The inaccurate model parameters in the predictive current control of surfacemounted permanent magnet synchronous motor (SMPMSM) affect the current dynamic response and steadystate error. This paper presents a model parameter correction algorithm based on the relationship between the errors of model parameters and the static errors of
dq
axis current. In this correction algorithm, the errors of inductance and flux are corrected in two steps. Resistance is ignored. First, the proportional relations between inductance and
d
axis static current errors are utilized to correct the error of model inductance. Second, the flux is corrected by utilizing the proportional relations between flux and
q
axis static current errors under the condition that inductance is corrected. An experimental study with a 100 W SMPMSM is performed to validate the proposed algorithm.
I. INTRODUCTION
Permanent magnet synchronous motors (PMSMs) are widely utilized in servo systems because of their high efficiency, high power density, and high torque current ratio
[1]
,
[2]
. A highperformance PMSM servo system requires a fastresponse current inner loop to ensure the high performance of speed and position loops. The traditional control methods of current loop include current hysteresis control and proportionalintegral (PI) control
[3]
,
[4]
. Current hysteresis control has the problems of variable switching frequency and large steady ripple. PI control is usually accompanied by overshoot because it requires tradeoffs between dynamic and steadystate performances. Both of them hardly meet the highperformance control requirements. With the development of highspeed digital signal processing technology, predictive current control, which requires complex computation, has been the focus of investigation in the high dynamic control of PMSM
[5]

[7]
.
By using a motor model, predictive current control can predict the future current behavior to select a proper voltage vector, under which the current can follow the reference current in an optimal trajectory
[8]

[12]
. Predictive current controls can be divided into (at least) three classes, namely, direct predictive control (DPC), twoconfiguration predictive control (2PC), and pulsewidth modulation (PWM) predictive control (PPC)
[13]
. DPC selects a voltage vector that minimizes a cost function and directly applies it to the inverter. Large current and torque ripples exist in DPC because the selected voltage vector is applied to the inverter in the entire sampling interval. A one–zero voltage vector is introduced in 2PC to overcome this major drawback of DPC. An active voltage vector and a zero voltage vector are applied to the inverter in a sampling interval to reduce the current and torque ripples. PPC calculates the ideal voltage vector, which is modulated through space vector PWM, and then applies it to the inverter. Two active voltage vectors and a zero voltage vector are applied to the inverter in one sampling interval to eliminate the current and torque errors. The control block diagrams of PPC and conventional PI control have the same structure; thus, the PPC controller can replace the PI controller. Therefore, PPC is investigated in this study.
Inaccurate model parameters affect the current dynamic response and steadystate error in predictive current control
[14]

[21]
. Many scholars have conducted extensive research to solve the problem of inaccurate parameters. In
[14]
, a constraintrelaxing deadbeat predictive control strategy was proposed, in which the current offset constraint and the output voltage prediction method were modified to enhance the system stability under the condition of inductance mismatch. A weighting factor was introduced in
[15]
to improve the system robustness. In
[16]
, two adjacent sampling interval prediction models were subtracted to eliminate constant items to achieve closeloop control, which can avoid the steadystate error and eliminate the influence of flux. In
[2]
, the predictive control method was improved by paralleling with an integrator to control
q
axis current to eliminate the static torque current error caused by the flux error. All these methods can improve the performance of predictive current control with inaccurate parameters but cannot eliminate the model parameter error. In
[17]
, the static current error was eliminated by introducing error integration in
d
axis current control, and the model flux was dynamically adjusted according to the
q
axis current error. However, the authors did not consider the influence of the inductance error on dynamic current performance. The model reference adaptive system (MRAS) was used in
[18]
and
[19]
to identify the model parameters of the motor online to eliminate the influence of the parameter error. However, MRAS requires an additional adjustable model, which increased the complexity of the system. In
[20]
and
[21]
, the function between inductance and flux with
i_{d}
and
i_{q}
was established and utilized for predictive current control to eliminate the problem of inaccurate parameters. However, inductancecurrent and fluxcurrent nonlinear maps were required. The maps were measured through repeated experimental procedures or with the help of software, which was not easy to obtain.
In this study, a model parameter correction algorithm for predictive current control of surfacemounted PMSM (SMPMSM) is proposed. The algorithm uses the proportional relation between inductance and
d
axis static errors to correct the inductance error. The proportional relation between flux and
q
axis static errors is utilized to correct the flux error under the condition that inductance is corrected. Inductance and flux converge to an actual value through the correction algorithm proposed in this study, and the problems of inaccurate inductance and flux are solved.
II. PREDICTIVE CURRENT CONTROL METHOD FOR SMPMSM
The stator voltage and statespace equations of the SMPMSM in the
d

q
rotor reference frame are provided by Equs. (1) and (2), respectively.
where
u_{d}
,
u_{q}
and
i_{d}
,
i_{q}
are the
d

q
frame voltages and currents, respectively;
L
is stator inductance;
R
is stator resistance;
Ψ_{f}
is the flux established by the permanent magnet;
ω_{e}
is the electrical angular velocity of the rotor.
Control period
T
in a servo system is small. Consequently,
ω_{e}
is considered constant during each sampling period
T
. Based on the forward Euler approximation method, Equs. (1) and (2) can be discretized into Equs. (3) and (4), respectively.
The PWM predictive current controller is built based on the voltage equation in Equ. (3). We suppose that
i_{dr}
and
i_{qr}
are the reference currents of
d
axis and
q
axis, respectively. The goal of predictive current control is for the actual currents to follow the reference currents after one modulation period. Therefore, we suppose that
i_{d}
[
k
+1] =
i_{dr}
and
i_{q}
[
k
+1] =
i_{qr}
and apply them to Equ. (3). Accordingly, we obtain the following equation.
The
u_{d}
[
k
] and
u_{q}
[
k
] calculated by Equ. (5) are the required voltage vectors that allow the current vectors to reach the reference currents after one modulation period.
Fig. 1
shows a block diagram of PWM predictive current control.
Block diagram of PWM predictive current control.
III. PARAMETER SENSITIVITY ANALYSIS
Predictive current control is based on a motor model to calculate the desired voltage vectors. Inaccurate model parameters force the voltage vectors to deviate from the expected ones and thus result in poor control performance.
In this study, we suppose that the actual motor parameters are
R
_{0}
,
L
_{0}
, and
Ψ_{f}
_{0}
, and the predictive model parameters are
R
,
L
, and
Ψ_{f}
. During one control period, the voltage vectors calculated by Equ. (5) are applied to the actual motor, and the actual motor current response can be presented by Equ. (4). The following equations can be obtained by applying Equ. (5) to Equ. (4).
where Δ
R
=
R
−
R
_{0}
, Δ
L
=
L
−
L
_{0}
, and Δ
Ψ_{f}
=
Ψ_{f}
−
Ψ_{f}
_{0}
are the errors between model and actual motor parameters.
In a practical system, the order of magnitude of
T
is generally 10
^{−4}
,
R
is 10
^{−1}
, and
L
is from 10
^{−3}
to 10
^{−2}
. Considering that
T
Δ
R
is much smaller than Δ
L
,
T
Δ
R
is ignored, and Equ. (6) can be simplified as
Considering that
i_{d}
[
k
+1] =
i_{d}
[
k
] at steadystate operation,
i_{d}
[
k
+1] =
i_{d}
[
k
] is applied to Equ. (7), and the static error of
d
axis current response can be obtained.
where Δ
i_{d}
=
i_{d}
[
k
+1] 
i_{dr}
.
Equ. (7) shows that
i_{q}
[
k
+1] receives the dual effects of Δ
L
and Δ
Ψ_{f}
. The current response of
q
axis is analyzed under the condition of accurate inductance. Considering that
i_{q}
[
k
+1] =
i_{q}
[
k
] at steadystate operation, Δ
L
=0 and
i_{q}
[
k
+1] =
i_{q}
[
k
] are applied to Equ. (7). The static error of
q
axis current response can also be obtained.
where Δ
i_{q}
=
i_{q}
[
k
+1] 
i_{qr}
.
Equ. (8) shows that Δ
i_{d}
is proportional to Δ
L
and is not related to flux. Equ. (9) shows that Δ
i_{q}
is proportional to Δ
Ψ_{f}
under the condition of accurate inductance. The current responses of different inductance and flux combinations are shown in
Table I
. A small inductance value causes the static current error of
d
axis to be greater than zero. A large inductance value causes the static current error of
d
axis to be less than zero. Under the condition of accurate inductance, a small flux value causes the static current error of
q
axis to be less than zero. A large flux value causes the static current error of
q
axis to be greater than zero.
CURRENT RESPONSES OF DIFFERENT INDUCTANCE AND FLUX COMBINATIONS IN THEORY
CURRENT RESPONSES OF DIFFERENT INDUCTANCE AND FLUX COMBINATIONS IN THEORY
IV. MODEL PARAMETER CORRECTION ALGORITHM
Based on the analysis in the preceding section, we propose a model parameter correction algorithm for predictive current control.
 A. Inductance Correction Algorithm
Equ. (8) shows that the response error of
d
axis static current error Δ
i_{d}
is proportional to Δ
L
and is not related to flux. Δ
i_{d}
and Δ
L
have an opposite sign (
ω_{e}i_{q}
> 0 at steadystate operation). Equ. (10) is defined to correct the inductance error (if Δ
i_{d}
> 0, then Δ
L
< 0; the inductance should be increased and vice versa).
where
C_{L}
is the inductance constant increment,
K_{IL}
is the inductance incremental integral coefficient, and
K_{PL}
is the inductance incremental proportionality coefficient.
In Equ. (10), Equ. (1) is the constant incremental mode, Equ. (2) is the integral incremental mode, and Equ. (3) is the proportional plus integral incremental mode. The first mode is relatively simple, but the second or the third mode is faster by selecting proper proportional and integral coefficients. The mode can be selected according to the control requirement of performance in a practical application.
If and only if Δ
L
= 0, then Δ
i_{d}
= 0. Equ. (10) converges with Δ
L
= 0 and Δ
i_{d}
= 0, which means that inductance
L
equals actual inductance
L
_{0}
.
 B. Flux Correction Algorithm
Equ. (9) shows that the response error of
q
axis static current Δ
i_{q}
is proportional to Δ
Ψ_{f}
under the condition of accurate inductance, and only one flux variable exists. The signs of Δ
i_{q}
and Δ
Ψ_{f}
are identical (at steadystate operation). Equ. (11) is defined to correct the flux error (If Δ
i_{q}
> 0, then Δ
Ψ_{f}
> 0; the flux should be decreased and vice versa).
where
C_{Ψ}
is the flux constant increment,
K_{IΨ}
is the flux incremental integral coefficient, and
K_{PΨ}
is the flux incremental proportionality coefficient.
In Equ. (11), Equ. (1) is the constant incremental mode, Equ. (2) is the integral incremental mode, and Equ. (3) is the proportional plus integral incremental mode. The mode can be selected according to the control requirement of performance in a practical application.
If and only if Δ
Ψ_{f}
= 0, then Δ
i_{q}
= 0. Equ. (11) converges with Δ
Ψ_{f}
= 0 and Δ
i_{q}
= 0, which means that flux
Ψ_{f}
equals actual flux Δ
Ψ_{f}
_{0}
.
 C. Flow of the Model Parameter Correction Algorithm
The preceding analysis indicates that the flux is corrected under the condition of accurate inductance. Hence, the flow of the model parameter correction algorithm must correct the inductance first and then the flux. The flow diagram of the model parameter correction algorithm is shown in
Fig. 2
. At steadystate operation (constant speed), the inductance correction algorithm is enabled first. Then, the flux correction algorithm is enabled after inductance convergence. The entire algorithm is completed when the flux converges.
Flow diagram of the modelparametersetting algorithm.
The block diagram of the PWM predictive current control with the model parameter correction algorithm is shown in
Fig. 3
. A block called parameter correction algorithm is added in
Fig. 3
unlike the diagram in
Fig. 1
. Reference and feedback currents are utilized to correct inductance and flux errors.
Block diagram of the model parameter correction algorithm for PWM predictive current control.
V. EXPERIMENTAL RESULTS AND ANALYSIS
An experimental platform is established to verify the correctness of the parameter sensitivity analysis for predictive current control and the proposed model parameter correction algorithm. This platform utilizes a Xilinx Spartan6 fieldprogrammable gate array (FPGA) as the main control chip and two identical SMPMSMs to build a drag system. The specifications of the SMPMSMs utilized in this research are listed in
Table II
.
MOTOR SPECIFICATION
The sampling period of the current loop is set to 100 μs. The highspeed computing performance of FPGA makes the time delay only 7.4 μs from current sampling to PWM updating, accounting for 7.4% of the sampling period. Therefore, the instant duty cycle update strategy mentioned in
[4]
is used in this experiment.
The experimental data are sent to the upper computer through the communication module for monitoring and processing. A total of 1000 N (N channels within 0–100 ms) data are stored in the FGPA randomaccess memory and then read and sent to ensure that the data in each sampling period can be sent without loss.
 A. Experiment on Parameter Sensitivity
Experiments are conducted for parameter sensitivity analysis. No speed loop exists in these experiments, and
q
axis reference current
i_{qr}
is set from 0 A to 4 A at 10 ms moment and set from 4 A to 2 A at 20 ms moment. The waveforms during 8–25 ms and the amplified waveform near 10 and 20 ms are recorded. The experimental results of five different conditions (see
Table I
) are provided (
Figs. 4
–
8
).
Experimental result when L=L_{0} and Ψ_{f}=Ψ_{f0}.
Experiment result when L=0.5L_{0} and Ψ_{f}=Ψ_{f}_{0}.
Experimental result when L=1.5L_{0} and Ψ_{f}=Ψ_{f}_{0}.
Experimental result when L=L_{0} and Ψ_{f}=0.5Ψ_{f}_{0}.
Experimental result when L=L_{0} and Ψ_{f}=1.5Ψ_{f}_{0}.
Fig. 4
shows the experimental results when
L
=
L
_{0}
and
Ψ_{f}
=
Ψ_{f}
_{0}
. The current rise time from 0 A to 4 A is 3
T
( three control periods).
u_{q}
(perunit value) reaches the limiting value in the first 2
T
. The falling time from 4 A to 2 A is 1
T
only, during which
u_{q}
is not saturated. The static errors of
d
axis and
q
axis are zero. The predictive current control with accurate parameters thus has a good control performance.
Figs. 5
and
6
show the experimental results when
L
= 0.5
L
_{0}
and
Ψ_{f}
=
Ψ_{f}
_{0}
and when
L
= 1.5
L
_{0}
and
Ψ_{f}
=
Ψ_{f}
_{0}
, respectively.
Figs. 7
and
8
show the experimental results when
L
=
L
_{0}
and
Ψ_{f}
= 0.5
Ψ_{f}
_{0}
and when
L
=
L
_{0}
and
Ψ_{f}
= 1.5
Ψ_{f}
_{0}
, respectively. The experimental results in
Figs. 4
–
8
are concluded in
Table III
.
CURRENT RESPONSES OF DIFFERENT INDUCTANCE AND FLUX COMBINATIONS
CURRENT RESPONSES OF DIFFERENT INDUCTANCE AND FLUX COMBINATIONS
Table III
verifies that the
d
axis static current error is only related to the inductance error. The
d
axis static current error is greater than zero with small model inductance (
Fig. 5
) and less than zero with large model inductance (
Fig. 6
). Under the condition of accurate inductance, the
q
axis static current error is only related to the flux error. The error is greater than zero with large model flux (
Fig. 7
) and less than zero with small model flux (
Fig. 8
). The experimental results (
Table III
) are consistent with the analysis results (
Table I
), and the correctness of the analysis is verified.
 B. Experiment on the Model Parameter Correction Algorithm
The constant incremental mode for the inductance and flux correction algorithm is selected to simplify the system and save on FPGA logic resources. A steadystate operation with a reference speed of 1500 r/min and
q
axis current of 4 A by loading is implemented for the experiment. The experimental results of the model parameter correction algorithm are provided (
Figs. 9

12
).
Experimental result when L=0.5L_{0} and Ψ_{f}=Ψ_{f}_{0}.
Experimental result when L=1.5L_{0} and Ψ_{f}=Ψ_{f}_{0}.
Experimental result when L=L_{0} and Ψ_{f}=0.5Ψ_{f}_{0}.
Experimental result when L=L_{0} and Ψ_{f}=1.5Ψ_{f}_{0}.
Inductance correction is not related to flux. Therefore, only the experimental results when
L
=0.5
L
_{0}
and
Ψ_{f}
=
Ψ_{f0}
and when
L
=1.5
L
_{0}
and
Ψ_{f}
=
Ψ_{f0}
are provided.
Figs. 9
and
10
present the experimental results of the inductance correction algorithm when
L
= 0.5
L
_{0}
and
L
= 1.5
L
_{0}
, respectively. The inductance converges to the actual value with a convergence time of approximately 15 ms and a convergence error of approximately 5%. The convergence error is not related to the initial value, and the inductance converges with
L
=
L
_{0}
and Δ
i_{d}
= 0.
Figs. 11
and
12
show the experimental results of the flux correction algorithm when
Ψ_{f}
= 0.5
Ψ_{f}
_{0}
and
Ψ_{f}
= 1.5
Ψ_{f}
_{0}
after inductance convergence, respectively. The flux converges to the actual value with a convergence time of approximately 12 ms and a convergence error of approximately 1.2%. The convergence error is not related to the initial flux value, and the flux converges with
Ψ_{f}
=
Ψ_{f}
_{0}
and Δ
i_{q}
= 0.
The experimental results (
Figs. 9
–
12
) verify the correctness and feasibility of the proposed model parameter correction algorithm.
VI. CONCLUSION
Given its excellent control performance, predictive current control is attractive in the highperformance control of SMPMSM. Predictive current control is sensitive to the errors of model parameters. Based on the analysis of the relations between model parameter and current static errors, a model parameter correction algorithm was developed for predictive current control. Considering that the
d
axis static current error is only related to the inductance error, the algorithm corrects the inductance and converges with Δ
L
= 0 and Δ
i_{d}
= 0. Under the condition that the inductance converges, the
q
axis static current error is proportional to the flux error. By utilizing this proportional relation, the flux is corrected and converges with Δ
Ψ_{f}
= 0 and Δ
i_{q}
= 0. The experimental results show that inductance and flux converge to actual values with convergence times of approximately 15 and 12 ms and with convergence errors of approximately 5% and 1.2%, respectively. These results verify the correctness and feasibility of the proposed algorithm.
Acknowledgements
This work was supported by the Doctoral Program of Higher Education of China under Grant 20113108110008 and the National Natural Science Foundation of China under Grant 51507097.
BIO
Yonggui Li was born in Guangxi, China, in 1991. He received his B.S. degree in automation from Shanghai University, Shanghai, China, in 2014. He is presently a postgraduate student at Shanghai University, Shanghai, China. His current research interests include new energy vehicles, intelligent control theory, power electronics, and highperformance servo control systems.
Shuang Wang was born in Jilin, China, in 1977. He received his B.S., M.S., and Ph.D. degrees in electrical engineering from Harbin Institute of Technology, Harbin, China, in 2000, 2005, and 2009, respectively. Since 2010, he has been with the School of Mechatronic Engineering and Automation, Shanghai University, Shanghai, China, where he is presently working as an assistant professor. His current research interests include intelligent control theory and its application to new energy vehicles, power electronics, and servo control systems.
Hua Ji was born in Qingdao, China, in 1977. She received her M.S. degree in electrical engineering from Nanjing University of Aeronautics and Astronautics, Nanjing, China, in 2004. Since 2004, she has been with the Department of Electrical and Electronic Engineering, Shandong University of Technology, Zibo, China, where she is presently an associate professor. Her current research interests include the study of highperformance servo control systems.
Jian Shi was born in Henan, China, in 1982. He received his M.S. degree in electrical engineering from Fuzhou University, Fuzhou, China, in 2007 and his Ph.D. degree in electrical engineering from Harbin Institute of Technology, Harbin, China, in 2013. From 2007 to 2010, he worked for ThyssenKrupp Elevator Co., China, as a research engineer. Since 2014, he has been a postdoctoral researcher at Shanghai University, Shanghai, China. His current research interests include electric machines, power electronics, and control systems.
Surong Huang was born in Shanghai, China. He received his diploma degree from Shanghai Institute of Mechanics, Shanghai, China, in 1977. In 1977, he joined Shanghai Institute of Mechanics, where he was promoted to associate professor and then professor in 1993 and 2001, respectively. He was a visiting faculty member at the Department of Electrical and Computer Engineering, University of Wisconsin, Madison, WI, USA, from 1995 to 1996 and from 1998 to 2000. He is currently a professor and doctoral supervisor at the Department of Automation, Shanghai University, Shanghai, China. He is engaged in the research and development of new types of electrical machines and drive systems. His current research interests include design, control, modeling, and simulation of electrical machines and AC drives and vibration and noise analyses of electrical machines. He has published more than 100 papers on these topics.
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