1. Introduction
As a successful application of magnetic suspension technology in the field of motor drives, bearingless motor technology achieves the integration of rotor rotation and magnetic suspension simultaneously [
1,
2,
3,
4,
5]. This not only facilitates the high-speed operation of the motor but also solves the problem that conventional bearings are difficult to maintain in extremely high- and low-temperature, toxic, and harmful situations [
6,
7,
8]. Bearingless motors can be divided into different types, such as the bearingless permanent magnet synchronous motor (BPMSM) [
1], bearingless synchronous reluctance motor [
2], bearingless induction motor [
3], and bearingless flux-switching permanent magnet motor [
4,
5]. Among the various bearingless motors, the BPMSM has the advantages of high efficiency and high torque density and therefore has application value in semiconductors, the chemical industry, biomedicine, aerospace, and so on [
9,
10,
11,
12]. Accurate rotor speed and radial displacement sensing is the basis for the stable rotation and suspension of the BPMSM. Generally, photoelectric encoders and Hall displacement sensors are applied to measure the speed and radial displacement of the rotor. However, these sensors increase the weight, volume, and cost of the whole system, and the detection results are easily affected by environmental factors and are not suitable for extreme environments [
13,
14,
15,
16]. Therefore, the research on the self-sensing technology of the speed and radial displacement of the BPMSM is expected to enhance the integrated level and reliability of the system, reduce the axial length of the motor, and provide more application fields for the BPMSM [
17,
18,
19,
20].
In the past decade, the self-sensing technology of bearingless motors has become one of the research focuses on bearingless motor technology. In [
13], a model-based rotor angle observer at zero and low BPMSM speeds is proposed. The radial displacements and suspension force model are used to observe the angular position. In [
14], in order to realize the speed sensorless control of the BPMSM pump, a method is proposed, which calculates the physical angle from the freewheeling current of a driving phase to synchronize with the estimated angle. The methods proposed in [
13,
14] require the radial displacement obtained by the displacement sensors to determine the rotor angular position, and the sensorless control of the displacement and speed cannot be realized simultaneously. In [
15], a BPMSM speed sensorless control method using a neural network inverse system is proposed. However, the neural network itself has some problems, such as slow convergence and being easy to fall into local optimal solutions. In [
16], a BPMSM speed sensorless control method based on an improved high-frequency signal injection method with a finite impulse-response filter is proposed. In [
17], a displacement self-sensing method of the bearingless induction motor using the mutual inductances affected by the displacement is proposed. In [
18], the displacement self-sensing of the BPMSM is realized by using the high-frequency signal injection method. References [
16,
17,
18] use high-frequency signal injection methods, where signal processing circuits are inevitably added and the complexity of the control system is increased. In [
19], four search coils of a bearingless induction motor are added and connected to a high-frequency voltage source. The rotor radial displacements are then obtained by processing the middle-point voltages between two search coils. The introduction of search coils takes up the space of the stator slots and reduces the performance of the motor. In [
20], a rotor displacement self-sensing method using the difference of symmetric-windings flux linkages based on the symmetrical structure of a six-phase single-winding bearingless flux-switching permanent magnet machine is proposed. This method is only applicable to the proposed motor.
Much of the research on the sensorless control methods of the bearingless motors listed in the previous paragraph only focus on one aspect; that is, either the speed sensorless control or the displacement sensorless control. However, both the photoelectric encoder and the probes of the eddy current displacement sensors are installed on the motor, increasing the volume and reducing reliability. Moreover, using different methods to realize the speed and displacement self-sensing of the BPMSM will greatly increase the complexity of the control system. For example, if the adaptive method and the high-frequency signal injection method are used to realize the speed and displacement self-sensing, respectively, the self-sensing program and the filter circuit for signal processing should be included in the control system, which will increase the complexity and reduce the reliability of the control system. According to the inverse system theory, by cascading the left inverse system with the original system, the composite system can be used as an input observer, meaning that the original input signal can be obtained from the output signal of the inverse system. Thus, the left inverse system theory can be utilized to realize the speed and displacement self-sensing of the BPMSM. As we know, the BPMSM is a nonlinear time-varying system, and its models of the speed subsystem and the displacement subsystem are different. To overcome the difficulty of obtaining accurate models, a least-squares support vector machine (LS-SVM) is used to construct the inverse systems of the speed and displacement subsystems. The proposed method can not only reduce the system cost and increase reliability but also does not require additional hardware circuits.
The structure of this paper is as follows. The operating principle and the mathematical model of the BPMSM are introduced in
Section 2. In
Section 3, the self-sensing principle of the left inverse system is explained, and the left reversibility of the speed and displacement subsystems is verified. In
Section 4, the LS-SVM is used to identify the speed and displacement left inverse systems. In
Section 5, a speed and displacement sensorless control system of the BPMSM is designed. The simulation and experiment results are given in
Section 6. Finally, the conclusion is drawn in
Section 7.
2. Operation Principle and Mathematics Model
In the BPMSM, the pole-pair numbers of torque windings
PM and suspension force windings
PB are different by ±1, and two sets of windings are arranged in the same stator slots to produce the torque and suspension force synchronously [
21].
Figure 1 shows the generation principle of the suspension force. The subscript B is relative to suspension force windings, and M is relative to torque windings.
The model established by finite-element analysis software is shown in
Figure 1. The stator slots are divided into two-pole suspension force windings in the inner layer and four-pole torque windings in the outer layer. It can be seen from
Figure 1a,b that the torque magnetic field is a uniformly distributed four-pole magnetic field, and the suspension force magnetic field is a uniformly distributed two-pole magnetic field. Since the magnetic flux density is directional, the composite magnetic field generated by the two magnetic fields is not uniformly distributed. As shown in
Figure 1c, the suspension force magnetic field and the torque magnetic field are of the same polarity in the positive direction of the
y-axis, thus the magnetic field in this area is enhanced. In the negative direction of the
y-axis, the suspension force magnetic field and the torque magnetic field have opposite polarities, thus the magnetic field in this area is decreased. Therefore, a magnetic-field-strengthening zone and a magnetic-field-weakening zone are formed in the air gap. According to the Maxwell force principle, the total force on the rotor at this time points to the positive
y-axis. A suspension force in the opposite direction can be produced by adding a negative current in the suspension force windings. The suspension force in the
x-direction can be produced similarly.
The expression of the radial suspension force can be obtained by the virtual displacement method. The flux linkages of the torque windings and the suspension force windings in the α-β coordinate can be written as [
21]
where
ΨMα,
ΨMβ,
ΨBα, and
ΨBβ are the flux linkages of torque windings and suspension force windings in the α-β coordinate,
LM and
LB are the self-inductances of torque windings and suspension force windings,
iMα,
iMβ,
iBα, and
iBβ are the currents of torque windings and suspension force windings in the α-β coordinate,
M is the derivative of mutual inductance with respect to the rotor radial displacement, and
x and
y are the radial displacements in the α-β coordinate.
The stored magnetic energy
Wm is then given by
The radial forces
Fx and
Fy can be derived from the partial derivatives of the stored magnetic energy
Wm, which is given as
3. Observer Design
3.1. Left Inverse System Theory
The left inverse system is the basic concept in the inverse system theory. However, unlike the right inverse system, the left inverse system focuses on the reappearance of the input signal of the system, and its basic concept is similar to “observability” in control theory. Therefore, the left inverse system theory can be utilized to realize the speed and displacement estimation of the BPMSM [
22,
23].
The principle of the left inverse system is given below. For a nonlinear system
where
is the
n-dimensional state variable,
is the
m-dimensional unobservable variable, and
is the
n −
m-dimensional observable variable.
It can be assumed that an “inner sensor subsystem” exists in the nonlinear system. The input variables of the inner sensor subsystem are nonobservable variables and the input variables of the original system, and the output variables are the observable variables of the original system. The inner sensor subsystem can then be written as
For such a subsystem, if the left inverse system of the system exists, the expression of the left inverse system can be written as follows
The general definition of the left inverse system is as follows. Assume that a nonlinear system Σ has the following mapping relationship
u →
y. For this system, there is such a system Σ
1. Its mapping relationship is
v →
w, and it satisfies the initial conditions of the system Σ. Meanwhile, if
v(
t) =
y(
t), then
w(
t) =
u(
t). Therefore the system Σ
1 is the left inverse system of the original system, thus the original system is left invertible. On the basis of proving that the subsystem is left invertible, the left inverse system of the inner sensor subsystem is connected to the right side of the nonlinear system. As shown in
Figure 2, a left inverse system observer is constructed, which can realize the observation of unobservable variables in the nonlinear system [
15,
23].
3.2. Speed Observer
When ignoring magnetic saturation and eddy current loss, the mathematical model of the speed subsystem of the BPMSM can be given as follows
where
uMd,
uMq,
iMd, and
iMq are the voltage components and current components of torque windings in the d-q coordinate,
PM is the pole-pair number of the torque windings,
LM and
RM are the inductance and resistance of the torque windings,
ω is the rotor speed,
Te and
TL are the electromagnetic torque, and load torque
J is the moment of inertia of the rotor.
State variables are chosen as X = [x1, x2, x3]T = [iMd, iMq, ω]T. Input variables are chosen as U = [u1, u2]T = [uMd, uMq]T. The state variables xm = [x1, x2]T = [iMd, iMq]T can be measured directly, and state variable xum = x3 = ω is the variable to be observed.
The auxiliary algorithm can be used to analyze the left reversibility of the speed subsystem. The derivatives of
x1 and
x2 can be given as
The Jacobin matrix can be calculated as
According to Equation (9), rank (A) = 1 can be obtained. Because the rank of Jacobi matrix A is equal to the number of the variable to be observed, the auxiliary algorithm ends. In the next step, the modeling algorithm is needed to establish the left inverse model of the speed subsystem.
To estimate
xum,
Z is established as follows
The speed inner sensor model is left invertible. Theoretically, the left inverse system can be expressed as follows
The left inverse speed observation system for the BPMSM is shown in
Figure 3.
3.3. Displacement Observer
The radial displacement of the rotor is mainly affected by the suspension force and further affected by the suspension force flux linkage. Therefore, when considering the self-sensing of the rotor radial displacement of the BPMSM, we use the suspension force flux linkage to establish the displacement inner sensor system. The displacement inner sensor system can be established by rewriting the voltage equation of suspension force windings, which is given as
where
fB is the current frequency of the suspension force windings, and
uBα,
uBβ are the components of the suspension force windings voltage.
The state variables are chosen as X = [x1, x2, x3, x4]T = [ΨBα, ΨBβ, x, y]T. The input variables are chosen as U = [u1, u2]T = [uBα, uBβ]T. The state variables xm = [x1, x2]T = [ΨBα, ΨBβ]T can be calculated by the flux observer, and state variable xum = [x3, x4]T = [x, y]T is the variable to be observed.
The state equation is then shown as
Thus, the Jacobi matrix can be calculated as
det(
A) is then calculated as
Obviously, det(
A) ≠ 0. Therefore, the system is left invertible, and the expression of the left inverse system is given as
The left inverse displacement observation system for the BPMSM is shown in
Figure 4.
4. LS-SVM Left Inverse System
According to Equations (12) and (17), the left inverse speed observer and the left inverse displacement observer can be constructed. However, the direct construction of the left inverse system is not only complex but also difficult to guarantee stability and robustness. Therefore, the LS-SVM is used to identify the speed left inverse system and the displacement left inverse system.
The support vector machine (SVM) has been successfully developed in control and signal processing in recent years to solve nonlinear problems. The basic theory is introduced in [
24]. SVM has the advantages of good generalization, strong robustness, and simple structure. LS-SVM is a reformulation of standard SVM. It uses the square term in the optimization index and only equality constraints.
The linear regression function in high dimensional space is given as
where
w is the weight vector in high dimensional space,
b is the threshold, and
φ(
x) is a nonlinear function that maps the input vector
xk to the high-dimensional space.
The estimation of
φ(
x) can be turned into solving the following constrained optimization problem
where
γ is the regularization parameter, and
ek is the fitting error of the loss function.
The optimization problem is solved by the Lagrange method. The following Lagrange functions are introduced
where
αk,
k = 1, 2,...,
l are the Lagrange multipliers, and
α = [
α1,α2,...αl,]
T∈
Rl.
According to the Karush-Kuhn-Tucker (KKT) conditions, the analytical solution of the optimization problem can be obtained as
where
1l×1 = [1, 1,..., 1]
T,
y = [
y1,
y2,...,
yl]
T,
I is the identity matrix, and
Ω = {
Ωij}
l×l =
K(
xi,
xj) =
φ(
xi)
T·
φ(
xj),
i,
j = 1, 2,...,
l.
K(
xi,
xj) is the kernel function, which satisfies the Mercer conditions.
The Gaussian radial basis function is selected as the kernel function, which is given as
where
σ is the width of the kernel.
The final result of the LS-SVM model for function estimation is given as
For the displacement subsystem, the suspension force windings flux linkages and their derivatives are determined as the input variables and the output variables are the rotor displacements {x, y}. Because the LS-SVM used here can only deal with the single output problem, two models are established to estimate the rotor displacements x and y, respectively.
For the torque subsystem, the torque windings currents and their derivatives are determined as the input variables and the output variable is the rotor speed .
To obtain the training sample of the LS-SVM inverse system, the closed-loop control system of the BPMSM is constructed. The random signal in the working range is used as the excitation of the closed-loop control system, and the response is used as the training and test sample of the LS-SVM inverse system.
The data are normalized by the following formula
where
D is the original sample data,
is the normalized sample data,
Dmax is the maximum value of the sample data, and
Dmin is the minimum value of the sample data. One thousand sets of data are selected as the training sample from the processed sample data at medium intervals, and 500 sets of data are selected as the test sample to test the identification accuracy and generalization ability of the LS-SVM inverse system.
The training process of LS-SVM comes down to the solving process of linear equations. Therefore, it is not necessary to solve a constrained convex quadratic programming such as SVM so that LS-SVM has less computational complexity than a standard SVM. Its topological structure is shown in
Figure 5.
5. Construction of Control System for BPMSM
The electromagnetic torque expression of the BPMSM can be given as
Decoupling control of electromagnetic torque can be realized when using the rotor magnetic field orientation control method; that is, under
iMd = 0, Equation (25) can be reduced to
As for the suspension force control, the radial suspension forces
Fx and
Fy in the d-q coordinate can be expressed as
The design idea of the suspension force control subsystem is as follows. The rotor displacement signal x, y obtained by the displacement self-sensing algorithm are compared with the displacement given signal x*, y*, respectively. The resulting error signals are converted to suspension force given signals Fx*, Fy* by PID controllers. After the force/current conversion (Equation (27)), the given current signals iBd*, iBq* of the suspension force windings are obtained. Comparing the given signals with the current feedback values iBd and iBq of the suspension force windings and obtaining the input instruction uBa*, uBb* of the SVPWM module after the obtained error signal is transformed by the PI controller and the coordinate transformation.
The BPMSM speed sensorless control system is constructed by combining the BPMSM rotor-flux-oriented control method with the speed self-sensing method based on the LS-SVM left inverse system. The BPMSM displacement sensorless control system is constructed by combining the suspension force control method mentioned above with the proposed displacement self-sensing method. The complete sensorless control block diagram is depicted in
Figure 6.
7. Conclusions
The mechanical sensors used in the BPMSM have some limitations, such as increasing the size, cost, and complexity of the motor, and the detection results are easily affected by environmental factors. For solving these problems, a speed and displacement sensorless control method based on the LS-SVM left inverse system is proposed. According to the simulation and experimental analysis results, the following conclusions can be drawn. The LS-SVM left inverse system observation method proposed in this paper can realize the detection of rotation speed and rotor radial displacement, which proves the feasibility and effectiveness of this method. In the case of variable speed, it maintains good speed estimation performance. In the case of force disturbance and change of the given displacement, it has good displacement estimation ability and robustness. Furthermore, because the LS-SVM left inverse system observation algorithm can be implemented by software, the cost of the system is reduced, and the reliability of the system is increased.