Voltage feedback based flux-weakening control of IPMSMs with fuzzy-PI controller

Abstract

As the traditional flux-weakening control strategies have some limitations, such as poor parameter robustness, non-ideal dynamic performance in the constant power region and so on, a new control strategy based on the formula-based feedforward control and the voltage feedback control is proposed. In this strategy, the given reference values of the flux-weakening currents are first calculated by the simplified formulas used in feedforward control, improving the processing speed of the controller by reducing the computational complexity. Then the reference value of the d axis current is compensated by the fuzzy-PI feedback controller, avoiding operating point drifts caused by the change or inaccurate of parameters in the actual. On this basis, a switching strategy is proposed to achieve a smooth switching between the constant torque and the constant power operation region. The experimental results show that the proposed control strategy has excellent performances, such as strong robustness to parameter perturbation, ideal dynamic performance in the flux-weakening region, smooth switching between the operation regions and so on.

Keywords

Interior Permanent Magnet Synchronous Motor (IPMSM)flux-weakening fuzzy-PI controller

1. Introduction

Interior permanent magnet synchronous motor (IPMSM) has been widely used in the electric vehicle, owing to its advantages of simple structure, high power density, high efficiency, and etc. [1–3]. Since the magnetomotive force of the rotor produced by the permanent magnet is fixed and constant, the voltage equation of the motor can only be balanced by adjusting the value of the stator current. In the IPMSM drive system, as the DC bus voltage of the voltage source inverter is always fixed, a stator current regulation method which named flux-weakening control strategy is adopted to realize the constant power operation of the motor under high speed, thus effectively expand the motor speed range [4–7]. Therefore, the research of the flux-weakening control strategy which’s performance is very important for the IPMSM drive system, has always been a hot issue in domestic and foreign scholars [8–10].

As the most common form of the flux-weakening control, the formula calculation method and the look-up table method are based on the mathematical model of the motor, that is the judgment of flux-weakening regions and the reference values of the flux-weakening currents totally relied on the formulas or the look-up tables [11–13]. The look-up table method is adopted in [11], in which the relationships among the stator currents, the electromagnetic torque and the flux are analysed to make them two-dimensional tables, and then the d and q axis reference value are obtained through real-time look-up table in the actual drive system. This method is obviously simple, and can effectively improve the dynamic performance of the motor in the constant power region. However, these two-dimensional tables are usually obtained by a series of experiments, and two-dimensional tables of the different motors are different, which leads to the need to do a lot of off-line experiments in the early stage when adopting this method. The formula calculation method is adopted in [12] and [13], in which the relationships between the stator currents, the electromagnetic torque and the flux are also analysed, well a formula in the constant torque region and the constant power region are derived respectively. And then the d and q axis current reference value are calculated by the formulas. Obviously, the flux-weakening running track is easy to plan and the dynamic response mostly closes to ideal, but the robustness to the motor parameters and operating conditions is inevitably poor in this method.

The flux-weakening control strategy based on a voltage feedback controller is also researched. In this kind of method, the judgment of the IPMSM operation region is always based on the voltage difference between the maximum output voltage of the inverter and the reference voltage outputted by the dq axis current controllers, and then the flux-weakening current is obtained by the voltage difference through a controller which named the voltage feedback controller [14–16]. As described in [14,15], whether the PMSM enters the constant power operation region is judged by the voltage difference. And as long as the motor enters that region, the reference value of the d axis current is amended through a PI regulator, making sure that the operating point of the motor is back to the range of voltage limit ellipse. Obviously, this method is not only simple and easy to implement, but also has good robustness to the motor parameters. However, the dynamic performance of this method is unsatisfactory. And the inner loop current regulator is easily saturated during the control process, causing the current to be temporarily out of control [16]. Thus the judgment of the IPMSM operation regions is determined by the angle between the constant torque curve and the voltage decline curve in [17]. Although the direction of flux-weakening can be adjusted in real time, the algorithm is relatively complex and sensitive.

A flux-weakening control strategy combing with the voltage feedback control method, the formula calculation method and the look-up table method is proposed in [18], and the judgment of the IPMSM operation regions is also determined by the voltage feedback instruction. However, this strategy can only be applied to the surface permanent magnet synchronous motors, and only the temperature and the stator phase resistance are taken into account when the influence of the motor parameters on the control algorithm is considered.

In order to improve the performance of the traditional flux-weakening control strategy always with poor parameter robustness and weak dynamic performance, a novel flux-weakening control strategy based on the formula calculation method and the voltage feedback control method with a fuzzy-PI controller is proposed in this paper. Thus the given reference value of the d axis current is firstly calculated by a simplified feed-forward control formula which is first proposed, and is secondly amended by the voltage feedback loop with a fuzzy-PI controller, in order to solve the problem of running point deviations generated by the changes of motor parameters in actual. Then a switching control strategy based on a weighted function is adopted to realize the smooth switching between the constant torque operation region and the constant power operation region.

2. The mathematical model of IPMSM

To simplify the analysis, make the following assumptions:

(1) Suppose that the induction electromotive force waveform is sine in the motor winding, and the rotor magnetic field is distributed as a standard sine wave in the air gap space; (2) Ignore the influence of stator ferromagnetic saturation and temperature change; (3) No eddy current and hysteresis losses. A mathematical model of d-q axis for PMSM is established. $\begin{eqnarray}\displaystyle & \displaystyle u_{d}=L_{d}\frac{\text{d}i_{d}}{\text{d}t}+R_{s}i_{d}-{\omega}_{r}L_{q}i_{q} & \displaystyle\end{eqnarray}$ (1) $\begin{eqnarray}\displaystyle & \displaystyle u_{q}=L_{q}\frac{\text{d}i_{q}}{\text{d}t}+R_{s}i_{q}+{\omega}_{r}L_{d}i_{d}+{\omega}_{r}{\psi}_{f} & \displaystyle\end{eqnarray}$ (2) $\begin{eqnarray}\displaystyle & \displaystyle T_{e}=\frac{3}{2}p[{\psi}_{f}i_{q}+(L_{d}-L_{q})i_{d}i_{q}] & \displaystyle\end{eqnarray}$ (3) $\begin{eqnarray}\displaystyle & \displaystyle T_{e}=T_{L}+B_{m}{\Omega}_{r}+J\frac{\text{d}{\Omega}_{r}}{\text{d}t} & \displaystyle\end{eqnarray}$ (4) Where u _d, u _q are dq-axis voltages, i _d, i _q are dq-axis currents, L _d, L _q are dq-axis inductances, 𝜓_f is the rotor magnetic flux, R _s is stator resistance, ω_r is the electrical angular velocity, 𝛺_r is the mechanical angular velocity, T _e is electromagnetic torque, B _m is the viscous friction coefficient, J is the moment of inertia of motor and load, p is the motor pole pair.

It is assumed that the voltage drop on the stator resistor is less than the back-EMF when the motor is operated in the constant power region. That is the voltage and current limit relation of the motor in the full range is obtained as follows, ignoring the differential term. $\begin{eqnarray}\displaystyle & \displaystyle u_{d}^{2}+u_{q}^{2}=u_{s}^{2}\leq u_{smax}^{2} & \displaystyle\end{eqnarray}$ (5) $\begin{eqnarray}\displaystyle & \displaystyle (L_{q}i_{q})^{2}+(L_{d}i_{d}+{\psi}_{f})^{2}\leq \left(\frac{u_{smax}}{{\omega}_{r}}\right)^{2} & \displaystyle\end{eqnarray}$ (6) $\begin{eqnarray}\displaystyle & \displaystyle i_{d}^{2}+i_{q}^{2}=I_{s}^{2}\leq I_{smax}^{2} & \displaystyle\end{eqnarray}$ (7) Where u _s is the vector amplitude of the stator voltage, u _smax is the maximum output voltage of the inverter, I _s is the vector amplitude of the stator current, I _smax is the maximum output current of the inverter.

The Voltage and current restriction relationships of the IPMSM in the whole region are shown in Fig. 1, in which T _e1 and T _e2 are the constant torque curves; the curve OA express the curve of MTPA (maximum torque per ampere); the ellipses are limit ellipse of the voltage at different electrical angular velocities, and the relationship among these electrical angular velocities are ω₁ < ω₂ < ω₃ < ω₄ < ω₅; the circle is limit circle of the current.

It is necessary to explain that the center of the voltage limit ellipse may be located inside or outside the current limit circle, which is depended on the motor parameters. In this paper, the voltage limit ellipse is centred on (−𝜓_f∕L _d, 0), which is outside the current limit circle, as shown in Fig. 1, due to the parameters of the experiment motor.

Fig. 1.

The voltage and current restriction relationships of IPMSM.

3. Traditional flux-weakening control strategy

3.1. The formula calculation method

The block diagram of the flux-weakening control of the IPMSM based on the formula calculation method is shown in Fig. 2.

Fig. 2.

The diagram of the formula calculation method.

As shown in Fig. 2, the reference value of the d-axis current $i_{d}^{\ast }$ in the constant torque region and the constant power region are calculated by the corresponding formula. And the operating region is judged by the actual speed of the motor n. In order to make full use of the reluctance torque and improve the efficiency, the MTPA control strategy is usually used in the constant torque region. And in this strategy, the reference value of the d axis current is calculated by the traditional formula as follows. $\begin{eqnarray}\displaystyle i_{d}^{\ast }=\frac{{\psi}_{f}}{2(L_{q}-L_{d})}-\sqrt{\frac{{\psi}_{f}^{2}}{4(L_{q}-L_{d})^{2}}+i_{q}^{\ast 2}} & & \displaystyle\end{eqnarray}$ (8) Where $i_{q}^{\ast }$ is the reference value of the q axis current.

The flux-weakening control strategy must be used to make the motor run above the base speed, when the terminal voltage of the IPMSM reaches the maximum output voltage of the inverter. And the intersection of the constant torque curve and the voltage limit ellipsis is chosen as the operation point of the IPMSM in order to minimize the motor losses, as shown in Fig. 1, operating point D. That is the reference value of d axis current in constant power region can be expressed as $\begin{eqnarray}\displaystyle i_{d}^{\ast }=-\frac{{\psi}_{f}}{L_{d}}+\frac{1}{L_{d}}\sqrt{\left(\frac{u_{\text{smax}}}{{\omega}_{r}}\right)^{2}-(L_{q}i_{q}^{\ast })^{2}}. & & \displaystyle\end{eqnarray}$ (9)

Ideally, the IPMSM runs at operating point D according to this method, when its operating conditions are ω₄ and T _e2, as shown in Fig. 1. However, the motor parameters always change with the variation of the magnetic field saturation, environment temperature and other factors, resulting in the operation point not at D point any more, drift to E point or F point in actual. Thus the motor is difficult to run accurately on the optimal curve, and the performance and quality of the control system will be affected. Therefore, it is necessary to take into account the influence of motor parameters on control system in the flux-weakening control.

3.2. The voltage feedback method

The block diagram of the IPMSM based on voltage feedback method is shown in Fig. 3.

The MTPA control strategy is also used in the constant torque region in this method in order to make the motor copper loss minimum, as shown in Fig. 3. In this region, the voltage difference Δu between the maximum output voltage of the inverter u _smax and the reference value of the dq axis voltage $u_{s}^{\ast }=\sqrt{u_{d}^{\ast 2}+u_{q}^{\ast 2}}$ is positive, i.e. ${\rm\Delta}u=u_{\text{smax}}-u_{s}^{\ast }\geq 0$ . As a result, the output of the PI controller in the voltage feedback loop is clipped to zero, because its output maximum limit is set to zero. That is the voltage feedback loop does not work in the constant torque region. The voltage difference Δu is getting smaller and smaller with the increase of speed until turning from positive to negative. That is the motor enters the constant power region from the constant torque region. At the same time, the output of the PI controller, which is no longer zero, is used to adjust the reference value of the d-axis current, thereby achieving the flux-weakening control.

Fig. 3.

The diagram of the voltage feedback method.

In this method, the flux-weakening current is obtained by the PI controller of the voltage feedback loop, no longer need calculations by the formula. Thus it is easy to implement and not sensitive to motor parameters, and but its dynamic performance may be unsatisfactory due to the introduction of the voltage control loops.

Furthermore, the outputs of the dq axis current regulator always fluctuate in practice, causing the voltage difference Δu fluctuates near zero when the motor is running near the switching point. These will cause a false switching of the IPMSM operation region, and further affect the stability of the system, since the switching is triggered as long as the voltage difference Δu is negative.

4. The improved flux-weakening control strategy

Aiming at the problems existing in the traditional methods, an improved flux-weakening control strategy is proposed, and the block diagram is shown in Fig. 4. It can be seen that the improved control strategy consists of three parts, namely ‘Part I: simplified formula calculations’, ‘Part II: voltage feedback fuzzy-PI control’, and ‘Part III: operation region switching’.

Fig. 4.

The diagram of the improved flux-weakening control strategy.

Firstly, the reference value of the d-axis current $i_{d}^{\ast }$ in the MTPA control and the flux-weakening control are respectively calculated by the corresponding formulas, as shown in Part I. And the formulas are simplified to reduce the computation amount of the algorithm, so as to shorten the calculation time of the controller, conducing to a good dynamic performance. Secondly, the calculated $i_{d}^{\ast }$ is amended by the voltage feedback loop with a fuzzy-PI controller as shown in Part II, in order to reduce the influence of parameter mismatches. Finally, a switching principle between the constant torque region and the constant power region only based on the motor actual speed is adopted, as shown in Part III.

4.1. Part I: Simplified formula calculations

As shown in the formula ((8)) and formula ((9)), complex mathematical operations such as square and root operations, are contained in the formulas, resulting in a large amount of computation. And a controller with much higher operation speed is always required in actual. Thus these formulas in the feedforward current control have to be simplified to reduce the requirement for the controller. Taking the MTPA formula i.e. formula ((8)) as an example, the Taylor series is used to expand it as follows. $\begin{eqnarray}\displaystyle i_{d}^{\ast }=-\frac{(L_{q}-L_{d})}{{\psi}_{f}}i_{q}^{\ast 2}+\frac{(L_{q}-L_{d})^{3}}{{\psi}_{f}^{3}}i_{q}^{\ast 4}+\cdots. & & \displaystyle\end{eqnarray}$ (10)

As the coefficients decrease with the increase of the order of the current i _q, the fourth and higher order terms which are much smaller than the first term, can be ignored. Therefore, the MTPA formula can be simplified as $\begin{eqnarray}\displaystyle i_{d}^{\ast }=-\frac{(L_{q}-L_{d})}{{\psi}_{f}}i_{q}^{\ast 2}. & & \displaystyle\end{eqnarray}$ (11)

As the same, the Flux-weakening formula i.e. formula (9) can be simplified as follows, in which the fourth and higher order terms are ignored. $\begin{eqnarray}\displaystyle i_{d}^{\ast }=-\frac{{\psi}_{f}}{L_{d}}+\frac{u_{\text{smax}}}{L_{d}{\omega}_{r}}-\frac{L_{q}^{2}{\omega}_{r}}{2L_{d}u_{\text{smax}}}i_{q}^{\ast 2} & & \displaystyle\end{eqnarray}$ (12)

4.2. Part II: Voltage feedback fuzzy-PI control

The fuzzy-PI controller is composed of a PI controller and a fuzzy controller, as shown in Fig. 4, in which the parameters of the PI controller can be self-adjusted by the fuzzy controller, improving the anti-interference performance of the control system. Thus the input variables of the fuzzy controller are the difference voltage Δu and its change rate Δu ^′, as well as the output variables are the adjustment value of the PI controller parameters Δk _p and Δk _i. In this paper, the fuzzy set of input and output variables are all $\{\text{NB},\text{NM},\text{NS},\text{ZO},\text{PS},\text{PM},\text{PB}\}$ , representing negative large, negative medium, negative small, unchanged, positive small, positive medium, positive large, respectively. And the fuzzy domain of input and output variables are all $\{-3,-2,-1,0,1,2,3\}$ . So the membership functions of input and output variables are shown in Fig. 5.

Fig. 5.

Membership functions of input and output variables.

Table 1

Fuzzy control-rule table of Δkp

	Δu ^′
Δu	NB	NM	NS	ZO	PS	PM	PB
NB	PB	PB	PM	PM	PS	ZO	ZO
NM	PB	PM	PM	PS	PS	ZO	NS
NS	PM	PM	PM	PS	ZO	NS	NS
ZO	PM	PS	PS	ZO	NS	NM	NM
PS	PS	PS	ZO	NS	NS	NM	NM
PM	PS	ZO	NS	NM	NM	NM	NB
PB	ZO	ZO	NM	NM	NM	NB	NB

The fuzzy control-rules of Δk _p and Δk _i are shown in Table 1 and Table 2, respectively. By these fuzzy control-rule tables and the Mamdani reasoning algorithm, the fuzzy output is obtained, and the fuzzy quantity is obtained through the solution, so as to realize the dynamic adjustment of PI parameters.

In this paper, the center of gravity method is used to ambiguity resolution, shown as follows. $\begin{eqnarray}\displaystyle g=\frac{\displaystyle \mathop{\sum }_{j=1}^{7}g_{j}A(g_{j})}{\displaystyle \mathop{\sum }_{j=1}^{7}A(g_{j})} & & \displaystyle\end{eqnarray}$ (13) Where g is obtained after solving fuzzy definiton Δk _p and Δk _i, g _j is the weight of the elements in each group, A (g _j) is the membership department of g _j.

Table 2

Fuzzy control-rule table of Δki

	Δu ^′
Δu	NB	NM	NS	ZO	PS	PM	PB
NB	NB	NB	NM	NM	NS	ZO	ZO
NM	NB	NB	NM	NS	NS	ZO	ZO
NS	NB	NM	NS	NS	ZO	PS	PS
ZO	NM	NM	NS	ZO	PS	PM	PM
PS	NM	NS	ZO	PS	PS	PM	PB
PM	ZO	ZO	PS	PS	PM	PB	PB
PB	ZO	ZO	PS	PM	PM	PB	PB

4.3. Part III: Operation region switching

For the IPMSM, the reference value of the d-axis current $i_{d}^{\ast }$ in the constant torque region and the constant power region are different, which makes the switching rules should be carefully designed to achieve a smoothly switch between two regions, as shown in Fig. 4. Therefore, the actual speed is chosen as the judgement condition of the operation region switching. And a smooth switching rule is designed based on the weighted function to avoid frequent switching between the two regions mentioned above caused by some factors, such as speed fluctuation.

That is the reference value of the d axis current is given as follow. $\begin{eqnarray}\displaystyle & \displaystyle i_{d}^{\ast }=ki_{d1}^{\ast }+(1-k)i_{d3}^{\ast } & \displaystyle\end{eqnarray}$ (14) $\begin{eqnarray}\displaystyle & \displaystyle \left\{\begin{array}{@{}ll@{}}k=1 & n<n_{1}\\ 0<k<1 & n_{1}<n<n_{2}\\ k=0 & n_{2}<n\end{array}\right. & \displaystyle\end{eqnarray}$ (15) Where n is the motor speed, [n ₁, n ₂] is the switching interval, k and (1 − k) are the coefficients of the reference values of the d axis current in the constant torque region and the constant power region. When the motor runs in the constant torque region, i.e. n < n ₁, then k = 1, that is the MTPA control strategy is adopted. When the motor runs in the constant power region, i.e. n > n ₂, then k = 0, that is the flux-weakening control strategy is adopted. When the motor runs near the switching point, i.e. n ₁ < n < n ₂, then 0 < k < 1, that is the switching rule works to achieve a smooth switch.

5. Experimental results

A test-rig has been set up to verify the proposed control strategy in this paper, as shown in Fig. 6. The target machine is an IPMSM which is designed for electric vehicle applications and its parameters are listed in Table 3.

Fig. 6.

Physical diagram of experimental system.

Table 3

Parameters of the IPMSM

Quantity	Value	Unit
DC side voltage U _DC	320	V
Rated current I _N	94	A
Rated torque T _N	64	N ⋅ m
Rated speed n _N	3000	r/min
Pole pairs p	4	–
Flux linkage 𝜓_f	0.0727	Wb
d axis inductance L _d	0.2	mH
q axis inductance L _q	0.555	mH
Stator resistance R _s	11.4	mΩ

5.1. Parameter robustness test

The motor parameters used in the control system are carefully designed in the tests, in order to simulate the motor parameters mismatches in actual operation, that is, the d, q-axis inductance and the flux linkage produced by permanent magnet are decreased or increased by 30%, respectively. The motor starts at no-load and finally runs at 6000 r/min, which is two times the rated speed. The results of traditional formula calculation strategy and the proposed strategy are shown in Fig. 7–Fig. 10.

Fig. 7.

The waveform of the voltage difference Δu when the d-axis and q-axis inductance are decreased by 30%. (a) The traditional strategy; (b) The proposed strategy.

Fig. 8.

The waveform of the voltage difference Δu when the d-axis and q-axis inductance are increased by 30%. (a) The traditional strategy; (b) The proposed strategy.

Fig. 9.

The waveform of the voltage difference Δu when the flux linkage produced by permanent magnet is decreased by 30%. (a) The traditional strategy; (b) The proposed strategy.

Fig. 10.

The waveform of the voltage difference Δu when the flux linkage produced by permanent magnet is increased by 30%. (a) The traditional strategy; (b) The proposed strategy.

As shown in Fig. 7–Fig. 10, the voltage difference Δu is no longer zero under the traditional strategy, but it is still adjusted to zero under the proposed strategy, no matter which parameter has changed, and no matter the parameter becomes larger or smaller.

The results show that the actual operating point of the motor is departed from the voltage limit ellipse when the parameter changes, resulting in a poor flux-weakening operation performance of the system. And the parameter robustness of the system is effectively improved by the proposed strategy, in which the corrected value of the d-axis current obtained by the fuzzy PI feedback link is used to offset the deviation of operation point caused by the change of parameters.

5.2. Response performance test

A slope function is selected as the reference speed in the test, in order to verify the speed response performance of the proposed strategy. That is the motor starts at no-load and accelerates to 6000 r/min in 4 s. And the speed response waveform and the current response waveforms are shown in Fig. 11.

Fig. 11.

The speed response and the current response of the control system. (a) The speed response; (b) The d, q-axis current response.

As shown in Fig. 11, the motor speed follows the change of reference speed quickly and accurately, which shows that the proposed strategy has good dynamic and steady speed response characteristics.

The load torque response performance of the proposed strategy in the constant power region is also tested. That is the motor is running at a steady state of 6000 r/min, and the load torque is increased from 13 N ⋅ m to 38 N ⋅ m at 2.5 s, as well as decreased from 38 N ⋅ m to 18 N ⋅ m at 7 s. And the torque response waveform and the current response waveforms for sudden increase and sudden reduction of a step load are shown in Fig. 12.

Fig. 12.

The torque response and the current response of the control system. (a) The torque response; (b) The d, q-axis current response.

As shown in Fig. 12, the d and q axis current controller are not saturated during the whole step dynamic process of the torque. The d and q axis current follows the reference value well as well as the torque regulation is fast and achieves the stable value quickly.

The results show that the proposed strategy can deal with the sudden change of the load torque, effectively avoid the temporary out-of-control of the inner loop current controllers, and has good dynamic and steady-state performance.

5.3. Switching method test

In addition, the proposed switching method is also tested. And the same slope function is selected as the reference speed in the test, etc. the motor accelerates to 6000 r/min in 4 s. The speed switching interval is set to [4350 r/min, 4650 r/min], and the response waveforms with the proposed switching method are shown in Fig. 13.

Fig. 13.

Response waveforms of the control system during the switching region. (a) The speed response; (b) The torque response; (c) The d, q-axis current response.

As can be seen from Fig. 13, the d and q axis current is continuous at the switching point when the proposed switching method is adopted. That is a smooth switching can be achieved from the constant torque region to the constant power region by the proposed switching method.

6. Conclusion

As the traditional flux-weakening control strategy has the problem of poor parameter robustness and low dynamic performance of the constant power region. A new control strategy based on feedforward control and fuzzy-PI feedback control strategy is presented in this paper. Firstly, the flux-weakening calculation formula is simplified, thus the processing speed of the controller is improved in practice. Secondly, the d-axis current corrected value obtained from a fuzzy-PI feedback controller is used to update d axis current reference value in real time, effectively eliminating the effect of parameter changes on the control system during the flux-weakening region. Thirdly, a smooth switching strategy is proposed to realize a smooth switching between the constant torque region and the constant power region. The correctness and validity of the proposed strategy are verified by the experimental results.

Footnotes

Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grant 51507111.

References

Chen

and Ma

, A harmonic current injection method for electromagnetic torque ripple suppression in permanent-magnet synchronous machines, International Journal of Applied Electromagnetics and Mechanics 53(2) (2017), 327–336.

Liu

Zhu

Pride

Deodhar

and Sasaki

, Efficiency improvement of switched flux PM memory machine over interior PM machine for EV/HEV applications, IEEE Transactions on Magnetics 50(11) (2014), 8202104.

Chen

Deng

and Deng

, An analytical model of unbalanced magnetic pull for PMSM used in electric vehicle: Numerical and experimental validation, International Journal of Applied Electromagnetics and Mechanics 54(4) (2017), 583–596.

Jiang

and Mu

, Novel composite sliding mode control for PMSM drive system based on disturbance observer, IEEE Transactions on Applied Superconductivity 26(7) (2016), 0612905.

Miyajima

Fujimoto

and Fujitsuna

, A precise model-based design of voltage phase controller for IPMSM, IEEE Transactions on Power Electronics 28(12) (2013), 5655–5664.

Schoonhoven

and Uddin

, MTPA- and FW-based robust nonlinear speed control of IPMSM drive using Lyapunov stability criterion, IEEE Transactions on Industry Applications 52(5) (2016), 4365–4374.

Guo

Xia

Wang

and Shi

, Improving rotor geometry to strengthen anti-demagnetization ability of PM, International Journal of Applied Electromagnetics and Mechanics 56(2) (2018), 263–274.

Rahman

Osheiba

Kurihara

Jabbar

Ping

Wang

and Zubayer

, Advances on single-phase line-start high efficiency interior permanent magnet motors, IEEE Transactions on Industrial Electronics 59(3) (2012), 1333–1345.

Pellegrino

Armando

and Guglielmi

, Direct flux vector control of IPM motor drives in the maximum torque per voltage speed range, IEEE Transactions on Industrial Electronics 59(10) (2012), 3780–3788.

10.

Wang

Xia

Yan

and Shi

, Current harmonic suppression in the flux-weakening control of surface permanent magnet synchronous motors using resonant controllers, Transactions of China Electrotechnical Society 29(9) (2014), 83–91.

11.

Lenke

De Doncker

Kwak

Kwon

and Sul

, Field weakening control of interior permanent magnet machine using improved current interpolation technique, in: 37th IEEE Power Electronics Specialists Conference (PESC), 2006, pp. 1–5.

12.

Pan

and Sue

, A linear maximum torque per ampere control for IPMSM drives over full-speed range, IEEE Transactions on Industry Applications 20(2) (2005), 359–366.

13.

Bai

Tang

and Wu

, Speed control of flux weakening on interior permanent magnet synchronous motors, Transactions of China Electrotechnical Society 26(9) (2011), 54–59.

14.

Zhou

Chen

Liu

and Jiang

, Flux-weakening strategy for improving PMSM dynamic performance, Electric Machines and Control 18(9) (2014), 23–29.

15.

Stojan

Drevensek

Plantic

Grcar

and Stumberger

, Novel field-weakening control scheme for permanent-magnet synchronous machines based on voltage angle control, IEEE Transactions on Industry Applications 48(6) (2013), 2390–2401.

16.

Zhu

Wen

Zhao

and Kong

, Control policies to prevent PMSMs from losing control under field-weakening operation, Proceedings of the Chinese Society of Electrical Engineering 31(18) (2011), 67–72.

17.

Liu

and Zhang

, Optimized flux weakening control of IPMSM based on gradient descent method with single current regulator, Transactions of China Electrotechnical Society 31(15) (2016), 8–15.

18.

Tursini

Chiricozzi

and Petrella

, Feedforward flux-weakening control of surface-mounted permanent magnet synchronous motors accounting for resistive voltage drop, IEEE Transactions on Industrial Electronics 57(1) (2010), 440–448.