1448 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 54, NO.

2, MARCH/APRIL 2018

Model-Based Sensorless Control of an IPMSM With

Enhanced Robustness Against Load Disturbances
Based on Position and Speed Estimator Using a
Speed Error
Younggi Lee , Student Member, IEEE, and Seung-Ki Sul , Fellow, IEEE

Abstract—In this paper, new model-based sensorless control Rs Stator winding resistance.
methods are proposed that include independent estimations of po- λf Flux linkage from permanent magnets.
sition and speed errors, and compatible position and speed esti-
Lds , Lq s Synchronous inductances on the d and q axes.
mators. Using the proposed position estimator, a unity transfer
function can be achieved from the actual position to the estimated Jm Moment of inertia.
position, eliminating the effects of load disturbances. This implies Bm Friction coefficient.
that the position error would ideally be zero, even in a transient Te , Tload Electric torque and load torque.
state. In addition, the effects of parameter and voltage synthesis P Number of poles.
errors on the steady-state position error are analyzed. Experimen- p Differential operator.
tal results verify both the analysis and the effectiveness of the
proposed methods under severe load disturbances such as speed
transient (20 000 r/min/s) and load torque transient (10 p.u./s).
With the proposed methods, position estimation errors are signif- I. INTRODUCTION
icantly reduced by more than 50% during speed and load torque
transients at identical dominant-pole placements, verifying the en- ENSORLESS control of an interior permanent magnet syn-
hanced tracking capability and robustness of the proposed methods
against external load disturbances. S chronous motor (IPMSM) has been widely used in various
drive applications, including home appliances, robots, and trac-
Index Terms—Interior permanent magnet synchronous motor tion systems, due to the various studies and commercialization
(IPMSM), model-based sensorless control, position and speed esti- efforts that have taken place. Considering that the advantages of
mator, robustness, speed error. sensorless control method, such as costs and volumes, and the
NOMENCLATURE increased reliability are attractive, many approaches to estimate
rotor position and speed without a position sensor have been
Superscript “r” Rotor reference frame. developed over the past few decades [1]–[34].
Superscript “r̂” Estimated rotor reference frame. Sensorless control methods of an AC motor can generally be
θ r , ωr Rotor position and speed in electrical angle. divided into two categories: 1) high-frequency signal injection
θ̂r , ω̂r Estimated values of θr and ωr . methods [1]–[11]; and 2) model-based methods [8]–[34]. The
θ̃r , ω̃r Position and speed estimation errors. former are based on magnetic saliency and are commonly used
r
vds , vqr s d and q components of the stator input voltage in standstill and low-speed operations. However, because the
in the rotor reference frame. operating speed is limited and an additional loss is imposed
irds , irq s d and q components of the input current in the due to the injection voltage, the latter methods are preferred in
rotor reference frame. high-speed operations.
In both of these sensorless methods, accurate estimation per-
Manuscript received July 3, 2017; revised September 27, 2017; accepted formance and increased control bandwidths have always been
November 14, 2017. Date of publication November 22, 2017; date of current
version March 19, 2018. Paper 2017-IDC-0631.R1, presented at the 2016 IEEE the important issues. For an accurate estimation of the ro-
Symposium on Sensorless Control for Electrical Drives, Nadi, Fiji, Jun. 5–6, tor position, research has shown that the voltage distortion
and approved for publication in the IEEE TRANSACTIONS ON INDUSTRY APPLI- by the inverter should be compensated, as the distorted volt-
CATIONS by the Industrial Drives Committee of the IEEE Industry Applications
Society. This work was supported by the Seoul National University Electric age leads to position estimation errors [6]–[8], [12]–[16]. In
Power Research Institute. (Corresponding author: Younggi Lee.) addition, improved estimation performance can be realized if
Y. Lee is with Department of Electrical and Computer Engineering, Seoul cross coupling and nonlinearly varying inductances from sat-
National University, Seoul 08826, South Korea (e-mail: younggi@snu.ac.kr).
S.-K. Sul is with Department of Electrical and Computer Engineering, Seoul uration and the mechanical structure are considered [2]–[4],
National University, Seoul 08826, South Korea (e-mail: sulsk@plaza.snu.ac.kr). [15]–[17]. Elaborate machine model using flux or the con-
Color versions of one or more of the figures in this paper are available online cept of the extended electromotive force (EEMF) [18]–[21]
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TIA.2017.2777390 and online parameter identification schemes [12], [22], [35]

0093-9994 © 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
See http://www.ieee.org/publications standards/publications/rights/index.html for more information.
LEE AND SUL: MODEL-BASED SENSORLESS CONTROL OF AN IPMSM WITH ENHANCED ROBUSTNESS AGAINST LOAD DISTURBANCES 1449

separate the position error from the estimator gains. In addi-

tion, the effects of parameter and voltage synthesis errors on the
steady-state position error are analyzed and verified by experi-
mental results. Although the classical papers presented sensor-
less methods using the speed (or EMF) error information [28],
[29], their concepts regarding the speed error are different from
that in this paper. From this reason, several terms that are impor-
tant in highly dynamic operations were neglected in previous
research.
Fig. 1. Position and speed estimator with modified speed output. Using the proposed position estimator, a unity transfer func-
tion can be achieved from the actual position to the estimated
would be other viable solutions to ensure accurate position position, eliminating the effects of load disturbances. This in-
estimation. dicates that the position error would ideally be zero, even in
Meanwhile, regarding the increased control bandwidth, the a transient state. Experimental results verify the effectiveness
position and speed estimator that reflects mechanical relation- of the proposed methods under severe load disturbances such
ship has been proposed [23]. In [23], by inserting a torque as speed transient (20000 r/min/s) and load torque transient
feedforward input into the estimator, the estimation bandwidth (10 p.u./s). With the proposed methods, the maximum position
can be increased in sensorless speed control. This scheme can estimation error is conspicuously reduced by 70% during speed
be applied to the high-frequency signal injection methods as transient and by 50% during load torque transient under identi-
well in the same manner [5], [6]. In [5], the estimator in [23] cal dominant-pole placements, verifying the enhanced tracking
was modified such that the estimated speed is represented as capability and robustness of the proposed methods in high- and
the output of an integrator, as shown in Fig. 1. In [5], along low-speed conditions.
with square-wave signal injection, the overall bandwidth of the
system, including the position, speed, and current control, can II. CONVENTIONAL MODEL BASED SENSORLESS CONTROL
be extended by removing two low-pass filters.
Despite continuous efforts to enhance the control perfor- In medium- and high-speed regions, the rotor position and
mance, sensorless control methods remain vulnerable to external speed are mainly estimated by model-based methods because
disturbances such as speed variations in the current control mode the required information is included in the back-EMF voltages.
and load torque variations in the speed control mode. Even if In this section, conventional model-based sensorless methods
robustness against external loads is one of the most important are introduced. The first uses an EEMF and the second uses
characteristics in actual sensorless control, especially for servo a speed error, which is similar to the proposed method in this
applications, it remains as a weak point because external dis- paper.
turbances are not known in most cases. Naturally, there have The fundamental model of a PMSM can be represented in the
been approaches to neutralize load disturbances as well [11], rotor reference frame by (1), where d-axis is aligned with the
[24], [36]. However, the estimated loads in [24] and [36] have direction of the magnetic north pole of the rotor (= θr )
been used to improve the speed control performance by revis-     
r
vds Rs −ωr Lq s irds
ing the current reference rather than estimating the position and =
speed. Meanwhile, effective disturbance rejection performance vqr s ωr Lds Rs irq s
was presented in [11], but the transient was not severe as much    r   
Lds 0 ids 0
as that in this paper. Other studies in [25], [26], and [35] have + p r + . (1)
0 Lq s iq s ωr λf
shown that robustness against parameter variations and voltage
synthesis errors can be increased. In (1), it is assumed that the cross-coupling inductances be-
This paper, therefore, focuses on robust estimations of the tween d and q axes are sufficiently small.
position and speed against external load disturbances based on For sensorless control, (1) should be expressed in the esti-
a model-based sensorless control. The key contributions of this mated rotor reference frame because every control is based on
paper are independent estimations of position and speed errors the estimated position. However, it can be noticed from (15)
and the proposal of two compatible position and speed estima- in the appendix that much complicated terms are induced from
tors. This topic has already been covered in [27], and it has coupling terms due to the saliency in a motor, i.e., Lds ࣔ Lqs .
been shown that the estimation performance in a severe tran- To simplify the expression in the estimated frame, (1) can be
sient state can be improved. However, the estimator proposed in rearranged as (2) with the introduction of the EEMF concept,
[27] is prone to steady-state speed error with parameter errors. Eex [20], [21]
Moreover, because the position estimation error is dependent  r    r 
on the estimator gain, it should be considered as well as the vds Rs −ωr Lq s ids
=
dynamic characteristics during the gain-setting procedure. vqr s ωr Lq s Rs irq s
Considering these issues, the position and speed estima-    r   
Lds 0 i 0
tor proposed in this paper is improved from that in [27] by + p ds + (2)
adding an additional state, which can nullify the speed error and 0 Lds irq s Eex
1450 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 54, NO. 2, MARCH/APRIL 2018

speed (or EMF) errors were estimated based on the difference

between the measured and estimated current variations at two
consecutive sampling points, i.e., Δir̂dq s − Δir̂dq s,est . The exact
expression of the difference can be deduced as follows:
   
Δir̂ds Δir̂ds,est
−
Δir̂q s Δir̂q s,est
⎛   ⎞
ωr sin θ̃r
⎜λf ⎟
⎜ ω̂r − ωr cos θ̃r ⎟
⎜ ⎟
⎜    ⎟
⎜
Tsam p ⎜ 0 −1 ids r̂ ⎟
⎟
= ⎜+Ls pθ̃r ⎟ . (5)
Ls ⎜ 1 0 r̂
iq s ⎟
⎜ ⎟
⎜    ⎟
⎜ 0 −ω̂r + ωr ids ⎟
r̂
⎝ ⎠
+Ls
ω̂r − ωr 0 r̂
iq s
Fig. 2. Conventional EEMF estimator. (a) Using a disturbance estimator with
a low-pass filter. (b) Using a PI-type state filter. In this equation, no parameter errors and ideal voltage synthe-
sis were assumed. However, despite these assumptions, previous
studies estimated position and speed errors while neglecting the
where Eex ≡ ωr λf + 2ωr ΔLs irds − 2ΔLs pirq s and ΔLs is de- last two terms or considering only one of them in (5). For the
fined as (Lds − Lq s )/2. The voltage equation of the IPMSM is first case of neglecting the last two terms, even if they appear
then expressed as if the inductance matrices were symmetric. to cancel out each other, this is not true in all cases, such as
Similarly, (2) can be expressed in the estimated frame by the the case shown in Fig. 1, where differentiation of the estimated
following equation: position differs from the estimated speed. Thus, excluding these
    
r̂
vds Rs −ωr Lq s ir̂ds terms leads to inaccurate estimations during dynamic operation
= such as rapid speed variations or abrupt load disturbances.
vqr̂ s ωr Lq s Rs ir̂q s
     
Lds 0 ir̂ er̂ds III. PROPOSED ESTIMATION METHOD FOR POSITION AND
+ p ds + (3) SPEED ERRORS
0 Lds ir̂q s er̂q s
In this paper, position and speed errors are estimated based on
      (16) in the appendix, where (16) is the linearized form of (15)
er̂ds − sin θ̃r ir̂q s under the assumptions that θ̃r is less than 30 °E. Additionally, the
where = Eex + Lds (ωr − ω̂r ) .
er̂q s cos θ̃r −ir̂ds speed term in (15) has been separated into the estimated and error
terms, i.e., ωr = ω̂r + ω̃r , which is the most important concept
Based on (3), various types of EEMF estimators have been in this paper, as the actual speed as well as the position cannot
proposed under the assumption that ω̃r ≡ ωr − ω̂r ≈ 0 [12], be used in sensorless operations. By means of this separation
[20], [21], [23]. Among them are the two classical estimators step, θ̃r and ω̃r can be utilized for improved sensorless control
shown in Fig. 2, where it has been disclosed that the dynamic after they are estimated from ẽr̂dq s in (16).
characteristics of the estimators in Fig. 2(a) and (b) are identi- The errors are estimated by (6), which can be easily derived
cal [15]. In this paper, for straightforward implementation, the from (16)
estimated EEMF er̂dq s,est is determined by Fig. 2(a).      
For estimation of the position and speed, a Proportional– ω̃r,est 1 λpθ ,q −λpθ ,d ẽr̂
= · s ds
Integral–Derivative (PID)-type position and speed estimator in- θ̃r,est D 0 0 ẽr̂q s
cluding a torque feedforward input in Fig. 1 is commonly used    
[5], [6]. The estimator in Fig. 1 obtains input θ̃r,est from er̂dq s,est eθ ,q −eθ ,d ẽr̂ds
after the simple algebraic manipulation expressed in the follow- + (6)
−λω ,q λω ,d ẽr̂q s
ing equation [20]:
 r̂  where D = (λω ,d · λpθ ,q – λω ,q · λpθ ,d )·s+(λω ,d · eθ ,q – λω ,q
−1 e er̂ds · eθ ,d ). In addition, eθ ,d , eθ ,q , λpθ ,d , λpθ ,q , λω ,d , and λω ,q are
θ̃r,est = tan − ds or − . (4)
er̂q s Êex presented in detail in (16) in the appendix. In the mathemat-
Subsequently, the position and speed estimator adjusts the ical model in (6), because the derivative of ẽr̂dq s is required
estimated position θ̂r and speed ω̂r in the direction that θ̃r,est is to calculate θ̃r,est and ω̃r,est , a simple estimation method is
nullified. introduced in Fig. 3, rather than direct differentiation. In this
r̂ 
On the other hand, classical papers discussed sensorless way, a filtered form of er̂dq s,est , i.e., edq s,est and its derivative,
r̂ 
methods using speed or EMF error for nonsalient machines s · edq s,est , are obtained. In the experiment, Le , referring to the
(Ls = Lds = Lq s ) [28], [29]. In these papers, the position and cut-off frequency of the estimator, was set to 2π · 100. How-
LEE AND SUL: MODEL-BASED SENSORLESS CONTROL OF AN IPMSM WITH ENHANCED ROBUSTNESS AGAINST LOAD DISTURBANCES 1451

Fig. 3. Block diagram of the proposed estimation method for position and Fig. 5. Proposed position and speed estimator.
speed errors.

Fig. 6. Position and speed estimator using speed error in [27].

errors in Fig. 3 because they use only one input θ̃r,est . However,
in the proposed method, ω̃r,est is extracted as well as θ̃r,est
simultaneously and independently. Therefore, in this paper, a
new position and speed estimator that is compatible with the
estimated errors in Fig. 3 is devised. For the design of this es-
Fig. 4. Comparison of the coefficients in D at 1000 r/min. (a) Coefficient of
the first-order term. (b) Coefficient of the zero-order term.
timator, the state equation of (7) is applied, and its structure is
shown in Fig. 5. In this figure, L indicates −P Lγ I /2Jˆm
γI
TABLE I  
PARAMETERS OF THE IPMSM θ̃r
d
x = Ax + BT̂e + L (7)
dt ω̃r
IPMSM parameter Value

Rated power 300 W ⎡ ⎤

Rated current 2.85 Arms 0 1 0 Lγ θ
Pole number 6 ⎢ ⎥ ⎡ ⎤
⎢ ⎥ Lθ θ Lγ θ
Back-EMF constant 0.0626 V·s ⎢ ⎥ ⎢ ⎥
Winding resistance (R s + R inv ) 0.675 Ω ⎢ 0 − B̂ −
P
Lγ P ⎥ ⎢ Lθ P Lγ P ⎥
⎢ ⎥
Synchronous inductances Ld s : 7.15 mH, L q s : 10.6 mH where A = ⎢ Jˆm 2Jˆm ⎥ L=⎢
⎢L
⎥
⎢
⎢0
⎥
⎥ ⎣ θI Lγ I ⎥
⎦
⎢ 0 0 Lγ I ⎥
⎣ ⎦ Lθ γ 0
ever, the dynamic characteristics in the denominator are ignored 0 0 0 0
because it can cause stability problems depending on the oper-
ating condition. For this reason, only the zero-order term in D x = [θ̂r ω̂r T̂load γin  ]T and B = [0 P/2Jˆm 0 0]T . As shown in
is considered. This scheme can be justified by Fig. 4, which (7) and in Fig. 5, the input stage of the estimator has been mod-
shows that the first-order term is sufficiently small in all operat- ified from the estimator in Fig. 6 [27]. The position and speed
ing areas of the motor described in Table I, even under dynamic estimator in Fig. 6 is based on a lower order state equation than
operations of several tens of Hz. (7), i.e., x = [θ̂r ω̂r T̂load ]T . Therefore, the position and speed
One noticeable point in the estimation is that the minimum are adjusted in the direction such that ω̃r,est is nullified rather
speed is always required for the estimation, which is common in than θ̃r,est , as shown in Fig. 6. In case where an error arises in
model-based sensorless control methods, because ω̂r is included ω̃r,est , however the estimator in Fig. 6 results in a steady-state
in both eθ ,d and eθ ,q in the denominator. Therefore, a more speed error. Moreover, gain setting becomes difficult because
suitable sensorless control method is preferred, such as a high- the position estimation error is influenced by Lθ θ and by param-
frequency signal injection method under certain speeds. eter errors. Therefore, in the proposed estimator in Fig. 5, the

additional state variable γin has been added from Fig. 6. Even if
IV. PROPOSED POSITION AND SPEED ESTIMATOR one pole to be placed has been increased, the position and speed
can now be estimated in the direction such that θ̃r,est is nullified,
A. Design of the Estimator
meaning that the steady-state speed error can be eliminated and
Conventional position and speed estimators such as that the steady-state position error becomes independent of the gain
shown in Fig. 1 are not appropriate for using the estimated as well.
1452 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 54, NO. 2, MARCH/APRIL 2018

Fig. 7. Modified position and speed estimator including an auxiliary speed estimator. (a) Auxiliary speed estimator. (b) Modified position estimator.

In determination of the gains in (7), it is assumed that θ̃r,est operates together with the auxiliary estimator in Fig. 7(a), which
and ω̃r,est are extracted without error, that is, θ̃r,est and ω̃r,est is based on the state equation in the following equation:
⎡ ⎤⎡ ⎤
are identical to θr − θ̂r and ωr − ω̂r , respectively. The transfer   B̂m P ω̂r
functions from θr to θ̂r and from ωr to ω̂r can then be derived d ω̂r ⎢− ˆ − ˆ ⎥ ⎣ ⎦
= ⎣ Jm 2Jm ⎦
by the following equation: dt T̂L
0 0 T̂L
θ̂r = Hθ θ (s) θr + Hθ T (s) ΔT ⎡ P ⎤  
ω̂r = Hω ω (s) ωr + Hω T (s) ΔT (8) LauxP
+ ⎣ 2Jˆm ⎦ T̂e + (γin  + ω̃r ) . (10)
where ΔT is defined as the disturbance torque, which is the dif- LauxI
0
ference between the feedforward torque T̂e and the net torque
applied to the motor, i.e., Te − Tload . Hθ θ , Hθ T , Hω ω , and Hω T The auxiliary estimator has a structure similar to that of the
are described in detail in the appendix. It should be noted that original estimator, but it is entirely separate from the position
Lγ θ = 1 can eliminate the effect of ΔT on θ̂r from Hθ T . There- estimator in Fig. 7(b). With the proposed auxiliary estimator, the
fore, by introducing ω̃r,est into the estimator, it is expected that dynamics of the speed estimation can be enhanced while main-
robustness of the position estimation to load torque variations taining the characteristics of the position estimation process.
can be enhanced. In addition, in (17) in the appendix, it should Similar to the gain setting method in (9), LauxP and LauxI
also be noted that the two gains of θ̃r in the first column in can be determined by the second-order transfer function in the
(7), i.e., Lθ P and Lθ I , are disabled by Lγ θ = 1, where Lθ I is following equation:
defined as −P Lθ I /2Jˆm . This is already reflected in Fig. 5.    
Jm 2 Bm 
Additionally, even if it appears that (8) is a second-order Jˆm
s + LauxP + Jˆm
s + L auxI γin + 2JPˆ sΔT
system, it becomes a fourth-order system if there is an error ω̂r =   m

s2 + LauxP + B̂Jˆm s + LauxI 

in θ̃r,est or ω̃r,est . Therefore, all gains should be set and can be m

determined by the multiplication of two characteristic equations (11)

of the general second-order system in the following equation: where LauxI is defined as −P LauxI /2Jˆm . Also, in (11),
  ω̃r,est = ω̃r and γin  ≈ 0 are assumed because γin 
varies slowly
B̂m compared to ω̃r,est and the steady-state value is zero. For the
2 2
s + 2ζn ,1 ωn ,1 s + ωn ,1 = s + 2
+ Lγ P s + Lγ I 
Jˆm auxiliary speed estimator to reduce the effects of ΔT on ω̂r or
enhance the dynamics, it should have a higher bandwidth than
s2 + 2ζn ,2 ωn ,2 s + ωn ,2 2 = s2 + Lθ θ s + Lθ γ . (9) that of the original estimator. In this way, more robust speed-
Using the relationship in (9), four gains in addition to Lγ θ = 1 estimation performance can be achieved while minimizing the
can be determined in accordance with the desirable natural effects of noise on θ̂r .
frequency ωn and damping factor ζn . On the other hand, for the design of the original estimator

to remain valid, the speed error input ω̃r,est to the estimator in
B. Auxiliary Speed Estimator Fig. 7(b) should be the difference between the actual speed ωr
and the unused state variable ω̂r in Fig. 7(b). For calculation of
Although the response of θ̂r to ΔT can be enhanced by the ωr − ω̂r , the following calculation method can be used:
estimator in Fig. 5, that of ω̂r is still affected by ΔT . Even if

the increased gains would naturally reduce the effects, the gains ω̃r,est = γin + ω̂r − ω̂r
cannot be increased above certain limit in some cases, such as ≈ (ωr − ω̂r ) + ω̂r − ω̂r ≈ ωr − ω̂r . (12)
when large harmonic components exist in the back-EMF volt-
age and inductances [30] or in the case of fixed gains for a wide In (12), γin can be approximated as ωr − ω̂r if parameter
speed range. Therefore, in this paper, apart from the position es- errors and voltage distortion are not considered. Therefore, using
timator, an auxiliary speed estimator is also proposed, as shown the input stage in Fig. 7(b), the auxiliary speed estimator can
in Fig. 7(a). Moreover, the original estimator in Fig. 5 is changed be linked to the original estimator without any effects on the
to Fig. 7(b) with a modified input stage. The modified estimator characteristics of the original estimator.
LEE AND SUL: MODEL-BASED SENSORLESS CONTROL OF AN IPMSM WITH ENHANCED ROBUSTNESS AGAINST LOAD DISTURBANCES 1453

10 ms in the worst case at 500 r/min, as shown in Fig. 10, which is

inevitable because the effect of the ignored term in (6) increases
at a low speed. Therefore, considering the experimental results,
a more suitable sensorless control method is preferred, such as
a high-frequency signal injection method at speeds of less than
500 r/min. However, at speeds higher than or equal to 500 r/min,
the assumption in (6) is reasonable and the estimated errors can
be used as input for the proposed estimators in Figs. 5 and 7.
In Figs. 9 and 10, the estimation and current control per-
formances of the proposed methods are compared with those
by the conventional method applying the position and speed
Fig. 8. Voltage distortion of the prototype inverter by the phase current. estimator and the estimation method shown in Figs. 1 and 2,
respectively. In this experiment, the low-pass filter in Fig. 2(a)
V. EXPERIMENTAL RESULTS has been removed such that only the dynamics of the estimator
in Fig. 1 is taken into account. In Figs. 9 and 10, the exper-
The test setup used to evaluate the proposed sen-
imental results for the conventional method are presented in
sorless method and the estimator was constructed based
the first row and the results for the proposed methods with-
on a TMS320F28377 digital signal processor (DSP). The
out and with the auxiliary estimator are shown in the second
power device used in the prototype inverter was a Mit-
and third rows, respectively. Even when the dominant pole
subishi PSS10S51F6 IPM. Also, fsw = 10 kHz, fsam p =
of the estimators has been set equal to 4 Hz in both cases,
20 kHz, Vdc = 311 V, and Tdead was set to 1 μs to minimize
which is the edge of the stable response for the conventional
the voltage distortion by the inverter. In addition to the small
method at 500 r/min, it is clear that the estimation perfor-
dead time, the compensation voltage in Fig. 8 was superimposed
mances in Figs. 9(c) and 10(c) have been enhanced remark-
onto the output voltages of the controller by each phase current
ably in comparison with the conventional method. In this
[37]. The target motor under test is an IPMSM of the type used in
experiment, the poles are set as follows: pn = 14 Hz, ωn =
servo applications, the parameters of which are specified in Ta-
4 Hz, and ζn = 1.1 for the characteristic equation of the con-
ble I. In addition, in the experimental setup, a motor–generator
ventional estimator, i.e., (s +pn )(s2 +2ζn ωn s + ωn2 ) = 0; ωn ,1
set was established to emulate the load.
= 4 Hz, ζn ,1 = 1.1, ωn ,2 = 4 Hz, and ζn ,2 = 2.3 for the pro-
In the model-based sensorless method, the accuracy of the
posed estimator in (9); ωn ,aux = 5 Hz, ζn ,aux = 1.4 for the aux-
parameters is directly related to the estimation performance.
iliary estimator. In the gain-setting procedure, L1 in Fig. 1 and
Regarding Lq s in the test motor, it is nearly constant if Te is
Lθ θ in Figs. 5 and 7(b) have identical values, adapting ζn ,2 for the
in the range of 0.3–0.7 p.u. Therefore, for convenience, the
same effect of the feedforward terms. Also, in the experiments,
experiments are designed in this range.
the low gains selected for low-speed operation so as to reject
For the verification of the proposed estimation method and
harmonic disturbances are assumed to be held equal except for
the estimators, the following experiments are designed, indi-
ωn ,aux = 7 Hz at high speeds to show that the performance can
cating that the estimated errors, i.e., θ̃r,est and ω̃r,est , are in
be enhanced even more.
satisfactory agreement with the actual errors and that the sen-
In Fig. 10(a), the position error exceeds the stability limit of
sorless control performance can be significantly enhanced by
90 °E, and it can be seen that the applied load torque changes
the proposed methods in the current and speed control modes.
abruptly to regulate the speed. Meanwhile, in Figs. 9(b) and (c)
and 10(b) and (c), the maximum position error is less than 10 °E
A. Rapid Speed Variation (20000 r/min /s) in the Current and 20 °E, respectively, even under transient conditions. Even
Control Mode if Hθ T in (17) is nullified by setting Lγ θ equal to 1, the actual
In Figs. 9 and 10, the performances of position and speed experiment returned a position estimation error. It is likely to
estimations are presented when the test motor is in the current have been induced from the errors in the parameters in (6).
control mode (Te∗ = 0.5 p.u.) and with rapid changes in the Specifically, the errors in (6) undermine the assumptions in (8),
operating speed of the load machine. The rotating speed was i.e., θ̃r,est = θr − θ̂r and ω̃r,est = ωr − ω̂r .
varied by the load machine from 2000 to 2500 r/min, and back Figs. 9(c) and 10(c), where the auxiliary speed estimator is
to 2000 r/min again in Fig. 9. It was also varied from 500 to 1000 included, show that response time has been reduced by more
and then to 500 r/min again in Fig. 10 at a rate of 20000 r/min/s. than 100 ms in both speed cases, with a reduction in the max-
First, from the second and third rows in Figs. 9 and 10, the imum speed error by more than 30% in the high-speed case as
proposed estimation method by (6) can be verified by comparing compared to the outcomes in Figs. 9(a) and (b) and 10(a) and
θ̃r,est and ω̃r,est with the actual errors, i.e., θ̃r and ω̃r , from a (b). The speed response can be more clearly compared by ex-
position sensor. In this experiment, the test motor was operated amining Figs. 9(d) and 10(d). From these figures, it is clear that
without the aid of the position sensor, which was used only for the estimated speed as well as the estimated position track the
observing the actual errors. From the figure, it can be seen that actual values much more rapidly under highly dynamic opera-
θ̃r,est and ω̃r,est are accurately estimated even during the highly tions, even if the convergence has not been shortened without
dynamic operation, although they are delayed by approximately the auxiliary estimator. In Figs. 9(a) and (b) and 10(a) and (b),
1454 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 54, NO. 2, MARCH/APRIL 2018

Fig. 10. Estimation performance during speed variation (500 ↔ 1000 r/min).
Fig. 9. Estimation performance during speed variation (2000 ↔ 2500 r/min). (a) Conventional method. (b) Proposed method without an auxiliary speed esti-
(a) Conventional method. (b) Proposed method without an auxiliary speed esti- mator. (c) Proposed method with an auxiliary speed estimator. (d) Comparison
mator. (c) Proposed method with an auxiliary speed estimator. (d) Comparison of the estimated speed. (e) Comparison of the current response.
of the estimated speed. (e) Comparison of the current response.

At the bottom of Fig. 10(a), an additional trace for the

the convergence time of the overall estimation is approximately conventional method is plotted, where higher gains were as-
250 ms. However, with regard to the position estimation, the sessed for a similar magnitude of the position error with the
convergence time has been reduced by more than 70% in both proposed methods. For this waveform, the poles are placed at
speed cases by applying the proposed methods. pn = 15 Hz, ωn = 10 Hz, and ζn = 1.1, and it should be noted
In Figs. 9(e) and 10(e), the current response is presented. that the dominant pole is more than two times that in the original
In these figures, the conventional and the proposed methods case. Naturally, it would be more sensitive to external noise and
without the auxiliary estimator are compared. As noted earlier, harmonic components.
because the current reference is determined by the fixed torque
reference (Te∗ = 0.5 p.u.), the d and q axes currents should main-
B. Abrupt Load Torque Disturbance (10 p.u./s) in the Speed
tain their values against external variations. In Fig. 9(e), where
the operating speed is sufficiently high and the position estima- Control Mode
tion error is considerably reduced, the performance is clearly In this experiment, robustness is tested in the speed con-
enhanced compared to that in Fig. 10(e). Meanwhile, at lower trol mode when the load torque Tload is changed from 0.3 to
speed where the position estimation is not remarkably reduced, 0.7 p.u. and then back to 0.3 p.u. at a rate of 10 p.u./s, as shown
the average variation shows no improvements from the conven- in Fig. 11(d). The operating speed is regulated by the test motor,
tional case despite the fact that it is much better during the second which operates at 2000 r/min in Fig. 11 and at 500 r/min in
transient, where the conventional method loses the tracking. Fig. 12.
LEE AND SUL: MODEL-BASED SENSORLESS CONTROL OF AN IPMSM WITH ENHANCED ROBUSTNESS AGAINST LOAD DISTURBANCES 1455

Fig. 11. Estimation performance during load torque variation (0.3 ↔ 0.7 p.u., Fig. 12. Estimation performance during load torque variation (0.3 ↔ 0.7 p.u.,
speed reference: 2000 r/min). (a) Conventional method. (b) Proposed method speed reference: 500 r/min). (a) Conventional method. (b) Proposed method
without an auxiliary speed estimator. (c) Proposed method with an auxiliary without an auxiliary speed estimator. (c) Proposed method with an auxiliary
speed estimator. (d) Comparison of the actual speeds. speed estimator. (d) Comparison of the actual speeds.

Similar to the first experiment, the gains were set as the edge concluded that the robustness against load variations has been
for the conventional method to keep the speed at 500 r/min under remarkably enhanced by the proposed methods.
a severe load torque transient. That is, pn = 14 Hz, ωn = 5 Hz, In Fig. 12, where the test motor regulates the speed as
and ζn = 1.1 were used for the conventional estimator; ωn ,1 = 500 r/min, it can be observed that while the maximum posi-
5 Hz, ζn ,1 = 1.1, ωn ,2 = 5 Hz, and ζn ,2 = 2.1 for the pro- tion and speed errors during the first transient are reduced by
posed estimator; and ωn ,aux = 6 Hz and ζn ,aux = 1.1 for half and by 60%, respectively, when using the proposed meth-
the auxiliary estimator at 500 r/min and ωn ,aux = 7 Hz at ods, the response becomes oscillatory. Regarding the oscillatory
2000 r/min. The bandwidth of the speed controller was set to response, it was verified by the simulation including the inverter
35 Hz. model that the oscillatory response comes from imperfect com-
Figs. 11 and 12 show the position and speed estimation per- pensation of the voltage distortion in Fig. 8, as the effect of
formances of the conventional and proposed methods when they voltage error increases at lower speeds. In Fig. 12(b) and (c),
are exposed to load variations. In Fig. 11, where the test motor the voltage error is represented as the estimated speed error
regulates the speed as 2000 r/min, it can be observed that while with a dc value. Because the proposed methods estimate the po-
the maximum position error in Fig. 11(a) is about 35 °E during sition and speed by nullifying the estimated position error, the
the first transient, the errors in Fig. 11(b) and (c) are less than estimated speed error can have a dc value if the parameter infor-
3 °E, which is less than one tenth of that in the first case. Sim- mation or voltage compensation is inaccurate. Therefore, even
ilarly, the maximum speed error of the conventional method is if the compensation of the voltage distortion is not the focus
300 r/min during the first transient. However, it can be seen that in this paper, a more appropriate compensation method would
the speed error was reduced to about 100 r/min when applying be preferable to eliminate the oscillation, especially in the low-
the proposed methods. In Fig. 11(d), where the actual speeds speed control mode. Despite the harmonic oscillation, however,
of each method are compared, it is shown that not only has the Fig. 12 indicates that the low-speed response can also be more
magnitude of the errors been decreased but that the convergence robust against load torque transients when using the proposed
time has also been reduced by 100 ms during the first transient methods, reducing the possibility of the failure of the sensorless
when the auxiliary estimator is employed. Therefore, it can be control.
1456 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 54, NO. 2, MARCH/APRIL 2018

Fig. 13. Estimation performances from previous research [27]. (a) During Fig. 14. Comparison of performances with previous research. (a) Current re-
speed variation (2000 ↔ 2500 r/min). (b) During load torque variation (0.3 ↔ sponse against speed variation. (b) Speed response against load torque variation.
0.7 p.u., speed reference: 2000 r/min).

At the bottom of Figs. 11(a) and 12(a), an additional trace

for the conventional method is shown. In these waveforms, in
the same way of Fig. 10(a), higher gains were assessed for the
position estimation performance similar to that of the proposed
methods. Here, the poles are placed at pn = 28 Hz, ωn = 14 Hz,
and ζn = 1.1 in Fig. 11(a) and at pn = 14 Hz and ωn = 6 Hz
with the same value of ζn in Fig. 12(a). Fig. 11(a) indicates
that the position estimation error is comparable to the errors in
Fig. 11(b) and (c) with the reduced speed error and convergence
time although the dominant pole in this case is nearly three times
larger than in the first case. Meanwhile, in the low-speed case, a
small increase of the gain resulted in a reduction of the position
error by 50%. However, the increased load began to bring about Fig. 15. Comparison of estimation performances under nonideal conditions.
mechanical vibration at this or higher gains. Therefore, these (a) Variation of estimated L q s . (b) Variation of L θ θ .
results indicate that the position estimation performance of the
conventional method is very sensitive to the gain and that the be concluded that the dynamic characteristics in two methods
possible range of the gains is quite limited in the low-speed are identical.
control mode. Meanwhile, in Fig. 15, practical issues are considered. As
aforementioned, the reasons for including the state variable,
 
γin , contrast to the earlier study [27], where no γin is, are be-
C. Comparison With the Previous Research [27] cause: 1) the steady-state speed error can be eliminated, and
Because the proposed method is an improvement over an 2) the steady-state position error becomes independent of the
earlier method [27], comparison results from the previous study gain. Fig. 15(a) illustrates the first reason. In this experiment,
[27] are provided in addition to the results given in Figs. 9–12. the test motor operates in the speed control mode with a speed
First, Fig. 13 shows the estimation performance from the reference of 2000 r/min. In addition, L̂q s was intentionally var-
earlier work [27] in terms of speed variation and load torque ied from 0.8 Lq s to Lq s at 0.6 s, and then to 1.2Lq s at 1.4 s. In
variation, which corresponds to Figs. 9(b) and Fig. 11(b), re- Fig. 15(a), it can be observed that the estimated speed by earlier
spectively. Here, all gains are equal to those of the correspond- method [27] differs from the reference, whereas the proposed
ing experimental conditions. In these figures, it can be observed method tracks the reference well. Moreover, when L̂q s = Lq s ,
that the responses by two methods are nearly identical during it is clear that there is a speed error in the previous method
a rapid transient, indicating that newly the designed state γin  even if no intentional parameter error existed and the compen-
in (7) and Fig. 5 rarely affects the dynamic characteristics as sation voltage in Fig. 8 was applied, which means that ω̃r,est is
expected. not entirely accurate. However, accurate speed control is pos-
Fig. 14 shows similar results in the current and speed con- sible with the proposed method regardless of the position error
trol modes. In Fig. 14(a), the current responses of the method from L̂q s .
under comparison [27] and of the proposed method without an On the other hand, Fig. 15(b) shows the effect of Lθ θ . In
auxiliary estimator are overlapped. The overlapped speed re- Fig. 15(b), the test motor is controlled in the current control
sponses are also given in Fig. 14(b). From these figures, it can mode with Te∗ = 0.5 p.u., and Lθ θ was varied from 60% of the
LEE AND SUL: MODEL-BASED SENSORLESS CONTROL OF AN IPMSM WITH ENHANCED ROBUSTNESS AGAINST LOAD DISTURBANCES 1457

as λ̃f increases in the positive direction, whereas the errors from

others are linear. Additionally, in Fig. 16, the position estimation
error from Lq s , when the conventional method in Figs. 1 and 2
is applied, has been calculated [16] and shown for comparison,
as denoted by the black dotted line. As shown in Fig. 16, the
effects of Lq s are similar in the two methods. Meanwhile, λf
is irrelevant with regard to the position estimation when using
the conventional method with EEMF, as the main error signal
er̂ds,est is independent of λf , as shown in (14), while the proposed
method is significantly affected by the error of λf
r̂ ∗
er̂ds,est = v ds − R̂s ir̂ds + ω̂r L̂q s ir̂q s ∝ θ̃r . (14)

Therefore, accurate estimations of the parameters including

λf are the utmost important when using the proposed methods.
To verify the calculation, the estimated parameters were in-
tentionally varied in the experiment, and the resulting position
errors were then measured and the results are overlapped onto
Fig. 16. In the experiment, L̃q s and λ̃f among the main sources
of error were changed because the voltage error is difficult to
Fig. 16. Calculated and experimental results for the position estimation error
induced by parameter errors and voltage distortion.
define due to voltage distortion by the inverter. As shown in
Fig. 16, the position errors from the experiment are in good
agreement with the calculated results. From this comparison, it
original value to 100% at 0.6 s, and then to 200% at 1.4 s. In has been verified that the position estimation error in a steady
this figure, it can be observed that the position error is affected state is only affected by θ̃r,est and that ω̃r,est simply assists with
by the variation of Lθ θ , as ω̃r,est is nullified by the estimator the estimation to achieve enhanced robustness.
in Fig. 6, rather than θ̃r,est , while the proposed method is not
affected by Lθ θ at all.
VI. CONCLUSION
D. Effects of Parameter Errors on the Steady-State In this paper, a new estimation method for position and speed
Position Error errors and compatible position and speed estimators were pro-
posed and the effects of parameter errors on the position esti-
Theoretically, the resulting position error can be calculated by
mation error were analyzed. In the proposed method, the speed
substituting θ̃r,est in (6) with 0, from which (13) can be deduced
error and the position error were extracted by splitting the actual
as follows:
⎛      ⎞ speed in the voltage equation into the estimated and error terms.
r̂
ṽds R̃s −ωr L̃q s ir̂ds 0 In addition, the position and speed estimator using the esti-
⎜ r̂ + + ⎟
⎜ ṽq s ir̂q s ωr λ̃f ⎟
mated errors were designed to nullify the speed error in a steady
⎜ ωr L̃ds R̃s ⎟
T ⎜ ⎟
⎡ ⎤ state, and a guideline by which to set the gains of the estimator
 ⎜ ⎟
−ωq ,est ⎜ − sin θ̃r ⎟ was described. Additionally, an auxiliary speed estimator was
⎜+ω λ ⎣ ⎦ ⎟= 0.
⎜ r f ⎟ devised and combined with the first estimator to enhance the ro-
ωd,est ⎜ cos θ̃r − 1 ⎟
⎜ ⎟ bustness and the speed-estimation capabilities by the proposed
⎜    r̂  ⎟
⎜ − sin 2θ̃ cos 2θ̃ − 1 i ⎟ methods.
⎝+ω ΔL r r ds ⎠
All of the proposed methods and the analyses were veri-
r s
cos 2θ̃r − 1 sin 2θ̃r ir̂q s fied by experiments. The experimental results demonstrated
(13) the effectiveness of the proposed methods under highly dy-
namic operating conditions, such as speed variation at a rate of
s , R̃s , L̃ds , L̃q s , and λ̃f indicate the voltage error
r̂
In (13), ṽdq 20000 r/min/s in the current control mode and load disturbance
on the d and q axes and the parameter errors of the resistance, at a rate of 10 p.u./s in the speed control mode. With the pro-
inductances, and back-EMF constant, respectively. Using (13), posed methods, maximum position errors were conspicuously
the position estimation error induced from each element was reduced by 70% during speed transient and by 50% under load
calculated. Calculated results are plotted in Fig. 16, where the disturbance with identical dominant-pole placements, verifying
error ratio is defined as the error divided by the actual value. the enhanced tracking capability and robustness offered by these
In the calculation, 50% load and 1000 r/min conditions were novel methods.
assumed, and because the relationship in (13) is not linear or
analytic, the estimation error was obtained recursively with an
APPENDIX
initial value of θ̃r = 0. From the figure, it can be observed that
the effects of L̃q s , λ̃f , and ṽds r̂
on the position error are signif- Here, the long and complex equations that cannot be placed
icant. Specifically, the estimation error increases exponentially in the main text are presented as follows.
1458 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 54, NO. 2, MARCH/APRIL 2018

 r̂   r̂       
vds ids Lds cos2 θ̃r + Lq s sin2 θ̃r (Lds − Lq s ) sin θ̃r cos θ̃r ir̂ds − sin θ̃r
= Rs + p + ωr λf
vqr̂ s ir̂q s (Lds − Lq s ) sin θ̃r cos θ̃r Lds sin2 θ̃r + Lq s cos2 θ̃r ir̂q s cos θ̃r
   r̂ 
−(Lds − Lq s ) sin θ̃r cos θ̃r Lds cos2 θ̃r + Lq s sin2 θ̃r ids
+ pθ̃r
−Lds sin2 θ̃r − Lq s cos2 θ̃r (Lds − Lq s ) sin θ̃r cos θ̃r ir̂q s
   r̂ 
−(Lds − Lq s ) sin θ̃r cos θ̃r −Lds sin2 θ̃r − Lq s cos2 θ̃r ids
+ ωr (15)
Lds cos2 θ̃r + Lq s sin2 θ̃r (Lds − Lq s ) sin θ̃r cos θ̃r ir̂q s
 r̂   r̂            r̂ 
vds ids Lds 0 ir̂ds 0 −Lq s ir̂ds 0 ẽds
= Rs + p + ω̂r + + (16)
vqr̂ s ir̂q s 0 Lq s ir̂q s Lds 0 ir̂q s ω̂r λf ẽr̂q s
               
ẽr̂ds λω ,d λpθ ,d eθ ,d λω ,d −Lq s ir̂q s λpθ ,d Lds ir̂q s
where = ω̃r + pθ̃r + θ̃r , = , = ,
ẽr̂q s λω ,q λpθ ,q eθ ,q λω ,q Lds ir̂ds + λf λpθ ,q −Lq s ir̂ds
   
eθ ,d −ω̂r λf − 2ΔLs (ω̂r ir̂ds − pir̂q s )
= .
eθ ,q 2ΔLs (ω̂r ir̂q s + pir̂ds )

      
Lγ θ s2 + Lθ θ s + Lθ γ Lγ θ H2 (s) + (1 − Lγ θ ) H3 (s) + s2 JJˆm s2 + BJˆm + Lγ P s + Lγ I 
m m
Hθ θ =
H1 (s) H2 (s) + (1 − Lγ θ ) H3 (s)
   
(1 − Lγ θ ) 2JPˆ s2 H1 (s) Jm
Jˆm
s2 + Bm
Jˆm
+ Lγ P s + Lγ I  + (1 − Lγ θ ) H3 (s)
m
Hθ T = , Hω ω = ,
H1 (s) H2 (s) + (1 − Lγ θ ) H3 (s) H1 (s) H2 (s) + (1 − Lγ θ ) H3 (s)

H1 (s) 2JPˆ s
m
Hω T = (17)
H1 (s) H2 (s) + (1 − Lγ θ ) H3 (s)
 
B̂m
2
where H1 (s) = s + Lθ θ s + Lθ γ Lγ θ , H2 (s) = s + 2
+ Lγ P s + Lγ I  ,
Jˆm

H3 (s) = Lθ P s2 + (Lγ P Lθ γ + Lθ I  )s + Lγ I  Lθ γ .

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[23] H. Kim, M. C. Harke, and R. D. Lorenz, “Sensorless control of in- neering from Seoul National University, Seoul, South
terior permanent-magnet machine drives with zero-phase lag position Korea, in 1980, 1983, and 1986, respectively.
estimation,” IEEE Trans. Ind. Appl., vol. 39, no. 6, pp. 1726–1733, From 1986 to 1988, he was an Associate
Nov./Dec. 2003. Researcher in the Department of Electrical and
[24] T. Senjyu, T. Shingaki, and K. Uezato, “Sensorless vector control of Computer Engineering, University of Wisconsin,
synchronous reluctance motors with disturbance torque observer,” IEEE Madison, WI, USA. From 1988 to 1990, he was a
Trans. Ind. Electron., vol. 48, no. 2, pp. 402–407, Apr. 2001. Principal Research Engineer at LG Industrial Sys-
[25] W. Sun, J. Gao, Y. Yu, G. Wang, and D. Xu, “Robustness improvement of tems Company, Seoul. Since 1991, he has been a
speed estimation in speed-sensorless induction motor drives,” IEEE Trans. Member of Faculty with the School of the Electri-
Ind. Appl., vol. 52, no. 3, pp. 2525–2536, May/Jun. 2016. cal and Computer Engineering, Seoul National University, where he is cur-
[26] J. Choi, K. Nam, A. A. Bobtsov, A. Pyrkin, and R. Ortega, “Robust rently a Professor. He has authored or co-authored more than 150 IEEE journal
adaptive sensorless control for permanent-magnet synchronous motors,” papers and a total of more than 340 international conference papers in the
IEEE Trans. Power Electron., vol. 32, no. 5, pp. 3989–3997, May 2017. area of power electronics. He holds 14 U.S. patents, seven Japanese patents,
[27] Y. Lee and S.-K. Sul, “Model-based sensorless control of IPMSM enhanc- 11 Korean patents, and has supervised 43 Ph.D. students. His research interests
ing robustness based on the estimation of speed error,” in Proc. 2016 IEEE include power electronic control of electrical machines, electric/hybrid vehicles
Symp. Sensorless Control Elect. Drives, Jun. 5–6, 2016, pp. 46–53. and ship drives, high-voltage dc transmission based on modular multilevel con-
[28] N. Matsui, “Sensorless PM brushless DC motor drives,” IEEE Trans. Ind. verter, and power-converter circuits for renewal energy sources.
Appl., vol. 43, no. 2, pp. 300–308, Apr. 1996. Dr. Sul was the Program Chair of the IEEE Power Electronics Specialists
[29] N. Matsui, T. Takeshita, and K. Yasuda, “A new sensorless drive of Conference’06 and the General Chair of the IEEE Energy Conversion Congress
brushless DC motor,” in Proc. Int. Conf. Ind. Electron. Control Instrum., and Exposition-Asia and the International Conference on Power Electronics,
Nov. 1992, pp. 430–435. 2011. In 2015, he was the President of the Korean Institute of Power Electron-
[30] R. W. Hejny and R. D. Lorenz, “Evaluating the practical low-speed limits ics. He was the recipient of the 2015 IEEE TRANSACTIONS first and second paper
for back-EMF tracking-based sensorless speed control using drive stiffness awards on industrial applications, simultaneously. He was also the recipient of
as a key metric,” IEEE Trans. Ind. Appl., vol. 47, no. 3, pp. 1337–1343, the 2016 Outstanding Achievement Award from the IEEE Industrial Application
May/Jun. 2011. Society and the 2017 Newell Award from the IEEE Power Electronics Society.