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Zhao, S., Wallmark, O., Leksell, M. [Year unknown!]
Damping of Torsional Drive-Train Oscillationsusing a Position Sensorless PMSynRel Drive.
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1
Damping of Torsional Drive-Train Oscillations
using a Position Sensorless PMSynRel Drive
Shuang Zhao, Student Member, IEEE, Oskar Wallmark, Member, IEEE, and Mats Leksell, Member, IEEE
Abstract—Due to their high torque density and the potential
to operate without using a mechanical rotor position sensor,
permanent-magnet-assisted synchronous reluctance (PMSynRel)
machines are well suited for propulsion in hybrid-electric vehicles
(HEVs) and pure electric vehicles (EVs). In this paper, a
methodology for tuning a controller for damping torsional drivetrain oscillations in an EV/HEV are presented for the case when
the rotor position is not measured but rather estimated, commonly referred to as position sensorless control. Specifically, the
drive-train oscillations, approximated using a three-mass model
representation, and the control tuning process evaluated in an offvehicle environment are described. The proposed methodology
can be used to determine how well a certain electric drive (electric
machine and corresponding power electronic converter) is suited
for position sensorless control when used in a specified drivetrain.
Index Terms—Carrier-signal injection, drive-train oscillations,
permanent magnet-assisted synchronous reluctance machines,
position estimation, position sensorless control.
I. I NTRODUCTION
P
ERMANENT-MAGNET assisted synchronous reluctance
(PMSynRel) machines are attractive in automotive traction applications due to their high torque density which can
be realized with relatively little amount of expensive rare-earth
permanent material [1]–[3].
By exploiting the inherent magnetic saliency, PMSynRel
machines can operate without the need of a position sensor,
commonly referred to as position sensorless control, at all
speeds, including standstill. In the low-speed region, carriersignal injection methods are used for rotor position estimation
[4]–[7]. These methods (and more recent variants thereof)
rely on tracking a rotor position-dependent magnetic saliency
which, unfortunately, can be significantly affected by magnetic
saturation arising during loaded conditions [8]–[11]. A number
of approaches have been developed to mitigate this problem,
thereby realizing improved position sensorless performance in
the low-speed region [12]–[15]. However, since the carriersignal is generated by the power electronic converter (typically
switching at 5–20 kHz) and since filtering is typically required
in the demodulation process used to extract the position
information superposed on the phase currents, the resulting
bandwidth of the speed and position estimates are, typically,
significantly reduced compared to operation using a position
This research has been supported by the Swedish Hybrid Vehicle Center
(SHC) and the STandUP for Energy Research Collaboration.
S. Zhao, O. Wallmark and M. Leksell are with the Department of
Electrical Energy Conversion (E2C), KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden (e-mail:{shuang.zhao, oskar.wallmark,
mats.leksell}@ee.kth.se).
sensor or position sensorless operation above the low-speed
range where information found in the back-electromotive force
(EMF) can be used to extract position information.
Attention has been put on developing novel rotor designs
aiming to improve the capability for low-speed sensorless
control without affecting other machine performance characteristics negatively [16]–[19]. As of today, the concept of
position sensorless control has matured to a stage that it has
been implemented in certain industrial products available in
the market [20].
When hybridizing a vehicle with an electric machine, torsional drive-train oscillations, caused by the elastic coupling
between the electric machine and the wheels, may result in
unwanted mechanical stresses and vibrations. To mitigate such
problems, different control schemes, often referred to as active
oscillation damping, have been presented in the literature, see,
e.g., [21]–[24]. Generally, the mean to dampen the torsional
oscillations in these schemes is to manipulate the commanded
electric torque with a corrective signal. Disregarding how this
corrective signal is obtained, information about the rotor speed
is typically used.
If the rotor position is measured with high accuracy, using,
e.g., a resolver or a high-resolution position encoder, the rotor
speed can be estimated with a high bandwidth. In this case, the
impact of limited speed estimation dynamics can generally be
neglected when designing the active damping control scheme.
However, if the bandwidth of the speed estimation dynamics
is limited, e.g., during position sensorless operation in the
low-speed region, this effect should be considered during the
control design process.
At the design stage of active damping schemes, validation
and experimental tests in the laboratory environment (with
easier access to the measurements) are normally preferred.
Emulation of the mechanical drive-train by means of electrical
drive is often considered and general control strategies can be
found in many publications [25]–[27]. However, the design of
an emulator is project-specific due to the different mechanical
systems and testing requirements [28]–[31]. In this paper,
an experimental test bench, developed to emulate torsional
drive-train oscillations and associated damping schemes in
[32], is used to investigate the possibilities and limitations
of using active oscillation damping control in a position
sensorless PMSynRel developed for propulsion in a hybrid
electric vehicle (HEV). Particular focus is put on low-speed
carrier-signal injection based position sensorless control. A
method is proposed where the parameters associated with
speed and position estimation are tuned while controlling the
fundamental current using the measured rotor position. In this
2
way, optimized parameter settings can be obtained relatively
fast without risking to render the system unstable during
the control tuning process. By incorporating the reduced
bandwidth of the speed estimate from the position sensorless
algorithm, the active damping scheme considered is tuned
using an optimization approach.
This paper is organized as follows. In Section II, the
torsional oscillation problem and the active damping scheme in
consideration are reviewed. Section III presents an analysis of
the impact of reduced speed estimation dynamics on the active
damping control scheme. The proposed tuning methods for the
position sensorless algorithm and the active damping control
scheme are presented in Section IV. Finally, conclusions are
reported in Section V.
Fig. 2.
Three-mass representation of the considered drive-train.
to generate a small corrective torque signal Tctrl manipulating
the commanded torque T1 to eliminate the oscillation. In this
T1
Tctrl
II. T ORSIONAL D RIVE -T RAIN O SCILLATIONS
In HEV drive-trains, the coupling between the electric machine and the wheels typically can, in general, not be assumed
to have an infinite stiffness. Hence, torsional oscillations can
appear and may cause a poor driving comfort [23]. Fig. 1
shows the principal layout of an electric rear axle drive
(ERAD) used in a commercial HEV. The electric machine
(ERAD) is linked to the wheels via several elastic components (gears, drive shaft, universal joints and suspensions). In
this work, the mechanical parts of the drive-train have been
modeled using the ADAMS/Car1 software and the parameters
of the three-mass model shown in Fig. 2 have been fit to
correspond with the ADAMS/Car model2 . Two dominating
low-frequency resonances are found (7.7 Hz and 12.2 Hz);
both with relatively poor damping (7.6% and 10.2%).
In [32], the linearized drive-train model (except for the
ERAD) has been sucessfully emulated using a servo motor.
Thereby, different control schemes for the electric machine
(ERAD) can be evaluated outside a vehicle environment. The
same approach is used in this paper.
Machine mounts
Suspension
Shaft
Electrical
machine
Universal
joint
Fig. 1.
HEV.
Universal
joint
Principal layout of an electric rear axle drive used in a commercial
To eliminate torsional torque oscillations, different active
damping schemes have been proposed in the literature [21]–
[24]. The principal schematics of such active damping schemes
are depicted in Fig. 3 where the rotor speed ω̂1 (measured
or estimated) and the wheel speed ω̂w (often provided by the
ABS sensor) are fed back to the oscillation damping controller
1 ADAMS/Car
is a registered trademark of MSC Software Corporation,
Santa Ana, CA.
2 The authors would like to acknowledge Dr. Anders Hägglund, formerly at
Volvo Cars Corporation, for carrying out this model approximation.
El.
mach.
ωw
Tb
Drive
shafts
Estimator
Oscillation
damping
controller
Fig. 3.
ω1 θ 1
^1
ω
^w
ω
ABS
sensor
Principal layout of active damping control.
paper, a high-pass filter based oscillation damping controller
similar to what presented in [33] is considered. Denoting s as
the Laplace variable, the controller can be expressed as
Tctrl =
Ks
(ω̂1 − ω̂w )
s+α
(1)
where K and α are the gain and bandwidth of the controller, respectively. If the rotor position is measured with
high accuracy, using, e.g., a resolver or a high-resolution
position encoder, the rotor speed can be estimated with a high
bandwidth and ω̂1 ≈ ω1 can be assumed.
Tuning the controller expressed in (1) for the plant model
depicted in Fig. 2, i.e., selecting suitable values for K and
α, can be done in numerous ways. In this paper, K and α
are determined using the optimization procedure illustrated in
Fig. 4. The Matlab3 function “lsqnonlin” is used to minimize
the difference between the shaft torque Td and the desired
shaft torque Td∗ when a step from zero to rated torque is
commanded. Fig. 5 depicts the corresponding experimental
results with and without the controller activated and, as seen,
good damping is realized.
III. I MPACT OF R EDUCED S PEED E STIMATION DYNAMICS
In this paper, the phase-locked loop type estimator proposed
in [34] is used to estimate the rotor speed ω̂1 from the
measured rotor position θ1 . The PLL-type estimator can be
expressed as
ω̂˙ 1 = 2ρe
˙
θ̂1 = ω̂1 + ρ2 e
(2)
(3)
where the dot ( ˙ ) denotes the time derivative, ρ is the
bandwidth (in rad/s) and e = θ−θ̂ is the error signal containing
3 Matlab
is a registered trademark of The Mathworks Inc. Natick, MA.
3
Initial
control
parameters
in the right-hand-plane (RHP), known as non-minimum phase
zeros. Disregarding how the oscillation damping controller is
designed, the poles of the closed-loop system move towards
to the nonminimum-phase zeros as the loop gain increases,
and, thus, may render the closed-loop system unstable [35],
[36]. In essence, the feedback control system has a limited
gain margin which limits the performance of the oscillation
damping controller unless the rotor speed can be estimated
with a higher bandwidth. Fig. 7 shows the pole-zero map of
the closed-loop feedback system for different gains K (α is
kept constant). As seen, the poles move to the RHP when K
increases rendering the system unstable.
Plant
model
Update
control
parameters
Optimized
control
parameters
80
Fig. 4. Adopted optimization procedure for tuning the oscillation damping
scheme [32].
0.5
0.3
0.2 0.1
60 0.7
50
Imaginary axis
40
Td (Nm)
200
100
0
ωw (rad/s)
0
0.1
0.2
0.3
0.4
0.5
0.9
20
0
−20
0.9
−40
40
−60 0.7
20
−80
−60
50
0.5
−50
−40
0
0.3
−30
0.2 0.1
−20
−10
0
10
Real axis
−20
0
0.1
0.2
0.3
0.4
Fig. 6. Pole-zero map of the transfer function from T1 to ω̂ − ωw zoomed
in around the imaginary axis.
0.5
t (s)
ω1 (rad/s)
100
50
100
100
0.3
0
0.2
0.1
80 0.5
−50
0
0.1
0.2
0.3
0.4
60
0.5
Fig. 5. Experimental measurements of the shaft torque Td , wheel speed ωw
and rotor speed ω1 without (solid line) and with (dashed line) active damping
control.
Imaginary axis
t (s)
20
0
−20
−40
position information. The transfer function from the actual
speed ω1 to the estimated speed ω̂1 can be expressed as
−60
ρ2
ω̂1 =
ω1 .
(s + ρ)2
−100
(4)
4
Fig. 6 shows the pole-zero map of the transfer function
from T1 to ω̂1 −ωw , without active damping control, when the
bandwidth of the PLL-type speed estimator is selected (deliberately low) to ρ = 50 rad/s (8 Hz). As seen, three zeros appear
4 The pole-zero maps illustrated in Fig. 6–8 are zoomed in around the
imaginary axis. The rest poles and zeros locate far outside the figures and
have minor impacts to the system stability.
50
40
50
−80 0.5
−40
0.3
0.2
0.1
−30
−20
−10
1000
10
Real axis
Fig. 7. Pole-zero map of the closed-loop transfer function from T1 to ω̂−ωw ,
zoomed in around the imaginary axis, when α = 0.026 rad/s and 0.27 ≤ K ≤
2.56. The arrows indicate the direction of the pole movements when gain K
is increased.
Figs. 8–10 show the pole-zero map of the closed-loop transfer function for different values of ρ together with correspond-
4
ing experimental results for α = 0.026 rad/s and K = 2.73. As
seen, selecting ρ ≥ 200 rad/s (32 Hz) results in a relatively
well dampened system and increasing ρ further provides only
minor improvements in the resulting speed deviations.
80
Td (Nm)
200
0
0.1
60
50
0.2
ωw (rad/s)
0.3
20 0.5
0
0.1
0.2
0
0.1
0.2
0.3
0.4
0.5
0.3
0.4
0.5
10
0
−10
−20 0.5
0.3
−40
0.2
t (s)
50
−60
−80
−10
0
20
40
Imaginary axis
100
Fig. 10. Experimental measurements of the shaft torque Td and the wheel
speed ωw with oscillation damping control. The solid line corresponds to
when ρ = 200 rad/s (32 Hz) and the dashed line when ρ = 4000 rad/s
(637 Hz).
0.1
−5
0
5
Real axis
Fig. 8. Pole-zero map of the closed-loop transfer function from T1 to ω̂−ωw ,
zoomed in around the imaginary axis, for ρ = 50, 100, and 200 rad/s (8, 16,
and 32 Hz). The arrows indicate the direction of the movements of the zeros
and pole when ρ increases.
20
Td (Nm)
dic
(5)
dt
where Ldq is the differential inductance matrix, representing the inductance for small deviations around a certain
operating point. The carrier voltage can be expressed as
v̂c = [Vc cos ωc t 0]T in the estimated rotor reference frame
(governed by the rotor position estimate θ̂). The resulting
current îc = [îcd îcq ]T can be expressed as
Z
−1
v̂c dt
(6)
îc = T(θ̃)L−1
dq T(θ̃)
vc = Ldq
30
10
0
−10
0
0.5
1
1.5
2
10
ωw (rad/s)
is small in magnitude and the resistive voltage drop due to
the resulting high frequency current is considerably smaller
than the corresponding inductive voltage drop. Hence, for the
carrier current ic and the carrier voltage vc , the machine model
can be simplified to an inductive load as
5
where
0
cos θ̃
T(θ̃) =
sin θ̃
−5
0
0.5
1
1.5
2
t (s)
Fig. 9. Experimental measurements of the shaft torque Td and the wheel
speed ωw with oscillation damping control for ρ = 50 rad/s (8 Hz).
IV. P OSITION S ENSORLESS C ONTROL AT L OW S PEEDS
A. Basic Principles
As is well known, by adding a high-frequency carrier to
the voltage references sent to the modulator, position information can be detected in the resulting high-frequency carrier
current provided that the machine possesses rotor saliency.
In this paper, the pulsating voltage vector injection method
proposed in [5] is considered. At low speeds, the back EMF
− sin θ̃
cos θ̃
(7)
and θ̃ = θ − θ̂ is the rotor position estimation error. In general,
the inductance matrix Ldq can be expressed as
′
Ld L′dq
.
(8)
Ldq = ′
Lqd L′q
Assumnig a conservative system (neglecting iron losses) the
condition L′qd = L′dq can be assumed [37]. With this assumption, the resulting current vector in the estimated rotor
reference frame can be expressed as


2(L′d sin2 θ̃ + L′q cos2 θ̃ + L′dq sin 2θ̃)

L′d L′q − L′2
Vc sin ωc t


dq
îc =
 . (9)

′
′
∆L sin 2θ̃ − 2Ldq cos 2θ̃
2ωc 

′2
′
′
Ld Lq − Ldq
5
where LPF and HPF denote low- and high-pass filtering5 ,
respectively (the high-pass filtering is carried out in order to
remove the dc component caused by the fundamental phase
current). From (9), ε is found as
q
∆L′2 + 4L′2
dq
Vc
sin
2
θ̃
−
ϕ
ε=
4ωc L′d L′q − L′2
dq
, Ke sin 2θ̃ − ϕ
(11)
i
h
q
where ϕ = cos−1 ∆L′ / ∆L′2 + 4L′2
dq represents the phase
shift induced by cross-saturation (since L′dq 6= 0). Now, ε is
normalized (i.e., multiplied with 1/Ke ) and fed to the speed
and position estimator reviewed in Section III which forces
this error signal to zero. Letting ε → 0, it is evident from (11)
that ϕ 6= 0 introduces a steady-state position estimation error
θ̃∗ = ϕ/2 which must be compensated for. In this work, the
compensation method proposed in [12] is adopted. Further,
ref
the current reference trajectories (iref
d and iq ) are selected so
that a minimum differential saliency is fulfilled at all operating
points according to [15].
and for rotor position detection, respectively. In this way,
the converter is never tripped due to over current caused by
too large speed and position estimation errors. The maximum
rotor-position estimation error is then measured for different
cutoff frequencies for the case when a 100 Nm torque step is
commanded, corresponding approximately to the rated torque
of the PMSynRel in consideration. The experimental results
are plotted with corresponding simulations6 in Fig. 11 and 12
for ρ = 30 rad/s (5 Hz) and ρ = 50 rad/s (8 Hz), respectively. As seen, reasonable agreement is achieved between the
experimental and corresponding simulation results. Increasing
ρ further significantly increases both the noise content and
the speed and position estimation errors rendering a closedloop position sensorless control unfeasible. When the cutoff
frequency of the high-pass filter is higher than 25 Hz, the
maximum position estimation error becomes larger. Further,
the maximum position estimation error does not reduce significantly more when the cutoff frequency of the low-pass filter
is higher than 100 Hz.
(a)
23
3
5
15
25
35
22
21
θ̃ (deg.)
Now, the error signal ε (containing position information) is
formed by demodulating and filtering îcq as
o
n
n o
(10)
ε = LPF HPF îcq sin ωc t
20
19
18
B. Tuning Approach
17
Due to noise always found in the error signal ε, the
bandwidth of the PLL-type speed and position estimator must
be reduced significantly when the carried-signal injection
method is adopted for low-speed position sensorless control
compared to the situation when the rotor position is measured.
As highlighted in Section III, ρ must be selected above a
certain value to achieve sufficient damping. By means of
electromagnetic design, the magnitude of the error signal ε
can be manipulated [16]–[19]. The magnitude and frequency
of the carrier signal (Vc and ωc ) can also be changed but
these quantities are essentially limited by the available dc-bus
voltage and the maximum switching frequency of the power
electronic converter. In this work, Vc is selected to Vc = 115 V
which is well below the limit dictated by the available dcbus (battery) voltage (in this case, 400 V) and the carrier
frequency is selected to 500 Hz (corresponding to one tenth
of the switching frequency).
Selecting the proper cut-off frequencies of the low-pass and
high-pass filters as well as selecting the bandwidth ρ of the
speed and position estimator is not straight forward process.
To fulfill the requirements governed by damping the torsional
oscillations, the test bench presented in [32] and reviewed in
Section II is used. For the tuning process, a resolver is used
to provide the rotor position for fundamental current control.
However, similar as in [17], a carrier voltage signal is injected
to the estimated d-axis and the currents are projected to the
real and estimated rotor reference frame for current control
From the experimental results, the cutoff frequencies of the
low- and high-pass filter are selected to 100 Hz and 15 Hz,
respectively and ρ is selected to ρ = 50 rad/s (8 Hz). Fig. 13
shows a sample experimental result when closed-loop position
sensorless control is adopted. As seen, the system is stable but
with significant torsional oscillations. Tuning the oscillation
damping controller when operating position sensorless will be
covered below.
5 In this work, second-order Butterworth filters, implemented digitally, are
used.
6 The simulations are implemented in a Matlab/Simulink simulation environment where the machine model is linearized around the operating points.
16
30
40
50
60
70
80
90
100
(b)
30
3
5
15
25
35
θ̃ (deg.)
25
20
15
10
30
40
50
60
70
fc,lp (Hz)
80
90
100
Fig. 11. Maximum rotor position estimation error for ρ = 30 rad/s (5 Hz) for
different cutoff frequencies fc,lp of the low-pass filter. The legend denotes
the cutoff frequency of the high-pass filter: (a) Simulation. (b) Experimental
results.
6
(a)
computed by (4) and the corresponding experimental result
during a 100 Nm torque step. As seen, a good agreement is
achieved.
20
3
5
15
25
35
θ̃ (deg.)
18
16
25
14
20
12
10
40
50
60
70
80
90
100
(b)
30
3
5
15
25
35
θ̃ (deg.)
25
15
ω̂1 (rad/s)
30
10
5
0
20
−5
15
0
0.5
1
1.5
2
t
10
30
40
50
60
70
fc,lp (Hz)
80
90
100
Fig. 12. Maximum rotor position estimation error for ρ = 50 rad/s (8 Hz) for
different cutoff frequencies fc,lp of the low-pass filter. The legend denotes
the cutoff frequency of the high-pass filter: (a) Simulation. (b) Experimental
results.
Td (Nm)
200
100
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0
0.1
0.2
0.3
0.4
0.5
0.6
θ̃ (elec.deg)
20
10
0
−10
t (s)
Fig. 13.
Experimental position senorless torque step response without
oscillation damping control.
C. Position Sensorless Oscillation Damping Control
From the results above, it is reasonable to assume a relatively small θ̃ at all operating points, including transients.
Hence, the electrical torque can be assumed to correspond
well with the commanded torque reference and only the impact
of the dynamics between ω1 and ω̂1 , governed by (4), need
to be taken into consideration when the oscillation damping
controller is tuned. This is verified in Fig. 14 showing ω̂1
Fig. 14. Comparison between measurement and (4) (expressed in the time
domain) when T1 = 100 Nm.
The approximate speed estimation dynamics, governed by
(4), are now incorporated in the design process outlined in Section II, i.e., instead of using ω1 , ω̂1 , governed by (4), is used
in the optimization procedure depicted in Fig. 4. Thereby, new
values for K and α are obtained. Corresponding experimental
results are reported in Fig. 15 with and without oscillation
damping control. As seen, the torsional oscillation is damped
by the oscillation controller also when the rotor speed and
position are estimated. Essentially due to the reduced dynamics
of the speed estimation, the performance of the oscillation
damping control is reduced compared to the case when the
rotor position is measured.
In an actual HEV, the wheel speed is often computed using
ABS sensor and, due to limited resolution of this sensor,
accurate speed wheel speed estimates are not available at
low speeds. In this situation, only ω̂1 is available for the
active damping control. The oscillation damping controller
in consideration is evaluated in this situation by changing
the input of (1) from ω̂1 − ω̂w to ω̂1 . The corresponding
experimental measurements are reported in Fig. 16.
Comparing the results in Fig. 10 with the results reported
above, it is evident that the oscillation damping control is
significantly worsened when position sensorless control is
adopted for the specific PMSynRel and drive-train setup.
However, due to the detailed tuning process of the controller,
the obtained results highlights the limits of what can be
achieved with the specified hardware.
V.
CONCLUSION
In this paper, guidelines for tuning a controller for damping
torsional drive-train oscillations in an EV/HEV application
have been presented when the position is not measured but
rather estimated, commonly referred to as position sensorless
7
T1∗ = 20 (Nm)
Td (Nm)
40
[3]
20
[4]
0
0
0.1
0.3
0.4
0.5
[5]
T1∗ = 100 (Nm)
200
Td (Nm)
0.2
[6]
100
[7]
0
0
0.1
0.2
0.3
0.4
0.5
[8]
t (s)
Fig. 15. Oscillation damping control using the estimated rotor position and
speed (position sensorless control). The shaft torque without and with active
damping control is plotted as a solid and dashed line, respectively.
[9]
[10]
T1∗ = 20 (Nm)
Td (Nm)
40
[11]
20
[12]
0
0
0.1
T1∗
200
Td (Nm)
0.2
0.3
0.4
0.5
[13]
= 100 (Nm)
[14]
100
[15]
0
0
0.1
0.2
0.3
0.4
0.5
[16]
t (s)
[17]
Fig. 16. Oscillation damping control using the estimated rotor position and
speed (position sensorless control) when only ω̂1 is available. The shaft torque
without and with active damping control is plotted as a solid and dashed line,
respectively.
control. The drive-train oscillations were modeled using a
three-mass representation and a tuning process evaluated in
an off-vehicle environment were described. The proposed
technique can be used to determine whether a specified electric
drive (electric machine and corresponding power electronic
converter) can be suited for position sensorless control in a
given vehicle.
[18]
[19]
[20]
[21]
[22]
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