Decoupling Control Strategy of BLIM considering Rotor Mass Eccentricity

A BLIM, i.e., bearingless inductionmotor, is a multivariable, nonlinear, and strong coupling object; to achieve its high performance magnetic suspension operation control and overcome the influence of the rotor mass eccentricity, a decoupling control strategy considering the rotor mass eccentricity is proposed. Firstly, the mathematical model of the torque system based on the rotor flux orientation and the mathematical model of the magnetic suspension system based on the air gap flux orientation are presented; on this basis, the inverse system decoupling controlmethod of the BLIM is researched.Then, according to the frequency characteristics of the unbalanced displacement, an unbalance vibration compensator is designed, which can generate a compensation force to suppress or eliminate the unbalanced displacement. Simulation experimental results have shown that the decoupling control among the rotor flux-linkage, motor speed, and two radial displacement components can be achieved; in the steady state, the unbalanced displacement can be basically eliminated, while, during the mutation process of motor speed, the unbalanced displacement can be suppressed effectively.


Introduction
The motor supported by the mechanical bearings cannot meet the need of high-speed and long-time operation [1,2].Then the motor supported by the magnetic bearing is widely developed [3,4], but it still has some disadvantages, such as the more power consumption for magnetic suspension and the overspeed difficulty [5].The bearingless motor is a new type of AC motor, which is proposed based on the structure similarity between the magnetic bearing and the common AC motor stator [3,4].In the common bearingless motor, there are two sets of stator windings, i.e., torque windings (pole-pair number  1 , angle frequency  1 of current) and suspended windings (pole-pair number  2 , angle frequency  2 of current).When the two sets of stator windings meet the qualifications of " 1 =  2 ± 1" and " 1 =  2 ," a stable and controllable magnetic suspension force can be generated.Figure 1 shows the schematic diagram of the generation principle of the controllable magnetic suspension force [6].
The bearingless motor has a series of advantages, such as no friction, no mechanical noise, and higher critical speed; then it is more suitable for high-speed operation, and it has become a research hotspot [5][6][7][8][9][10][11][12].
The bearingless induction motor (BLIM) is a multivariable, nonlinear, and strong coupling object.To realize its high performance control, it is necessary to achieve the dynamic decoupling between the controlled variables.As for a nonlinear system, the inverse system method is an effective one that can be used for the liberalization decoupling [12,13].In addition, to improve the control performance of the torque system, the rotor flux orientation is usually adopted.About the inverse system decoupling control strategy of the BLIM based on the rotor flux orientation, there has been some research progress [13][14][15].But for the magnetic suspension system, the air gap flux-linkage of the torque system is required in the model of magnetic suspension force.In the coordinate system orientated by the rotor flux-linkage of the torque system, the air gap flux-linkage components of the torque system are time-varying, which will inevitably lead to the calculation complexity of the suspension control current.In this paper, the rotor flux orientation is still applied to the torque system; but in the magnetic suspension system, the magnetic suspension force will be controlled based on the synthetic air gap flux-linkage of the torque system; the inverse system decoupling will be applied to the overall BLIM system; the required air gap flux-linkage of the torque system will be obtained by an independent air gap flux-linkage observer.
In addition, due to the uneven material, machining accuracy, assembly error, and other reasons, the mass eccentricity of rotor is unavoidable, and, during the rotor rotation, an unbalanced excitation force acting on the rotor is generated.The unbalanced excitation force has the same angle frequency with the rotational speed, which will lead to the unbalance vibration [16][17][18][19].In view of the uncertainties of the mass eccentric direction and the mass eccentricity, it is difficult to establish an accurate model of unbalanced excitation force, and then it is difficult to solve the unbalance vibration problem by an analytic inverse system method.But the unbalanced excitation force will generate an unbalanced displacement.In this paper, the radial displacement would be divided into two parts, i.e., the random displacement and the unbalanced displacement; the inverse system decoupling control strategy is adopted to control the random displacement, and a vibration compensation controller is adopted to suppress or eliminate the unbalanced displacement.Simulation experimental results have shown that the decoupling control between the motor speed, rotor flux-linkage, and two radial displacement components can be realized; the unbalanced displacement can be effectively suppressed.

Model of Torque System
Definition. d-q is the rotor flux orientation coordinate system of the torque system.Then in the d-q coordinate system, the motion equation and the rotor flux-linkage equation can be expressed as follows: where  r is the mechanical angular velocity of rotor;  1 is the electrical angular velocity of the d-q coordinate system;  r and  1 are the amplitude and space position angle of the rotor flux-linkage vector;  s1d and  s1q are the current components of torque windings along the d-and q-coordinate axes;  m1 is the equivalent two-phase mutual inductance of the torque system;  푟 is the rotor time constant;  푟 =  푟 / 푟 ;  푟 is the rotor's self-inductance and  푟 is the rotor resistance; J is the inertia moment;  1 is the pole-pair number of torque windings;  퐿 is the load torque.

Model of Magnetic Suspension
System.The controllable magnetic suspension force acting on the rotor is the interaction result between the air gap flux-linkage of the torque system and that of the magnetic suspension system.In the decoupling calculation of the magnetic suspension system, the air gap flux-linkage information of the torque system is required.Defining - as the two-phase stationary orthogonal coordinate system, its coordinate origin locates at the center of stator.Then the air gap flux-linkage components of the torque system in the  and  direction, i.e.,  1훼 and  1훽 , can be expressed as follows: where  s1훼 and  s1훽 are the voltage components of torque windings along the and -coordinate axes;  s1훼 and  s1훽 are the current components of torque windings along the and -coordinate axes;  푠1 is the stator resistance of torque windings;  푠1푙 is the leakage inductance of torque windings.
Definition.M-T is the air gap flux orientation coordinate system of the torque system.Then the amplitude  1m and space position angle  2 of the air gap flux-linkage of the torque system can be expressed as follows: According to the air gap flux-linkage amplitude of the torque system, the controllable magnetic suspension force components in the static - coordinate system can be controlled by the suspension current components  s2m and  s2t ; the specific models can be expressed as follows [18]: where F 훼 and F 훽 are the controllable magnetic suspension force components along the and -coordinate axes; is the magnetic suspension force coefficient determined by the BLIM structure;  s2m and  s2t are the control current components of magnetic suspension force along the M-and T-coordinate axes.
When the rotor deviates from the stator axis, the unilateral electromagnetic pull will be generated, and the specific expressions are as follows: where  and  are radial displacement components;  s is the radial displacement stiffness coefficient;  s훼 and  s훽 are the unilateral electromagnetic pull components along the and -coordinate axes, respectively.
Figure 2 presents the rotor mass eccentricity diagram.In Figure 2, u-v is the rotor synchronous rotating coordinate system; M is the centroid of rotor; ( m ,  m ) is the axis of rotor;  is the eccentric distance of rotor mass,  is the initial mass eccentricity angle in the u-v coordinate system.
During the rotation of the mass eccentric rotor, an unbalanced excitation force  a is generated in the mass eccentricity direction, whose angular frequency is the same with the angular velocity of rotor.The unbalanced excitation force components along the and -coordinate axes can be expressed as follows [20]: where m is the rotor mass.
The rotor's radial suspension motion equation can be expressed as follows:

Control System of BLIM
The decoupling control based on the inverse system method requires the analytical model of the BLIM system.But it is difficult to measure the eccentric distance  and the initial eccentricity angle  of rotor mass.But the research results show that the unbalanced excitation force will generate periodic unbalanced displacement, and it has no effect on the random displacement [12,13].The unbalanced displacement components along the and -coordinate axes are sine and cosine signals, respectively, whose frequencies are the same with the angular velocity of rotor, and they can be expressed as follows: where "A" is the amplitude of unbalanced displacement, and it is proportional to the square of motor speed;  is the initial phase angle of unbalanced displacement.
In this paper, the radial displacement is divided into a random displacement component and an unbalanced displacement component.Then by the vibration compensation controller, a vibration compensation force is generated, which is used to suppress or eliminate the unbalanced displacement.As for the random displacement, an inverse system decoupling control strategy is adopted.

3.1.
Inverse System Decoupling Control.Select the system state variable x, input variable u, and output variables  as follows: Substituting ( 12), (13), and ( 14) into (1), (2), and (10), then the state equations of the BLIM system can be derived as follows: where the rotor flux orientation is adopted for the torque system; based on the air gap flux-linkage of the torque system, the magnetic suspension force and the radial suspension motion of the rotor are controlled by the control current components of the magnetic suspension system, and the required air gap flux-linkage information is obtained in real time from (4)∼( 6).
Select the input of inverse system as follows: Substituting ( 22) into ( 16)∼( 19), then the inverse system model of the BLIM can be derived as follows: From ( 23) and ( 24), it can be seen that when the magnetic suspension system is controlled based on the air gap fluxlinkage of the torque system, it only needs to calculate the air gap flux-linkage of the torque system in real time; in addition, there is no obvious coupling between the inverse model of the torque system and that of the magnetic suspension system; this facilitates the "independent inverse system decoupling" between the torque system and the suspension system.
By connecting the inverse system in series before the original BLIM system, the torque system is dynamically decoupled into two first-order linear integral subsystems, including a rotor flux-linkage subsystem and a motor speed subsystem; meanwhile the magnetic suspension system is dynamically decoupled into two second-order linear integral subsystems, including the  random displacement component subsystem and the  random displacement component subsystem.Then by adding an appropriate regulator for each subsystem, the dynamic decoupling control among the rotor flux-linkage, motor speed, and  and  random displacement components can be realized.

Compensation Control of Unbalance Vibration.
The radial displacement components caused by the mass eccentricity along the and -coordinate axes are sine and cosine signals, whose frequencies are the same with the angular speed of rotor.From these characteristics, the extraction of unbalanced displacement and the compensation of unbalanced excitation force can be realized.The processes can be summed up as follows: Firstly, define u-v as the synchronous coordinate system rotating with the rotor.From the rotor synchronous rotation transformation, the radial displacement components measured in the - coordinate system are transformed to those in the u-v coordinate system.Then in the u-v coordinate system, the DC variable signals of unbalanced displacement, i.e.,  푚 and V 푚 , are extracted by relevant low pass filters.Secondly, in the u-v coordinate system, the given signals of unbalanced displacement components are set to zero; then through the closed-loop controls of  푚 and V 푚 , the given signals of vibration compensation force components in the u-v coordinate system, i.e.,  cu Air gap fluxlinkage ( 4)-( 6) Replacing F 훼 and F 훽 in (7) with  c훼 * and  c훽 * , then the analytical formula of vibration compensation current can be derived as follows: In (25),  * c2m and  * c2t are the given signals of the vibration compensation current components along the mand t-coordinate axes.
Figure 4 is the schematic diagram of the unbalance vibration compensation control system.

Decoupling Control Performance of BLIM System.
According to Figure 4, a four-pole BLIM with twopole suspension windings is taken as the object; system simulation is carried out by Matlab/Simulink.In order to verify the decoupling performance between the motor speed, rotor flux-linkage, and two displacement components, the influence of rotor unbalance vibration is temporarily ignored; the given values of related variables are suddenly changed at different times: the given rotor flux-linkage is reduced from 0.95 Wb to 0.5 Wb at 0.5 s; the given motor speed is increased to 3000 r/min from 1500 r/min at 0.75 s; the given  displacement is increased to 0.03 mm from  0.0 mm at 1.0 s and recovered to 0.0 mm at 1.2 s; the given  displacement is increased to 0.03 mm from 0.0 mm at 1.4 s and recovered to 0.0 mm at 1.6 s; a 10 N.m load torque is suddenly added at 1.5 s.
Figure 5 shows the response curves of the BLIM control system.From Figure 5, it can be seen that when any one of the motor speed, rotor flux-linkage, and  and  radial displacement components is suddenly changed, the other controlled variables basically remain unchanged or vary very little.The simulation results have shown that a good decoupling control between relevant controlled variables is achieved.In addition, the control system has some advantages, such as a faster response speed and a stronger resistance to load.
Figure 6 shows the displacement response curves before vibration compensation.From Figure 6, (1) without vibration compensation, under the action of the displacement closed-loop control, the rotor unbalance vibration occurs; the radial displacement components along the and -coordinate axes present periodic fluctuations (2) after the BLIM control system enters its steady state, the rotor axis trajectory is a circle with an amplitude of about 6 m.The rotor's suspension control precision is greatly influenced Figure 7 presents the displacement response curves after vibration compensation.From Figure 7, (1) in the initial starting stage, because the vibration compensation force has not been completely established, there exist some fluctuations with a smaller amplitude in the and -displacement curves; however, with the establishment of vibration compensation force, the unbalanced displacement is quickly suppressed (2) after the BLIM control system enters its steady state, under the combined action of the random displacement controller and the vibration compensation controller, the unbalanced displacement components along the and coordinate axes gradually decrease to zero; the rotor axis trajectory almost shrinks to the stator center; the suspension control precision of rotor has been greatly improved Taking the horizontal  direction as an example, Figure 8 shows the comparison curves between the unbalanced excitation force and the vibration compensation force.From Figure 8,  (1) in the initial starting stage, the amplitude of vibration compensation force increases with the unbalanced excitation force (2) after the BLIM control system enters its steady state, the vibration compensation force is the same as that of the unbalanced excitation force in amplitude, and their direction is always opposite.And then, the vibration compensation force can be used to overcome the influence of the unbalanced excitation force The comparison curves between the unbalanced excitation force and the vibration compensation force in the vertical  direction are similar to that in the horizontal  direction and will not be introduced.

Conclusions
To solve the strong coupling problem of the BLIM system and to solve the unbalance vibration problem caused by the rotor eccentric mass, a decoupling control strategy of the BLIM system considering the rotor mass eccentricity is studied.In order to achieve a better speed control performance, the rotor flux orientation is adopted for the torque system; to simplify the computational complexity of the magnetic suspension system, the required air gap flux-linkage of the torque system is obtained through independent and realtime calculation, and the rotor suspension motion control is achieved based on the air gap flux-linkage of the torque system.In order to improve the overall dynamic decoupling performance of the BLIM system, the whole inverse system decoupling is adopted.In order to overcome the unbalance vibration problem caused by the rotor mass eccentricity, the compensation controller of unbalance vibration is designed, and the vibration compensation force is used to suppress or eliminate the unbalanced excitation force that acts on the rotor.According to the simulation experimental results, there are the following conclusions: (1) Adopting the presented control strategy, the dynamic decoupling control between the rotor flux-linkage, motor speed, and two radial displacement components can be achieved (2) During the change of motor speed, the unbalanced displacement components of rotor can be suppressed effectively; meanwhile in the steady state of the BLIM control system, the unbalanced displacement components can be eliminated, and the magnetic suspension control accuracy of rotor can be greatly improved

Figure 1 :
Figure 1: Schematic diagram of controllable magnetic suspension force generation.

Figure 5 :
Figure 5: Response curves of decoupling control system.