Predictive current control of a bearingless induction motor model based on fuzzy dynamic objective function

Abstract

A control strategy for a bearingless induction motor (BL-IM) based on the fuzzy dynamic objective function is proposed in this paper. Firstly, based on the discrete mathematical model of the BL-IM, the stator current and flux linkage are predicted according to the given stator current and flux linkage, the objective function of model predictive current control (MPCC) is designed. Secondly, the fuzzy control algorithm is introduced in the objective function of the MPCC to dynamically assign the weighting factors before the current component on the d-q axis and the influence of the objective function on the performance of the BL-IM is analyzed under different weighting factors. By discretizing the rotational speed deviation Δω and rotational speed deviation rate, fuzzy reasoning is performed to obtain the optimal fuzzy dynamic function. Finally, the optimal fuzzy dynamic function is selected as the objective function of the MPCC to perform the simulations and experiments. The results show that the performance of the BL-IM under the MPCC strategy based on the fuzzy dynamic objective function is improved compared with the traditional MPCC and the vector control based on the fuzzy PID, due to its better dynamic and suspension performance. Meanwhile, the stability of rotor current component is enhanced.

Keywords

Bearingless induction motor (BL-IM)model predictive control (MPC)current control objective function fuzzy control

Introduction

Induction motors are widely used in the fields of agricultural production and machinery manufacturing (Leite et al., 2015), due to their low price, simple structure (Lim et al., 2014) and convenient maintenance. With the continuous development of society and technology, the market has higher and higher requirements for the performance of the motors. Ordinary induction (Oguz and Dede, 2011) motors generate much noise and mechanical loss during the operation, which directly affects their service life and dynamic performance. These deficiencies limit their application in high speed and high precision fields (Sun et al., 2017). Thus, the bearingless induction motor (BL-IM) (Bu et al., 2019a) becomes the study focus. It not only has the advantages of simple structure and low cost, but also has the advantages of no friction. Therefore, the BL-IM is widely applied in the fields of aerospace, artificial heart pumps, flywheel energy storage, biomedicine and other special electric fields. (Sun et al., 2013a).

Because of the wide application of the BL-IM, its control methods are also developing rapidly. The most common control methods for the BL-IM are vector control and direct torque control (Sun et al., 2019). In order to improve the dynamic response speed and suspension performance of the BL-IM, the stator current is decomposed into two components on the d-axis and q-axis to control the flux linkage and torque of the BL-IM respectively (Bu et al., 2019b). However, this method leads to the incomplete decoupling of the current components on the d-q axis and low anti-interference. The direct torque control method is proposed to control the torque and flux linkage of the motor directly by using space vector coordinates. However, it is easy to produce large torque ripples and the steady-state performance is poor.

With the rapid development of the power electronics technology and digital signal processing technology, the control methods of the BL-IM are also increasingly diversified (Rodriguez and Santisteban, 2011), such as neural network control (Sun et al., 2016), unbalance compensation control (Xu et al., 2018), sliding mode control, optimal parameter estimation and control, et al. Although these algorithms improve the accuracy of calculation, they also have many limitations on the application to the BL-IM. The neural network algorithm can increase the accuracy of the control system. However, it has a long training time for the BP network and its calculation is too large. Although sliding mode control can improve the robustness of the motor to the external interference, the chattering phenomena in the sliding mode control is very serious. To avoid the chattering phenomenon in the conventional sliding mode control methods, a novel reaching law is designed based on hyperbolic functions to guarantee that the sliding mode variable infinitely approaches to the equilibrium point instead of crossing it (Wang et al., 2020). However, if this reaching law based on hyperbolic functions is applied to the control system of the BL-IM, it will complicate the control system and reduce the real-time performance of the motor. To improve the transient and steady-state performance, a modified funnel variable, which relaxes the limitation of the original funnel control, is developed by using the tracking error to replace the scaling factor (Wang et al., 2019). Adding this method to the control system of the BL-IM can increase the control accuracy of the current component, but a tracking error applied to the closed-loop suspension current control will increase the calculation and decrease the response speed of the control system. The model predictive control (MPC) is a novel control method, which is widely used in the motion control of various motors (Liu and Li, 2012) for its accurate and reliable control performance (Geyer et al., 2009) among these algorithms. Model predictive current control (MPCC) is one MPC. Compared with the above control algorithm, the MPCC has low chattering phenomena and no pulse width modulation. Moreover, the optimal voltage vector can be used for the control system directly, which needs less computation. (Ku et al., 2017). So, it is widely used in the field of high speed and high precision (Ren et al., 2018).

In recent years, relevant scholars have achieved remarkable results in the field of improving the accuracy of MPCC. In order to enhance the tracking ability of the current component on the d-q axis, a current controller by using model predictive control (Papafotiou et al., 2009) method is proposed and verified by simulations. But this method reduces the start-up performance of the motor. Also, the influence of each current component in the objective function is not specifically analyzed. The prediction algorithm of the current loop is compared and analyzed from the theoretical derivation, model construction and experimental simulation. And it is designed to improve the start-up performance of the motor, but the design process is too cumbersome (Morel et al., 2009).

To save the limitations on the tradition MPCC, a control strategy for the BL-IM based on the fuzzy dynamic objective function is proposed in this paper. In the traditional MPCC, when the objective function is optimal, the dynamic performance of the motor is optimal. However, the stability of the rotor current is greatly reduced. To solve this limitation, a fuzzy function is introduced to determine the optimal state between the dynamic performance and the stability of the rotor current. At this time, while the motor has good dynamic performance, the rotor current is also more stable. In order to achieve this state, a MPCC method based on fuzzy dynamic objective function is proposed in this paper. The “dynamic” in this objective function is used to explore the relationship between the dynamic performance of the BL-IM and stability of the current component. This is an optimization algorithm for the objective function in the MPCC. The fuzzy control algorithm is introduced in this paper to add the weight factors before the current component on the d-q axis. Through Simulink simulation, the better dynamic performance the BL-IM has, the more unstable the rotor current component is. By introducing fuzzy control (Li et al., 2013) into the objective function, the rotational speed deviation and the rotational speed change rate are discretized to obtain the fuzzy inference table. Through solving the ambiguity, the optimal fuzzy dynamic function (Lam et al., 2014) is obtained and used as the objective function of the MPCC. The BL-IM is performed under the MPCC based on the fuzzy dynamic objective function. The results of the simulation and experiment prove that the MPCC strategy based on the fuzzy dynamic objective function improves the dynamic performance and suspension stability of the BL-IM.

Mathematical model of a BL-IM

The stator of the BL-IM (Sun et al., 2013b) is sheathed with two groups of windings. They are torque winding and suspension winding. The pole pairs are P₁ and P₂ and the electrical angular frequency are ω₁ and ω₂, respectively. The controllable radial force of the BL-IM is generated by controlling the non-uniformity of the air-gap magnetic field through the interaction of the current in the two sets of windings (Ren and Fang, 2012). Only when the following three conditions are met, the BL-IM can stably suspend and rotate:

P₁ = P₂± 1;

ω₁ = ω₂;

The direction of the generated rotating magnetic field is the same.

The mathematical model of the BL-IM consists of two parts. They are rotating part and suspension part.

Mathematical model of the rotating part

The mathematical model of the rotating part of the BL-IM is established as follows (Yang et al., 2019)

{\begin{matrix} \frac{d i_{1 s α}}{dt} = \frac{1}{σ L_{1 s} T_{1 r}} ψ_{1 s α} + \frac{1}{σ L_{1 s}} ω_{r} ψ_{1 s β} - ω_{r} i_{1 s β} \\ + \frac{u_{1 s α}}{σ L_{1 s}} - \frac{R_{1 s} L_{1 r} + R_{1 r} L_{1 s}}{σ L_{1 s} L_{1 r}} i_{1 s α} \\ \frac{d i_{1 s β}}{dt} = \frac{1}{σ L_{1 s} T_{1 r}} ψ_{1 s β} - \frac{1}{σ L_{1 s}} ω_{r} ψ_{1 s α} + ω_{r} i_{1 s α} \\ + \frac{u_{1 s β}}{σ L_{1 s}} - \frac{R_{1 s} L_{1 r} + R_{1 r} L_{1 s}}{σ L_{1 s} L_{1 r}} i_{1 s β} \\ \frac{d ψ_{1 s α}}{dt} = - R_{1 s} i_{1 s α} + u_{1 s α} \\ \frac{d ψ_{1 s β}}{dt} = - R_{1 s} i_{1 s β} + u_{1 s β} \end{matrix}

(1)

The equation of the magnetic flux leakage coefficient σ and the rotor time constant T_r is expressed as

{\begin{matrix} σ = \frac{1 - L_{1 m}^{2}}{L_{1 s} L_{1 r}} \\ T_{r} = \frac{L_{r}}{R_{r}} \end{matrix}

(2)

The torque equation for the BL-IM is expressed as

T_{e} = P_{1} (ψ_{1 s α} i_{1 s β} + ψ_{1 s β} i_{1 s α})

(3)

where i_1sd and i_1sq are the current component of the torque winding stator current on the d-axis and q-axis; L_1s, L_1r, L_1m are the stator self-inductance, rotor self-inductance and stator-rotor mutual inductance of the torque winding; ω_r is the rotor angular velocity; R_1s and R_1r are the stator resistance and rotor resistance of the torque winding; ψ_1sd and ψ_1sq are the flux linkages component of the torque winding stator flux linkage on the d-axis and q-axis; U_1sd and U_1sq are the voltage component of the stator voltage on the d-axis and q-axis (Yang et al., 2018).

Mathematical model of the suspension part

The mathematical model of the suspension part of the BL-IM is established as follow (Sun et al., 2018)

{\begin{matrix} F_{x} = K (i_{2 s α} ψ_{1 α} + i_{2 s β} ψ_{1 β}) \\ F_{y} = K (i_{2 s β} ψ_{1 α} - i_{2 s α} ψ_{1 β}) \end{matrix}

(4)

where F_x and F_y are the radial suspension force (Ye et al., 2019) component on the x, y coordinate axes, ψ_1d and ψ_1q are the flux linkage component of the torque winding air gap flux linkage on the d-axis and q-axis (Zhang and Zhu, 2015)

K = \frac{π P_{1} P_{2} L_{2 m}}{18 lr μ_{0} W_{1} W_{2}}

(5)

In equation (5), the mutual inductance of the suspension winding is L_2m. The effective length of the core is l. The vacuum permeability is μ₀ (Zhang et al., 2016). Also, the effective turns of the torque winding and the suspension winding are W₁ and W₂.

MPCC

Basic principle

The MPCC (Guzman et al., 2014) is a common control method, which is used in the field of induction motors. Its main idea is to predict the future behavior by applying the mathematical model of the system in a predetermined period of time (Shi et al., 2014). In this paper, the predictive current control of the BL-IM based on fuzzy dynamic objective function is adopted dynamically to explore the relationship between the stability of the current component and the dynamic performance of the BL-IM. The BL-IM is driven by a two-level voltage inverter (Cortes et al., 2012). Due to the special characteristics of the bearingless motor, it requires both rotation and suspension. Thus, the control system is divided into two parts. They are the rotating part and the suspension part.

The torque prediction part consists of two parts: outer ring and inner ring. The function of the outer ring is to obtain the given value of the stator current (Xia et al., 2014). The output speed of the motor is compared with the rated speed by PID error and decoupled by the air gap magnetic field to obtain the component current of the stator current on the α-axis and β-axis . The set value of the stator current is obtained by 2/3 transformation. The stator current value at time k+1 is obtained by the model prediction and delay compensation in the inner ring. The fuzzy objective function is used to replace the traditional rolling optimization method. The objective function is optimized, and the optimal objective function is used as the objective function of the MPCC to control the BL-IM. The voltage vector under fuzzy control is selected as the optimal voltage vector. The switch state corresponding to the optimal voltage vector is output at time k+1.

In the suspension part, the air gap flux linkage is calculated based on the stator current and stator flux linkage predicted by the torque part. Through the force/current converter, 2/3 conversion and CRPWM-inverter, a stable current is obtained to control the stable suspension of the BL-IM. This paper is to optimize the prediction algorithm of the torque part, so the prediction model of the suspension part will not be expanded in the following paragraphs. The control block diagram of the BL-IM under the MPCC is shown in Figure 1.

Mathematical model of the MPCC for rotating part

Rewrite equation (1) into the form of state space equation

{\begin{matrix} \overset{\cdot}{x} = Ax + Bu \\ y = Cx \end{matrix}

(6)

A, B, and C are in matrix form

A = [\begin{matrix} - \frac{R_{1 s} L_{1 r} + R_{1 r} L_{1 s}}{σ L_{1 s} L_{1 r}} & - ω_{r} & \frac{1}{σ L_{1 s} T_{1 r}} & \frac{ω_{r}}{σ L_{1 s}} \\ ω_{r} & - \frac{R_{1 s} L_{1 r} + R_{1 r} L_{1 s}}{σ L_{1 s} L_{1 r}} & - \frac{ω_{r}}{σ L_{1 s}} & \frac{1}{σ L_{1 s} T_{1 r}} \\ - R_{1 s} & 0 & 0 & 0 \\ 0 & - R_{1 s} & 0 & 0 \end{matrix}],

B = [\begin{matrix} \frac{1}{σ L_{1 s}} & 0 \\ 0 & \frac{1}{σ L_{1 s}} \\ 1 & 0 \\ 0 & 1 \end{matrix}],

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

x = [\begin{matrix} i_{1 s α} \\ i_{1 s β} \\ ψ_{1 s α} \\ ψ_{1 s β} \end{matrix}] .

The predictive equation of the stator current at the next moment is obtained by discretizing the first-order discrete Euler method of equation (6)

{\begin{matrix} i_{1 s α} (k + 1) = \frac{T_{s}}{σ L_{1 s} T_{1 r}} ψ_{1 s α} + \frac{T_{s}}{σ L_{1 s}} ω_{r} ψ_{1 s β} + [1 - \frac{T_{s} (R_{1 s} L_{1 r} + R_{1 r} L_{1 s)}}{σ L_{1 s} L_{1 r}}] i_{1 s α} \\ - T_{s} ω_{r} i_{1 s β} + \frac{u_{1 s α} T_{s}}{σ L_{1 s}} \\ i_{1 s β} (k + 1) = \frac{T_{s}}{σ L_{1 s} T_{1 r}} ψ_{1 s β} - \frac{T_{s}}{σ L_{1 s}} ω_{r} ψ_{1 s α} + [1 - \frac{T_{s} (R_{1 s} L_{1 r} + R_{1 r} L_{1 s)}}{σ L_{1 s} L_{1 r}}] i_{1 s β} \\ + T_{s} ω_{r} i_{1 s α} + \frac{u_{1 s β} T_{s}}{σ L_{1 s}} \end{matrix}

(7)

In order to make the predicted value of the stator current track the given value more accurately, the concept of fuzzy dynamic function is introduced in this paper. To minimize the error between the actual current and the reference current, the designed objective function can be expressed as

\begin{matrix} g = {(i_{d}^{*} - i_{d} (k + 1))}^{2} + {(i_{q}^{*} - i_{q} (k + 1))}^{2} \\ + \overset{\land}{f} (i_{d} (k + 1), i_{q} (k + 1)) \end{matrix}

(8)

In order to determine the optimal voltage vector, a nonlinear function that defines the amplitude of the stator current can be expressed as

\begin{matrix} \overset{\land}{f} (i_{d} (k + 1), i_{q} (k + 1)) \\ = {\begin{matrix} \infty | i_{d} (k + 1) | > i_{max} or | i_{q} (k + 1) | > i_{max} \\ 0 | i_{d} (k + 1) | \leq i_{max} and | i_{q} (k + 1) | \leq i_{max} \end{matrix} \end{matrix}

(9)

where i_max is the maximum value of the stator current. If one of i_d(k+1) or i_q(k+1) is greater than i_max, the objective function g=∞, the voltage vector does not meet the condition. If both i_d(k+1) and i_q(k+1) are less than i_max at the same time, the third item of equation (8) is 0, and the value of the objective function g depends on the first two items.

Design of the objective function

It can be seen from equation (8) that i_d and i_q belong to one type of controlled variable, so the fuzzy control (Chen et al., 2012) can be introduced to optimize the objective function. As for the single fuzzy control for the BL-IM, this is not suitable. Because the BL-IM is a novel motor of high efficiency, it has a high demand for real-time performance because of its suspension performance. However, if the accuracy of fuzzy control requires to be improved, it is necessary to increase the fuzzy set level. This will lead to the expanded rule search scope and reduce the speed of calculation. In the end, the real-time control ability of the BL-IM will be reduced. Thus, the fuzzy control should be introduced into the optimization of the objective function of the MPCC in this paper, which can improve the control accuracy of the motor without reducing the real-time performance. The fuzzy control is used to adjust the weighting factor before each variable in the objective function (Qiu et al., 2016) to make the MPCC meet the need for control. However, i_d and i_q have their own physical meanings. If they are randomly distributed, the dynamic and suspension performance of the BL-IM will be greatly affected. Therefore, it is necessary to determine the optimal weighting factors of i_d and i_q through the fuzzy function. When the objective function is optimal, the voltage vector is also the optimal voltage vector.

A two-level three-phase inverter is used in this paper, so there are eight kinds of switch states, which are (0,0,0),…,(1,1,1). In order to optimize the switching frequency, only one switch is allowed to jump once at a time. Such as (0,0,0) to (0,0,1) or (0,0,0) to (0,1,0). The improved voltage vector switching mode is shown in Figure 2.

The weighting factors are introduced in equation (8), and the fuzzy objective function of the MPCC can be expressed as

\begin{matrix} g = Q_{1} {(i_{d}^{*} - i_{d} (k + 1))}^{2} + Q_{2} {(i_{q}^{*} - i_{q} (k + 1))}^{2} \\ + \overset{\land}{f} (i_{d} (k + 1), i_{q} (k + 1)) \end{matrix}

(10)

where Q₁ and Q₂ are the weighting factors of the current component on the d-axis and q-axis.

Design of the fuzzy control

Based on equation (10), the influence of the dynamic and suspension performance of the BL-IM under different weight factors are analyzed in this section. As for the hardware parameters, such as the switching frequency of the external inverter, the application of these parameters to the objective function will lead to excessive calculation, and these parameters do not have large effect on the performance of the BL-IM. Thus, these hardware parameters can be disregarded. The model of the BL-IM under the MPCC based on fuzzy function is built in Matlab/Simulink for the analysis of simulation. The parameters of the BL-IM are shown in Table 1.

Table 1.

Parameters of the BL-IM.

Parameters (unit)	Torque winding	Suspension winding
Rated power (kW)	1	0.5
Rated current (A)	2.86	2.86
Stator resistance (Ω)	2.01	1.03
Rotor resistance (Ω)	11.48	0.075
Mutual inductance of stator and rotor (mH)	158.56	9.32
Stator leakage inductance (mH)	4.45	2.67
Rotor leakage inductance (mH)	9.22	5.42
Rotational inertia (kg·m²)	0.00769	0.00769
Rotor mass (kg)	2.85	2.85
Stator inner diameter (mm)	98	98
Core length (mm)	105	105
Pole pairs	1	2

Performance of the BL-IM under different weighting factors

In order to analyze the performance of the BL-IM under different weight factors, the distribution of the weighting factors is shown as follows. The weight factors are selected equidistantly to make the fuzzy set easy to observe and calculate. Through isometric selection, the direct relationship between dynamic performance, stability of the current component and weight factors can be explored: (1) weighting factors are the same (Q₁=Q₂=0.5); (2) focus on the current component on the d-axis (Q₁ = 0.7, Q₂ = 0.3); (3) focus on the current component on the q-axis (Q₁ = 0.3, Q₂ = 0.7). The BL-IM under three groups of different weighting factors are simulated, and the simulation results are compared as follows.

It can be seen from Figure 3 and Figure 4 that the smaller the Q₁ is, the better the start-up performance of the motor will be. Conversely, an increase in Q₁ will impair the dynamics performance of the motor. It can be seen from Figure 5 and Figure 6 that the larger the Q₁ is, the worse the control accuracy of the current component on the q-axis will be, and the fluctuation of the i_q becomes large. Conversely, when Q₁ is reduced, the control precision of i_d is reduced. In order to improve the start-up performance of the motor, Q₁ should be decreased, but the control precision of the i_d is influenced. In order to improve the control accuracy of i_d, Q₁ can be increased, but the start-up performance of the motor and the control precision of the i_q are sacrificed. Therefore, it is necessary to determine the most suitable Q₁ and Q₂ by the fuzzy dynamic objective function, so that the motor can obtain a good start-up performance while losing less control accuracy on the current component.

Fuzzy dynamic objective function

According to the analysis of the performance of the BL-IM under different weighting factors, it can be known that Q₁ and Q₂ interact with each other and restrict each other. Therefore, the fuzzy function is introduced in this paper. The fuzzy conditional statement and algorithm are used to describe the complex functional relationship among targets. Therefore, an objective function based on the fuzzy control is applied to the MPCC.

Through the comparison of the simulation of the previous three isometric weighting factors, it can be seen that the dynamic performance of the motor increases with the increase of Q₂. The maximum value of Q₂ appears at the maximum of the speed curve slope. Thus, the maximum and minimum of Q₁, Q₂, the speed deviation △ω and the speed deviation rate can be determined. In Figure 7, the deviation rate of the rotational speed calculated by the slope is [10000, 16666], and the rotational speed deviation Δω is [0, 3000]. At the same time, Q_1max = 0.4, Q_1min = 0.2; Q_2max = 0.85, Q_2min = 0.45. Therefore, the input range for defining Q₁ is [0.2, 0.4], and the input range for Q₂ is [0.45, 0.85].

The points in the fuzzy set are divided according to the equidistance, which facilitates the induction and iteration of the MPCC based on fuzzy control. The rotational speed deviation Δω and the rotational speed deviation rate are taken as the equidistant input variables of the fuzzy controller. Discretize the input and output ranges and define the fuzzy set: the fuzzy set of the input variable is {NB (negative big), NM (negative medium), NS (negative small)}, and the fuzzy set of the output variable is {ZO(zero), PS (positive small), PM (positive medium), PB (positive big)}, the definition of fuzzy domain is shown in Table 2.

Table 2.

Definition of the fuzzy domain.

Variable/Set	Domain
	NB	NM	NS	ZO	PS	PM	PB
Speed deviation △ω	−3000	−1500	−750	0	750	1500	3000
Speed deviation rate	−16666	13333	−10000	0	10000	13333	16666
Q₁	—	—	—	0.4	0.4	0.32	0.2
Q₂	—	—	—	0.4	0.55	0.68	0.85

The fuzzy function is obtained by defining the table according to the fuzzy theory domain and the fuzzy inference rules, as shown in Table 3.

Table 3.

Fuzzy reasoning rules.

Rate of revolution	Speed deviation Δω
	NB	NM	NS	ZO	PS	PM	PB
NB	PS PB	PS PB	PS PB	PS PB	PS PB	PS PB	PS PB
NM	PS PB	PM PM	PM PM	PM PM	PM PM	PM PM	PS PB
NS	PS PB	PM PM	PB PS	PB PS	PB PS	PM PM	PS PB
ZO	PS PB	PM PM	PB PS	ZO ZO	PB PS	PM PM	PS PB
PS	PS PB	PM PM	PB PS	PB PS	PB PS	PM PM	PS PB
PM	PS PB	PM PM	PM PM	PM PM	PM PM	PM PM	PS PB
PB	PS PB	PS PB	PS PB	PS PB	PS PB	PS PB	PS PB

As for the BL-IM, the predicted current can be fed back to the input for the next process of the prediction. It is also an anti-interference optimization for the inaccurate parameters of the motor and the external disturbances. After the optimization of the MPCC based on fuzzy control, the optimal objective function is obtained to achieve the best balance between the dynamic performance and the stability of the current component. The control method proposed in this paper improves the dynamic performance and suspension performance of the BL-IM. This will be verified in the simulation and experiment.

Simulation and experiment

Analysis of simulation results

Through calculation and solving ambiguity, the optimal objective function can be obtained. The model of the BL-IM is built in the Matlab/Simulink for the analysis of simulation. The parameters of the BL-IM are shown in Table 1. And the control period T is determined according to the trend of the waveform. In order to test the performance of the BL-IM under the MPCC based on fuzzy dynamic objective function, three groups of objects were set up: (1) fuzzy dynamic objective function; (2) traditional rolling optimization objective function; (3) vector control based on fuzzy PID.

It can be seen from Figure 8 that the BL-IM under MPCC based on the fuzzy dynamic objective function has better response speed and can reach the rated speed faster than the other two methods. Because the BL-IM under MPCC based on the fuzzy dynamic objective function can perform directly under the optimal objective function determined by fuzzy control. A load of 7N·M is added to the BL-IM at 0.3s, it can return to the rated speed faster. Therefore, the BL-IM under MPCC based on the fuzzy dynamic objective function has better anti-interference.

It can be seen from Figure 9 and Figure 10 that compared with other two control methods, the rotor of the BL-IM under MPCC based on the fuzzy dynamic objective function has smaller vibration. Also, the BL-IM under this control method has better suspension performance

In Figure 11, the BL-IM starts without load and the load is suddenly added at 0.3s. It can be shown in the figure that the BL-IM under MPCC based on the fuzzy dynamic objective function has higher dynamic response speed and better stability, which further illustrates the reliability of the control method proposed in this paper.

It can be seen from Figure 8 to Figure 11 that the use of the MPCC based on the fuzzy dynamic objective function can improve the dynamic performance and suspension performance of the BL-IM. In Figure 12, compared with the other two control methods, the BL-IM under MPCC based on the fuzzy dynamic objective function has a better control accuracy of the current component, and the peak fluctuation is reduced by 30%. The problem mentioned above that the dynamic performance of the motor and the control accuracy of the current component are mutually restricted is solved. The simulation results verify the feasibility and effectiveness of the control method proposed in this paper.

Analysis of experimental results

In order to further verify the feasibility of the control strategy proposed in this paper, the experiment is carried out on the 2-dof test prototype designed and manufactured by the research group. The experiment platform is shown in Figure 13. In the experiment, TMS320F2812 is used as the control chip, the rotor radial displacement is detected by eddy current sensor, and the speed of the motor is detected by photoelectric encoder. The inverter device used in this article is IPM PS21265 produced by Mitsubishi, which integrates six IGBT power switching devices. Its rated parameters are 20A/600V. And the turning on and off time of the IGBT is 1.25μs and 1.5μs. When a fault occurs, the F₀ terminal of the IPM will output a low-level pulse signal. The experimental parameters of the motor are the same as those of the simulation. In order to compare with the simulation research, the given speed is also set to 3000r/min. The control block diagram of the experiment is shown in Figure 14.

It can be seen from Figure 15 that the BL-IM under the traditional vector control based on fuzzy PID, traditional MPCC and MPCC based on fuzzy dynamic function can all reach the maximum speed of 3000r/min without overshoot. The BL-IM under traditional MPCC has better dynamic performance and stability than the BL-IM under the vector control based on fuzzy PID. The BL-IM under the MPCC based on fuzzy dynamic function further improves the stability while losing little dynamic performance.

Figure 16 is the experimental diagram of Phase A current when a load is added to the motor at the speed of 3000r/min. Figures 16(a), (b) and (c) are the comparison diagrams of Phase A current of the BL-IM under the traditional vector control based on fuzzy PID, the traditional MPCC and the MPCC based on fuzzy dynamic function. The results show that the Phase A current under the MPCC based on fuzzy dynamic function has the best stability and sinusoidal shape. The rotor current of the motor is positively correlated with the three-phase current of the motor. Therefore, the control method proposed in this paper increases the stability of the rotor current.

Figures 17(a), (b) and (c) are the rotor trajectory diagrams of the BL-IM under the traditional vector control based on fuzzy PID, the traditional MPCC and the MPCC based on fuzzy dynamic function. Since the air gap of the motor is set at 0.4mm, the maximum radial displacement of the rotor under the three control method is 25 microns, 20 microns and 10 microns. Therefore, the BL-IM under these three control methods can all achieve a stable suspension. By comparing these three figures, it can be seen that the radial displacement in Figure 17(c) is the smallest, so the BL-IM under the MPCC based on fuzzy dynamic function has the best suspension performance. The experimental results can verify the feasibility and effectiveness of the control method proposed in this paper.

Conclusion

In this paper, an optimization method based on the fuzzy function is applied to the objective function of the MPCC. Compared with the traditional control strategy, the advantages of the control method proposed can be summarized as follows:

The fuzzy dynamic objective function is applied to the MPCC, which improves the control precision of the current component compared with the traditional rolling optimization objective function.

Compared with the traditional vector control based on fuzzy PID, the MPCC is adopted to omit the coordinate transformation link, reduce the burden of DSP operation, and increase the control accuracy and start-up speed of the BL-IM.

The simulation and experimental results show that the BL-IM under the MPCC based on fuzzy dynamic objective function has better dynamic performance and suspension performance.

Footnotes

Appendix

Declaration of conflicting interests

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 under Projects 51475214 and 51875261, the Natural Science Foundation of Jiangsu Province of China under Projects BK20170071 and 20180046, the “333 project” of Jiangsu Province under Project BRA2017441, and the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD).

ORCID iD

Jiajie Wu

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