Research of a Six-Pole Active Magnetic Bearing System Based on a Fuzzy Active Controller

Abstract

Magnetic bearings have a series of excellent qualities, such as no friction and abrasions, high speed, high accuracy, and so on, which have fundamentally innovated traditional forms of support. In order to solve the problems of the large volume, low power density and high coupling coefficient of three-pole magnetic bearings, a six-pole AC active magnetic bearing is designed. Firstly, the basic structure and working principle of a two-degree-of-freedom (2-DOF) six-pole active magnetic bearing is introduced. Secondly, a suspension force modeling method of a 2-DOF AC active magnetic bearing based on the Maxwell tensor method is proposed, and the mathematical model of active magnetic bearing is established. Considering the fact that AC active magnetic bearing is essentially a nonlinear system, a fuzzy active disturbance rejection control (ADRC) method is designed based on fuzzy control and ADRC theory. Its control algorithm and control block diagram are given, and the fuzzy ADRC method is simulated and verified. Finally, the control block diagram of an experimental system based on the 2-DOF six-pole active magnetic bearing is given, and the experimental platform is constructed. The experimental results show that the mechanical and magnetic circuit structure of the 2-DOF six-pole active magnetic bearing is reasonable, and the fuzzy controllers can realize the stable suspension of the rotor.

Introduction

A magnetic bearing is supported by the stator, ensuring that the rotor moves in a fixed range. The stator and the rotor have no mechanical contact through the magnetic levitation force [1,2]; therefore, it has the advantages of no lubrication, no friction, high speed, high accuracy, etc. [3]. At present, it is widely known that the radial magnetic bearings with an 8-pole structure are usually driven by 4 unipolar power amplifiers or 2 bipolar power amplifiers [4], which leads to the disadvantages of a large size and high cost of magnetic bearing power amplifiers [5]. The literature [6] has put forward a three-pole AC active magnetic bearing, which has significant advantages in reducing power losses and saving costs. The six-pole magnetic bearing is similar to the three-pole structure, which has six magnetic poles and is evenly distributed. Compared with the three-pole magnetic bearing with the same volume, the maximum bearing capacity of the six-pole magnetic bearing is increased, and the force of the rotor is uniform.

The characteristics between the suspension force and the control current are nonlinear in the traditional method, and mathematical modeling cannot be accurately established. The control methods commonly used at present are PID control, fuzzy control, neural network control, variable structure control, and so on. A fuzzy controller is used to control the experimental platform in the literature [7], which has certain anti-interference. In the literature [8], a linear/nonlinear ADRC is proposed to decouple the magnetic bearing system.

By establishing an extended state observer, the uncertain link in the system can be estimated and compensated as the total disturbance. The structure design of the extended state observer is simple and easy to implement in engineering, but it is difficult to use because of its many parameters to be adjusted. By using the fuzzy controller, the parameters in the ADRC can be adaptively adjusted, and the system can quickly achieve control stability [9]. Based on the basic structure of six-pole magnetic bearings, a radial suspension forces a modeling method of AC active magnetic bearing based on the Maxwell tensor method, adopted in this manuscript [10].

In this manuscript, the fuzzy ADRC is proposed to solve the coupling problems between the degrees of freedom of a six-pole active magnetic bearing. The structure and the working principle are introduced in Section 2. In Section 3, the mathematical model of the six-pole AMB is built. In Section 4, the fuzzy ADRC of the six-pole AMB is simulated and verified. In Section 5, the experiments are carried out. Finally, the conclusion is drawn in Section 6.

2. Basic Structure and Working Principle of a 2-DOF Six-Pole Active Magnetic Bearing

2.1. Fundamental Structure

Figure 1 shows the combination of the magnetic bearing. Figure 1a is an explosion diagram of the six-pole AMB, and Figure 1b is the axial and radial views of the six-pole AMB, composed of six radial control coils, a rotor, a rotating shaft and a stator. The outer coaxial sleeve of the rotor has a circular stator core, and six radial protruding stator poles are arranged at the inner edge of the stator core. These six stator poles are arranged uniformly along the circumferential direction, and the control coils are wound around them. The six control coils are all connected to AC power through a three-phase inverter; two opposite ones are in series and have the same winding direction as the one phase, and the three-phase coil is in a star link.

2.2. Working Principle

The radial suspension force in AC magnetic bearings is mainly Maxwell forces. The magnitude of the suspension force is determined by the magnetic pole area of the rotor surface, and the direction is a medium vertical surface facing outward. The Maxwell force in 2-DOF active magnetic bearings is composed of two parts [11,12]: one part is the force produced by the eccentricity of the rotor subjected to an external disturbance without applying the controlled current, and the other part is the Maxwell force produced by the control current when the rotor is in a balanced position in the bias magnetic field.

In order to satisfy the condition that a symmetric magnetic pole can counteract the quadratic term of the current, taking phase A (magnetic poles A1 and A2) as an example, it is required that the control current generated in the air gap on both sides of phase A is equal in amplitude. When the control current is positive, the suspension force is generated in the A1 direction and then in the A2 direction. When the control current is negative, phases B and C are the same way.

3. Mathematical Model of the Suspension Force of a Six-Pole Active Magnetic Bearing

3.1. Proposed Method of Constructing a Suspension Force Model Based on the Maxwell Tensor Method

The suspension principle of AC magnetic bearings is similar to that of a bearingless motor [13]. The condition for the production of radial suspension force, PB = PM ± 1, is satisfied, and a three-phase inverter drives the radial control coil. This special type of AC magnetic bearing can be considered a special kind of bearingless motor. Compared with the traditional radial suspension force modeling method, this method has three advantages in the modeling process:

• Accuracy: the drawback of the error caused by the simplified Maxwell force formula is avoided, and the integral calculation of the Maxwell force on the rotor surface can be carried out;
• Directness: the drawback of a complicated calculation process caused by a large amount of detailed equivalent magnetic circuit analysis and formula derivation in the process of modeling is avoided. It is only necessary to modify the expression of Maxwell’s tension directly according to the magnetic circuit characteristics and structure of the modeling object;
• Universality: the drawback of so much magnetic circuit analysis based on different types and structures of modeling objects is avoided. The key expressions in the modeling process can be modified only by referring to the suspension force modeling method introduced in manuscripts.

3.2. Mathematical Model of Suspension Force for a 2-DOF Six-Pole Active Magnetic Bearing

For the convenience of analysis, the leakage inductance between the stator magnetic poles, the leakage inductance at the end of the winding and the magnetic saturation effect, as well as the nonlinear magnetic saturation, eddy current and core loss of ferromagnetic materials, are all ignored in this manuscript [14].

According to the Maxwell tensor method [15,16], the Maxwell force on the rotor surface per unit area dS along the spatial angle θ can be obtained:

$$\mathrm{d}F(\theta )=\frac{B^2(\theta ,t)\cdot \mathrm{d}S}{2{\mu }_0}=\frac{B^2(\theta ,t)}{2{\mu }_0}\cdot (lr\mathrm{d}\theta )$$

(1)

where l denotes the equivalent length of the rotor, r denotes the rotor radius, and μ₀ denotes the permeability of the vacuum.

When the rotor is eccentrically subjected to the external disturbance force f, the resultant force F₀ on the rotor surface per unit area dS along the spatial angle θ is

$$\begin{array}{ll}\mathrm{d}{F}_0(\theta )& =\mathrm{d}(\frac{B^2(\theta ,t)}{2{\mu }_0}\cdot S+f)\\ & =\frac{B^2(\theta ,t)\cdot \mathrm{d}S}{2{\mu }_0}+f_0\cdot \mathrm{d}S\\ & =\frac{B^2(\theta ,t)}{2{\mu }_0}\cdot (lr\mathrm{d}\theta )+f_0\cdot \mathrm{d}S\end{array}$$

(2)

where f₀ denotes the external disturbance force acting on the unit area of the rotor surface.

The Maxwell force applied to the rotor is decomposed along the x, y-axis into:

$$\{\begin{array}{l}\mathrm{d}{F}_x(\theta )=\mathrm{d}F(\theta )\cos \theta =\frac{B^2(\theta ,t)\cdot lr\cos \theta }{2{\mu }_0}\mathrm{d}\theta \\ \mathrm{d}{F}_y(\theta )=\mathrm{d}F(\theta )\sin \theta =\frac{B^2(\theta ,t)\cdot lr\sin \theta }{2{\mu }_0}\mathrm{d}\theta \end{array}$$

(3)

For the designed AC magnetic bearing system, both l and r are fixed values. Assuming that f_x, f_y is the component of f on the X, Y axis, x₀, and y₀ is the offset component of the center of the rotor after eccentricity relative to the original balance position, the expression of rotor eccentricity angle α is

$$\alpha =\arctan (y_0/x_0)$$

(4)

where the direction of eccentricity is the same as the external disturbance force f of the rotor.

If the higher harmonic is ignored, the fundamental wave of the air gap magnetomotive force(MMF) generated by the control coil is

$$f_2(\theta ,t)=F_2\cos (\theta -\omega t-\phi )$$

(5)

where F₂ denotes the amplitude of the fundamental wave of the MMF, φ denotes the MMF vector space phase angle at rotor balance time, θ denotes the air gap spatial angle, and ω denotes the electric angle of the current variation.

Thus, the flux density of the control coil at any angle in the air gap at t time is as follows:

$$B_2(\theta ,t)=F_2\cos (\theta -\omega t-\phi )\cdot \frac{{\mu }_0}{2{\delta }_0(1-\epsilon \cos (\theta -\alpha ))}$$

(6)

and

$$F_2=\frac{3}{2}{F}_{2A}=\frac{3}{2}{F}_{2B}=\frac{3}{2}{F}_{2C}=\frac{3}{2}\cdot \frac{N_{3}I}{p}$$

(7)

where F_2A, F_2B, F_2C denotes the amplitude of air gap MMF generated by the three-phase current of phase A, B, and C, respectively. N₃ denotes the effective turns per phase of three-phase coils. The pole logarithm of the rotating magnetic field of AC 2-DOF active magnetic bearing p is 1 in this manuscript.

The amplitude of the air gap flux density generated by the control coil is

$$B_2=\frac{{\mu }_0\cdot F_2}{2{\delta }_0}$$

(8)

Substituting Equation (8) into Equation (6), the value of ε is too small, so its square term is neglected in the course of calculation

$$B_2(\theta ,t)=B_2\cos (\theta -\omega t-\phi )(1+\epsilon \cos (\theta -\alpha ))$$

(9)

The MMF generated by the bias magnetic field is

$$F_1=N_3{I}_0$$

(10)

where I₀ denotes the amplitude of the bias current on the coil per phase (fixed value).

The expression of bias flux density is

$$B_1(\theta ,t)=B_1\cdot (1+\epsilon \cos (\theta -\alpha ))$$

(11)

According to the principle of magnetic flux superposition, the total flux density B(θ, t) produced by the bias coil and the control coil in the air gap is

$$\begin{array}{ll}B(\theta ,t)& =B_1(\theta ,t)+B_2(\theta ,t)\\ & =(1+\epsilon \cos (\theta -\alpha ))\cdot \left(B_1+B_2\cos (\theta -\omega t-\phi )\right)\end{array}$$

(12)

Therefore, the component of the Maxwell force on the X, Y-axis is given as

$$\{\begin{array}{ll}{F}_{\mathrm{x}}& ={\displaystyle {\int }_0^{2\pi }\frac{B^2(\theta ,t)\cdot lr\cos \theta \mathrm{d}\theta }{2{\mu }_0}}\\ & ={\displaystyle {\int }_{-\frac{5\pi }{36}}^{\frac{5\pi }{36}}\frac{B^2(\theta ,t)\cdot lr\cos \theta \mathrm{d}\theta }{2{\mu }_0}}+{\displaystyle {\int }_{\frac{7\pi }{36}}^{\frac{17\pi }{36}}\frac{B^2(\theta ,t)\cdot lr\cos \theta \mathrm{d}\theta }{2{\mu }_0}}\\ & ={\displaystyle {\int }_{\frac{19\pi }{36}}^{\frac{29\pi }{36}}\frac{B^2(\theta ,t)\cdot lr\cos \theta \mathrm{d}\theta }{2{\mu }_0}}+{\displaystyle {\int }_{\frac{31\pi }{36}}^{\frac{41\pi }{36}}\frac{B^2(\theta ,t)\cdot lr\cos \theta \mathrm{d}\theta }{2{\mu }_0}}\\ & ={\displaystyle {\int }_{\frac{43\pi }{36}}^{\frac{53\pi }{36}}\frac{B^2(\theta ,t)\cdot lr\cos \theta \mathrm{d}\theta }{2{\mu }_0}}+{\displaystyle {\int }_{\frac{55\pi }{36}}^{\frac{65\pi }{36}}\frac{B^2(\theta ,t)\cdot lr\cos \theta \mathrm{d}\theta }{2{\mu }_0}}\\ F_{\mathrm{y}}& ={\displaystyle {\int }_0^{2\pi }\frac{B^2(\theta ,t)\cdot lr\sin \theta \mathrm{d}\theta }{2{\mu }_0}}\\ & ={\displaystyle {\int }_{-\frac{5\pi }{36}}^{\frac{5\pi }{36}}\frac{B^2(\theta ,t)\cdot lr\sin \theta \mathrm{d}\theta }{2{\mu }_0}}+{\displaystyle {\int }_{\frac{7\pi }{36}}^{\frac{17\pi }{36}}\frac{B^2(\theta ,t)\cdot lr\sin \theta \mathrm{d}\theta }{2{\mu }_0}}\\ & ={\displaystyle {\int }_{\frac{19\pi }{36}}^{\frac{29\pi }{36}}\frac{B^2(\theta ,t)\cdot lr\sin \theta \mathrm{d}\theta }{2{\mu }_0}}+{\displaystyle {\int }_{\frac{31\pi }{36}}^{\frac{41\pi }{36}}\frac{B^2(\theta ,t)\cdot lr\sin \theta \mathrm{d}\theta }{2{\mu }_0}}\\ & ={\displaystyle {\int }_{\frac{43\pi }{36}0}^{\frac{53\pi }{36}}\frac{B^2(\theta ,t)\cdot lr\sin \theta \mathrm{d}\theta }{2{\mu }_0}}+{\displaystyle {\int }_{\frac{55\pi }{36}}^{\frac{65\pi }{36}}\frac{B^2(\theta ,t)\cdot lr\sin \theta \mathrm{d}\theta }{2{\mu }_0}}\end{array}$$

(13)

Because the value of ε and B₂ is very small, their square term is neglected for convenient calculation. Equation (13) can be simplified as follows:

$$\{\begin{array}{l}{F}_x=\frac{lr}{2{\mu }_0}(\frac{5\pi }{3}{B}_1{B}_2\cos \phi +\frac{5\pi }{3}\epsilon B_1{}^2\cos \alpha )\\ F_y=\frac{lr}{2{\mu }_0}(\frac{5\pi }{3}{B}_1{B}_2\sin \phi +\frac{5\pi }{3}\epsilon B_1{}^2\sin \alpha )\end{array}$$

(14)

The former term in Equation (14) is the Maxwell force generated by the current through the control coil, decomposed to the components on the X, Y-axis and transformed by Clark coordinates. It can be seen that the amplitude of the three-phase composite current is 3/2 times of the current of each phase coil, so the model of controllable radial suspension force can be obtained:

$$\{\begin{array}{l}{F}_{1x}=\frac{5\pi lr{\mu }_0{N}_3^2{I}_0}{24{\delta }_0{}^2}\cdot i_{xc}\\ F_{1y}=\frac{5\pi lr{\mu }_0{N}_3^2{I}_0}{24{\delta }_0{}^2}\cdot i_{yc}\end{array}$$

(15)

where i_xc and i_yc denote the current component of the three-phase composite current converted by Clark transformation to the X, Y axis.

The transformation of the three-phase control current from a three-phase stationary coordinate system to a two-phase is given by

$$N_3\left[\begin{array}{c}{i}_{\mathrm{A}}\\ i_{\mathrm{B}}\\ i_{\mathrm{C}}\end{array}\right]=N_2\left[\begin{array}{cc}1& 0\\ -\frac{1}{2}& \frac{\sqrt{3}}{2}\\ -\frac{1}{2}& -\frac{\sqrt{3}}{2}\end{array}\right]\left[\begin{array}{c}{i}_x\\ i_y\end{array}\right]$$

(16)

where i_A, i_B, and i_C denote the three-phase control current, N₂ denotes the equivalent number of control coils in the two-phase coordinate system, and i_x, i_y denotes the three-phase control current on the X, Y-axis, which is converted from 3/2 transformation, in the case of total power invariant.

Equation (17) can be obtained from Equation (16) as follows:

$$\frac{N_3}{N_2}=\sqrt{\frac{2}{3}}$$

(17)

Thus,

$$\left[\begin{array}{l}{i}_{xc}\\ i_y{}_c\end{array}\right]=\frac{N_2}{N_3}\cdot \left[\begin{array}{l}{i}_x\\ i_y\end{array}\right]=\left[\begin{array}{l}\frac{\sqrt{3}}{\sqrt{2}}{i}_x\\ \frac{\sqrt{3}}{\sqrt{2}}{i}_y\end{array}\right]$$

(18)

Equation (18) is substituted into Equation (15); then, the controllable radial suspension force model is obtained:

$$\{\begin{array}{l}{F}_{1x}=\frac{\sqrt{3}}{\sqrt{2}}\cdot \frac{5\pi lr{\mu }_0{N}_3^2{I}_0}{24{\delta }_0{}^2}\cdot i_x\\ F_{1y}=\frac{\sqrt{3}}{\sqrt{2}}\cdot \frac{5\pi lr{\mu }_0{N}_3^2{I}_0}{24{\delta }_0{}^2}\cdot i_y\end{array}$$

(19)

Similarly, the latter term in Equation (14) is the Maxwell force produced by rotor eccentricity. Equation (10) is substituted into Equation (14), and its component force on the X, Y-axis can be obtained, whose direction is consistent with the direction of rotor eccentricity

$$\{\begin{array}{ll}{F}_{2x}& =\frac{5\pi lr\epsilon N_3{}^2{I}_0{}^2{\mu }_0}{24{\delta }_0{}^2}\cdot \cos \alpha \\ & =\frac{5\pi lr{N}_3{}^2{I}_0{}^2{\mu }_0}{24{\delta }_0{}^3}\cdot x\\ F_{2y}& =\frac{5\pi lr\epsilon N_3{}^2{I}_0{}^2{\mu }_0}{24{\delta }_0{}^2}\cdot \sin \alpha \\ & =\frac{5\pi lr{N}_3{}^2{I}_0{}^2{\mu }_0}{24{\delta }_0{}^3}\cdot y\end{array}$$

(20)

Therefore, the suspension force model of the 2-DOF six-pole active magnetic bearing is as follows:

$$\{\begin{array}{l}{F}_x=F_{1x}+F_{2x}=\frac{\sqrt{3}}{\sqrt{2}}\cdot \frac{5\pi lr{\mu }_0{N}_3^2{I}_0}{24{\delta }_0{}^2}\cdot i_x+\frac{5\pi lr{N}_3{}^2{I}_0{}^2{\mu }_0}{24{\delta }_0{}^3}\cdot x\\ F_y=F_{1y}+F_{2y}=\frac{\sqrt{3}}{\sqrt{2}}\cdot \frac{5\pi lr{\mu }_0{N}_3^2{I}_0}{24{\delta }_0{}^2}\cdot i_y+\frac{5\pi lr{N}_3{}^2{I}_0{}^2{\mu }_0}{24{\delta }_0{}^3}\cdot y\end{array}$$

(21)

Because the levitation force generated by the bias magnetic EMF is used to balance the gravity of the rotor when the rotor is in the balance position and suspends stably, the F_x = 0, F_y = G (gravity of rotor) is required, so the corresponding equivalent current i_x and i_y are respectively equal to the offset current i_x0 and i_y0, which can be obtained as

$$\{\begin{array}{l}{i}_{\mathrm{x}0}=0\\ i_{\mathrm{y}0}=\pm \sqrt{\frac{12(\sqrt{6}-\sqrt{2}){\delta }_0^{2}G}{5\pi lr{\mu }_0{N}_3^2}}\end{array}$$

(22)

4. Study on Control Strategy of Six-Pole Active Magnetic Bearing

4.1. Structure of Active Disturbance Rejection Controller

Magnetic bearings have strong nonlinear characteristics and are easily disturbed by external factors such as vibration and shock [17], which are more complicated to control. The ADRC is a new controller designed based on feedback linearization. It is composed of a nonlinear tracking differentiator (TD), extended state observer (ESO) and nonlinear state error feedback rule (NLSEF).

4.2. Design of Controller

(1) Organize the transition process (TD):

$$\{\begin{array}{l}\stackrel{\cdot }{{\upsilon }_1}={\upsilon }_2\\ \stackrel{\cdot }{{\upsilon }_2}=-R\cdot \mathrm{sat}(A,\det 1)\end{array}$$

(23)

$$\mathrm{sat}(A,\det 1)=\{\begin{array}{l}\mathrm{sign}(A),\left|A\right|\ge \det 1\\ A/\det 1,\left|A\right|<\det 1\end{array}$$

(24)

$$A={\upsilon }_1-u_{\mathrm{r}}+{\upsilon }_2\left|{\upsilon }_2\right|/2R$$

(25)

(2) Extended state observer (ESO) [18]:

$$\{\begin{array}{l}e=z_1-y\\ \stackrel{\cdot }{z_1}=z_2-\mathrm{bet}1\cdot \mathrm{fal}(e,a_1,\det 2)\\ \stackrel{\cdot }{z_2}=z_3-\mathrm{bet}2\cdot \mathrm{fal}(e,a_2,\det 2)+\mathrm{bu}\\ \stackrel{\cdot }{z_3}=-\mathrm{bet}3\cdot \mathrm{fal}(e,a_3,\det 2)\end{array}$$

(26)

(3) Nonlinear state error feedback (NLSEF):

$$\{\begin{array}{l}{e}_1={\upsilon }_1-z_1\\ e_2={\upsilon }_2-z_2\\ u_0=\mathrm{bt}1\cdot \mathrm{fal}(e_1,a_4,\det 3)\\ \hspace{1em}+\mathrm{bt}2\cdot \mathrm{fal}(e_2,a_5,\det 3)\\ u=u_0-z_3/b\end{array}$$

(27)

$$\mathrm{fal}(e,a,\det )=\{\begin{array}{l}{\left|e\right|}^a\cdot sign(e),\left|e\right|\ge \det \\ e/{\det }^{1-a},\left|e\right|<\det \end{array}$$

(28)

where R, det1, det2, det3, bet1, bet2, bet3, a₁, a₂, a₃, a₄, a₅, b, bt1, and bt2 are all undetermined parameters and need to be adjusted to determine.

4.3. Fuzzy Active Disturbance Rejection Controller

A fuzzy algorithm is mainly composed of fuzzy logic, fuzzification, fuzzy reasoning and fuzzy solution [19], as shown in Figure 2. It has strong fault tolerance and adaptability. It can solve the problems of system delay and nonlinearity, but it also has the problem of not having integral elements and low control precision.

Figure 3 is a schematic diagram of the active magnetic bearing fuzzy ADRC, composed of a second-order TD, nonlinear state error feedback rule (NLSEF), third-order ESO and the fuzzy controller. In practical application, the nonlinear feedback parameter β₁ is the proportional coefficient, and β₂ is the differential coefficient, which is similar to the tuning of parameters of the PD controller, K_p and K_d [20]. Therefore, a fuzzy controller is designed, e₁ and e₂ are used as input, and the parameters of ADRC are modified online by using fuzzy control rules to meet the requirements of β₁ and β₂ self-tuning at different times.

4.4. Parameter Tuning of Active Disturbance Rejection Control

Because the three components of ADRC, TD, ESO and NLSEF are designed independently, their parameters can be adjusted independently [21]. In the process of parameter tuning, it can be found that the parameters of TD can be immobilized. The parameter of ESO is proportional to the amplitude of disturbance, and the larger the amplitude of disturbance is, the larger the parameter is [22]. Therefore, using the difference value e and the change rate of the difference value e_c as input, the fuzzy control theory is used to modify parameters β₁ and β₂ online.

The variation range of e, e_c, and output U₁, U₂ is defined as a domain on a fuzzy set e, e_c, U₁, U₂ = −3, −2, −1, 0, 1, 2, 3, and its member functions are all trigonometric functions with high sensitivity.

The fuzzy control subset in this controller is e, e_c, U₁, U₂ = {NB, NM, NS ZO, PS, PM, PB}, establishing the fuzzy control rule table for β₁ and β₂, as shown in

Table 1. Fuzzy control rule table of β₁.

e/ec	NB	NM	NS	ZO	PS	PM	PB
NB	PB	PB	PB	PM	PS	ZO	ZO
NM	PB	PB	PM	PM	PS	ZO	NS
NS	PM	PM	PM	PS	ZO	NS	NS
ZO	PM	PM	PS	ZO	NS	NM	NM
PS	PS	PS	ZO	NS	NS	NM	NB
PM	PS	ZO	NS	NS	NM	NB	NB
PB	ZO	NS	NM	NB	NB	NB	NB

and

Table 2. Fuzzy control rule table of β₂.

e/ec	NB	NM	NS	ZO	PS	PM	PB
NB	NB	NB	NB	NM	NM	NS	ZO
NM	NM	NS	NS	PM	PS	ZO	PS
NS	PB	PM	PM	PS	ZO	PS	PM
ZO	ZO	NM	NS	ZO	PS	PS	PM
PS	NS	ZO	ZO	PS	PS	PM	PM
PM	NM	NM	NS	ZO	NS	PS	PM
PB	ZO	PS	ZO	PM	PM	PB	PB

.

4.5. Simulation of the Fuzzy ADRC

The control diagram of the AMB based on the fuzzy ADRC is shown in Figure 4. The displacement sensors detect the displacement {x, y}. After being compared to the desired displacement {x*, y*}, the displacement {e_x, e_y} enters the fuzzy ADRC. They convert to the desired suspension force {F_x*, F_y*}, and then change to the desired currents {i_x*, i_y*} according to the force-current transformation module. The desired currents change to the three-phase desired currents {i_a*, i_b*, i_c*}, and the AMB is controlled by the control current outputted from the three-phase inverter.

To verify the superiority of the fuzzy ADRC in this manuscript, it is compared with the traditional ADRC control method.

Figure 5a is the displacement curve of the fuzzy ADRC in the case of external disturbance. The abscissa is the time coordinates, the ordinate is the displacement coordinates, the black dashed line is the x-direction displacement, and the blue dashed line is the y-direction displacement. At 0 s, an external interference of 60 N is applied to the rotor in the x-direction. At this time, the x-direction displacement is 0.02 mm, and the y-direction displacement is 0.008 mm. At 0.08 s, an external interference of 120 N is applied to the rotor in the x-direction, and at this time, the x-direction displacement is 0.059 mm, and the y-direction displacement is 0.025 mm.

The ADRC is the same controller as the fuzzy ADRC, just without the fuzzy control portion. Figure 5b is the displacement curve of the ADRC in the case of external disturbance. The abscissa is the time coordinates, and the ordinate is the displacement coordinates. The black dashed line is x-direction displacement, and the blue dashed line is y-direction displacement. At 0 s, an external interference of 60 N is applied to the rotor in the x-direction. At this time, the x-direction displacement is 0.025 mm, and the y-direction displacement is 0.012 mm. At 0.08 s, an external interference of 120 N is applied to the rotor in the x-direction; at this time, the x-direction displacement is 0.077 mm, and the y-direction displacement is 0.033 mm. By comparing the displacement curves of the external disturbance controller, the fuzzy ADRC has better anti-interference ability than the ADRC in the case of external interference of different sizes.

5. Experimental Research

The parameters of a 2-DOF six-pole active magnetic bearing are shown in

Table 3. Specification of six-pole active magnetic bearing.

Parameter	Value
Length of radial air gap/mm	0.5
Saturation induction density/T	0.8
Radial magnetic pole area/mm2	320
Turns of radial control coils/At	160
Outer diameter of radial stator/mm	149
Inner diameter of radial stator/mm	115
Radial width of radial pole/mm	22
Angle of the stator poles/°	40

.

In order to verify the correctness of the six-pole AMB, the floating and disturbance experiments are carried out on the experimental prototype, as shown in Figure 6. The DSP is TMS320F2812 of TI, whose maximum frequency is 150 MHz, and the instruction cycle can be shorted to 6.67 ns.

The experimental results shown in Figure 7 are the floating waveform of the rotor of the 2-DOF magnetic bearing from the static status to stable suspension, showing the position signal of the rotor in the direction of the X, Y-axis. After floating, the rotor can quickly return to the balance position under the action of the control current and realize the stable suspension, which indicates that the system has good floating performance.

When the rotor suspends stably, a disturbance is applied to one direction of the rotor, and the change in the displacement waveform of the rotor is observed. Figure 8 shows the displacement waveform in the X- and Y-direction subjected to a 120 N disturbance force. The displacement fluctuations in the x-direction are 46 μm, and the displacement fluctuations in the y-direction are 39 μm. The position is automatically returned to the equilibrium position in 0.2 s without the disturbance force. Comparing the results of Figure 5a and Figure 8, the simulation results and the experimental results are similar, which verifies the calibration of the simulation results. The fuzzy ADRC has a strong anti-interference ability.

6. Conclusions

Aiming at the asymmetry of a three-pole AMB, a new structure scheme of a six-pole AMB is proposed, which greatly eliminates the coupling between two degrees of freedom for six-pole AMBs. A modeling method based on the Maxwell tensor method is proposed to establish the radial suspension force model of the six-pole magnetic bearing. This method is more direct and accurate. Aiming at the coupling characteristic of the magnetic bearing, it is difficult to achieve high control precision by using traditional control methods. The fuzzy control and ADRC theory are applied to the control of six-pole active magnetic bearings to realize the real-time online adjustment of parameters through simulation analysis. The fuzzy ADRC has better anti-interference ability than the ADRC in the case of external interference of different sizes. Experimental analysis shows that the structure has good suspension force current characteristics and a higher space utilization ratio. The fuzzy ADRC has good stability, fast response speed and strong anti-interference ability.