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7/27/2019 The Comparison of Sensorless Estimation Techniques for PMSM Between Extended Kalman Filter and Flux-linkage
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The comparison of Sensorless Estimation
Techniques for PMSM between Extended
Kalman Filter and Flux-linkage Observer
M.C. Huang1 and A.J.Moses1 and F. Anayi11 Cardiff University/School of Engineering, Cardiff, UK
AbstractThis paper compares two main sensorless control
techniques for permanent magnet synchronous motors
(PMSM): Flux-linkage observer (FLO) and Extended
Kalman Filter(EKF). This paper investigates experimentally
five aspects via FLO and EKF: the error of estimated
position, speed response, variable speed tuning , the ability
of speed zero crossing and convergence in the estimation for
position transition in four-quadrants.
I.
II.
INTRODUCTIONHigh performance industrial servo drives mostly employpermanent magnet synchronous motors with position
sensors such as optical encoders or revolvers for feedback.
Although sensorless technique has been developing for 15
years, it is still difficult for them to be applied in drivesbecause of some inherent drawbacks: initial position
judgment at start-up, operation in four-quadrant region,
etc. Flux linkage observer (FLO) [1, 2] is a sensorless
estimation for permanent magnet synchronous motors(PMSM) based on the integration of back EMF, with a
simple start-up method, an explicit and fast computing
scheme, and reliable precision of position. However,
initially the voltage vector must be applied to terminals of
the motor winding [3, 4], and FLO is only used onsurface-mounted permanent magnet synchronous motors(SMPMSM), not on salient-pole permanent magnet
motors such as interior PM motors. However, surface-mounted PMSMs are mostly used in AC servo drives;
therefore FLO is still the low cost and reliable scheme for
motor drive manufactures. The Extended Kalman Filter
(EKF) [5, 6] is another stochastic state observation basedon leastsquare variance estimation. Due to automatic on-
line adjustment for leastsquare variance from system
state variables, EKF has strong auto-search capability of
the initial position [7-8], which can correct the wrongconvergence in the start-up. The initial voltage vector
needs not be applied to the motor. As EKF can directly
estimate the angular velocity, sensorless operation based
on EKF for PMSM certainly gives a more rapid speedresponse than any other sensorless methods based on
incremental calculations. However, EKF requires heavy
on-line matrix computing, it is huge challenge for most
motor drive manufacturers which are using fix-pointprocessors. Besides, although EKF can actively search for
initial position, it inevitably has slower response than
FLO. In addition, incorrect convergences occasionally
happens when sensorless control based on EKF for PMmotor start up in the anticlockwise direction or operation
in four-quadrants. This paper compares the two classicsensorless methods in start-up, speed response and zero
crossing in four-quadrants.
SENSORLESS CONTROL BASED ON FLUX LINKAGE
OBSERVER
The flux linkage observer is based on the phasor diagram
of a PM synchronous motor shown in figure 1, where Uand I are voltage and current vector, s is stator flux-
linkage, f is the excitation flux linkage by PM fixed onthe d-axis, L and R are phase inductance and resistance
respectively.
Figure 1 The phasor diagram for PMSM [9]
The angle between d-axis and -axis is the estimated
rotor oriented angle. The first step is to obtain froms
)0(0)( s
t
s dtIRU &&&& += (1)
The investigation of stator flux linkage needs to be
executed in a stationary coordinate frame - while the
rotor flux linkage is certainly studied in d-q coordinates.
The angle between d-axis and -axis is the estimated rotor
position angle. Obviously, position estimation relied on
f, f. Those components of f are calculated by theintegration of the respective EMFs, which are
correspondingly obtained by subtracting the voltage dropson the winding resistance from the voltages U, U
computed and transformed from the DC link terminal
voltage based on space vector PWM theory.
s(0) represents the initial value for the synthesized fluxlinkage, because at the initial instant the rotor is aligned
by applying one voltage vector on the stator. s(0) isconsidered equal to the initial stator flux linkage vector
since at that moment the current in the stator winding is
zero. The pure integration from (1) will accumulate any dc
6540-7803-9547-6/06/$20.00 2006 IEEE.
7/27/2019 The Comparison of Sensorless Estimation Techniques for PMSM Between Extended Kalman Filter and Flux-linkage
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offset and error from the measured current so as s enters
the saturation region. Thus, the result cannot keep the
deduced stable. A high-pass filter [10] whose transferfunction is s/(s+0), can be inserted at the front of the
integrated quantities. Since the integration can be
represented in the form of 1/s as a Laplace
transformation, the eventual transfer function is 1/(s+0)
which is actually equivalent to a low pass filter. Thereby,the corresponding integration equation is amended to (2):
f
)0(00
)]([ sst
s dtIRU &&&&& ++= (2)
where 0 is cutoff frequency.
The second step is to calculate f. From figure 1, f isobtained by subtracting the voltage drop in phase
inductance from s. If the difference between inductanceL and L can be ignored and L= L= Ls, the equation for
f can be simplified to:
ILssf&&& = (3)
Therefore, in the - coordinate, (3) can be used to
calculate f, f components of f, as shown in figure 1.
Finally, the rotor positionr
can be computed by equation
(4) as:
)arctan(
f
f
r = (4)
The final deduced experimental components of fluxlinkage are displayed in figure 2. The average flux linkage
is 0.081Wb. It illustrates in figure 1that the locus of flux
linkage of PM body is almost circular as it should be
theoretically.
Flux linkage in the stat ionary co-ordination
-0.10
-0.08
-0.06
-0.04
-0.02
0.00
0.02
0.04
0.06
0.08
0.10
0 0.02 0.04 0.06 0.08 0.1
Time(s)
Fluxlinkag
e(Wb)
Flux linkage of alpha axis flux linkage of beta linkage
(a)
Flux linkage cycle by PM body
-0.10
-0.08
-0.06
-0.04
-0.02
0.00
0.02
0.04
0.06
0.08
0.10
-0.10 -0.05 0.00 0.05 0.10
flux linkage(Wb)
fluxlinkage(wb)
Flux linkage cycle by PM body
(b)
Figure 2 Components f, f (a) and the locus of vectorf(b)
III. SENSORLESS CONTROL BASED ON EXTENDED KALMANFILTER
The EKF is an optimal estimator in the least squarevariance for estimating the states of dynamic nonlinear
systems. In order to straightforwardly apply the EKF to
sensorless operation for PMSM, the PM motor nonlinear
state equations are written as (5):
+=
++=
)())(()(
)()()]([)(
ttxhty
ttBvtxftx
kk
&(5)
where (t) is the system noise, (t) is the measured noise ,
both (t) and (t) have a zero-mean white Gaussiandistribution; Q(t) is the covariance of (t); R(tk) is the
covariance of (t). PMSM has been modeled in a two-axis stationary reference frame (, ). The system state
variable ][ iix , Input vector v isis
][ vv and the system output variable is [y ] ii .
The system matrices can be written as:
=
=
+
=
=
i
ixh
L
L
BLL
iR
LL
iR
f
f
f
f
xfs
s
ss
s
ss
s
)(
,
00
00
10
01
,
0
cos
sin
)(
4
3
2
1
where RS is the motor phase resistance, Ls is motor
synchronous inductance, is the PM flux linkage.Obviously, this system is nonlinear and the variables in
the system state are closely coupled. If EKF is used to
rewrite this nonlinear model, the Jacobian matrices are :
)6(0010
0001))((
0100
0000
sincos0
cossin0
))((
)(
)(
=
=
=
=
=
=
txx
sss
s
sss
s
txx
x
htxH
LLL
R
LLL
R
x
ftxF
After relative linearization of the system equation (5),there is a partial linear relationship between states and
outputs as given in equation (7) and (8), which can be
discretized into equation (9) with constant period TC.
(7))()())(()( ttxtxFtx +=&
)()()( ttxHty += (8)
))()())(,,()( 1111 += kkkkkk ttxtxtttx (9)
is the state transfer matrix for the linear system (7) and
it is an exponential matrix. It approximately can be
simplified in [5] to be given by:ckkk FTItxtt + ))(,,( 11 (10)
For a given sampling time tk, both the optimal state
estimated sequencekk
x~ and its covariance matrixkk
P~
are
generated by EKF through two steps: The first one is aprediction step, performing a prediction of both quantities
based on the previous estimates11
~ kk
x and mean voltage
vector vk-1 actually applied to the system in the periodfrom tk-1 to tk.
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A simple rectangular integration technique is used to
produce the following recursive difference equations:
)11()
~~(
~
)]()~([~
111111111
11111
+++=
++=
dckkkkkkkkkk
ckkkkkkk
QTFPPFPP
TBxfxx
where Fk-1 is computed for x= 11~
kkx .
The second step is an innovation step, correcting the
predicted state estimate and its covariance matrix througha feedback correction scheme that makes use of the actual
measured quantities; it is realized by the following
equations .
1
)(
)12(~)(~
11
11
11
+=
=
+=
RHPHHPK
PHKPP
xHyKxx
kkkkk
kkkkkkk
kkkkkkkk
IV. EXPERIMENTAL SETUP
Figure 3 DSP-control PMSM experimental rig
Sensorless algorithms of EKF and RLKF are implemented
in the experimental set-up including : PMSM ACM2n320-
4/2-3, TMS320C31-50 DSP DSK , MOSFET inverter andDC shunt motor as a load generator. PMSM ACM2n320-
4/2-3 has the specification given in table 1:
TABLE I. PMSM SPECIFICATION
Rated power 1.4kW
Rated DC-link voltage 350V
Pole pair number 3
Phase resistance 1
Phase inductance 5.5mH
Rated speed 4000rpm
The controller chosen is TMS320C31DSK which embeds
DSP TMS320C31-50 on board; The peripheral out-data/acquisition cards are respectively based on 12- bit
Analogue-to-Digital converter AD678 and Digital-to-
Analogue converter DA767, the system sub-rack frameobeys the Euro-card standard. The operation frequency ofDSP is 50MHz. The position sensor is an optical encoder
which has a precision of 2048 pulse per turn.
V. EXPERIMENTAL RESULTA. Estimation Error Comparision:
The average error for rotor position by FLO is 8.09/cycle
shown in figure 4. The error analysis shows that the angleerror via FLO is less than 10 in absolute position angle [-
180, 10] while the error via FLO falls into range [10,
20] in the absolute position angle [10, 180]. Inevitably,the maximum error in the sensorless controlled PMSM
always occurs at the start and end of the cycle when the
error can be up to 360. This type of edge error
occasionally occupies about 10 of a cycle for FLO.
position estimation via flux linkage observer (FLO)
-360.0
-310.0
-260.0
-210.0
-160.0
-110.0
-60.0
-10.0
40.0
90.0
140.0
190.0
0 0.005 0.01 0.015 0.02
time(s)
position(degree
real position es timated pos ition error
position estimated error in flux linkage observer (FLO)
-30.0
-20.0
-10.0
0.0
10.0
20.0
30.0
0.00 0.01 0.01 0.02 0.02
time(s)
position(degree)
Real position Estimated position error
Figure 4 Estimated position (top) and error (bottom) via FLO
Figure 5 illustrates the position estimation via EKF anderrors. P40 and Q40 represent the initial value of state
covariance P0(4) and noise covariance Q0(4), which
determines the convergence precision for EKF. Explicitly
the estimation error via EKF is distributed quite even in
amplitude other than that of FLO. The average error is
about 12.4/ cycle. The error analysis denotes that mosterrors fall in the range [10, 20], a minority of errors
occur beyond the average value. The edge error shift
where maximum error occurs is about 13.
Position estimation via EKF P40=2E-12,Q40=-12
-400.0
-300.0
-200.0
-100.0
0.0
100.0
200.0
0.000 0.005 0.010 0.015 0.020 0.025 0.030
time(s)
position(degree)
rea l pos it ion est ima ted pos it ion error
(a)
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estimated pos ition error via EKF P40=2E-12,Q40=5E-12
-30.0
-20.0
-10.0
0.0
10.0
20.0
30.0
0.002 0.007 0.012 0.017 0.022
time(s)
position(degree)
real pos ition es timated position error
(b)
Figure 5 Estimated position (a) and error (b) via EKF
B. Speed Response to unit step instruction:
Because FLO estimates angular velocity through
incremental computation, obviously EKF implies a faster
speed response than FLO as shown in figure 6. The reason
is apparent that EKF can directly estimate speed throughthird state variable in equation (5) and (7). Figure 6
especially verified that the speed estimated by EKF cantake longer than that of FLO to reach steady-state. Figure
6 illustrates the overshoot for FLO would last longer thanthat of EKF.
Speed Response to unit step via Flux-linkage
observer through estimated position incrementation
-100.0
100.0
300.0
500.0
700.0
900.0
1100.0
1300.0
0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00
time(s)
speed(rpm)
Speed Command
estimated speed via FLO
Real Speed via encoder
(a)
Speed Res ponse to Unit Step Command via EKF
0.0
200.0
400.0
600.0
800.0
1000.0
1200.0
1400.0
3.15 4.15 5.15 6.15 7.15 8.15
time(s)
speed(rpm)
S p ee d Co mma n d S pe e d Re s po ns e fro m EKF
(b)
Figure 6 Speed response via FLO(top) and EKF(bottom)
C. Unilateral Speed adjustment ability comparison :
speed response via Flux-linkage observer through
estimated position incrementation
0.0
300.0
600.0
900.0
1200.0
1500.0
0 5 10 15 20
time(s)
speed(rpm)
Sp eed command es timated s peed via F LO real s peed v ia encoder
(a)
Speed Res ponse to Unilater al command via EKF
0.0
200.0
400.0
600.0
800.0
1000.0
1200.0
0.00 5.00 10.00 15.00 20.00
Time(s)
Speed(rpm
estimated spee d
unilateral speed command
real spee d
(b)Figure 7 Dynamic response to unilateral command
As shown in figure 7 , the unilateral speed command is asquare wave whose top instruction is 1100rpm and bottom
instruction is 550rpm, and load torque is 0.9 N.m.
Experimental results indicate that FLO and EKF both can
be applied in unilateral velocity adjustment. Figure 7 (a)
denotes that there are serious ripples and offsets levels inthe estimated speed through increment via FLO while
speed response via EKF explicitly tracks the real speed
locus without much high frequency ripples or offset as
shown in figure 7 (b).
D.Speed cross-zero ability comparision:
Speed response via flux-linkage observer in 4quadrant operation
-1500.0
-1000.0
-500.0
0.0
500.0
1000.0
1500.0
0 5 10 15 20
time(s)
speed(rp
m)
Speed Command Speed Response
(a)
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Figure 9 indicates that FLO and EKF both have the
capability of tracking instantaneous position transition
caused by operation in four-quadrant. However, FLOwould produce fugacious oscillation when speed
switching in 4-quadrant occurs in the angular positionwhich is not only the peak value (+) in the clockwise
direction but also the same value (-) in the anticlockwise
direction. When speed via FLO passes zero-speed, theinitial error caused via FLO is relatively larger than thatvia EKF. This also proves that EKF has a very strong self
start-up capability. The phase shift caused via FLO is still
obviously smaller than that via EKF regardless of the error
analysis mentioned above.
Speed Response in 4 QUADRA EKF Operation
-1500.00
-1000.00
-500.00
0.00
500.00
1000.00
1500.00
16 21 26 31
time(s)
Speed
(rpm)
spe ed c omma nd S pee d Respo nse via EK F
(b) Figure 10 shows that FLO and EKF both track real
position transition at the negative peak value where
PMSM passes zero-speed from clockwise to anticlockwisedirection. Fugacious oscillation happened in figure 9 (a)
also can be seen in figure 10 (a). EKF causes phase lag for
positive jump to while FLO causes no phase lag.However, FLO takes a little longer time to get close to
real track while EKF has no lag to converge into real track
promptly.
Figure 8 Speed response in four-quadrant region by FLO(a) and EKF(b)
The modern servo industry requires actuators such as
PMSM to acquire the capability of speed zero crossing,which allow actuators to switch among 4 quadrants.
Actuators should go through zero velocity promptly
without any delay from the switch instructs from
microprocessor. Figure 8 compares the characteristics of
the velocity zero-crossing ability of FLO and EKF.Bidirectional speed command would lead PMSM to pass
zero from +1000rpm to -1000rpm or from -1000rpm to
+1000rpm.Figure 8 (b) shows that estimated speed via EKF smoothly
crosses zero-speed and quickly reaches state-steady
command without any overshooting, or severe ripples,
even minor offset while estimated speed via FLO passeszero with high ripple and finally enter the state-steady
situation with a large offset.
E. Estimated Position transition for cross zero-speed:
Estimated position transition for rising speed
cross-zero via FLO
-250.0
-200.0
-150.0
-100.0
-50.0
0.0
50.0
100.0
150.0
200.0
250.0
0.125 0.175 0.225 0.275 0.325 0.375
time(s)
Position(degree)
Real Estimated
(a)
Estimated position transition for rising speed cross-zero
via EKF
-250.0
-200.0
-150.0
-100.0
-50.0
0.0
50.0
100.0
150.0
200.0
250.0
0.7 0.9 1.1 1.3 1.5
Time(s)
Position(degree)
Real Estimated
(b)
Figure 9 Estimated position transition for rising speed cross-zero
Estimated position transition for falling
spee d cross-zero via FLO
-250.0
-200.0
-150.0
-100.0
-50.0
0.0
50.0
100.0
150.0
200.0
250.0
0 .03 0 .08 0.13 0 .18 0.23
Time(s)
Position(degree)
R ea l P o sit io n E st ima te d P o sit io n via FLO
(a)
Estimated position transition for falling speed cross-
zero via EKF
-250.0
-200.0
-150.0
-100.0
-50.0
0.0
50.0
100.0
150.0
200.0
250.0
0.3 0.35 0.4 0.45 0.5 0.55 0.6
Time(s)
Position(degree)
Real Estimated
(b)
Figure 10 Estimated Position Transition for the falling speed cross-zero
VI. CONCLUSION
Sensorless control based on FLO and EKF both can be
applied in AC servo drives, EKF needs to solve parameter
tuning so as to match the requirement in the four-quadrant. EKF has a faster speed response than FLO. FLO
implemented by cheap floating point processors is also an
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alternative method for time tight development tasks. In
four quadrant regions, position error from FLO obviously
is smaller than EKF. Generally, the estimated positionprecision via FLO is a little higher than that of EKF.
REFERENCES
[1] T. Senjyu, , Vector control of permanent magnet synchronousmotors without position and speed sensors,Power ElectronicsSpecialists Conference, 1995. PESC '95 Record., 26th Annual
IEEEVolume 2, 18-22 June 1995 Page(s):759 - 765 vol.2 ;
[2] Longya Xu, Implementation and experimental investigation ofsensorless control schemes for PMSM in super-high variablespeed operation,Industry Applications Conference, 1998. Thirty-Third IAS Annual Meeting. The 1998 IEEEVolume 1, 12-15 Oct.1998 Page(s):483 - 489 vol.1;
[3] D, Yousfi, Position and speed estimation with improvedintegrator for synchronous motor, Industrial Electronics Society,1999. IECON '99 Proceedings. The 25th Annual Conference of the
IEEEVolume 3, 29 Nov.-3 Dec. 1999 Page(s):1097 - 1102 vol.3;
[4] R.. Lagerquist,,Sensorless-control of the synchronous reluctancemotor, IEEE Transactions on industry Applications, Volume30, Issue 3, May-June 1994 Page(s):673 682;
[5] S.Bolognani, Extended Kalman Filter tuning in sensorlessPMSM drives, IEEE Trans.on Industry Application, vol.39,
November/December 2003.
[6] O.Seta, an EKF based solution for the compensation of load,Friction and torque ripple in direct drive systems, Industrial
Electronics Society, 2003. IECON '03. The 29th AnnualConference of the IEEEVolume 2, 2-6 Nov. 2003 Page(s):1414 -1418 Vol.2;
[7] A.Qiu, sensorless control of permanent magnet synchronousmotor using extended kalman filter, CCECE 2004- CCGEI 2004,
Niagara Falls, May/mai 2004;
[8] Z.Xu, An extended kalman filter observer for the direct torquecontrolled interior permanent magnet synchronous motor drive,
Power Electronics and Drive Systems, 2003. PEDS 2003. TheFifth International Conference on Volume 1, 17-20 Nov. 2003Page(s):686 - 691 Vol.1;
[9] J.X. Shen, Improved speed estimation in sensorless PM brushlessAC drives,IEEE Transactionson Industry Applications Volume38, Issue 4, July-Aug. 2002 Page(s):1072 1080;
[10] H. Tajima,; Y. Hori, ,Speed sensorless field-orientation control ofthe induction machine, IEEE Transactions on Industry
Applications, Volume 29, Issue 1, Part 1, Jan.-Feb. 1993Page(s):175 - 180
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