A sensorless control method was recently investigated in the robot and automation industry. This method can solve problems related to the rise of manufacturing costs and system volume. In a vector control method, the rotor position estimated in the sensorless control method is generally used. This study is based on a conventional fullorder flux observer. The proposed fullorder flux observer estimates both currents and fluxes. Estimated d and qaxis currents and fluxes are used to estimate the rotor position. In selecting the gains, the proposed fullorder flux observer substitutes gain
k
for the speed information in the denominator of the gain for fast convergence. Therefore, accurate speed control in a lowspeed region can be obtained because gains do not influence the estimation of the rotor position. The stability of the proposed fullorder flux observer is confirmed through a rootlocus method, and the validity of the proposed observer is experimentally verified using a surface permanentmagnet synchronous motor.
I. INTRODUCTION
Permanentmagnet synchronous motors (PMSMs) have high efficiency because rotor winding is not required to generate magnetic flux. Industries have, therefore, recently begun to use PMSMs, which are operated through a vector control method. The vector control method requires speed information and rotor position for the independent control of torque and flux. Speed information and rotor position can be obtained from sensors, such as an encoder and a resolver. However, sensors have disadvantages because of noise, increased volume of the system, and cost from sensor malfunction
[1]
,
[3]
,
[13]
.
The present study uses fullorder flux observer among many sensorless control methods. Fullorder flux observer is a method to estimate both currents and fluxes. In a fullorder flux observer, errors between estimated currents and real currents are used as inputs, and estimated fluxes are used to calculate the rotor angle. Conventional fullorder flux observer has gains to estimate currents and fluxes of d and qaxis. However, gains to estimate fluxes include speed information in the denominator, which has the disadvantage of increasing gain values in a lowspeed region. Hence, the conventional fullorder flux observer includes a ripple because of the increased gain values, which generate a speed error value at low speed
[1]

[12]
.
Therefore, in this study, the proposed fullorder flux observer removes speed information from the denominator of the gain to estimate fluxes, and gain
k
for fast convergence characteristic is added instead. Gain
k
without speed information in the denominator does not influence a lowspeed region. The estimated fluxes of the d and qaxis are used to estimate the rotor angle, and the estimated speed obtained from the rotor angle does not have a speed error value in the lowspeed region.
The estimated speed is calculated through the speed estimation method through a proportionalintegral (PI) controller. A PI controller generally has a simple structure and is commonly used because of its short calculation time. The input of a PI controller uses the error value of the feedback rotor angle, which is changed from the integral value by the estimated speed. Therefore, the speed error value of the PI speed controller converges to values near zero. The output, which is an angular speed, is the sum of the values generated through proportion and integration
[10]
.
In the current study, the stability of the proposed fullorder flux observer with the improved gain is verified through a rootlocus method. The simulation of the rootlocus method is performed using a MATLAB tool. Furthermore, the estimation performance of the proposed algorithm over a wide range is experimentally confirmed.
II. FLUX MODELING OF SPMSM
The stator winding of a surface permanentmagnet synchronous motor (SPMSM) generates a rotating magnetic field. The rotor rotates synchronously with the magnetic field. The rotor of an SPMSM is also a permanent cylindrical magnet. Therefore, the effective air gap of SPMSM is constant, and the inductance values of the d and qaxis are equal. The daxis of the stationary coordinate system is in the pole direction of the permanent magnet, and the qaxis forms a right angle to the daxis. The d and qaxis of the synchronous coordinate system rotate along with the rotating magnetic field.
Fig. 1
shows the structure and equivalent circuit of SPMSM
[1]

[3]
.
Interior structure and equivalent circuit of the SPMSM.
III. FULLORDER FLUX OBSERVER
The d and qaxis flux values of the fullorder flux observer are directly affected by current
[1]

[3]
. Therefore, detected current values are only used in error information; estimated currents are used directly to estimate the d and qaxis fluxes.
However, the fullorder flux observer is only used for the detected currents to obtain error information; the observer uses estimated currents to estimate the d and qaxis fluxes. Therefore, the fullorder flux observer does not generate distortion by estimating currents and fluxes. It also has advantages in terms of load change and disturbances when estimating currents and fluxes
[1]

[6]
.
 A. FullOrder Flux Observer
The voltage equation of the stationary coordinate system of SPMSM is expressed as
The d and qaxis fluxes, namely,
λ_{ds}
and
λ_{qs}
respectively, are defined as
where
λ_{f}
is the permanent magnet flux, and the current and voltage use values of the synchronous reference frame
[1]
–
[3]
.
Hence, Eq. (3) shows the d and qaxis matrix structures of the fullorder flux observer from Eqs. (1) and (2):
where
h_{11}
,
h_{12}
,
h_{21}
, and
h_{22}
are the gains of the fullorder flux observer.
From the d and qaxis fluxes, the information on the rotor angle is expressed as
 B. Conventional Gain of the FullOrder Flux Observer
The gain of the fullorder flux observer is determined from the flux modeling of SPMSM. Hence, the system characteristic equation is defined as
The roots of the secondorder equation are determined by
α_{1}
+
jβ_{1}
and
α_{2}
+
jβ_{2}
, and are expressed as
The secondorder equation, which is Eq. (6), can be defined by dividing the real and imaginary numbers from Eqs. (7) and (8)
[1]

[6]
,
[8]
,
[9]
. Equation (7) is used to obtain the gain of the current, while Eq. (8) is used to obtain the gain of the flux:
Coefficients
β1
and
β2
are defined as 0 from Eq. (9) for a stable pole placement
[1]
–
[3]
. Hence,
h_{11}
,
h_{12}
,
h_{21}
, and
h_{22}
are defined as
Thus, Eq. (13) includes the speed information from the denominator.
When the speed information is included in the denominator, the gain increases because of the low value of the speed and becomes an important factor in the deterioration of estimation performance
[1]

[3]
. Consequently, at low speed, the estimated currents and fluxes containing ripples decrease the estimation performance. The estimated rotor angle is the distorted swing to the decrease in estimation performance. Hence, the rotor cannot perform precise speed control at low speed.
 C. Proposed Gain of the FullOrder Flux Observer
To solve the problem at low speed in a conventional fullorder flux observer, the speed information of the denominator is eliminated and gain
k
is added for fast offset convergence. The proposed gain is not affected by the estimation performance at low speed. Precise sensorless speed control becomes feasible by obtaining accurate position information on the rotor from the estimated flux, which does not contain ripples. The gain variables
h_{11}
,
h_{12}
,
h_{21}
, and
h_{22}
of the fullorder flux observer are defined as
respectively, where the coefficients
β_{1}
and
β_{2}
are 0 for stable pole placement. Equation (17) eliminates the speed information in the denominator
[1]
,
[3]
,
[8]
,
[9]
.
Hence, the gain value does not increase owing to the absence of the speed information when gain
k
is added for fast offset convergence and stable pole placement.
 D. Stability of the Proposed Observer
Equation (18) is used to obtain the gain of the estimated current, while Eq. (19) is used to obtain the gain of the estimated flux.
where
h_{11}
and
h_{21}
are the real parts, and
h_{21}
and
h_{22}
are the imaginary parts. Furthermore,
α_{1}
and
α_{2}
are the values of −75 and −1,400.
In accordance with Eq. (20), the statespace equation is redefined as
[
sI

A
]
^{1}
is defined as
Hence, the characteristic equation using Eq. (22) is defined as
Gains are added
k
for quick error convergence of the speed.
Fig. 2
shows the pole placement depending on the value of
k
. The stability of the pole placement can be checked according to the pole direction. If the pole is positive, the system is unstable, whereas the system is stable if the pole is negative. A smaller negative value of the pole means a more stable system
[1]

[3]
,
[8]
,
[9]
.
Rootlocus stability criterion method by changed gain k.: 10–300 rpm.
he result in
Fig. 2
(a) shows the pole placement when
k
= 0.001. In
Fig. 2
(a), the pole placement is composed narrowly by the speed change. The area of the narrowly composed pole placement means an almost unstable system in the lowspeed range. The result in
Fig. 2
(b) shows the pole placement when
k
= 0.01. In
Fig. 2
(b), the pole placement is composed widely by speed change. The area of the widely composed pole placement means a stable system in the lowspeed and highspeed ranges. The result in
Fig. 2
(c) shows the pole placement when
k
= 0.1. The pole placement is composed near the value of 0 by the speed change from
Fig. 2
(c). The narrowly composed pole placement area means an almost unstable system. The result in
Fig. 2
(d) shows the pole placement when
k
= 1. From
Fig. 2
(d), the pole placement is composed narrowly by the speed change. Similarly, the narrowly composed pole placement area means an almost unstable system. The result in
Fig. 2
(e) shows the pole placement when
k
= −0.01. The pole placement is an unstable system because of the positive value. An appropriate value of gain
k
should be used depending on the system. Hence, a value of 0.01 is used for the proposed gain
k
.
 E. Speed Estimation
In general, a PI controller is used for speed estimation
[1]

[3]
,
[10]
. A PI controller has a simple structure and can control speed through the control gain.
A PI controller uses the estimated rotor angle to calculate the estimated angular speed. As the estimated angular speed is integrated into the rotor angle through the integral term, the error value between the estimated rotor angle and the integrated rotor angle is used as an input.
Fig. 3
shows the block diagram of the PI controller, and
Fig. 4
shows the overall block diagram of the proposed fullorder flux observer.
Block diagram of the PI controller.
Overall block diagram of the proposed fullorder flux observer.
In
Fig. 3
, the P term generates the estimated speed, which is proportional to the error value of the rotor angle, and the P gain determines the response rise and delay times. The I term generates the estimated speed proportional to the accumulated error value, and the I gain reduces the steadystate error. However, because the high control gain of the PI controller vibrates according to the speed, selecting the control gain value is difficult, and the ripple of the estimated speed can increase. If P and I are selected appropriately, the error of the rotor angle will converge to zero, and the output will be the estimated speed
[1]

[3]
,
[10]
. The rotor angle is estimated by the estimated fluxes of the d and qaxis via the proposed fullorder flux observer. Hence, accurate speed control is required for the accurate d and qaxis estimated fluxes.
IV. EXPERIMENTAL RESULTS
Table I
shows the gains
α_{1}
,
α_{2}
, and
k
proposed in this study, while
Table II
shows the SPMSM parameters used in the experiment. The control period was 100 μs, the switching frequency was 10 kHz, and the dclink voltage was 550 V. The experiment in this study was performed using the SPMSM parameters in
Tables I
and
II
.
VALUES OF THE PROPOSED GAIN
VALUES OF THE PROPOSED GAIN
PARAMETERS OF THE SPMSM
Fig. 6
shows the actual rotor angle, estimated rotor angle and estimated d and qaxis fluxes at 10 rpm. From the experimental results shown in
Fig. 6
, an improved performance is confirmed.
Fig. 7
is the estimated performance of the proposed fullorder flux observer. The conventional fullorder flux observer, which generates distortion in the rotor angle, included ripples in the estimated values because the estimated d and qaxis fluxes were formed by
h_{11}
and
h_{22}
. The conventional gain of the fullorder flux observer described in Section III included the angular speed in the denominator of the gain. Therefore, the gain at low speed could be greatly increased, which generated distortion in the estimated rotor angle.
Experimental setup. (a) SPMSM. (b) Control board and threelevel inverter.
Conventional fullorder flux observer: 10 rpm.
Proposed fullorder flux observer: 10 rpm.
The experimental conditions of
Fig. 7
were equivalent to those of
Fig. 6
. The proposed gain did not include speed information. Therefore, the system estimated the d and qaxis fluxes accurately. The estimated rotor angle also reflected the performance of the rotor angle estimation without the distortion by the d and qaxis fluxes.
Fig. 8
shows the actual rotor angle, estimated rotor angle, and estimated speed at 10 rpm, illustrating the estimated performances of the conventional fullorder flux observer and the proposed fullorder flux observer. The estimated speed of the conventional fullorder flux observer included ripples from the gain. Furthermore, the estimated speed of the proposed fullorder flux observer did not include the ripples from the gain because speed information was not included in the denominator.
Performance comparison of fullorder flux observers.
Fig. 9
shows the actual rotor angle, estimated rotor angle, actual speed, and Aphase current at 10 rpm under a fullload condition. The current was increased by increasing the load and creating the d and qaxis fluxes. Therefore, the fluxes could be estimated more accurately by the large currents, and the proposed fullorder flux observer shows accurate estimated performance with a full load.
Proposed fullorder flux observer at 10 rpm with full load.
Figs. 10
and
11
show the characteristics when the speed of the conventional fullorder flux observer is changed. The speed changed from 10 rpm to 300 rpm.
Fig. 11
shows the change in speed from 300 rpm to 10 rpm.
Figs. 10
and
11
show the actual rotor angle, speed, and estimated speed. The enlarged waveforms in the lowspeed range are shown at the bottom of
Figs. 10
and
11
.
Speed variable of the conventional fullorder flux observer: 10–300 rpm.
Speed variable of the conventional fullorder flux observer: 300–10 rpm.
The bottom part of
Fig. 10
shows an enlarged waveform of 10 rpm. The estimated performance of the conventional fullorder flux observer includes a ripple of changed speed.
The bottom part of
Fig. 11
is similar to that of
Fig. 10
, and
Fig. 11
shows an enlarged waveform of 10 rpm in the changed speed range from 300 rpm to 10 rpm. The conventional fullorder flux observer showed the estimated performance over a wide range of speed regions, but a ripple was included in the lowspeed range.
Figs. 12
and
13
also show the characteristics when the speed of the proposed fullorder flux observer is changed. The speed changed from 10 rpm to 300 rpm in
Fig. 12
, whereas the speed changed from 300 rpm to 10 rpm in
Fig. 13
.
Figs. 12
and
13
show the actual rotor angle, speed, and estimated speed. The enlarged waveforms in the lowspeed range are shown at the bottom of
Figs. 12
and
13
.
Speed variable of the proposed fullorder flux observer: 10–300 rpm.
Speed variable of the proposed fullorder flux observer: 300–10 rpm.
The bottom part of
Fig. 12
shows an enlarged waveform at 10 rpm. The proposed fullorder flux observer indicated a stable estimated performance with changed speed.
In addition, the waveform of the estimated performance in the lowspeed region did not include a ripple. The bottom part of
Fig. 13
is similar to
Fig. 12
, which shows an enlarged waveform of 10 rpm in the changed speed range from 300 rpm to 10 rpm. The proposed fullorder flux observer showed a stable estimated performance over a wide range of speed regions.
The fullorder flux observer experienced a gain from the changed speed. Therefore, the gain from the changed speed indicated a changed value in
Figs. 14
and
15
.
Fig. 14
shows the changed value using the conventional gain. The experiment results of the conventional gain showed a ripple in the lowspeed region because the denominator of the conventional gain included speed information.
Dependence of conventional gain on speed.
Fig. 15
shows the changed value using the proposed gain. The experiment results of the proposed gain showed stable estimated speed in the lowspeed region. Furthermore, the experimental results of the proposed gain did not include a ripple because of the changed speed. Hence, the proposed gain was able to estimate over a wide range of low and high speeds.
Dependence of proposed gain on speed.
V. CONCLUSION
The conventional fullorder flux observer includes speed information in the gain denominator. Hence, the estimated performance is reduced in the lowspeed region. However, the proposed fullorder flux observer can achieve stable estimation performance for the d and qaxis fluxes in the lowspeed range. Therefore, the proposed fullorder flux observer uses accurate estimated flux values that estimate the exact rotor angle. In this study, gain
k
was added for fast convergence of the error value, and the speed information of the gain denominator was removed. In addition, stable performance of the proposed gain was verified by comparing the proposed and conventional gains.
Therefore, the performance of the accurate speed control of the proposed fullorder flux observer was experimentally confirmed.
Acknowledgements
This work was supported by 20134030200310 of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Korea government Ministry of Knowledge Economy.This research was also supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2013R1A1A2A10006090).
BIO
KyoungGu Lee received his B.S. degree in Mechatronic Engineering from Korea Polytechnic University, Siheung, Korea, in 2012. He is currently working toward his M.S. degree at Ajou University, Suwon, Korea. His research interests include electric machine drives and switched reluctance motor drives.
JuneSeok Lee received his B.S. and M.S. degrees in Electrical and Computer Engineering from Ajou University, Korea, in 2011 and 2013, respectively. He is currently working toward his Ph.D. degree at Ajou University, Korea. His research interests include gridconnected systems, multilevel inverter, and reliability.
KyoBeum Lee received his B.S. and M.S. degrees in Electrical and Electronic Engineering from Ajou University, Korea, in 1997 and 1999, respectively. He received his Ph.D. degree in Electrical Engineering from Korea University, Korea, in 2003. From 2003 to 2006, he worked at the Institute of Energy Technology, Aalborg University, Aalborg, Denmark. From 2006 to 2007, he was with the Division of Electronics and Information Engineering, Chonbuk National University, Jeonju, Korea. In 2007, he joined the School of Electrical and Computer Engineering, Ajou University, Suwon, Korea. He is an associated editor of the Institute of Electrical and Electronics Engineers Transactions on Power Electronics and Journal of Power Electronics. His research interests include electric machine drives, renewable power generations, and electric vehicles.
Sim H. W.
,
Lee J. S.
,
Lee K. B.
2014
“A simple strategy for sensorless speed control for an IPMSM during startup and over wide speed range,”
Journal of Electrical Engineering & Technology
9
(5)
1582 
1591
DOI : 10.5370/JEET.2014.9.5.1582
Matsumoto A.
,
Hasegawa M.
,
Tomita M.
,
Matsui K.
“Algebraic design of fullorder flux observer for IPMSM position sensorless control,”
in Proc. IEEE IEMDC.
2011
1276 
1281
Lee K. G.
,
Lee J. S.
,
Lee K. B.
“SPMSM sensorless control for wide speed range using fullorder flux observer,”
in Proc. IEEE ICIT.
2014
164 
168
Quang N. K.
,
Hieu N. T.
,
Ha Q. P.
2014
“FPGAbased sensorless PMSM speed control using reducedorder extended Kalman filters,”
IEEE Trans. Ind. Electron.
61
(12)
6574 
6582
DOI : 10.1109/TIE.2014.2320215
Zhifeng Z.
,
Enyuyan T. R
,
Boadong B.
,
Dexin X.
2010
“Novel direct torque control based on space vector modulation with adaptive stator flux observer for induction motors,”
IEEE Trans. Magn.
48
(8)
3133 
3136
Qu Z.
,
Hinkkanen M.
,
Harnefors L.
2014
“Gain scheduling of a fullorder observer for sensorless induction motor drives,”
IEEE Trans. Ind. Appl.
50
(6)
3834 
3845
DOI : 10.1109/TIA.2014.2323482
Hasegawa M.
,
Matsui K.
2009
“Position sensorless control for interior permanent magnet synchronous motor using adaptive flux observer with inductance identification,”
IET Elect. Power Appl.
3
(3)
209 
217
DOI : 10.1049/ietepa.2008.0086
Pongam S.
,
Sangwongwanich S.
2012
“Stability and dynamic performance improvement of adaptive fullorder observers for sensorless PMSM drive,”
IEEE Trans. Power Electron.
27
(2)
588 
600
DOI : 10.1109/TPEL.2011.2153212
Ji X.
,
Wang F.
,
Zhang D.
“Robustness analysis of pole placement method with single degree of freedom in full order flux observer,”
in Proc. IEEE APPEEC.
2012
1 
4
Harnefors L.
,
Saarakkala S. E.
,
Hinkkanen M.
2013
“Speed control of electrixcal drives using classical control methods,”
IEEE Trans. Ind. Appl.
49
(2)
889 
898
DOI : 10.1109/TIA.2013.2244194
Kim J. S.
2014
“Development of the zerophaseerror speed controller for high performance PMSM drives,”
Transactions of Korean Institute of Power Electronics (KIPE)
19
(2)
184 
193
DOI : 10.6113/TKPE.2014.19.2.184
Park H. S.
,
Lee Y. I.
2014
“Torque tracking and ripple reduction of permanent magnet synchronous motor using finite control setmodel predictive control (FCSMPC),”
Transactions of Korean Institute of Power Electronics (KIPE)
19
(3)
249 
256
DOI : 10.6113/TKPE.2014.19.3.249
Kim S. H.
2014
“Maximum torque operating strategy based on stator flux analysis for direct torque and flux control of a SPMSM,”
Transactions of Korean Institute of Power Electronics (KIPE)
19
(5)
463 
469
DOI : 10.6113/TKPE.2014.19.5.463