proposed method, experimental verification was performed on the in-house-designed MSTMP, as shown in Figure 16. Figure 1 shows one of the components of the MSTMP, the turbo rotor. It is a component of the MSTMP shown in Figure 16. The piece of equipment depicted in Figure 16 is a complete MSTMP, with the rotor, stator blades, and turbo rotor as its key components. It is situated within the vacuum test system, suspended in the vacuum chamber, facilitating tests of pumping speed,  
Figure 14: Stability analysis of the ANF with different bandwidth factor \(\varepsilon\) at different speed \(\Omega\). (**a**) \(\Omega=70\) Hz. (**b**) \(\Omega=110\) Hz. (**c**) \(\Omega=120\) Hz. (**d**) \(\Omega=200\) Hz. (**e**) Root-locus dominant pole diagram of the ANF. Figure 15: Stability analysis of the novel integrated AMB controller with parallel ANF and cascaded improved DTNFs. (**a**) Nyquist frequency characteristic of the open-loop system. (**b**) Pole-zero map of the characteristic polynomial. ultimate vacuum, compression ratio, and other performance metrics. The backing pump is used to obtain the pre-vacuum environment, and the chiller cools the stator assembly during the operation of the MSTMP. In Figure 16, nitrogen is used for testing, and the test equipment includes a flowmeter and vacuum gauge. The algorithm is implemented in a digital signal processor (TMS320F28335). The ANF in [23] enables synchronous current elimination, but the notch parameters must be adjusted to obtain a trade-off between steady-state accuracy and transient convergence speed. Importantly, [PERSON] et al. suggest that further research should focus on the criteria for selecting the optimal value of the notch parameter. Therefore, this study continues to improve the structure of notch filters and multi-parameter optimization algorithms. So, the proposed algorithm is compared with the ANF-PSF algorithm in [23]. Blade mode suppression is a prerequisite for the stability of an AMB-rotor system. During the static suspension (\(\Omega=0\) rpm), multi-stage-blade modal vibration will occur when the turbo rotor is subjected to a current step signal or an imbalanced exciting force. This phenomenon causes rotor displacement divergence. The stable state cannot be achieved solely by adjusting incomplete differential PID control parameters. So, it is necessary to cascade multi-frequency improved DTNFs to filter the current component at the multi-stage-blade modal frequencies. The displacement (in the AX, AY, BX, and BY channels) and the Fast Fourier Transform (FFT) (in the BY channel) of the AMB-rotor with or without DTNFs during static suspension (\(\Omega=0\) rpm) are shown in Figure 17a. The rotor displacement without DTNFs diverges significantly, almost to half of the unilateral protective gap of the auxiliary bearing 0.1 mm. With DTNFs, the system quickly obtains a stable state and maintains stable suspension. With the bandwidth factor rectification method of parallel co-frequency ANF based on displacement stiffness perturbation and the multi-objective constrained optimization method of cascaded multi-frequency improved DTNFs based on linear normalization, the MSTMP can steadily speed up to 24,000 rpm. The proposed algorithm is compared with the general ANF-PSF algorithm recently proposed in this research field. When the MSTMP works at the rated speed of 24,000 rpm, the rotor displacement in the four radial channels of AX, AY, BX, and BY with different algorithms are shown in Figure 17b. Figure 16: Main components of the MSTMP experimental platform. (**a**) Control components. (**b**) Mechanical components. At the rated speed of 24,000 rpm, the time-domain waveforms of End-B rotor displacement are shown in Figure 18. End-B axis trajectory diagrams with different algorithms are shown in Figure 18a. With the ANF-PSF algorithm, the value of the rotor displacement converges from 0.17 mm to 0.08 mm, a reduction of 52.94%.