1. 978-1-4244-2800-7/09/$25.00 ©2009 IEEE ICIEA 2009
PMSM Speed Sensorless Direct Torque Control
Based on EKF
Yingpei Liu, Jianru Wan, Hong Shen, Guangye Li, Chenhu Yuan
School of Electric Engineering and Automation
Tianjin University
Tianjin, 300072, China
Email:liuyingpei_123@126.com
Abstract— System with mechanical speed sensor has lower
reliability and higher cost. Traditional direct torque control
method has disadvantages of big ripples on current and flux
linkage and torque. To solve these problems, the Extended
Kalman Filter is established to estimate both stator flux linkage
and rotor speed. Therefore, speed sensorless direct torque control
for surface permanent magnet synchronous motor is realized.
Simulation results have shown that the advantage of direct
torque control method such as rapid torque response is
maintained, at the same time, the system based on EKF is robust
to motor parameters and load disturbance. The dynamic and
static performances are dramatically improved.
Index Terms—Direct torque control, Permanent magnet
synchronous motor (PMSM), Extended kalman filter (EKF),
Stator flux linkage observation, Speed sensorless control
I. INTRODUCTION
Direct torque controlled permanent magnet synchronous
motor (PMSM) has rapid response and good static and
dynamic performance; so many scholars have conducted
research to this field and achieved certain results [1-4]. In the
direct torque control method, the observation accuracy of
stator flux linkage directly determines the performance of the
entire system. System with mechanic speed sensor has lower
reliability and higher system cost. So how to get stator flux
linkage and speed information has become the research
hotspot.
In the traditional DTC control, flux linkage is observed
through pure voltage integration. The model is very simple,
but in practical applications, the disadvantages impact of
integrator such as the sensitivity to initial value and DC offset
has influenced stator flux linkage observation accuracy. In
order to solve these problems, in literature [5], an improved
integration is applied to observe stator flux linkage in
induction motor, which could only get accurate phase
information. In literature [6], low-pass filter is applied instead
of integrator to observe flux linkage, which results in flux
linkage phase ahead and its amplitude smaller. Amplitude and
phase compensation is researched in literature [7], which can
not fundamentally resolve the shortcomings of pure integrator.
Aiming at speed sensorless DTC controlled surface
permanent magnet synchronous motor (SPMSM), Extended
This paper is supported by National Science Foundation of China (60874077)
and Specialized Research Fund for the Doctoral Program of Higher Education
of China (20060056054).
Kalman filter (EKF) observer is researched to estimate both
stator flux linkage and rotor speed in the paper. The
disadvantage of pure integrator has been overcome, and the
advantages of DTC method such as rapid torque response and
strong robustness are still maintained. In the meantime, the
problems resulting from mechanical speed sensor has been
resolved.
II. MODEL OF SPMSM
βα − Coordinate is chosen in order to design EKF
observer. The voltage equation of SPMSM is as following.
sin
cos
s s r r r
s s r r r
di
u R i L
dt
di
u R i L
dt
α
α α
β
β β
ωψ θ
ωψ θ
⎧
= + −⎪⎪
⎨
⎪ = + +
⎪⎩
(1)
Equation (2) is gotten from (1).
s
s
d
u R i
dt
d
u R i
dt
α
α α
β
β β
ψ
ψ
⎧
= +⎪⎪
⎨
⎪ = +
⎪⎩
(2)
Where uα and uβ
are stator voltage ,α β coordinate
components, iα 、 iβ are stator current components, αψ and
βψ are stator flux linkage components, sR is stator resistance,
sL is stator inductance, rω is rotor speed, rθ is rotor position
angle.
III. DESIGN OF EKF OBSERVOR
The state and output equations of discrete linear system
with random interference is as following [8].
1k k k k k k
k k k k
x A x B u w
y C x v
+ = + +⎧
⎨
= +⎩
(3)
Where system noise ( )kw takes into account the system
disturbances and model errors, while ( )kv represents the
measurement noise, which takes into account all measure
noise and measure errors. Both ( )kv and ( )kw are zero-mean
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2. white noise with covariance of R and Q , respectively and
independent from each other.
Both the optimal state and its covariance are computed in a
two-step loop. The first one (prediction step) performs a
prediction of both quantities based on the previous estimates.
The equations are as the following:
/ 1 1/ 1 / 1 1k k kk k k k kx A x B U
∧ ∧
− −− − −= + (4)
/ 1 / 1 1 / 1
T
k k k k k k kP A P A Q
∧
− − − −= + (5)
The second step (innovation step) corrects the predicted
state and estimation and its covariance matrix through a
feedback correction scheme that makes use of the actual
measured quantities, which are realized by the following
equations.
1
/ 1 / 1
T T
k k k kk k k kK P C C P C R
−∧ ∧
− −
⎛ ⎞= +⎜ ⎟
⎝ ⎠
(6)
/ / 1 / 1k k k k k kk k kx x K y C x
∧ ∧ ∧
− −
⎛ ⎞
= + −⎜ ⎟
⎝ ⎠
(7)
( ) / 1/ k kk k k kP I K C P
∧
−= − (8)
The flux linkage equation of SPMSM is as the following.
s
s
cos
sin
f r
f r
i
i
α α
β β
ψ ψ θ
ψ ψ θ
+⎧⎪
⎨
+⎪⎩
＝L
＝L
(9)
Combined with (2), (10) can be obtained.
cos
sin
s s
f r
s s
s s
f r
s s
d R R
u
dt L L
d R R
u
dt L L
α
α α
β
β β
ψ
ψ ψ θ
ψ
ψ ψ θ
⎧
= − + +⎪
⎪
⎨
⎪ = − + +
⎪⎩
(10)
Compared to the dynamic process of SPMSM, sampling
interval time is so small that speed can be considered
unchanged, which is 0=
dt
dω . Also, we have the
equation 0=
dt
d rθ .
( ) [ ]T
rrtx θωψψ βα= is selected as state variable,
[ ]T
uuu βα= and [ ]T
iiy βα= are chosen as input and
output variable respectively which can easily obtained from
the measurements. Thus the dynamic state model for SPMSM
was as following.
( )
( )⎪⎩
⎪
⎨
⎧
=
+=
•
xgy
Buxfx (11)
Where
( )
cos
sin
0
s s
f r
s s
s s
f r
s s
r
R R
L L
R R
f x
L L
α
β
ψ ψ θ
ψ ψ θ
ω
⎡ ⎤
− +⎢ ⎥
⎢ ⎥
⎢ ⎥
− += ⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎣ ⎦
，
( )
( )
( )
1
cos
1
sin
f r
s
f r
s
L
g x
L
α
β
ψ ψ θ
ψ ψ θ
⎡ ⎤
−⎢ ⎥
⎢ ⎥=
⎢ ⎥
−⎢ ⎥
⎣ ⎦
,
1 0
0 1
0 0
0 0
B
⎡ ⎤
⎢ ⎥
⎢ ⎥=
⎢ ⎥
⎢ ⎥
⎣ ⎦
Equation (12) is obtained by linearization and discretization of
(11).
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
1x k A k x k H k u k w k
y k C k x k v k
+ = + +⎧⎪
⎨
= +⎪⎩
(12)
Where
0 0 sin
0 0 cos
0 0 0 0
0 0 1 0
s s
f r
s s
s s
f r
s s
R R
L L
R R
A
L L
ψ θ
ψ θ
⎡ ⎤
− −⎢ ⎥
⎢ ⎥
⎢ ⎥
−= ⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎣ ⎦
, ( )
0
0
0 0
0 0
T
T
H k BT
⎡ ⎤
⎢ ⎥
⎢ ⎥≈ =
⎢ ⎥
⎢ ⎥
⎣ ⎦
1
0 0 sin
1
0 0 cos
f
r
s s
f
r
s s
L L
C
L L
ψ
θ
ψ
θ
⎡ ⎤
⎢ ⎥
⎢ ⎥=
⎢ ⎥
−⎢ ⎥
⎢ ⎥⎣ ⎦
System noise ( )kw represents the error caused by
parameters changes and linearization and discretization, while
( )kv represents errors caused by input and output
measurements.
Given system initial state, state estimates can be got
through recursive operation according to (4) to (8).
IV. SPEED SENSORLESS DTC CONTROL
Schematic diagram of speed sensorless DTC control for
SPMSM based on EKF is shown as Figure 1. Switch voltage
vector selection is shown as TABLE Ⅰ.
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3. Figure 1. Schematic diagram of speed sensorless DTC control for SPMSM
based on EKF
TABLE I. SWITCH VOLTAGE VECTOR SELECTION TABLE
△
ψ
△
T
① ② ③ ④ ⑤ ⑥
1 1 Us2 Us3 Us4 Us5 Us6 Us1
0 Us7 Us8 Us7 Us8 Us7 Us8
-1 Us6 Us1 Us2 Us3 Us4 Us5
-1 1 Us3 Us4 Us5 Us6 Us1 Us2
0 Us8 Us7 Us8 Us7 Us8 Us7
-1 Us5 Us6 Us1 Us2 Us3 Us4
V.SIMULATION RESULT
Simulation model is established using
MATLAB/SIMULINK tools. EKF algorithm is achieved by S-
function. Parameters of SPMSM are shown is TABLE Ⅱ.
TABLE II. MOTOR PARAMETERS
Motor Parameters value
Ω/sR 1.26
nP 4
HLs / 6.5
Wbr /ψ 0.175
)./( 2
mkgJ 0.0008
Parameters of EKF are as following to ensure filter not
divergent.
( ) ( )
0.3 0 0 0
0 0.3 0 0 0.2 0
; ;
0 0 3 0 0 0.2
0 0 0 0.3
0.1 0 0 0 0
0 0.1 0 0 0
0 ; 0
0 0 200 0 0
0 0 0 10 0
Q R
P x
⎡ ⎤
⎢ ⎥
⎡ ⎤⎢ ⎥= = ⎢ ⎥⎢ ⎥ ⎣ ⎦
⎢ ⎥
⎣ ⎦
⎡ ⎤ ⎡ ⎤
⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥= =
⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥
⎣ ⎦ ⎣ ⎦
(1) Given rotor speed is 500rpm, at 0.2 seconds, load torque
decreases from 30N.m to 0N.m. Corresponding stator flux
linkage and electromagnetic torque waves using traditional
DTC method and DTC-based on EKF method are respectively
shown in Figure 2-3. Flux linkage under traditional DTC is not
smooth and has sharp fluctuation especially in the start-up,
while, under DTC-based on EKF flux linkage has no obvious
ripples due to more accurate EKF observer. Torque dynamic
response time is basically the same, which indicates EKF
control method do not affect the DTC dynamic performance.
The torque ripples using DTC-based on EKF method is
significantly reduced and steady performance has been greatly
improved.
(a) DTC-based on EKF method
(b) Traditional DTC method
Figure 2. Stator flux linkage wave
(a) DTC-based on EKF method
(b) Traditional DTC method
Figure 3. Electromagnet torque wave
(2) Given rotor speed is 100rpm, at 0.1 seconds, load torque
increases from 0N.m to 6N.m. Corresponding rotor speed and
current waves using traditional DTC method and DTC-based
on EKF method are respectively shown in Figure 4-5. The
system robustness to the load disturbance using DTC-based on
EKF method has been greatly enhanced. When disturbance
occurred, speed and current is regulated fast to avoid large
overstrike and oscillation. Their waves are more smoothing,
especially in the start-up.
(a) DTC-based on EKF method
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4. (b) Traditional DTC method
Figure 4. Rotor speed wave
(a) DTC-based on EKF method
(b) Traditional DTC method
Figure 5. A phase current wave
(3) At 0.5 seconds, given rotor speed increases from 1000rpm
to 1600rpm, and Load torque is 5N.m. Corresponding rotor
speed and rotor angle waves are shown in Figure 6-7. The
estimated speed and rotor angle lag behind real speed and
angle respectively when motor starts and given speed mutates,
but it also satisfies performance demand for motor control,
while, at steady operation the estimated speed follows up real
speed only with small error, which reflects EKF algorithm has
good filtering effect.
Figure 6. Estimated speed and real speed waves
(a) Real rotor angle wave
(b) Estimated rotor angle wave
Figure 7. Rotor angle wave
VI. CONCLUSION
Speed sensorless DTC control based on EKF for SPMSM
is proposed. For a relatively accurate SPMSM model, flux
linkage and rotor speed and rotor position can be estimated
more precisely by EKF algorithm. Ripples on torque and stator
flux are reduced. The motor start problems are solved as EKF
do not need accurate initial rotor position information to
achieve observer stability convergence. DTC control for
PMSM based on EKF has just begun. Many places are
imperfect and need for further study.
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