Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 2409179, 11 pages

https://doi.org/10.1155/2017/2409179

## A Bearingless Induction Motor Direct Torque Control and Suspension Force Control Based on Sliding Mode Variable Structure

^{1}School of Electrical and Information Engineering, Jiangsu University, Zhenjiang 212013, China^{2}Automotive Engineering Research Institute, Jiangsu University, Zhenjiang 212013, China

Correspondence should be addressed to Xiaodong Sun

Received 31 March 2017; Accepted 14 June 2017; Published 1 August 2017

Academic Editor: Jun M. Wang

Copyright © 2017 Zebin Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Aiming at the problems of the large torque ripple and unstable suspension performance in traditional direct torque control (DTC) for a bearingless induction motor (BIM), a new method of DTC is proposed based on sliding mode variable structure (SMVS). The sliding mode switching surface of the torque and flux linkage controller are constructed by torque error and flux error, and the exponential reaching law is used to design the SMVS direct torque controller. On the basis of the radial suspension force mathematical model of the BIM, a radial suspension force closed-loop control method is proposed by utilizing the inverse system theory and SMVS. The simulation models of traditional DTC and the new DTC method based on SMVS of the BIM are set up in the MATLAB/Simulink toolbox. On this basis, the experiments are carried out. Simulation and experiment results showed that the stable suspension operation of the BIM can be achieved with small torque ripple and flux ripple. Besides, the dynamic response and suspension performance of the motor are improved by the proposed method.

#### 1. Introduction

Bearingless motors inherited the characteristics of the traditional magnetic bearing motors, such as being without lubrication, having no wear, and having no mechanical noise. Compared with the traditional magnetic bearing motor, the space utilization rate, electromagnetic efficiency, and other aspects have been improved. It is of great significance to the development of the field of the biochemical medicine and industrial field such as the heart blood pump, the turbo molecular pump, and the high speed flywheel energy storage system [1, 2]. The levitation windings of bearingless motors, breaking the balance of the original state of the air gap magnetic field, changing the distribution of the air gap magnetic field, and producing the Maxwell force which have an effect on the rotor were to realize the stable suspension. Many scholars at home and abroad have studied the bearingless permanent magnet motor. However, the bearingless induction motor (BIM) has the advantages of low loss, low cost, simple structure, and withstanding high temperature, and it is easy to realize the open-loop speed control by the voltage source inverter, which people are increasingly concerned with [3].

The BIM is a strongly coupled nonlinear system. In order to realize the decoupling between the radial suspension force and the torque, the rotor flux oriented vector control is introduced into the BIM in [4]. However, the use of the rotor flux instead of the air gap flux in this method has bad influence on the rotor stable suspension. Reference [5] overcame the limitation of rotor flux orientation, introduced the air gap flux oriented of the BIM, and deduced the suspension force expression of the air gap flux oriented control, but the control algorithm is much more complicated and highly nonlinear, and the torque response will be instable, which makes the application of this method limited.

Direct torque control (DTC) is another way of AC speed regulation after field oriented vector control. The method is simple and does not need the complex vector transform and current control, which improves the dynamic response of the system. From the retrieved papers, there are not many literatures researching on the DTC of bearingless motor. The DTC is introduced into the bearingless permanent magnet synchronous motor and the bearingless permanent magnet slice motor in [6, 7]. The DTC is applied to the BIM in [8, 9] and the DTC based on space vector pulse width modulation is proposed. The experimental results showed that the method can realize the stable suspension of the motor. However, the traditional DTC has the problems of high torque ripple and nonfixed switching frequency. Because the stator flux estimation directly affects the performance of DTC, a large number of papers have been studied on the stator flux estimation [10–12]. Some papers committed to improving the structure of the controller [13–16], but these controllers have increased the complexity of the system, which make the control system of the BIM more complicated and influence the motor stable suspension operation.

In order to reduce the torque ripple and improve the performance of the traditional DTC of the BIM, sliding mode variable structure (SMVS) control is introduced to the DTC system. Two hysteresis controllers in the traditional DTC are replaced by the sliding mode controller. The switching table is no longer used to select the space voltage vector; instead, the voltage vectors of the two-phase static coordinate system are output by the sliding mode controller, which is constructed according to the torque error and flux linkage error. The choice of inverter switching value based on space voltage pulse width modulation (SVPWM) reduces the torque ripple fundamentally.

The BIM control system includes torque control and suspension force control. For the part of suspension force control, the traditional method is by detecting the radial displacement of the rotor, and the given value of suspension force is obtained by the PID modulation, which can realize the closed-loop control of the suspension force [17]. The method is relevant to air gap flux identification, and its dynamic response and the anti-interference performance are not good. To overcome this problem, the SMVS suspension force control method based on inverse system is proposed. On the basis of the radial suspension force mathematical model of the BIM, the reversibility of the inverse system is analyzed; the SMVS control is introduced into the inverse system and realized suspension force control. The BIM DTC and suspension force control system based on SMVS are built in MATLAB/Simulink toolbox. The dynamic performance of the torque response and the rotor suspension performance are compared and analyzed for two cases that are based on traditional DTC and SMVS-DTC, respectively. On this basis, the experiments were carried out. Simulation and experiment results showed that the torque ripple has greatly reduced and the performance of the rotor suspension is improved.

#### 2. Sliding Mode Variable Structure Direct Torque Control of the BIM

##### 2.1. Principle of SMVS

SMVS control is a special kind of nonlinear control method, which is a kind of control strategy of variable structure control system. The difference between this control strategy and the traditional control strategy is that the control is not continuous. It is a kind of switching characteristic that makes the system structure change with time. The control strategy makes the system pass through the prescribed state locus back and forth in high frequency and small amplitude. Because the state locus can be designed, and not related to the system parameters and outside disturbance, it has a strong robustness [18].

Sliding mode variable structure control has three basic problems: firstly, the sliding mode should exist; secondly, the reach-ability condition should be satisfied; finally, the stability of the sliding mode should be analyzed [19–21].

##### 2.2. Design of SMVS-DTC Controller

SMVS controller is used to replace the traditional hysteresis comparators. The torque winding reference voltage vector should be calculated by torque error and flux error, and thus the controller includes flux controller and torque controller.

The mathematical model of the BIM in the two-phase stationary* α-β*coordinate system and with the torque winding stator current and stator flux as state variables is as follows:where ; and , respectively, represent two components of the torque winding stator current and rotor current in coordinate system. and , respectively, represent two components of the torque winding stator flux and rotor flux in coordinate system. and are stator and rotor self-inductance of torque winding. and are stator and rotor resistance of torque winding; is mutual inductance between stator and rotor of torque winding. and , respectively, represent two components of the torque winding stator voltage and rotor voltage in coordinate system. is the rotor angular velocity.

The electromagnetic torque equation of the BIM is as follows:where is the number of the pole-pairs of torque winding.

The voltage-current model is used to estimate the stator flux of the torque winding in the two-phase stationary coordinate system ().

In order to track the desired trajectory of the torque and flux, the sliding mode switching surface is selected as follows:where and are the given values of torque and the square of flux. and are the calculated values of torque and the square of flux.

To make the system have a good dynamic performance, the reaching law is selected as the exponential reaching law:where and are positive constants. When the value of is very small and the value of is great, the speed of the reaching law away from the switching surface is fast and near the switching surface is slow. This can effectively reduce the chattering and shorten the transition time [22].

Calculating the derivation of and ,where .

in which* C*_{1},* D*_{1},* C*_{2},* D*_{2}, and* u *are as follows:

Combining (5) and (6) one can obtain

Substituting (4) into (7) one can obtain the controller formula:and , where

##### 2.3. Analyses of the Reach-Ability Condition of the Sliding Mode

If the sliding mode exists, the moving point beyond the switching surface will reach the switching surface in finite time and the sliding mode motion is stable. The following reach-ability condition is satisfied:

The global reach-condition is . From the exponential reaching law, it can be obtained that

Because and are positive constant, . The sliding mode satisfies the reach-ability condition.

##### 2.4. Analyses of the Stability of SMVS-DTC Controller

Selecting Lyapunov function: , the derivation of it can be calculated:

in which , , , and are positive constant, so it can be obtained that

It can be concluded that this SMVS-DTC controller satisfies the stability conditions.

#### 3. SMVS Suspension Force Control of the BIM

##### 3.1. Radial Suspension Force Mathematical Model of the BIM

According to the relationship between the suspension force and the suspension winding stator current of the BIM in the two-phase stationary coordinate system () [23],where . and are the numbers of pole-pairs of torque winding and suspension winding, respectively. and are the component of air gap flux in axis, respectively. and are the suspension force of radial and . is the mutual inductance between stator and rotor of suspension winding. is the permeability of vacuum. is the effective length of the rotor. is the outside diameter of rotor. and are the torque winding turns and suspension winding turns.

When the rotor deviates from the center of the motor stator, it will cause the motor flux distribution. Then the Maxwell force is not zero, and the direction of its action and the direction of the rotor eccentricity are consistent, pointing to the direction of minimum air gap; it is called unbalanced magnetic force and is showed aswhere and and are the unbalanced magnetic force in the direction of and , respectively. and are the displacement in the direction of and , respectively, and is the air gap length.

The motion equations of the rotor are expressed aswhere is the mass of the rotor and is the gravity acceleration. By (16)–(18), it can be obtained that

##### 3.2. SMVS Radial Suspension Force Control of the BIM Based on Inverse System

The basic principle of inverse system method is to compensate the controlled object as a system with a linear transfer relationship by using the inverse system of the controlled object, then, to synthesize the system according to the linear system theory, and to realize the decoupling in nonlinear systems [24, 25].

Theorem 1. *For a system of dimensional input, , and, for dimensional output, , with a set of initial states , and it can be expressed as the following state equation:*

The necessary and sufficient condition for the system to be reversible in the neighborhood of () has vector relative order in this neighborhood.

Select the state variables in (19).

The input variable is

The output variable is

The state equation of the composite controlled object is obtained by (19) and(21):

Taking the derivative of the output function until each component in include input explicit.

The Jacobi matrix of the input can be expressed as

Because , according to Theorem 1, the relative order of the system is , , so the system is reversible.

Assuming , (25) can be calculated as

In order to realize closed-loop displacement control and improve the performance of anti-interference, SMVS is introduced into the inverse system. The sliding mode switching surface of the sliding mode controller is defined as follows:where and are the given displacement of and direction, and are the measured displacement, and and are constants.

Exponential reaching law is adopted in this paper and the expression is as follows:

According to (14)-(15), the sliding mode controller is stable. It can be obtained by (28)-(29) that

Torque winding and suspension winding of the BIM are coupled by air gap flux of torque winding, and torque winding can be obtained by torque winding stator flux subtracting stator leakage inductance.

It can be known from (3) that the stator flux identification in the DTC has the pure integral part. The initial value of integration, the accumulated error, and other factors will make the flux identification inaccurate. Therefore, the traditional suspension force control method cannot meet the requirements of the stable suspension of the BIM. Because of the introduction of SMVS to the suspension force control, the robustness of the controller is improved, the dependence of the accuracy of stator flux estimation is reduced, and the performance of stator stable suspension is improved.

Through the separate control of torque and suspension force, the SMVS-DTC system of the BIM can be constructed, and the structure diagram of the system is showed as Figure 1.