Research on the least squares support vector machine displacement observer of a bearingless induction motor

Abstract

To achieve the radial displacement self-sensing detection of a bearingless induction motor, an observation method based on the LS-SVM (least squares support vector machine) is proposed. The state-space model of a magnetic suspension system is derived firstly. Then the LS-SVM is introduced to the radial displacement observer of the bearingless induction motor, the design principle and Lyapunov stability of the LS-SVM displacement observer are analysed in detail, and the construction method of the LS-SVM displacement observer is presented also. Simulation and verification results show that both in the starting suspension stage and in the process of stable suspension operation, the LS-SVM displacement observer can quickly track the radial displacement with high accuracy. Then, a new method is found for the radial displacement self-sensing detection of the bearingless induction motor.

Keywords

Bearingless induction motor state-space model least squares support vector machine displacement observer Lyapunov stability simulation verification and analysis

Introduction

The conventional alternating current (AC) motor supported by a mechanical bearing is widely used in various industrial fields (Sun et al., 2013; Sun et al., 2014), but it is difficult to meet the requirement of long-time and high-speed running (Bu et al., 2015; Sun et al., 2012); then, a motor supported by magnetic bearings was developed (Kim et al., 2014), but it still has some disadvantages, such as more magnetic suspensions cost, over speed difficulty and limited critical speed (Bu et al., 2009; Chiba and Santisteban, 2012; Huang et al., 2014; Wang et al., 2015). Based on the comparability in structure between the magnetic bearing and common AC motor, a bearingless motor is proposed. The bearingless motor is a new type of AC motor that is suitable for high-speed and long-time operation (Bu et al., 2014a, 2014b; Sun et al., 2013). Compared with the motor supported by a magnetic bearing, the bearingless motor has a series of advantages, such as shorter rotor shaft, higher critical speed and lower magnetic suspensions cost (Bu et al., 2014b; Chiba and Santisteban, 2012), and has become a new research hotspot. However, the magnetic suspension system of the bearingless motor has a negative displacement stiffness coefficient, so it is unstable. To stabilize the stable suspension control of the rotor, it is necessary that the radial displacement of the rotor should be detected in real time, and negative feedback control should be introduced. An eddy current sensor can be used to detect the radial displacement of the rotor, but the displacement sensor occupies a certain shaft length, and then limits the improvement of critical speed to a certain degree. In addition, the high cost of the displacement sensor greatly increases the cost of the control system. Therefore, the self-sensing observation technology of radial displacement has become a key problem of the bearingless motor in popularization and application.

The position tracking control of the common AC motor has been widely covered in the literature (Ha et al., 2015; Xia et al., 2005). Meanwhile, the radial displacement observation of the bearingless motor is an ongoing research topic. Based on the approximate linear relationship between the radial displacement of the rotor and the self-inductance of suspension windings, a high-frequency excitation signal is injected into the suspension windings of the bearingless induction motor by Tera et al. (2005), and by detecting the differential voltage variation of suspension windings, the radial displacement components are extracted. When the rotor deviates from the stator centre, the mutual inductance coupling between torque windings and suspension windings is generated; based on the mutual inductance coupling relationship, a high-frequency excitation signal is injected into the torque windings of a bearingless induction motor by Kuwajima et al. (2001), and the induced current in suspension windings is detected. After a series of signal processing, the radial displacement components of the rotor are extracted. The two methods are not affected by the fundamental physical phenomenon and are not sensitive to the parameters of the bearingless motor and of strong detection robustness; however, they require sustained high-frequency excitation, and depend on the complex signal processing technology, and then have difficulty in implementation. When the rotor deviates from the stator centre of the bearingless permanent magnet (PM) motor, based on the mutual inductance coupling relationship between torque windings and suspension windings, and by calculating in real time the coupling flux linkage of torque winding to suspension winding, Bai et al. (2009) proposed a displacement estimation method; this method does not need a high-frequency excitation signal, and is easy to realize; however, the accuracy of the mutual inductance parameter directly influences the observation precision of radial displacement. Then, how to overcome the influence of the motor parameter on the observation performance by machine learning theory is still a problem to be further studied. Based on the neural network, Xia et al. (2005) try to design the position observer of the switched reluctance motor and a better observation effect is obtained; however, some problems of the neural network itself, such as slower convergence rate, easily falling into local minima and greater dependence of network structure design on expert experience, limit the application range of the presented method. The support vector machine (SVM) is a new machine learning method based on statistical learning theory (Vapnik, 1999); it can minimize the dependence of the neural network on experience knowledge, and has some outstanding characteristics, such as small sample learning, stronger generalization ability, global optimization ability and fixed topological structure. The stronger self-learning ability and generalization ability of the SVM facilitate the motor model identification and parameter estimation. The LS-SVM, that is, the least squares SVM, is a further expansion of the conventional SVM (Suykens, 1999, 2001); it changes the inequality constraint of the SVM to equality constraint, and switches the problem of solving quadratic programming to that of solving a linear equation, effectively simplifying the complex computation process, and then makes the LS-SVM applicable to the modelling and control of the bearingless motor (Sun et al., 2012). Based on the model analysis of the magnetic suspension switched reluctance motor (Zhu et al., 2012), the estimation method of displacement and position are investigated based on the LS-SVM. Based on the regression theory of the relevance vector machine (RVM), Xiang et al. (2013) constructs the forecasting model of a magnetic suspension switched reluctance motor, and then the radial displacement is estimated based on the generalization ability of the RVM. When machine learning is used to detect motor parameters, the uncertainty and nonlinearity of the system can be effectively solved, and then an effective way of thinking has been provided to observe the displacement of the bearingless motor.

In this article, a new displacement observation method is presented for a bearingless induction motor. On the basis of existing researches, the LS-SVM is first introduced to the displacement observer of the bearingless induction motor. That is, the LS-SVM is used to approximate the nonlinear state-space characteristics of the magnetic suspension system, and then the radial displacement components of the bearingless induction motor are estimated.

Magnetic suspension system modelling of the bearingless induction motor

Working principle of the bearingless motor

From the electromagnetic field theory, the Maxwell electromagnetic force is generated on the interface between the motor iron core and air gap, and it is approximately vertical to the interface (Bu et al., 2009). When the air gap flux density is distributed symmetrically, the resultant radial force that acts on the rotor equals zero. However, when the rotor deviates from stator centre, the symmetric distribution of the air gap magnetic field is broken, and the resultant radial force does not equal zero again. The resultant radial force is entitled unilateral magnetic pull, which influences the stable suspension of the rotor. To achieve stable suspension of the rotor, it is necessary to produce a controllable radial force to counteract the unilateral magnetic pull and external radial load.

In the bearingless motor, there are two sets of windings in stator slots, namely torque windings and suspension control windings. Torque windings are the conventional AC motor windings, where the number of pole pairs is P₁ and the electrical angle frequency of current is ω₁. The suspension control windings are used to produce an additional suspension control magnetic field, where the number of pole pairs is P₂ and the electrical angle frequency of current is ω₂. When the suspension control magnetic field is superimposed on the original rotating magnetic field of the AC motor, the balance and symmetry of the air gap magnetic field is broken, the air gap magnetic field in some air gap area is enhanced and that the space symmetric area is weakened; then an additional radial force is produced in the enhancement direction of air gap magnetic field. When the two sets of windings meet the condition of “P₂ = P₁± 1, ω₂ = ω₁”, the produced additional radial force is controllable both in amplitude and in direction (Bu et al., 2009). The controllable radial force is the so-called controllable magnetic suspension force, which can be used to counteract the unilateral magnetic pull and external radial load, and to achieve stable suspension of the rotor.

Definition: αβ is the stationary coordinate system in mechanical space. Then, taking the four-pole bearingless motor with two-pole suspension control windings as an example, Figure 1 shows the generation principle of controllable magnetic suspension force. In Figure 1, N_a and N_b represent four-pole torque windings, that is, four-pole motor windings; N_α and N_β represent two-pole suspension control windings. At the moment that stator currents of the two sets of windings are as shown in Figure 1, the controllable magnetic suspension force along the β direction is produced. If the current direction of the suspension control windings is reversed, the controllable magnetic suspension force would be produced along the opposite β direction. The generation principle of controllable magnetic suspension force along the α direction is similar.

State-space model of the magnetic suspension system

Definition: dq is the synchronous coordinate system oriented by the rotor flux-linkage of the torque system. Then, according to the working principle of the bearingless induction motor, the mathematical model of controllable magnetic suspension force can be derived as follows (Bu et al., 2014b; Chiba and Santisteban, 2012):

{\begin{matrix} F_{α} = k_{m} (i_{s 2 d} ψ_{1 d} + i_{s 2 q} ψ_{1 q}) \\ F_{β} = k_{m} (i_{s 2 d} ψ_{1 q} - i_{s 2 q} ψ_{1 d}) \end{matrix}

(1)

In (1), F_α and F_β are the controllable magnetic suspension force components along the α and β directions, respectively; i_s_2d and i_s_2q are the current components of suspension control windings along the d and q coordinate axes, respectively; ψ_1d and ψ_1q are the air gap flux-linkage components of torque system along the d and q coordinate axes, respectively.

According to (1), it is clear that the controllable magnetic suspension force has a direct relationship with the air gap flux-linkage of torque system. According to the relationship between the air gap flux-linkage and rotor flux-linkage, the components ψ_1d and ψ_1q can be calculated in real time. The expressions of ψ_1d and ψ_1q are as follows (Bu et al., 2014a):

{\begin{matrix} ψ_{1 d} = (ψ_{r 1} + L_{r 1 l} i_{s 1 d}) L_{m 1} / L_{r 1} \\ ψ_{1 q} = L_{r 1 l} i_{s 1 q} \times L_{m 1} / L_{r 1} \end{matrix}

(2)

Based on the mechanical dynamics principles, and ignoring the unbalanced exciting force caused by rotor mass eccentricity, the suspension motion equation of the rotor can be defined as follows:

{\begin{matrix} m \overset{\cdot\cdot}{α} = F_{α} + f_{α} \\ m \overset{\cdot\cdot}{β} = F_{β} + f_{β} \end{matrix}

(3)

In (3), m is the mass of the rotor; $f_{α}$ and $f_{β}$ are the unilateral magnetic pull components along the α and β directions, respectively, which can be expressed as “ $f_{α} = k_{s} α$ ” and “ $f_{β} = k_{s} β$ ”. Here, $k_{s}$ is the radial displacement stiffness coefficient determined by the motor structure.

By substituting (1) and (2) into (3), the suspension motion equation of the rotor is rewritten as follows:

{\begin{matrix} \overset{\cdot\cdot}{α} = \frac{k_{m}}{m} \cdot \frac{L_{m 1}}{L_{r 1}} [(ψ_{r 1} + L_{r 1 l} i_{s 1 d}) i_{s 2 d} + L_{r 1 l} i_{s 1 q} i_{s 2 q}] + \frac{k_{s}}{m} α \\ \overset{\cdot\cdot}{β} = \frac{k_{m}}{m} \cdot \frac{L_{m 1}}{L_{r 1}} [L_{r 1 l} i_{s 1 q} i_{s 2 d} - (ψ_{r 1} + L_{r 1 l} i_{s 1 d}) i_{s 2 q}] + \frac{k_{s}}{m} β \end{matrix}

(4)

Definition: the input variables u, the state variables x and the output variables y of the magnetic suspension system are as follows:

{\begin{matrix} u = {(u_{1}, u_{2}, u_{3}, u_{4}, u_{5})}^{T} = {(i_{s 1 d}, i_{s 1 q}, i_{s 2 d}, i_{s 2 q}, ψ_{r 1})}^{T} \\ x = {(x_{1}, x_{2}, x_{3}, x_{4})}^{T} = {(α, \overset{\cdot}{α}, β, \overset{\cdot}{β})}^{T} \\ y = {(y_{1}, y_{2})}^{T} = {(α, β)}^{T} \end{matrix}

(5)

Assuming u_t1 and u_t2 as input components of the magnetic suspension system, the expressions of u_t1 and u_t2 are as follows:

u_{t} = [\begin{matrix} u_{t 1} \\ u_{t 2} \end{matrix}] = [\begin{matrix} \frac{k_{m}}{m} \cdot \frac{L_{m 1}}{L_{r 1}} [(u_{5} + L_{r 1 l} u_{1}) u_{3} + L_{r 1 l} u_{2} u_{4}] \\ \frac{k_{m}}{m} \cdot \frac{L_{m 1}}{L_{r 1}} [L_{r 1 l} u_{2} u_{3} - (u_{5} + L_{r 1 l} u_{1}) u_{4}] \end{matrix}]

(6)

Meanwhile, by substituting (5) and (6) into (4), the state-space model of the magnetic suspension system can be derived as follows:

{\begin{matrix} \overset{\cdot}{x} = A_{1} x + B_{1} u_{t} \\ y = C_{1} x \end{matrix}

(7)

where $A_{1} = [\begin{matrix} 0 & 1 & 0 & 0 \\ k_{s} / m & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & k_{s} / m & 0 \end{matrix}]$ , $B_{1} = [\begin{matrix} \begin{matrix} 0 & 0 \\ 1 & 0 \end{matrix} \\ \begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix} \end{matrix}]$ , $C_{1} = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{matrix}]$

According to (7), it is clear that if u_t1 and u_t2 are seen as independent input components, the magnetic suspension system is a linear system whose state variables x are observable, and the displacement components α and β can be estimated according to relevant state variables. Then, the observation problem of radial displacement can be transformed into that of state variables x.

Least squares support vector machine displacement observer

Design principle of the LS-SVM displacement observer

According to (6), u_t1 and u_t2 are not independent inputs, and they are nonlinear functions of u₁, u₂, u₃, u₄ and u₅. That is, the actual magnetic suspension system is nonlinear, and then the displacement observer cannot be designed simply by linear system theory.

Rewrite (7) as follows:

{\begin{matrix} \overset{\cdot}{x} = Ax + F (x, u) = F (x, u) \\ y = Cx \end{matrix}

(8)

where A is set to zero matrix; $F (x, u)$ is a function vector defined in $R^{4 \times 1}$ ; C is a constant matrix; and C is the same with C₁. They can be expressed as follows:

\begin{matrix} F (x, u) & = [\begin{matrix} F_{1} (x, u) \\ F_{2} (x, u) \\ F_{3} (x, u) \\ F_{4} (x, u) \end{matrix}] \\ = [\begin{matrix} x_{2} \\ \frac{k_{m}}{m} \cdot \frac{L_{m 1}}{L_{r 1}} [(u_{5} + L_{r 1 l} u_{1}) u_{3} + L_{r 1 l} u_{2} u_{4}] + \frac{k_{s} x_{1}}{m} \\ x_{4} \\ \frac{k_{m}}{m} \cdot \frac{L_{m 1}}{L_{r 1}} [L_{r 1 l} u_{2} u_{3} - (u_{5} + L_{r 1 l} u_{1}) u_{4}] + \frac{k_{s} x_{3}}{m} \end{matrix}] \end{matrix}

(9)

C = C_{1} = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{matrix}]

(10)

In the state-space model of magnetic suspension system as shown in (8), the function vector $F (x, u)$ represents all the system characteristics, such as uncertain or nonlinear characteristics; when the system parameters change or are not accurate, the accuracy of the state-space model will be influenced unavoidably, and then if the displacement observer is designed based on the model in (8), the observation precision of displacement will be influenced unavoidably by the system parameters. To avoid the dependence of the displacement observer on system parameters, the LS-SVM is considered to approximate the function vector $F (x, u)$ .

Figure 2 shows the designed topological structure of the LS-SVM; the principle and derivation process of the LS-SVM are described in related literature (Suykens and Vandewelle, 1999; Syukens et al., 2001), and will not be detailed here. From Figure 2, it is clear that the topological structure of the LS-SVM is similar to that of the three-layer neural network.

The first layer is the input layer of the LS-SVM. According to the relationship between the input and output of function vector $F (x, u)$ , the state-space control variables of the magnetic suspension system, that is, ${i_{s 1 d}, i_{s 1 q}, i_{s 2 d}, i_{s 2 q}, ψ_{r 1}}$ , are selected as the input variables of the LS-SVM.

The second layer is used for kernel operation of the LS-SVM. The function of the second layer is to reflect the limited input variables to high-dimensional feature space. All functions that meet the Mercer qualification can be used for kernel function $Ω (x_{i}, X)$ ; the common kernel functions include the sigmoid function, polynomial function, radial basis function, etc. Here, the radial basis function is selected as the kernel function, as shown in (11), where $σ$ is the kernel width:

Ω (x_{i}, X) = \exp [- | x_{i} - X |^{2} / (2 σ^{2})]

(11)

The third layer is the output layer of the LS-SVM. The output layer is used for the regression operation of the LS-SVM, so as to obtain the fitting expression of function vector $F (x, u)$ as follows:

F_{j} = Σ_{i = 1}^{N} W_{ji} Ω (x_{i}, X) + b_{j}

(12)

In (12), N is the training sample number; i is the serial number of the training sample, i = 1, 2, 3, …, N; j is the serial number of the LS-SVM learning machine, j = 1, 2, 3, 4; $W_{ji}$ is the weight of the no. j LS-SVM learning machine, and $b_{j}$ is the corresponding threshold; $x_{i}$ is the input value of the known training sample; and X is the input variable composed of all input values, that is, $X = u =$ $(i_{s 1 d}, i_{s 1 q}, i_{s 2 d}, i_{s 2 q}, ψ_{r 1})$ , so it can be written as $Ω (u_{i}) = Ω (x_{i}, X)$ .

Rewrite (12) in a matrix form:

F = W Ω (u)

(13)

In (13), W is the weight coefficient matrix and $Ω (u)$ is the input function vector composed of all input functions. They can be expressed as follows:

W = [\begin{matrix} b_{1} & W_{11} & \dots & W_{1 N} \\ b_{2} & W_{21} & \dots & W_{2 N} \\ b_{3} & ⋮ & ⋱ & ⋮ \\ b_{4} & W_{41} & \dots & W_{4 N} \end{matrix}]

(14)

Ω (u) = [1, Ω (u_{1}), Ω (u_{2}), \dots, Ω (u_{N})]^{T}

(15)

On the condition of ignoring the approximation error of the LS-SVM, where W is a constant matrix trained by the LS-SVM, the nonlinear system described by (8) is equivalent to the following:

{\begin{matrix} \overset{\cdot}{x} = Ax + W Ω (u) \\ y = Cx \end{matrix}

(16)

Then the observer designed for (8) can be transformed to that designed for (16); the designed observer can be expressed as follows:

{\begin{matrix} \overset{\cdot}{\hat{x}} = A \hat{x} + W Ω (u) + K (y - \hat{y}) \\ \hat{y} = C \hat{x} \end{matrix}

(17)

where K is a constant value gain matrix that makes “ $A_{m} = A - KC = - KC$ ” asymptotically stable. $\hat{x}$ and $\hat{y}$ are the observation values of $x$ and $y$ , respectively, that is, $\tilde{x} = x - \hat{x}$ , $\tilde{y} = y - \hat{y}$ .

After the difference between (16) and (17) is taken into account, the dynamic equation of state estimation error $\tilde{x}$ can be derived as follows:

\overset{\cdot}{\tilde{x}} = A_{m} \tilde{x}, \tilde{y} = C \tilde{x}

(18)

Stability analysis of the LS-SVM displacement observer

Select the Lyapunov function as follows:

V (\tilde{x}) = ({\tilde{x}}^{T} P \tilde{x}) / 2

(19)

where P is the positive definite solution of the Lyapunov equation “ ${A_{m}}^{T} P + P A_{m} = - Q$ ”; Q is a positive definite matrix. It is clear that V is positive definite, and $V (0)$ equals zero.

After taking the derivative for the Lyapunov function V, we get the following:

\begin{matrix} \overset{\cdot}{V} (\tilde{x}) & = ({\overset{\cdot}{\tilde{x}}}^{T} P \tilde{x} + {\tilde{x}}^{T} P \overset{\cdot}{\tilde{x}}) / 2 = ({\tilde{x}}^{T} {A_{m}}^{T} P \tilde{x} + {\tilde{x}}^{T} P A_{m} \tilde{x}) / 2 \\ = {\tilde{x}}^{T} ({A_{m}}^{T} P + P A_{m}) \tilde{x} / 2 = - {\tilde{x}}^{T} Q \tilde{x} / 2 \end{matrix}

(20)

From (20), it can be seen that $\overset{\cdot}{V} (\tilde{x})$ is negative definite, and $\overset{\cdot}{V} (0)$ equals zero. Then according to Lyapunov stability theory, it is clear that the system is asymptotically stable with large scale at equilibrium state $\tilde{x} = 0$ , and the state estimation error $\tilde{x}$ would approach zero. Therefore, the proposed observation method of radial displacement is effective.

Realization of the LS-SVM displacement observer

To realize the LS-SVM displacement observer, the offline training and fitting of function vector $F (x, u)$ and the displacement observer design based on the trained function vector $F (x, u)$ , etc., are required. The implementation steps of the LS-SVM displacement observer are summarized as follows.

The first step is the collection of a training sample. In this article, the training samples are obtained from the analytical inverse control system of the bearingless induction motor (Bu et al., 2014a). In the practical operation range, adopting the normal distribution random signal as input, it continues for 2 s, and the sampling period is 1 ms. During the sampling period, some signals are sampled, including $i_{s 1 d}, i_{s 1 q}, i_{s 2 d}, i_{s 2 q}, ψ_{r 1}$ , $α$ and $β$ . After the sampled datum are smooth filtered, the $\overset{\cdot}{α}, \overset{\cdot\cdot}{α}, \overset{\cdot}{β}, \overset{\cdot\cdot}{β}$ signals are derived based on the five-point numerical differentiation algorithm, and the training sample set ${i_{s 1 d}, i_{s 1 q}, i_{s 2 d}, i_{s 2 q}, ψ_{r 1}, \overset{\cdot}{α}, \overset{\cdot\cdot}{α}, β, \overset{\cdot\cdot}{β}}$ is obtained. From the 2000 sets of sample datum, 1000 sets are selected as the training sample set, and 500 sets are chosen as the test sample set.

The second step is offline training for the LS-SVM. Using $u =$ $(i_{s 1 d}, i_{s 1 q}, i_{s 2 d}, i_{s 2 q}, ψ_{r 1})$ as the training input of the LS-SVM, and using $(\overset{\cdot}{α}, \overset{\cdot\cdot}{α}, β, \overset{\cdot\cdot}{β})$ as the training output of the LS-SVM, four learning machines are established to train the LS-SVM. In the training process of the LS-SVM, the penalty factor $γ$ (a regularization parameter) and the kernel width $σ$ of the radial basis function determine the training accuracy and complexity of the LS-SVM. Here, the grid search method is used to obtain the best penalty factor and kernel width, and to obtain the LS-SVM model with higher training accuracy.

The third step is to construct a LS-SVM displacement observer. Based on the trained LS-SVM model and (17), the displacement observer is designed as shown in Figure 3. When the observed displacement deviates from the actual value, the differential quantity of observation error $\hat{\overset{\cdot}{x}}$ will be corrected in real time by gain matrix K, and then the offset between observation value and actual value would be gradually reduced.

Simulation verification and analysis

To verify the validity of the proposed LS-SVM displacement observer, the control system of the bearingless induction motor is constructed as shown in Figure 4. Based on MATLAB/Simulink, the proposed displacement observer is simulated and analysed.

In Figure 4, the control system structure is designed based on inverse dynamic decoupling control strategy (Bu et al., 2014a); due to the limit of paper length, it will not be introduced in detail. The parameters of the bearingless induction motor are listed in Table 1.

Table 1.

Parameters of the bearingless induction motor.

Parameters	Description	Value
r	Stator inner diameter (mm)	62
l	Effective core length (mm)	82
δ₁	Air gap of auxiliary bearing (mm)	0.2
P	Power of motor (kW)	2.2
R_s	Stator winding resistance (Ω)	1.6
R_r	Rotor winding resistance (Ω)	1.423
L_s1l	Leakage inductance of torque windings (H)	0.0043
L_r1l	Leakage inductance of rotor windings of torque system (H)	0.0043
L_m1	Mutual inductance of torque system (H)	0.0859
J	Rotation inertia of motor (kg.m²)	0.024
R_s2	Suspension stator winding resistance	2.7
L_s2l	Leakage inductance of suspension stator windings (H)	0.00398
L_m2	Mutual inductance of suspension system (H)	0.230

To validate the observation performance of the proposed LS-SVM displacement observer, we set the initial radial displacement of the bearingless induction motor as follows: α = −0.1 mm, β = −0.12 mm; on the conditions of adopting displacement sensor, the bearingless induction motor is started running at no-load. During the process from start suspension to stable suspension, the contrast response curves between the displacement sensor and the LS-SVM displacement observer are shown in Figures 5 and 6. In order to view the tracking accuracy of radial displacement more clearly, the displacement tracking error curves in Figures 5 and 6 have been enlarged.

From Figures 5 and 6, the following simulation results are analysed.

Before the suspension of the rotor, the observation values of displacement components are not equal to their actual values. Once the rotor starts to suspend, the observation values of displacement components quickly reach their actual values;

After a brief adjustment, the system is stable, then the outputs of the LS-SVM displacement observer are almost entirely consistent with that of radial displacement sensors; the tracking errors of displacement components stay within the range of 2 μm all the time.

The simulation results illustrate that the proposed LS-SVM displacement observer has excellent performance and higher precision. When the proposed LS-SVM displacement observer is adopted, apart from the short time transition process in starting the suspension stage, the radial displacement of the rotor can be observed with higher precision both in the stable starting process and in the stable suspension operation process.

To validate the effectiveness and feasibility of the proposed LS-SVM displacement observer, under the same simulation conditions, a simulation comparison study between two different cases has been made; one case is that the actual radial displacement sensor is adopted; another case is that the proposed LS-SVM displacement observer is adopted. In order to compare the dynamic response performance of the control system in two cases, the given signals of radial displacement components are suddenly changed at different times. The simulation response curves of α and β displacement components are as shown in Figures 7 and 8.

From Figures 7 and 8, the following simulation results are analysed.

Compared with the control system with the displacement sensor, when the proposed LS-SVM displacement observer is adopted, the observer needs to estimate the displacement according to relevant signals, and the establishments of relevant physical variables require a certain period of time, then at the moment of starting suspension, a certain observation delay is produced, which results in the fluctuation amplitude of the displacement component being slightly greater.

Based on the proposed LS-SVM displacement observer, stable suspension operation of the rotor can be achieved. When given signals of displacement components are suddenly changed at different times, the observed displacement components can quickly tack their given signals, and reach their stable status; the observed or estimated displacement response curves are almost entirely consistent with the outputs of actual displacement sensors.

In the output voltage of the actual pulse width modulation (PWM) inverter, there inevitably exists high-frequency harmonic voltage. In order to try to close the industrial reality, the high-frequency harmonic interference signal is added at the output end of torque winding; its frequency and amplitude are 10 KHz and ±10 V, respectively. Then the performance simulation of the LS-SVM observer is carried out. Figure 9 gives the observation waveform of displacement, and for the convenience of comparative analysis, the actual displacement waveform is given also. Figure 10 gives the comparison waveforms of observed displacement between before and after adding the high-frequency harmonic interference signal.

From Figures 5, 6, 9 and 10, the following simulation results are analysed.

After adding the high-frequency harmonic interference signal, the overshoots of observed displacement increase obviously and the displacement overshoots along the α and β directions reach about 25 and 30 μm, respectively. The reason for the overshoot increase is that the high-frequency harmonic voltage causes weak variations of stator current and motor flux linkage.

After adding the high-frequency harmonic interference signal, the radial displacements along the α and β directions can still can be tracked quickly by the proposed LS-SVM observer. Furthermore, there is no increase in the displacement tracking error; the tracking error is still kept within ±2 μm.

The validity and feasibility of the proposed LS-SVM observer have been further verified by the simulation results.

Discussion and conclusion

In this article, a LS-SVM observation method is proposed to achieve the displacement self- sensing detection of the bearingless induction motor, and to reduce the control system cost. Firstly, the state-space model of the magnetic suspension system is analysed, and then the LS-SVM is introduced to the displacement observer. By offline training the LS-SVM, the function vector F(x, u) in the state-space model is approximated by the LS-SVM; through the gain matrix K that is asymptotically stable, the estimated displacement components, that is, $\hat{α}$ and $\hat{β}$ , gradually reach their actual values, and then the radial displacement self-sensing detection of the bearingless induction motor is achieved. In the paper, based on the design principle of the LS-SVM displacement observer, the Lyapunov stability analysis is made and the construction method of the proposed LS-SVM displacement observer is introduced in detail also. Finally, simulation verification and analysis are made.

According to the simulation verification and analysis results, there are the following conclusions.

1) When the proposed LS-SVM displacement observer is adopted, the radial displacement of the rotor can be accurately estimated in the steady state; under the condition of no radial displacement sensors, the reliable magnetic suspension operation of the bearingless induction motor can be achieved.

2) When the proposed LS-SVM displacement observer is adopted, both in the starting suspension process of the rotor and in the sudden change process of the displacement component, a faster tracking rate and higher observation accuracy can be achieved.

3) The proposed LS-SVM displacement observer is effective and feasible; at the same time, the motor parameters are not necessary for the LS-SVM observer, and then the influence of motor parameters can be avoided effectively by the proposed LS-SVM displacement observer. In addition, a new technical method has been provided for the radial displacement self-sensing detection of the bearingless induction motor.

In this article, the LS-SVM displacement observer of the bearingless induction motor is discussed and investigated, although system simulation results have shown its good observation performance, effectiveness and feasibility, but in its subsequent application, there are still some problems that should be researched, mainly including the following aspects. Firstly, when the proposed LS-SVM displacement observer is offline trained, the displacement sensor is needed to supply training sample signals. Then how to get rid of the dependence on displacement sensors, and achieve the sensorless online training of function vector F(x, u), is an important problem in the next research. Secondly, if the training sample number is too small, the accuracy of the trained LS-SVM displacement observer cannot be guaranteed; however, if the training sample number is too large, the implement difficulty of the LS-SVM displacement observer will be increased; then the determination principle of the best training sample number for the LS-SVM is an important problem to solve. Thirdly, for the convenience of practical application, it is necessary to do further technical development for the digital realization algorithm of the proposed LS-SVM displacement observer and, during this period, there would be more repeated experimental studies. These problems will be gradually resolved in the follow-up studies.

Footnotes

Conflict of Interest

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (51277053), International Cooperation Project on Sci. & Tech. of Henan Province (114300510029) and Nature Sci. Fund of Henan Province Education Bureau (2010B510011).

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