Vibration Suppression of Multi-Stage-Blade AMB-Rotor Using Parallel Adaptive and Cascaded Multi-Frequency Notch Filters

Abstract

The application of active magnetic bearings (AMBs) in high-speed rotating machinery faces the challenge of micro-vibration. This research addresses the vibration control of a high-speed magnetically suspended turbo molecular pump (MSTMP) with rotor mass imbalance vibration and multi-stage-blade modal vibration. A novel integrated AMB controller consisting of parallel co-frequency adaptive notch filter (ANF) and cascaded multi-frequency improved double-T notch filters (DTNFs) is proposed. To suppress rotor mass imbalance vibration, a bandwidth factor rectification method of the ANF based on displacement stiffness perturbation is designed. To suppress multi-stage-blade modal vibration, a multi-objective constrained optimization method of cascaded improved DTNFs based on linear normalization is designed. Simulation and experimental results validate that the proposed structure improvement of the addition of an AMB controller and multi-parameter optimization of the algorithm can effectively improve not only the phase stability margin and the notch vibration performance of the magnetically suspended rotor (MSR) system but also the efficiency and practicability of the algorithm. At rotational speeds of 12,000 $\mathrm{rpm}$, 15,000 $\mathrm{rpm}$, 18,000 $\mathrm{rpm}$, and 21,000 $\mathrm{rpm}$, the suppression of co-frequency synchronous vibration is approximately maintained between −30.94 $\mathrm{dB}$ and −30.56 $\mathrm{dB}$. At the rated speed of 24,000 $\mathrm{rpm}$, compared with other algorithms, the value of the rotor displacement converges from 0.08 $\mathrm{mm}$ to 0.03 $\mathrm{mm}$, a reduction of 62.50%. The convergence time decreases from 3.67 $\mathrm{s}$ to 2.85 $\mathrm{s}$, a reduction of 22.34%.

1. Introduction

The quality of an ultra-high vacuum environment is increasingly becoming a limiting condition in the advancement of cutting-edge scientific research. Active magnetic bearings (AMBs), with the merits of being free of oil pollution, friction, micro-vibration, etc. [1,2,3,4], is particularly suitable for application scenarios which require high-speed, vacuum, and an ultra-clean environment [5,6,7,8]. As one kind of high-performance pumping equipment, the magnetically suspended turbo molecular pump (MSTMP) has become a key component for obtaining an ultra-high vacuum environment

In pursuit of high pumping speed and high compression ratio, the turbo rotor of the MSTMP is typically designed with multi-stage blades [9]. The dominant and prevalent vibrations of the turbo rotor are rotor mass imbalance vibration (co-frequency synchronous vibration) and multi-stage-blade modal vibration (multi-frequency mode vibration) [10], both of which significantly threaten the stability of the magnetically suspended rotor (MSR) system. Notably, consideration of the phase stability margin should not be neglected while performing vibration suppression. Otherwise, the impact of the high-speed AMB rotor on the auxiliary bearing during an unstable state will lead to severe wear and deformation of blades, which will result in irreversible mechanical structural damage to the turbo rotor [11,12], and will inevitably result in substantial economic losses.

The various disturbance factors that influence the stability of an MSTMP originate from different sources and exert different impacts. These disturbance factors are complicatedly related to electromagnetism, machinery, processing and assembly, and control methods [13]. For instance, even though complete balancing can theoretically be achieved through offline and online dynamic balancing corrections, residual imbalanced mass cannot be eliminated in practice [14]. In particular, the multi-stage blades of the turbo rotor are fixed at a single end of the elongated spindle, presenting a cantilevered structure. Such a cantilevered turbo rotor is highly susceptible to multi-stage-blade modal vibration when subjected to a current step signal during suspension or an imbalanced excitation force during high-speed rotation. Fortunately, the stiffness and damping of the AMB-rotor system are adjustable [15]. So, by improving the structure and optimizing the multi-parameters of the AMB controller, the balance between phase stability margin and notch vibration performance can be achieved.

In fact, the interplay between mechanical design and control methods in an MSTMP is complex. Mechanical design can provide the essential guidance for control methods, but its problem-solving scope is inherently limited, necessitating effective control strategies to address various vibration issues. The continuous improvement of the controller structure and the continuous optimization of the multi-parameters are thus of paramount importance. For instance, Zheng et al. investigated a feedforward approach to compensate for the displacement stiffness of high-speed magnetically suspended centrifugal compressors, which was used to suppress rotor mass imbalance vibration [16]. Peng et al. designed two-stage notch filters at low speed and high speed to suppress synchronous vibration caused by the gyroscopic effect [17]. An approach that involved adjusting the angular acceleration of the turbo rotor to make the rotor speed skip the natural frequency in a brief timeframe was proposed in [18] to avoid resonance. A current vibration suppression strategy based on the phase shift generalized integrator (PSGI) was proposed to filter out synchronous components in the displacement signal of the AMB-rotor [19]. However, it was complicated to select the appropriate phase shift angle according to the pole deviation angle of the imaginary axis. Chen et al. proposed a double-loop compensation design approach based on the generalized notch filter (GNF), but in which the sign of the convergence coefficient needed to be changed according to the rotational speed [20]. Xu et al. formulated a plural phase-shift notch filter (PSNF) that can identify synchronous components by tuning the amplitude of the controller and phase of the plural notch filter [21]. A parallel notch filter scheme was explored to enhance notch performance at high speeds in [22]. Although the synchronous current elimination was achieved by using an adaptive notch filter (ANF) in [23], a trade-off between steady-state accuracy and transient convergence speed must be established by adjusting notch parameters. Importantly, Chen et al. proposed that further research should focus on the criterion for selecting optimal values of the notch parameters. Therefore, this research continues to carry out the structural improvement and multi-parameter optimization of notch filters. To the best of our knowledge, there is limited research on the suppression of self-excited oscillations in multi-stage-blade modes of MSTMPs.

To sum up, an active vibration control strategy based on notch filters demonstrates strong engineering practicability, making it highly suitable for the micro-vibration suppression of compressors, turbines, and pumps. Addressing rotor mass imbalance vibration and multi-stage-blade modal vibration are critical for the engineering applications of MSTMPs. The innovative contributions of this research are primarily as follows:

• A novel integrated AMB controller consisting of parallel co-frequency ANF and cascaded multi-frequency improved double-T notch filters (DTNFs) is proposed. The structural improvement of the AMB controller and the multi-parameter optimization of the algorithm are carried out.
• To address the issue that the phase-lag angle introduced at open-loop cutoff frequency by the ANF with an inappropriate bandwidth factor can easily affect the phase stability margin of the MSR system, an ANF bandwidth factor rectification method based on displacement stiffness perturbation is designed.
• To mitigate the problems of an excessive notch (resulting in too large a phase-lag angle) and an incomplete notch (resulting in insufficient notch depth), taking the notch depth at the notch frequency, the notch width corresponding to −3 $\mathrm{dB}$, and the phase-lag angle introduced at the open-loop cutoff frequency by the DTNFs as constraints, a multi-objective constrained optimization method of cascaded improved DTNFs based on linear normalization is designed.

The rest of this paper is organized as follows. In Section 2, the AMB-rotor system is modeled, which considers multi-stage-blade modal vibration obtained through sinusoidal sweep excitation. In Section 3, the structural improvement and multi-parameter optimization algorithm of parallel co-frequency ANF and cascaded multi-frequency improved DTNFs are proposed, respectively. In Section 4, a comprehensive simulation analysis is conducted, followed by experimental validation on an MSTMP. Concluding remarks are provided in Section 5.

2. Modeling and Blade Modal Identification of AMB-Rotor System

2.1. Dynamic Modeling of AMB-Rotor System

A structural diagram of the MSTMP is shown in Figure 1. The object of this research is the MSTMP, which is supported by a magnetically suspended bearing and driven by a high-speed motor. It has the remarkable advantages of no lubricating oil pollution or friction loss, and active vibration control, and is very suitable for high-speed, vacuum, and ultra-clean applications. This MSTMP is self-developed in our laboratory and has a pumping speed of 4500 L/s (pumping speed). For the turbo rotor, which is composed of multi-stage turbo blades, a shaft, and a thrust disk, m is the mass of the turbo rotor, $J_r$ are the equatorial moments of inertia, $J_z$ are the polar moments of inertia, $\Omega$ is the rotation speed, $\alpha$ and $\beta$ are the tilting angles around the x-axis and the y-axis, and x and y are the displacement in the x-axis and the y-axis direction, respectively. If O is defined as the turbo rotor center of gravity, then $a_m$ and $b_m$ are the distances from O to the center of the radial magnetic bearing at End-A and End-B and $a_s$ or $b_s$ are the distances from O to the center of the radial sensor at End-A and End-B, respectively. Gyro technical equations are the core formulas of the dynamic model of the magnetically suspended rotor [3]. According to Newton’s law and Euler’s law, the dynamic equations of the AMB-rotor system can be obtained as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}m\ddot{x}=f_{\mathrm{AX}}+f_{\mathrm{BX}}\hfill \\ J_r\ddot{\beta }-J_z\Omega \dot{\alpha }=f_{\mathrm{AX}}{a}_m-f_{\mathrm{BX}}{b}_m\hfill \\ m\ddot{y}=f_{\mathrm{AY}}+f_{\mathrm{BY}}\hfill \\ J_r\ddot{\alpha }+J_z\Omega \dot{\beta }=-f_{\mathrm{AY}}{a}_m+f_{\mathrm{BY}}{b}_m\hfill \end{array}\right..\end{array}$$

(1)

The four radial channels, labeled AX, AY, BX, and BY, have the same control method, and the control is decentralized and independent, so the control method in the AX and BX channels (X direction) is analyzed as an example. According to [3], in the direction of X, the magnetic force $f_{\mathrm{X}}$ is the binary quadratic function of current $i_{\mathrm{X}}$ and displacement x

$$\begin{array}{c}\hfill f_{\mathrm{X}}={\mu }_0{N}^2{A}_{p}cos\left(22.5·{\displaystyle \frac{\pi }{180}}\right)\left[{\left({\displaystyle \frac{i_0+i_{\mathrm{X}}}{p_0-x}}\right)}^2-{\left({\displaystyle \frac{i_0-i_{\mathrm{X}}}{p_0+x}}\right)}^2\right]\end{array}$$

(2)

where ${\mu }_0$ is the magnetic permeability of a vacuum, N is the number of coils, $A_p$ is the magnetic pole cross-sectional area, $i_0$ is the bias current, and $p_0$ is the magnetic gap when the center of the rotor and the center of the radial magnetic bearing coincide.

For radial magnetic bearings, $k_{ia}$ and $k_{ib}$ are defined as the current stiffnesses of the radial magnetic bearing at End-A and End-B and $k_{ha}$ and $k_{hb}$ are defined as the displacement stiffnesses of the radial magnetic bearing at End-A and End-B, respectively. Under normal working conditions, the rotor center of the radial magnetic bearing is located in the geometric center of the stator. To simplify the expression of the magnetic force $f_{\mathrm{X}}$, it can be Taylor-expanded by ignoring high-order items at the equilibrium point $i_{\mathrm{X}}=0,x=0$ [3]. Therefore, the linearized equation of the magnetic force can be obtained as follows

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{f}_{\mathrm{AX}}=k_{ia}{i}_{\mathrm{AX}}+k_{ha}{m}_{\mathrm{AX}}\hfill \\ f_{\mathrm{BX}}=k_{ib}{i}_{\mathrm{BX}}+k_{hb}{m}_{\mathrm{BX}}\hfill \end{array}\right..\end{array}$$

(3)

Introducing the general linearized expression $f=k_{i}i+k_{h}m$ of the magnetic force $f_{\mathrm{AX}},f_{\mathrm{BX}}$ into (1), we can obtain

$$\begin{array}{c}\hfill \left\{\begin{array}{c}m\ddot{x}=\left(k_{ia}{i}_{\mathrm{AX}}+k_{ha}{m}_{\mathrm{AX}}\right)+\left(k_{ib}{i}_{\mathrm{BX}}+k_{hb}{m}_{\mathrm{BX}}\right)\hfill \\ J_r\ddot{\beta }-J_z\Omega \dot{\alpha }=\left(k_{ia}{i}_{\mathrm{AX}}+k_{ha}{m}_{\mathrm{AX}}\right)a_m-\left(k_{ib}{i}_{\mathrm{BX}}+k_{hb}{m}_{\mathrm{BX}}\right)b_m\hfill \end{array}\right..\end{array}$$

(4)

The rotor displacement in the magnetic bearing coordinate system ($m_{\mathrm{AX}},m_{\mathrm{AY}},m_{\mathrm{BX}},m_{\mathrm{BY}}$) and in the sensor coordinate system ($d_{\mathrm{AX}},d_{\mathrm{AY}},d_{\mathrm{BX}},d_{\mathrm{BY}}$) satisfy the following transformation relationships, respectively

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\alpha ={\displaystyle \frac{m_{\mathrm{BY}}-m_{\mathrm{AY}}}{a_m+b_m}}\hfill \\ \beta ={\displaystyle \frac{m_{\mathrm{AX}}-m_{\mathrm{BX}}}{a_m+b_m}}\hfill \\ x={\displaystyle \frac{b_m{m}_{\mathrm{AX}}+a_m{m}_{\mathrm{BX}}}{a_m+b_m}}\hfill \\ y={\displaystyle \frac{b_m{m}_{\mathrm{AY}}+a_m{m}_{\mathrm{BY}}}{a_m+b_m}}\hfill \end{array}\right.,\end{array}$$

(5)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\alpha ={\displaystyle \frac{d_{\mathrm{BY}}-d_{\mathrm{AY}}}{a_s+b_s}}\hfill \\ \beta ={\displaystyle \frac{d_{\mathrm{AX}}-d_{\mathrm{BX}}}{a_s+b_s}}\hfill \\ x={\displaystyle \frac{b_s{d}_{\mathrm{AX}}+a_s{d}_{\mathrm{BX}}}{a_s+b_s}}\hfill \\ y={\displaystyle \frac{b_s{d}_{\mathrm{AY}}+a_s{d}_{\mathrm{BY}}}{a_s+b_s}}\hfill \end{array}\right..\end{array}$$

(6)

By putting Equation (5) into (4), the following formula can be obtained

$$\begin{array}{c}\hfill \left\{\begin{array}{c}m\ddot{x}=\left(k_{ha}+k_{hb}\right)x+\left(k_{ha}{a}_m-k_{hb}{b}_m\right)\beta +k_{ia}{i}_{\mathrm{AX}}+k_{ib}{i}_{\mathrm{BX}}\hfill \\ J_r\ddot{\beta }-J_z\Omega \dot{\alpha }=\left(k_{ha}{a}_m-k_{hb}{b}_m\right)x+\left(k_{ha}{a}_m^2+k_{hb}{b}_m^2\right)\beta +k_{ia}{a}_m{i}_{\mathrm{AX}}-k_{ib}{b}_m{i}_{\mathrm{BX}}\hfill \end{array}\right..\end{array}$$

(7)

By putting Equation (6) into (7), and by Laplace transformation, the following formula can be obtained

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \frac{m{b}_s}{l_s}}{d}_{\mathrm{AX}}\left(s\right)s^2+{\displaystyle \frac{m{a}_s}{l_s}}{d}_{\mathrm{BX}}\left(s\right)s^2-{\displaystyle \frac{k_{ha}}{l_s}}e{d}_{\mathrm{AX}}\left(s\right)-{\displaystyle \frac{k_{ha}}{l_s}}f{d}_{\mathrm{BX}}\left(s\right)=k_{ia}{i}_{\mathrm{AX}}\left(s\right)+k_{ib}{i}_{\mathrm{BX}}\left(s\right)\hfill \\ {\displaystyle \frac{J_r}{l_s}}{d}_{\mathrm{AX}}\left(s\right)s^2-{\displaystyle \frac{J_r}{l_s}}{d}_{\mathrm{BX}}\left(s\right)s^2-{\displaystyle \frac{J_z\Omega }{l_s}}\left[d_{\mathrm{BY}}\left(s\right)s-d_{\mathrm{AY}}\left(s\right)s\right]-{\displaystyle \frac{k_{ha}}{l_s}}g{d}_{\mathrm{AX}}\left(s\right)-{\displaystyle \frac{k_{ha}}{l_s}}h{d}_{\mathrm{BX}}\left(s\right)=k_{ia}{a}_m{i}_{\mathrm{AX}}\left(s\right)-k_{ib}{b}_m{i}_{\mathrm{BX}}\left(s\right)\hfill \end{array}\right.\end{array}$$

(8)

where $l_s=a_s+b_s$, $e=2b_s+a_m-b_m$, $f=2a_s+b_m-a_m$, $g=a_m{b}_s-b_m{b}_s+a_m^2+b_m^2$, $h=a_m{a}_s-b_m{a}_s-a_m^2-b_m^2$.

A block diagram of the AMB-rotor control system with single input and single output (SISO) is shown in Figure 2. According to (8), the model of the AMB-rotor system $G_r\left(s\right)$ is

$$\begin{array}{c}\hfill G_r\left(s\right)={\displaystyle \frac{d_{\mathrm{AX}}\left(s\right)}{i_{\mathrm{AX}}\left(s\right)}}={\displaystyle \frac{k_{ia}A}{s^2-k_{ha}B}}\end{array}$$

(9)

where

$$\begin{array}{c}\hfill \begin{array}{c}A={\displaystyle \frac{1}{m}}+{\displaystyle \frac{a_m{a}_s}{J_r}},\hfill \\ B={\displaystyle \frac{a_s\left(a_m^2+b_m^2+a_m{b}_s-b_m{b}_s\right)}{J_r\left(a_s+b_s\right)}}+{\displaystyle \frac{2b_s+a_m-b_m}{m\left(a_s+b_s\right)}}.\hfill \end{array}\end{array}$$

For radial magnetic bearings, R is the coil resistance, and L is the coil inductance. The AMB-rotor system is driven by the switching power amplifier, $k_v$ is a quantity related to the mains voltage and the PWM duty cycle, $k_{ad}$ is the transfer factor of the analog-to-digital (A/D) converter, $k_{ic}$ is the enlargement factor of the current sensor, and $k_{amp}$ and $k_{ico}$ are the preamplification factor and feedback factor of the current controlled circuit, respectively. The model of the amplifier $G_w\left(s\right)$ is

$$\begin{array}{c}\hfill G_w\left(s\right)={\displaystyle \frac{k_{amp}{k}_v\frac{1}{Ls+R}}{1+k_{amp}{k}_v\frac{1}{Ls+R}·k_{ic}{k}_{ad}{k}_{ico}}}=K_w{\displaystyle \frac{W_w}{s+W_w}}\end{array}$$

(10)

where $K_w=k_{amp}{k}_v/\left(R+k_{amp}{k}_v{k}_{ic}{k}_{ad}{k}_{ico}\right)$, and $W_w=\left(R+k_{amp}{k}_v{k}_{ic}{k}_{ad}{k}_{ico}\right)/L$.

The incomplete differential PID controller is used as the base controller, and the model of the controller $G_c\left(s\right)$ is

$$\begin{array}{c}\hfill G_c\left(s\right)=k_p\left(1+{\displaystyle \frac{1}{T_{i}s}}+{\displaystyle \frac{T_{d}s}{T_{f}s+1}}\right)\end{array}$$

(11)

where $k_p$ is the scale factor, $T_i$ is the integration factor, $T_d$ is the incomplete differential factor, and $T_f$ is the time constant.

2.2. Blade Modal Identification by Sinusoidal Sweep Excitation

Modes are intrinsic vibrational characteristics of the turbo rotor, and each mode has a specific natural frequency, damping ratio, and vibration pattern. Finite element analysis (FEA) was performed using ANSYS Workbench. The turbo rotor model used 5-mm-sized elements, resulting in a mesh with 354,509 nodes and 173,934 elements. Th elastic modulus scaling factors were 0.01 for clearance fit, 0.1 for nut tightening, and 1 for interference fit. The first 40 natural frequencies were calculated to capture the main vibration characteristics. The following assumptions were made during the FEA:

• Material homogeneity and isotropy.
• Linear elastic behavior.
• Neglecting temperature effects.

Setting fluid model and boundary conditions for FEA of the MSTMP’s turbo rotor.

• Fluid model: The $k-\epsilon$ turbulence model is chosen, which is suitable for high Reynolds number flows.
• Inlet boundary condition: A velocity inlet is defined with a speed set at 573 m/s.
• Outlet boundary condition: A pressure outlet is specified with a pressure set at 0 Pa (relative pressure).
• Wall boundary condition: No-slip conditions are applied to simulate the actual interaction between the fluid and the wall surfaces.

The finite element simulation results of the turbo rotor are shown in Figure 3. The frequency response of the blade mode is characterized by second-order underdamping, and the multi-stage-blade modes have a superposition effect. So taking the multi-stage-blade modes into account, the AMB-rotor model $G_r\left(s\right)$ can be replaced by $G_m\left(s\right)$

$$\begin{array}{c}\hfill G_m\left(s\right)=\prod _{i=1}^n{\displaystyle \frac{k_{ia}A\left(s^2+{\sigma }_i{w}_{fi}s+w_{fi}^2\right)}{\left(s^2-k_{ha}B\right)\left(s^2+{\lambda }_i{\sigma }_i{w}_{fi}s+w_{fi}^2\right)}}\end{array}$$

(12)

where $w_{fi}$ is the i-th, $(i=1,2,3,4,5,6)$ blade modal frequency, and ${\lambda }_i$ and ${\sigma }_i$ are the peak amplitude factor and peak width factor of the i-th blade modes, respectively, which can be identified by sinusoidal sweep excitation.

In order to obtain the multi-stage-blade modal frequencies during static suspension, a sinusoidal excitation signal is added to the channel of BY. The sampling frequency is set as 10 $\mathrm{kHz}$, the sweep frequency range as 1–2 $\mathrm{kHz}$, and the excitation frequency interval as 1 $\mathrm{Hz}$. The test result of sinusoidal sweep excitation is shown in Figure 4a. The labels (419 $\mathrm{Hz}$, −63.87 $\mathrm{dB}$. 525 $\mathrm{Hz}$, −60.02 $\mathrm{dB}$. 583 $\mathrm{Hz}$, −37.06 $\mathrm{dB}$, 674 $\mathrm{Hz}$, −30.56 $\mathrm{dB}$, 710 $\mathrm{Hz}$, −38.15 $\mathrm{dB}$. 1598 $\mathrm{Hz}$, and −65.82 $\mathrm{dB}$) at the locations of the peaks are indicative of multi-stage-blade modes during static suspension (0 $\mathrm{rpm}$). Aiming for an attenuation to −80 $\mathrm{dB}$, the peak bandwidth $H_{\mathrm{wid}}$ and the peak amplitude $E_{\mathrm{amp}}$ are defined as shown in Figure 4a. According to $E_{\mathrm{amp}}$ and $H_{\mathrm{wid}}$, the peak amplitude factor ${\lambda }_i$ and the peak width factor ${\sigma }_i$ can be obtained, respectively

$$\begin{array}{c}\hfill {\lambda }_i={10}^{-{\displaystyle \frac{E_{\mathrm{amp}}}{20}}},\end{array}$$

(13)

$$\begin{array}{c}\hfill {\sigma }_i=\sqrt{{\displaystyle \frac{2-\sqrt{\frac{16{\pi }^2{\left(H_{\mathrm{wid}}\right)}^2}{w_{fi}^2}+4}}{2\times {\left({10}^{-\frac{E_{\mathrm{amp}}}{20}}\right)}^2-1}}}.\end{array}$$

(14)

Firstly, the theoretical frequencies of multi-stage-blade mode self-excited oscillations are obtained by finite element simulation. Secondly, the actual frequencies of blade modes are obtained by a sinusoidal sweeping excitation test. Finally, multiple superimposed second-order damped oscillation terms are introduced based on the AMB-rotor model $G_r\left(s\right)$ in Section 2.1. Thus, the whole process from simulation analysis to experimental verification and obtaining the transfer function model is realized for multi-stage-blade mode self-excited oscillations.

The rated speed of the MSTMP is 24,000 $\mathrm{rpm}$, which exposes the multi-stage blades to the comprehensive effect of thermal stresses, centrifugal forces, and alternating loads during acceleration and deceleration. These considerable forces generated by high-speed rotating machinery induce two effects: stress stiffening and spin softening. Consequently, the multi-stage-blade modal frequencies $w_{fi}$ are not constants but vary with the rotor speed. As shown in Figure 4b, multi-stage-blade modes bifurcate into forward (FW) modes and backward (BW) modes. The labels in Figure 4b show frequencies during static suspension (0 $\mathrm{rpm}$: 419 $\mathrm{Hz}$, 525 $\mathrm{Hz}$, 583 $\mathrm{Hz}$, 674 $\mathrm{Hz}$, 710 $\mathrm{Hz}$, 1598 $\mathrm{Hz}$) and after reaching the rated speed (24,000 $\mathrm{rpm}$: 350 $\mathrm{Hz}$, 463 $\mathrm{Hz}$, 530 $\mathrm{Hz}$, 628 $\mathrm{Hz}$, 745 $\mathrm{Hz}$, 980 $\mathrm{Hz}$). In short, the multi-stage-blade mode frequencies $w_{fi}$ are constantly changing in the full speed range.

Modeling and blade modal identification of the multi-stage-blade AMB-rotor are crucial to subsequently determine the precise notch frequency, select the appropriate notch width, and adjust the optimal notch depth of the parallel co-frequency ANF and cascaded multi-frequency improved DTNFs.

3. Suppression Strategy of Synchronous Vibration and Multi-Stage-Blade Modal Vibration

The dominant and prevalent vibrations of the multi-stage-blade AMB-rotor are rotor mass imbalance vibration (co-frequency synchronous vibration) and multi-stage-blade modal vibration (multi-frequency mode vibration). As shown in Figure 5, a novel integrated AMB controller combined with the parallel co-frequency ANF and cascaded multi-frequencies improved DTNFs is proposed. Structural improvement of the AMB controller and multi-parameter optimization of the algorithm are carried out.

3.1. Parameters and Characteristic of the ANF

An ANF is usually used to filter the co-frequency current component to suppress rotor mass imbalanced vibration. However, this may result in an instability of the closed-loop system: an inappropriate bandwidth factor $\epsilon$ of the ANF causes a phase-lag angle $\phi \left(\epsilon \right)$ at the open-loop cutoff frequency $w_c$, which reduces the phase stability margin $\gamma$ of the MSR system. So, an ANF bandwidth factor $\epsilon$ rectification method based on displacement stiffness $k_h$ perturbation is designed to solve this problem. The perturbation of $k_h$ highlights the innovation of the proposed method: the parameter uncertainty of the MSR system itself is taken into account.

Notch parameters and the notch characteristic are quantitatively calculated and qualitatively analyzed, respectively. The block diagram of the ANF is shown in Figure 6a. $\epsilon$ is the bandwidth factor and is one of the most important parameters in determining notch performance. The notch frequency w (with unit $\mathrm{rad}/\mathrm{s}$) of the ANF changes synchronously with the rotor speed $\Omega$ (with unit $\mathrm{rad}/\mathrm{s}$). According to [23], the transfer function $G_{\mathrm{ANF}}\left(s\right)$ of the ANF is

$$\begin{array}{c}\hfill G_{\mathrm{ANF}}\left(s\right)={\displaystyle \frac{s^2+w_{w=\Omega }^2}{s^2+\epsilon s+w_{w=\Omega }^2}}.\end{array}$$

(15)

A diagram of the definitions of the notch parameters is shown in Figure 7. $l_{\mathrm{depth}}^{\mathrm{ANF}}$ (with unit $\mathrm{dB}$) is the notch depth, which represents the attenuation of the magnitude at the notch frequency. $\Delta f^{\mathrm{ANF}}$ (with unit $\mathrm{Hz}$) is the notch width, and the amplitude of vibration in this frequency range is attenuated to different degrees. $B_w^{\mathrm{ANF}}$ (with unit $\mathrm{rad}/\mathrm{s}$) is the −3 $\mathrm{dB}$ bandwidth, defined as $B_w^{\mathrm{ANF}}=2\pi \Delta f^{\mathrm{ANF}}$. $Q^{\mathrm{ANF}}$ is the quality factor, defined as $Q^{\mathrm{ANF}}=\Omega /B_w^{\mathrm{ANF}}$.

The conversion relationship among the notch parameters such as $\Delta f^{\mathrm{ANF}}$, $Q^{\mathrm{ANF}}$, and $\epsilon$ is calculated below. With the notch frequency $w=\Omega$ as the center frequency, when the amplitude of the ANF attenuates to −3 $\mathrm{dB}$, the corresponding frequencies $w\in \left(w_1,w_2\right)$ are all within the range of the −3 $\mathrm{dB}$ bandwidth $B_w^{\mathrm{ANF}}$. $w_1$ is the upper limit frequency, and $w_2$ is the lower limit frequency, so we can obtain

$$\begin{array}{c}\hfill -20lg|G_{\mathrm{ANF}}\left(jw\right){{|}_{w=w_1}=-20lg|G_{\mathrm{ANF}}\left(jw\right)|}_{w=w_2}=-3,\end{array}$$

(16)

$$\begin{array}{c}\hfill \left|G_{\mathrm{ANF}}\left(jw\right)\right|=\left|{\displaystyle \frac{{\left(jw\right)}^2+{\Omega }^2}{{\left(jw\right)}^2+\epsilon ·jw+{\Omega }^2}}\right|={\displaystyle \frac{\sqrt{2}}{2}}.\end{array}$$

(17)

To obtain the solution of $w_1$ and $w_2$, further calculation of (17) produces

$$\begin{array}{c}\hfill w^4+\left(-2{\Omega }^2-{\epsilon }^2\right)w^2+{\Omega }^4=0.\end{array}$$

(18)

The results of solving (18) are

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{w}_1^2={\displaystyle \frac{2{\Omega }^2+{\epsilon }^2-\epsilon \sqrt{4{\Omega }^2+{\epsilon }^2}}{2}}\hfill \\ w_2^2={\displaystyle \frac{2{\Omega }^2+{\epsilon }^2+\epsilon \sqrt{4{\Omega }^2+{\epsilon }^2}}{2}}\hfill \end{array}\right..\end{array}$$

(19)

Because $B_w^{\mathrm{ANF}}=w_2-w_1=2\pi \Delta f^{\mathrm{ANF}}$, and $\Omega =\left(w_1+w_2\right)/2$ are already defined, the following formula can be obtained

$$\begin{array}{c}\hfill {\displaystyle \frac{w_2^2-w_1^2}{{\Omega }^2}}={\displaystyle \frac{4\pi \Delta f^{\mathrm{ANF}}}{\Omega }}.\end{array}$$

(20)

According to (19) and (20), the solution of the notch width $\Delta f^{\mathrm{ANF}}$ and the quality factor $Q^{\mathrm{ANF}}$ can be obtained

$$\begin{array}{c}\hfill \Delta f^{\mathrm{ANF}}={\displaystyle \frac{\epsilon \sqrt{4{\Omega }^2+{\epsilon }^2}}{4\pi \Omega }},\end{array}$$

(21)

$$\begin{array}{c}\hfill Q^{\mathrm{ANF}}={\displaystyle \frac{\Omega }{B_w^{\mathrm{ANF}}}}={\displaystyle \frac{\Omega }{2\pi \Delta f^{\mathrm{ANF}}}}={\displaystyle \frac{2{\Omega }^2}{\epsilon \sqrt{4{\Omega }^2+{\epsilon }^2}}}\stackrel{2\Omega \gg \epsilon }{\approx }{\displaystyle \frac{\Omega }{\epsilon }}.\end{array}$$

(22)

To obtain the solution of the bandwidth factor $\epsilon$, further calculation of (21) produces

$$\begin{array}{c}\hfill {\epsilon }^4+4{\Omega }^2{\epsilon }^2-16{\pi }^2{\Omega }^2{\left(\Delta f^{\mathrm{ANF}}\right)}^2=0.\end{array}$$

(23)

According to (23), the solution of the bandwidth factor $\epsilon$ can be obtained

$$\begin{array}{c}\hfill \epsilon =\sqrt{-2{\Omega }^2+2\Omega \sqrt{{\Omega }^2+4{\pi }^2{\left(\Delta f^{\mathrm{ANF}}\right)}^2}}.\end{array}$$

(24)

The magnitude-phase frequency characteristics of the ANF are shown in Figure 7, which represent the notch performance.

• At constant-speed $\Omega$, the variable bandwidth factor $\epsilon$ determines the convergence speed of the algorithm, the mismatch tolerance of notch frequency, and the degree of influence on the phase characteristic near the notch frequency.
• At variable-speed $\Omega$, with the constant bandwidth factor $\epsilon$, the effects of co-frequency synchronous vibration suppression decrease with increasing rotor speed $\Omega$.
• At variable-speed $\Omega$, with the variable bandwidth factor $\epsilon$, it is possible to ensure the consistent effect of co-frequency synchronous vibration suppression with increasing rotor speed $\Omega$.

3.2. Bandwidth Factor Rectification Method of the ANF

Generally speaking, the displacement stiffness $k_h$ used to suppress the co-frequency synchronous vibration is a constant value that is measured when the AMB-rotor is statically suspended. However, in fact, $k_h$ fluctuates with changing rotor displacement x. So, the perturbation should be considered on the basis of a constant value. The perturbation of displacement stiffness $k_h$ is compensated adaptively by adjusting the value of bandwidth factor $\epsilon$ in this research.

The constraint relationship between the notch parameter and the phase angle is calculated below. At the open-loop cutoff frequency $w_c\left(w_c<\Omega \right)$, the phase characteristic of the ANF exhibits hysteresis. According to (15), the phase-lag angle $\phi \left(\epsilon \right)$ ($\phi \left(\epsilon \right)<0$) introduced by an ANF is a function of the bandwidth factor $\epsilon$

$$\begin{array}{c}\hfill \phi \left(\epsilon \right)=-arctan\left({\displaystyle \frac{\epsilon w_c}{{\Omega }^2-w_c^2}}\right){\displaystyle \frac{180}{\pi }}.\end{array}$$

(25)

By constraining $\phi \left(\epsilon \right)$ at $w_c$, we ensure that $\left|\phi \left(\epsilon \right)\right|$ is less than the phase stability margin $\gamma \left(\gamma >0\right)$ of the open-loop frequency characteristics at $w_c$, $\left|\phi \left(\epsilon \right)\right|<\gamma$. So, according to (25), the constraint condition of $\epsilon$ can be obtained

$$\begin{array}{c}\hfill 0<\epsilon <{\displaystyle \frac{{\Omega }^2-w_c^2}{w_c}}tan\left(\gamma {\displaystyle \frac{\pi }{180}}\right).\end{array}$$

(26)

Equation (26) is the calculation formula based on the constraint condition of the bandwidth factor $\epsilon$. Similarly, the calculation formula based on the constraint condition of the notch width $\Delta f^{\mathrm{ANF}}$ can also be obtained

$$\begin{array}{c}\hfill 0<\Delta f^{\mathrm{ANF}}<\Delta f_{\max }^{\mathrm{ANF}}.\end{array}$$

(27)

According to (21) and (26), the maximum $\Delta f_{\max }^{\mathrm{ANF}}$ of the notch width can be obtained

$$\begin{array}{c}\hfill \Delta f_{\max }^{\mathrm{ANF}}={\displaystyle \frac{\left({\Omega }^2-w_c^2\right)tan\left(\gamma \frac{\pi }{180}\right)\sqrt{4{\Omega }^2+\frac{{\left({\Omega }^2-w_c^2\right)}^2}{w_c^2}{tan}^2\left(\gamma \frac{\pi }{180}\right)}}{4\pi \Omega w_c}}.\end{array}$$

(28)

Although the constraint conditions derived above can ensure the stability of an MSR system with an ANF, the bandwidth factor $\epsilon$ and the notch width $\Delta f^{\mathrm{ANF}}$ determined from (26)–(28) are a series of numerical values within a range, not a specific value. However, from the previous analysis in Section 3.1, it is obvious that the notch performance of the ANF is sensitive to changing $\epsilon$. Therefore, this research continues to search for an appropriate value of $\epsilon$.

An appropriate value of the bandwidth factor $\epsilon$ needs to be designed in conjunction with the rotor displacement x. The relationship between control current $i_{\mathrm{X}}$, rotor displacement x, and electromagnetic force $f_{\mathrm{X}}$ is shown in Figure 8a. The relationship between control current $i_{\mathrm{X}}$, rotor displacement x, and displacement stiffness $k_h$ is shown in Figure 8b. As can be seen, when the rotor displacement x changes, the perturbation of displacement stiffness $k_h$ will appear. According to (2), taking the parameter uncertainty of the MSR system itself into account, the dynamic displacement stiffness $k_h$ can be calculated according to the following formula [3]

$$\begin{array}{c}\hfill k_h={\displaystyle \frac{\partial f_{\mathrm{X}}\left(i_{\mathrm{X}},x\right)}{\partial x}}={\mu }_0{N}^2{A}_{p}cos\left(22.5·{\displaystyle \frac{\pi }{180}}\right)\left[{\displaystyle \frac{2{\left(i_0+i_{\mathrm{X}}\right)}^2}{{\left(p_0-x\right)}^3}}+{\displaystyle \frac{2{\left(i_0-i_{\mathrm{X}}\right)}^2}{{\left(p_0+x\right)}^3}}\right].\end{array}$$

(29)

The unilateral protective gap of the auxiliary bearing is defined as p. The stability description of the AMB-rotor system typically encompasses the following two aspects:

• There is no pole allocated on the right-half plane in the transfer function of the closed-loop system.
• The rotor displacement x is within a safe range that is allowed by the protective gap p.

According to the rotor displacement specified in the ISO14839-2 2004 standard, this research assumed that the amplitude of the rotor displacement x is 0.3, 0.4, 0.5, 0.75, and 1 times the protective gap p. This assumption is used to explain the displacement stiffness $k_h$ perturbation phenomenon caused by the change in rotor displacement x, which causes a change in open-loop cutoff frequency $w_c$ and brings instability to the phase margin $\gamma$ of the MSR system and is not the ultimate control target.

The open-loop transfer function $G_{\mathrm{ol}}\left(s\right)$ of the AMB-rotor system without an ANF is $G_{\mathrm{ol}}\left(s\right)=k_s{G}_c\left(s\right)G_w\left(s\right)G_m\left(s\right)$. Figure 9a shows the magnitude-phase frequency characteristics of the open-loop system with different rotor displacement x. Figure 9b shows the phase margin $\gamma$ at the open-loop cutoff frequency $w_c$.

As listed in

Table 1. Design process of ANF notch parameters based on the perturbation of displacement stiffness $k_h$ (With $\Omega$ = 120 $\mathrm{Hz}$).

x (m)	${\mathit{k}}_{\mathit{h}}$ ($\mathbf{N} {\mathbf{m}}^{-1}$)	${\mathit{w}}_{\mathit{c}}$ ($\mathbf{Hz}$)	$\mathit{\gamma }$ ($\mathbf{deg}$)	${\mathit{\epsilon }}_{max}$	$\mathbf{\Delta }{\mathit{f}}_{max}^{\mathbf{ANF}}$ ($\mathbf{Hz}$)	${\mathit{Q}}_{min}^{\mathbf{ANF}}$
$6\times {10}^{-5}$ ($0.3p$)	$9.7831\times {10}^5$	$92.42$	$21.91$	$160.194$	$25.639$	$4.680$
$8\times {10}^{-5}$ ($0.4p$)	$1.0836\times {10}^6$	$91.06$	$21.75$	$168.146$	$26.927$	$4.457$
$1\times {10}^{-4}$ ($0.5p$)	$1.2336\times {10}^6$	$89.10$	$21.52$	$179.662$	$28.796$	$4.167$
$1.5\times {10}^{-4}$ ($0.75p$)	$1.9177\times {10}^6$	$79.39$	$20.22$	$236.038$	$38.024$	$3.156$
$2\times {10}^{-4}$ (p)	$3.5507\times {10}^6$	$47.48$	$13.94$	$398.951$	$65.680$	$1.827$

, at a rotational speed of 120 $\mathrm{Hz}$, by the ANF bandwidth factor rectification method based on displacement stiffness $k_h$ perturbation, according to (22), (26), (28), and (29), with a change in x, $k_h$ is perturbed, so the distance between $w_c$ and $w=\Omega$ is obviously different, and $\gamma$ is different; finally, the critical values of ${\epsilon }_{max}$, $\Delta f_{max}^{\mathrm{ANF}}$, and $Q_{min}^{\mathrm{ANF}}$ are also different.

From the previous analysis in Section 3.1, although any value of $\epsilon$ obtained by this method less than ${\epsilon }_{max}$ can ensure the phase stability margin $\gamma$, a larger value of $\epsilon$ can improve the convergence speed of the algorithm and the mismatch tolerance of the notch frequency. Compared to existing algorithms with fixed $\epsilon$, this method not only can solve the problem that the consistent effect of co-frequency synchronous vibration suppression cannot be guaranteed with a change in rotor speed $\Omega$ but also can avoid the problem that the phase-lag angle $\phi \left(\epsilon \right)$ introduced by an inappropriate $\epsilon$ of the ANF affects the phase stability margin $\gamma$ of the MSR system.

3.3. Parameters and Characteristic of Improved DTNF

Multi-stage-blade mode suppression is the prerequisite for MSR system stability. Multi-stage-blade modal vibration will occur when the turbo rotor is subjected to a current step signal or an imbalanced exciting force, as shown in Figure 4a,b. Although rotor mass imbalance vibration (co-frequency synchronous vibration) is the main source of disturbances in MSTMPs, it is also important and indispensable to suppress multi-stage-blade modal vibration (multi-frequency mode vibration) by cascaded notch filters.

The transfer function $G_{\mathrm{BNF}}\left(s\right)$ of the traditional biquadratic notch filter (BNF) is

$$\begin{array}{c}\hfill G_{\mathrm{BNF}}\left(s\right)={\displaystyle \frac{w_a^2}{w_b^2}}{\displaystyle \frac{s^2+2{\xi }_b{w}_{b}s+w_b^2}{s^2+2{\xi }_a{w}_{a}s+w_a^2}}={\displaystyle \frac{\lambda s^2+\eta s+\nu }{\lambda s^2+\tau s+u}}\end{array}$$

(30)

where $\lambda =w_a^2/w_b^2,\tau =2{\xi }_a{w}_a^3/w_b^2,\eta =2{\xi }_b{w}_a^2/w_b,u=w_a^4/w_b^2,$ and $v=w_a^2$. As shown in Figure 10, the disadvantage of the traditional BNF is that the notch width is not adjustable, and the amplitude of the positive and negative resonance peaks cannot be adjusted separately.

In order to facilitate the calculation of multiple parameters and the adjustment of notch performance (such as notch depth $l_{\mathrm{depth}}^{\mathrm{DTNF}}$, notch width $\Delta f^{\mathrm{DTNF}}$, phase-lag angle, etc.), the structure of the traditional BNF is improved in this research. The block diagram of the improved DTNF is shown in Figure 6b. If we let $\lambda =1/w_f^2$, $\tau =k/w_f$, $\eta =\zeta k/w_f\left(0<\zeta <1\right)$, and $u=v=1$, the transfer function $G_{\mathrm{DTNF}}\left(s\right)$ of the improved DTNF can be obtained as follows

$$\begin{array}{c}\hfill G_{\mathrm{DTNF}}\left(s\right)={\displaystyle \frac{\lambda s^2+\eta s+\nu }{\lambda s^2+\tau s+u}}={\displaystyle \frac{s^2+\zeta k{w}_{f}s+w_f^2}{s^2+k{w}_{f}s+w_f^2}}\end{array}$$

(31)

where $w_f$ is the notch frequency (with unit $\mathrm{rad}/\mathrm{s}$), $\zeta$ is the depth factor, and k is the width factor.

The definitions of the notch parameters of the DTNF are similar to those of the ANF, and are shown in Figure 7. The conversion relationship between the notch parameters, such as $l_{\mathrm{depth}}^{\mathrm{DTNF}}$, $\Delta f^{\mathrm{DTNF}}$, $Q^{\mathrm{DTNF}}$, and k and $\zeta$, are calculated below. To obtain the solution for the notch depth $l_{\mathrm{depth}}^{\mathrm{DTNF}}$, if the frequency w of the input signal is equal to the notch frequency $w_f$ of the improved DTNF, ${\left|G_{\mathrm{DTNF}}\left(jw\right)\right|}_{w=w_f}$ will represent the value of the notch depth $l_{\mathrm{depth}}^{\mathrm{DTNF}}$, so we can obtain

$$\begin{array}{c}\hfill {\left|G_{\mathrm{DTNF}}\left(jw\right)\right|}_{w=w_f}=\left|{\displaystyle \frac{{\left(j{w}_f\right)}^2+\zeta k{w}_f·j{w}_f+w_f^2}{{\left(j{w}_f\right)}^2+k{w}_f·j{w}_f+w_f^2}}\right|=\zeta ,\end{array}$$

(32)

$$\begin{array}{c}\hfill l_{\mathrm{depth}}^{\mathrm{DTNF}}=-20lg\left|G_{\mathrm{DTNF}}\left(j{w}_f\right)\right|=-20lg\zeta .\end{array}$$

(33)

With the notch frequency $w_f$ as the center frequency, when the amplitude of improved DTNF attenuate to −3 $\mathrm{dB}$, the corresponding frequencies $w\in \left(w_3,w_4\right)$ are all within the range of the −3 $\mathrm{dB}$ bandwidth $B_w^{\mathrm{DTNF}}$. $w_3$ is the upper limit frequency, and $w_4$ is the lower limit frequency, so we can obtain

$$\begin{array}{c}\hfill -20lg|G_{\mathrm{DTNF}}\left(j{w}_3\right)|=-20lg|G_{\mathrm{DTNF}}\left(j{w}_4\right)|=-3,\end{array}$$

(34)

$$\begin{array}{c}\hfill \left|{\displaystyle \frac{{\left(jw\right)}^2+\zeta k{w}_f·jw+w_f^2}{{\left(jw\right)}^2+k{w}_f·jw+w_f^2}}\right|={\displaystyle \frac{\sqrt{2}}{2}}.\end{array}$$

(35)

To obtain the solution of $\Delta f^{\mathrm{DTNF}}$ and $Q^{\mathrm{DTNF}}$, further calculation of (35) produces

$$\begin{array}{c}\hfill {\left({\displaystyle \frac{w}{w_f}}\right)}^4+\left(-2+2{\zeta }^2{k}^2-k^2\right){\left({\displaystyle \frac{w}{w_f}}\right)}^2+1=0.\end{array}$$

(36)

The results of solving (36) are

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\left({\displaystyle \frac{w_3}{w_f}}\right)}^2={\displaystyle \frac{\left(-2+2{\zeta }^2{k}^2-k^2\right)-\sqrt{{\left(-2+2{\zeta }^2{k}^2-k^2\right)}^2-4}}{2}}\hfill \\ {\left({\displaystyle \frac{w_4}{w_f}}\right)}^2={\displaystyle \frac{\left(-2+2{\zeta }^2{k}^2-k^2\right)+\sqrt{{\left(-2+2{\zeta }^2{k}^2-k^2\right)}^2-4}}{2}}\hfill \end{array}\right..\end{array}$$

(37)

Because $B_w^{\mathrm{DTNF}}=w_4-w_3=2\pi \Delta f^{\mathrm{DTNF}}$ and $w_f=\left(w_3+w_4\right)/2$ are already defined, the following formula can be obtained

$$\begin{array}{c}\hfill {\displaystyle \frac{w_4^2-w_3^2}{w_f^2}}={\displaystyle \frac{4\pi \Delta f^{\mathrm{DTNF}}}{w_f}}.\end{array}$$

(38)

According to (37) and (38), the solution of the notch width $\Delta f^{\mathrm{DTNF}}$ and the quality factor $Q^{\mathrm{DTNF}}$ can be obtained

$$\begin{array}{c}\hfill \Delta f^{\mathrm{DTNF}}={\displaystyle \frac{w_f\sqrt{{\left(-2+2{\zeta }^2{k}^2-k^2\right)}^2-4}}{4\pi }},\end{array}$$

(39)

$$\begin{array}{c}\hfill Q^{\mathrm{DTNF}}={\displaystyle \frac{w_f}{B_w^{\mathrm{DTNF}}}}={\displaystyle \frac{w_f}{2\pi \Delta f^{\mathrm{DTNF}}}}={\displaystyle \frac{2}{\sqrt{{\left(-2+2{\zeta }^2{k}^2-k^2\right)}^2-4}}}.\end{array}$$

(40)

When the value of the depth factor $\zeta$ is very small, we can obtain $\Delta f^{\mathrm{DTNF}}\approx w_{f}k\sqrt{k^2+4}/\left(4\pi \right)$ and $Q^{\mathrm{DTNF}}\approx 2/(k\sqrt{k^2+4})$. To obtain the solution of the width factor k, according to (37) and (38), we can further obtain

$$\begin{array}{c}\hfill {\left(-2+2{\zeta }^2{k}^2-k^2\right)}^2-4={\displaystyle \frac{16{\pi }^2\Delta f^2}{w_f^2}}.\end{array}$$

(41)

According to (41), the solution of the width factor k is

$$\begin{array}{c}\hfill k=\sqrt{{\displaystyle \frac{2-\sqrt{\frac{16{\pi }^2{\left(\Delta f^{\mathrm{DTNF}}\right)}^2}{w_f^2}+4}}{2\times {\left({10}^{-\frac{l_{\mathrm{depth}}^{\mathrm{DTNF}}}{20}}\right)}^2-1}}}.\end{array}$$

(42)

When the value of the depth factor $\zeta$ is very small, we can obtain

$$\begin{array}{c}\hfill k\approx \sqrt{\sqrt{16{\pi }^2{\left(\Delta f^{\mathrm{DTNF}}\right)}^2/w_f^2+4}-2}.\end{array}$$

(43)

To obtain the solution of the depth factor $\zeta$, according to (33), the solution of the depth factor $\zeta$ is

$$\begin{array}{c}\hfill \zeta ={10}^{-{\displaystyle \frac{l_{\mathrm{depth}}^{\mathrm{DTNF}}}{20}}}.\end{array}$$

(44)

In summary, (33), (39), (40), (42), and (44) are the core formulas of the improved DTNF, which are important foundations of the multi-parameter optimization algorithm.

The magnitude-phase frequency characteristics of the improved DTNF are shown in Figure 10. Obviously, compared with the traditional BNF, the improved DTNF realizes independent regulation of the notch depth $l_{\mathrm{depth}}^{\mathrm{DTNF}}$ and the notch width $\Delta f^{\mathrm{DTNF}}$ by the depth factor $\zeta$ and the width factor k, respectively. This is also confirmed in (39) and (44). However, both the depth factor $\zeta$ and the width factor k have an impact on the phase-lag angle $\varphi$.

Therefore, a contradiction will inevitably occur in the process of multi-parameter adjustment, and multi-objective trade-offs are needed to minimize this contradiction. In other words, it is very important to adjust the depth factor $\zeta$ and the width factor k based on the multi-objective constraint optimization algorithm.

3.4. Multi-Objective Constrained Optimization Method of Cascaded Improved DTNFs

Taking the notch depth $l_{\mathrm{depth}}^{\mathrm{DTNF}}$ at the notch frequency $w_f$, the notch width $\Delta f^{\mathrm{DTNF}}$ corresponding to −3 $\mathrm{dB}$, and the phase-lag angle $\varphi$ introduced by the DTNF at the open-loop cutoff frequency $w_c$ as the multi-objective constraints, the multi-parameter optimization algorithm of cascaded multi-frequency improved DTNFs is as follows.

As shown in Figure 10, the phase-frequency characteristic of the improved DTNF at the open-loop cutoff frequency $w_c(w_c<w_f)$ exhibits hysteresis. According to (31), the phase-lag angle $\varphi <0$ introduced by the improved DTNF is a function of the depth factor $\zeta$ and width factor k

$$\begin{array}{c}\hfill \varphi \left(\zeta ,k\right)=\left(arctan{\displaystyle \frac{\zeta k{w}_{f}w}{w_f^2-w^2}}-arctan{\displaystyle \frac{k{w}_{f}w}{w_f^2-w^2}}\right){\displaystyle \frac{180}{\pi }}.\end{array}$$

(45)

The relationship between the depth factor $\zeta$, width factor k, and phase-lag angle $\varphi$ is shown in Figure 11a. Obviously, the phase lag angle $\varphi$ decreases with increasing depth factor $\zeta$, and the phase-lag angle $\varphi$ decreases with the decreasing width factor k.

However, the multi-objective optimization should consider multiple constraints, so the effects of the depth factor $\zeta$ and width factor k on the notch depth $l_{\mathrm{depth}}^{\mathrm{DTNF}}$ and the notch width $\Delta f^{\mathrm{DTNF}}$ should also be taken into account, instead of just considering the effects of depth factor $\zeta$ and width factor k on the phase-lag angle $\varphi$.

To consider the effect of the depth factor $\zeta$ on the notch depth $l_{\mathrm{depth}}^{\mathrm{DTNF}}$ and the phase-lag angle $\varphi$, an analysis is carried out. For the cascade i-stage improved DTNFs, $l_{{\mathrm{depth}}_i}^{\mathrm{DTNF}}$ can be determined by the peak amplitude $E_{{\mathrm{amp}}_i}$ at multi-stage-blade modal frequencies, as shown in Figure 4a. According to (44), the constraint condition of ${\zeta }_i$ can be obtained

$$\begin{array}{c}\hfill 0<{\zeta }_i\le {10}^{-{\displaystyle \frac{E_{{\mathrm{amp}}_i}}{20}}}.\end{array}$$

(46)

If the depth factor ${\zeta }_i$ is too large, the notch depth $l_{{\mathrm{depth}}_i}^{\mathrm{DTNF}}$ provided by the improved DTNFs will not be sufficient, thus, the amplitude attenuation will be incomplete. If the depth factor ${\zeta }_i$ is too small, not only will the amplitude attenuation be excessive but also a larger phase-lag angle ${\varphi }_i\left({\zeta }_i,k_i\right)$ will be introduced by the improved DTNFs.

So, to ensure that ${\varphi }_i\left({\zeta }_i,k_i\right)$ introduced by ${\zeta }_i$ is minimized, ${\zeta }_i$ is designed to be the maximum value ${10}^{-E_{{\mathrm{amp}}_i}/20}$. Furthermore, the maximum depth factor ${\zeta }_i={10}^{-E_{{\mathrm{amp}}_i}/20}$ is calculated when the notch depth $l_{{\mathrm{depth}}_i}^{\mathrm{DTNF}}$ is exactly equal to the peak amplitude $E_{{\mathrm{amp}}_i}$ of the multi-stage-blade modal vibration. This guarantees that the incomplete attenuation phenomenon will not occur.

To consider the effect of the width factor k on the notch width $\Delta f^{\mathrm{DTNF}}$ and the phase-lag angle $\varphi$, an analysis is carried out. As shown in Figure 11b, varying ${\zeta }_i$ in a small range ${\zeta }_i\in \left(0.01,0.10\right)$ mostly has very little impact on the phase-lag angle ${\varphi }_i\left({\zeta }_i,k_i\right)$. Especially when the value of the width factor $k_i$ is smaller, such as $k\in \left(0,7\right)$.

Therefore, after determining the depth factor ${\zeta }_i$, in order to select width factor $k_i$, the effect of $k_i$ on the notch width $\Delta f_i^{\mathrm{DTNF}}\left({\zeta }_i,k_i\right)$ and phase-lag angle ${\varphi }_i\left({\zeta }_i,k_i\right)$ is balanced and optimized by employing a linear normalization method.

The normalization of the notch width $\Delta f_i^{\mathrm{DTNF}}{\left({\zeta }_i,k_i\right)}_{\mathrm{normal}}$ is designed as follows

$$\begin{array}{c}\hfill \Delta f_i^{\mathrm{DTNF}}{\left({\zeta }_i,k_i\right)}_{\mathrm{normal}}={\displaystyle \frac{\Delta f_i^{\mathrm{DTNF}}\left({\zeta }_i,k_i\right)-min\left[\Delta f_i^{\mathrm{DTNF}}\left({\zeta }_i,k_i\right)\right]}{max\left[\Delta f_i^{\mathrm{DTNF}}\left({\zeta }_i,k_i\right)\right]-min\left[\Delta f_i^{\mathrm{DTNF}}\left({\zeta }_i,k_i\right)\right]}}\end{array}$$

(47)

where $\Delta f_i^{\mathrm{DTNF}}\left({\zeta }_i,k_i\right)=w_{f_i}\sqrt{{\left(-2+2{\zeta }_i^2{k}_i^2-k_i^2\right)}^2-4}/4\pi .$

Similarly, the normalization of the phase-lag angle ${\varphi }_i{\left({\zeta }_i,k_i\right)}_{\mathrm{normal}}$ is designed as follows

$$\begin{array}{c}\hfill {\varphi }_i{\left({\zeta }_i,k_i\right)}_{\mathrm{normal}}={\displaystyle \frac{{\varphi }_i\left({\zeta }_i,k_i\right)-min\left[{\varphi }_i\left({\zeta }_i,k_i\right)\right]}{max\left[{\varphi }_i\left({\zeta }_i,k_i\right)\right]-min\left[{\varphi }_i\left({\zeta }_i,k_i\right)\right]}}\end{array}$$

(48)

where ${\varphi }_i\left({\zeta }_i,k_i\right)=\left[arctan{\zeta }_i{k}_i{w}_{f_i}{w}_i/\left(w_{f_i}^2-w_i^2\right)-arctan{k}_i{w}_{f_i}{w}_i/\left(w_{f_i}^2-w_i^2\right)\right]180/\pi .$

To balance the effect of the width factor $k_i$ on the notch width $\Delta f_i^{\mathrm{DTNF}}\left({\zeta }_i,k_i\right)$ and the phase-lag angle ${\varphi }_i\left({\zeta }_i,k_i\right)$, $P_i$ is introduced as an adjustable weight. According to (46) and (47), the objective function $M_{\mathrm{optimal}}$ is designed as follows

$$\begin{array}{c}\hfill M_{\mathrm{optimal}}=P_i\Delta f_i^{\mathrm{DTNF}}{\left({\zeta }_i,k_i\right)}_{\mathrm{normal}}+\left(1-P_i\right){\varphi }_i{\left({\zeta }_i,k_i\right)}_{\mathrm{normal}}.\end{array}$$

(49)

When the MSTMP works at the rated speed of 24,000 $\mathrm{rpm}$, it can be seen from the analysis in Section 2.2 that there are six different frequencies (350 $\mathrm{Hz}$, 463 $\mathrm{Hz}$, 530 $\mathrm{Hz}$, 628 $\mathrm{Hz}$, 745 $\mathrm{Hz}$, 980 $\mathrm{Hz}$) of multi-stage-blade modal vibration. At six different blade mode frequencies, comprehensive normalized design results for the width factor k with multi-objective constraints are shown in Figure 12. Designed values of the notch depth $l_{{\mathrm{depth}}_i}^{\mathrm{DTNF}}$, the depth factor ${\zeta }_i$, the width factor $k_i$, and the phase-lag angle ${\varphi }_i\left(w_c\right)$ are shown in

Table 2. Designed values of improved DTNF based on multi-objective constrained optimization algorithm (With $\Omega$ = 24,000 $\mathrm{rpm}$).

${\mathit{w}}_{\mathit{f}}$ ($\mathbf{Hz}$)	${\mathit{l}}_{\mathbf{depth}}^{\mathbf{DTNF}}$ ($\mathbf{dB}$)	$\mathit{\zeta }$	$\mathit{P}$	${\mathit{k}}_{\mathbf{span}}$	$\mathit{k}$	$\mathit{\varphi }$(wc) ($\mathbf{deg}$)
350	2	$0.7943$	$0.85$	$\left(0,5\right)$	$1.4290$	$-2.3306$
463	14	$0.1995$	$0.84$	$\left(0,7\right)$	$0.7143$	$-3.9647$
530	51	$0.0028$	$0.83$	$\left(0,7\right)$	$0.2857$	$-1.0263$
628	29	$0.0355$	$0.82$	$\left(0,7\right)$	$0.4286$	$-3.6685$
745	5	$0.5623$	$0.81$	$\left(0,6\right)$	$0.8571$	$-2.7702$
980	11	$0.2818$	$0.80$	$\left(0,7\right)$	$0.4286$	$-1.7279$

. In actual operation, parameters are designed according to different mode frequencies at different speed stages.

Framework implementation of the integrated AMB controller and the proposed algorithm is shown in Figure 13.

4. Simulation and Experimental Results

4.1. Simulation Analysis

Using MATLAB R2020a to simulate and analyze stability, the parameters of the MSTMP system are listed in

Table 3. Structural and Control Parameters of the MSTMP system.

Parameter	Symbol	Value
Turbo rotor mass	m	$12.627\mathrm{kg}$
Equatorial moments of inertia	$J_r$	$0.0647\mathrm{kg} {\mathrm{m}}^2$
Polar moments of inertia	$J_z$	$0.0496\mathrm{kg} {\mathrm{m}}^2$
Distance from O to End-A sensor center	$a_s$	$0.0363\mathrm{m}$
Distance from O to End-B sensor center	$b_s$	$0.0897\mathrm{m}$
Distance from O to End-A magnetic bearing center	$a_m$	$0.0088\mathrm{m}$
Distance from O to End-B magnetic bearing center	$b_m$	$0.0692\mathrm{m}$
Current stiffness of End-A magnetic bearing	$k_{ia}$	$426\mathrm{N} {\mathrm{A}}^{-1}$
Current stiffness of End-B magnetic bearing	$k_{ib}$	$239\mathrm{N} {\mathrm{A}}^{-1}$
Displacement stiffness of End-A magnetic bearing	$k_{ha}$	$8.656\times {10}^5\mathrm{N} {\mathrm{m}}^{-1}$
Displacement stiffness of End-B magnetic bearing	$k_{hb}$	$4.869\times {10}^5\mathrm{N} {\mathrm{m}}^{-1}$
Protective gap	p	$0.0002\mathrm{m}$
Magnetic gap	$p_0$	$0.0004\mathrm{m}$
Displacement sensor magnification	$k_s$	$8.533\times {10}^6\mathrm{V} {\mathrm{m}}^{-1}$
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	$K_w$	$3.264\times {10}^{-4}$
Power amplifier bandwidth	$W_w$	$3.081\times {10}^3$

. 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. Rotor Assembly Material Constants of MSTMP.

Materials	Young’s Modulus ($\mathbf{Pa}$)	Density ($\mathbf{kg} {\mathbf{m}}^3$)	Poisson’s Ratio
1Cr18Ni9Ti	$1.84\times {10}^{11}$	7900	$0.31$
Sm2Co17	$1.08\times {10}^{11}$	8400	$0.30$
GH4169	$1.999\times {10}^{11}$	8240	$0.30$
Silicon steel	$2.06\times {10}^{11}$	7650	$0.30$
40CrNiMoA	$2.05\times {10}^{11}$	7850	$0.30$
Aluminum	$6.19\times {10}^{10}$	2710	$0.31$

.

The AMB-rotor system with a parallel co-frequency ANF results in a change in the closed-loop transfer function. Both the bandwidth factor $\epsilon$ 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 $\epsilon$ increases, the ANF has different critical instability of $\epsilon$ at different $\Omega$ (70 Hz, $\epsilon$ = 10, 110 Hz, $\epsilon$ = 100, and 120 Hz, $\epsilon$ = 200). Furthermore, as the root-locus dominant pole trajectory varies in the S-plane as $\Omega$ increases, the ANF with different $\epsilon$ has different critical instability of $\Omega$ ($\epsilon$ = 20, 90 Hz, $\epsilon$ = 50, 100 Hz, $\epsilon$ = 100, 110 Hz, $\epsilon$ = 200, 120 Hz, $\epsilon$ = 300, 140 Hz, $\epsilon$ = 400, 150 Hz, and $\epsilon$ = 500, 160 Hz). In short, dynamic rectification of the bandwidth factor $\epsilon$ based on displacement stiffness $k_h$ perturbation can avoid the instability introduced by the inappropriate choice of bandwidth factor $\epsilon$ of the ANF.

Nyquist frequency characteristic of the open-loop system with a parallel co-frequency ANF and six cascaded multi-frequency improved DTNFs is shown in Figure 15a. By the normalized optimization of the depth factor ${\zeta }_i$ and width factor $k_i$ based on the multi-objective constraint algorithm, as the number of DTNFs increases, the phase stability margin of the AMB-rotor system can still be maintained, although the phase margin is inevitably reduced.

The pole-zero map of the characteristic polynomial is shown in Figure 15b. Zeros of the characteristic polynomial represent closed-loop poles, and poles of the characteristic polynomial represent open-loop poles. The right half S-plane has an open-loop pole, that is, although the open-loop system is unstable, the closed-loop system is stable.

4.2. 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, 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, Chen 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 ($\Omega =0\mathrm{rpm}$), 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 ($\Omega =0\mathrm{rpm}$) 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 $\mathrm{mm}$. 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 $\mathrm{rpm}$. 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 $\mathrm{rpm}$, the rotor displacement in the four radial channels of AX, AY, BX, and BY with different algorithms are shown in Figure 17b.

At the rated speed of 24,000 $\mathrm{rpm}$, 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 $\mathrm{mm}$ to 0.08 $\mathrm{mm}$, a reduction of 52.94%. With the proposed algorithm, the value of the rotor displacement converges from 0.08 $\mathrm{mm}$ to 0.03 $\mathrm{mm}$, a reduction of 62.50%, which is far less than the unilateral protective gap of the auxiliary bearing (0.2 $\mathrm{mm}$). Moreover, in terms of the convergence time of different algorithms, the convergence time of the ANF-PSF algorithm is 3.67 $\mathrm{s}$ and the convergence time of the proposed algorithm is 2.85 $\mathrm{s}$. The latter is 0.82 $\mathrm{s}$ faster than the former, a reduction of 22.34%.

At the rated speed of 24,000 $\mathrm{rpm}$, the left view of the End-B axis trajectory diagram with different algorithms is shown in Figure 18b. Without any algorithm, with the ANF-PSF algorithm, and with the proposed algorithm, the maximum peak value of rotor displacement are 42.5%, 20%, and 7.5% of the double protective gap of the auxiliary bearing 0.4 $\mathrm{mm}$, respectively.

At rotational speeds of 12,000 $\mathrm{rpm}$, 15,000 $\mathrm{rpm}$, 18,000 $\mathrm{rpm}$, and 21,000 $\mathrm{rpm}$, FFTs of the rotor displacement in the BY channel with different algorithms are shown in Figure 19. With the ANF-PSF algorithm, the suppression of co-frequency synchronous vibration is approximately maintained between −21.57 $\mathrm{dB}$ and −17.39 $\mathrm{dB}$, and the fluctuation range varies greatly with the rotor speed. With the proposed algorithm, the co-frequency synchronous suppression is further attenuated, being approximately maintained between −30.94 $\mathrm{dB}$ and −30.56 $\mathrm{dB}$, and the fluctuation range varies relatively little with the rotor speed. Therefore, it is verified that the proposed algorithm can guarantee the consistency of vibration suppression at different rotational speeds.

At the rated speed of 24,000 rpm, FFTs of the rotor displacement in the BY channel with different algorithms are shown in Figure 20. With the ANF-PSF algorithm, the amplitude of the synchronous frequency (SF) is reduced from −15.82 dB to −17.36 dB, and the amplitude of the nutation frequency (NF) is reduced from −58.25 dB to −62.07 dB. With the proposed algorithm, the amplitude of the SF is further reduced to −30.74 dB, and the amplitude of the NF is further reduced to −71.65 dB. The reason for this phenomenon is that after optimizing multiple notch parameters such as the bandwidth factor $\epsilon$, the depth factor $\zeta$, and the width factor k, the phase-lag angle introduced by the parallel co-frequency ANF and the cascaded multi-frequency improved DTNFs is limited, which leads to an increase in the phase stability margin of the entire AMB-rotor system.

Structure improvement of notch filters and the multi-parameter optimization of the algorithm are not only limited to incomplete differential PID controllers but also can be combined with other advanced controllers, giving the approach strong engineering application and universality.

5. Conclusions

A novel integrated AMB controller consisting of parallel co-frequency ANF and cascaded multi-frequency improved DTNFs is proposed. To suppress rotor mass imbalance vibration, a bandwidth factor rectification method for the ANF based on displacement stiffness perturbations is designed. This addresses the issue of the phase-lag angle $\phi \left(\epsilon \right)$ introduced at the open-loop cutoff frequency by an ANF with an inappropriate choice of bandwidth factor $\epsilon$, which can adversely affect the phase stability margin $\gamma$ of the MSR system. To suppress multi-stage-blade modal vibration, a multi-objective constrained optimization method for cascaded improved DTNFs based on linear normalization is developed. This method addresses the problems of an excessive notch (resulting in too large a phase-lag angle) and an incomplete notch (insufficient notch depth).

The effects of the bandwidth factor $\epsilon$, depth factor $\zeta$, and width factor k on the stability of the MSR system are comprehensively simulated and analyzed in detail. Simulation and experimental results validate the effectiveness of the proposed method. The proposed algorithm is compared with an existing algorithm. At rotational speeds of 12,000 $\mathrm{rpm}$, 15,000 $\mathrm{rpm}$, 18,000 $\mathrm{rpm}$, and 21,000 $\mathrm{rpm}$, the spectrum only includes fundamental and harmonic components, with the suppression of co-frequency synchronous vibration maintained between −30.94 $\mathrm{dB}$ and −30.56 $\mathrm{dB}$, exhibiting minimal fluctuation across different rotor speeds. At the rated speed of 24,000 $\mathrm{rpm}$, the value of the rotor displacement converges from 0.08 $\mathrm{mm}$ (ANF-PSF algorithm) to 0.03 $\mathrm{mm}$ (proposed algorithm), a reduction of 62.50%. The convergence time of the algorithm decreases from 3.67 $\mathrm{s}$ (ANF-PSF algorithm) to 2.85 $\mathrm{s}$ (proposed algorithm), a reduction of 22.34%.