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Mode-switching-based active control of powertrain system with nonlinear backlash and flexibility for electric vehicle during regenerative deceleration

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Regenerative braking provided by an electric powertrain is very different from conventional friction braking with respect to the system dynamics. During regenerative decelerations, the powertrain backlash and flexibility excite driveline oscillations, causing the vehicle driveability and the blended brake performance to deteriorate. In this article, system models, including a powertrain model with non-linear backlash and flexibility, and a hydraulic brake system model, are developed. The effects of the powertrain backlash and the flexibility on the vehicle driveability during regenerative deceleration are analysed. To improve the driveability and the blended braking performance of an electric vehicle further, a mode-switching-based active control algorithm with a hierarchical architecture is developed providing compensation for the backlash and the flexibility. The proposed control algorithms are simulated and compared with the baseline strategy under the regeneration braking process. The simulation results show that the vehicle driveability and the blended braking performance can be significantly enhanced by the developed active control algorithm.
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Original Article
Proc IMechE Part D:
J Automobile Engineering
2015, Vol. 229(11) 1429–1442
ÓIMechE 2014
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DOI: 10.1177/0954407014563552
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Mode-switching-based active control of
a powertrain system with non-linear
backlash and flexibility for an electric
vehicle during regenerative
deceleration
Chen Lv, Junzhi Zhang, Yutong Li and Ye Yuan
Abstract
Regenerative braking provided by an electric powertrain is very different from conventional friction braking with respect
to the system dynamics. During regenerative decelerations, the powertrain backlash and flexibility excite driveline oscil-
lations, causing the vehicle driveability and the blended brake performance to deteriorate. In this article, system models,
including a powertrain model with non-linear backlash and flexibility, and a hydraulic brake system model, are developed.
The effects of the powertrain backlash and the flexibility on the vehicle driveability during regenerative deceleration are
analysed. To improve the driveability and the blended braking performance of an electric vehicle further, a mode-switch-
ing-based active control algorithm with a hierarchical architecture is developed providing compensation for the backlash
and the flexibility. The proposed control algorithms are simulated and compared with the baseline strategy under the
regeneration braking process. The simulation results show that the vehicle driveability and the blended braking perfor-
mance can be significantly enhanced by the developed active control algorithm.
Keywords
Active control, electric powertrain, backlash non-linearity, drivetrain flexibility, regenerative braking, vehicle driveability,
control performance
Date received: 7 July 2014; accepted: 17 November 2014
Introduction
In urban driving situations, about one third to one half
of the energy of the power plant is consumed during
deceleration processes.
1–3
As one of the key technolo-
gies of electrified vehicles, a regenerative braking sys-
tem (RBS), which has the ability to convert kinetic
energy into electrical energy during decelerations, can
effectively improve the energy efficiency of a vehicle.
Therefore, it has become a popular topic of research
and development.
4–8
Automobile manufacturers and parts manufacturers
worldwide have proposed several implementations of
an RBS. Most manufactured electrified vehicles,
including the Toyota Prius, Nissan Leaf and Tesla
Model S, are now equipped with regenerative brak-
ing.
9–11
Control strategies have also been widely investi-
gated by researchers around the world.
12–14
However, the introduction of regenerative braking
into deceleration operations not only provides great
potential for improving the vehicle’s energy efficiency
but also poses tremendous challenges to existing brake
theories and control methods.
7
Compared with a con-
ventional internal-combustion engine (ICE) vehicle, an
electrified vehicle equipped with a regenerative brake
has three different braking states, i.e. friction braking,
regenerative braking and blended braking.
2
These three
braking states may occur independently or switch
between each other frequently during one braking pro-
cedure. A regenerative braking torque, which is pro-
vided by the electric motor and transmitted via the
drivetrain, is very different from conventional hydraulic
State Key Laboratory of Automotive Safety and Energy, Tsinghua
University, Beijing, People’s Republic of China
Corresponding author:
Junzhi Zhang, State Key Laboratory of Automotive Safety and Energy,
Tsinghua University, Beijing 100084, People’s Republic of China.
Email: jzhzhang@mail.tsinghua.edu.cn
at Tsinghua University on August 25, 2015pid.sagepub.comDownloaded from
friction braking, which is just mounted and modulated
on the sides of the wheels. Considering the powertrain
dynamics, the application of a motor torque during
deceleration may excite driveline oscillations and may
cause the braking performance and the driveability to
deteriorate.
15
Also, dynamic regulation of the frictional
brake may also cause pressure fluctuations in the brake
circuits, resulting in negative impacts on the blended
brake performance and the brake comfort.
8
Therefore, it is of great importance to take the
dynamic behaviour of the powertrain system into
account in developing an advanced blended braking
control system for electric vehicles.
There are mainly two aspects of a powertrain that
greatly affect the dynamic performance of a vehicle:
backlash and flexibility.
Backlash introduces a hard non-linearity into the
powertrain control loop for torque generation and
transmission. The main source of backlash is the play
between gears in the final drive and the gearbox.
16–18
Flexibility, which is mainly contributed by the elastic
and damping properties of the propeller and half-shafts,
is another excitation source of powertrain oscilla-
tions.
19,20
As with the tip-in tip-out manoeuvres of con-
ventional ICE vehicles, in an electrified vehicle, when
the driver quickly depresses and releases the accelerator
pedal or vice versa, requesting a transition from accel-
eration to regenerative braking or vice versa, driveline
backlash is first traversed, exciting the non-linearity,
and no torque is transmitted through the shaft during
this period. After contact is achieved, torque transmis-
sion is recovered, but the impact (referred to as shunt)
leads to a large half-shaft torque, which, combined with
the flexibility of the powertrain, causes unexpected dri-
veline oscillations (referred to as shuffle).
21
The problems of powertrain backlash and flexibility,
and the solutions for compensating for them, have been
investigated for conventional ICE vehicles.
22–24
However, for an electric vehicle, this problem becomes
more complicated. The response of an electric motor is
much faster than that of an ICE,
15
and the motor’s
regenerative braking torque is usually far larger than
that of an engine braking in a conventional car, which
results in more severe oscillations. Moreover, during a
process of blended regenerative and friction braking,
the application of friction braking adds a disturbance
to the load side, further affecting the backlash beha-
viour and increasing the control complexity. The drive-
train flexibility also results in power loss during torque
transmission, which affects the control performance
and the energy efficiency. Therefore, compensation for
the powertrain backlash and flexibility are important
for improving the vehicle driveability and the control
performance during regenerative deceleration and are
worthwhile researching.
In existing studies, the driveability of electric vehicles
has been discussed from various perspectives. In the
paper by Kawamura et al.,
15
a shaking vibration con-
trol system was developed and successfully
implemented in a Nissan Leaf, achieving good perfor-
mances in acceleration and driveability. In the paper by
Bottiglione et al.,
19
a feedback control system was
implemented to compensate for the effect of the half-
shaft dynamics. In the paper by Bayar et al.,
25
a differ-
ential braking strategy was developed for a hybrid
electric vehicle. However, in the above studies,
although the shaft flexibility was considered, the non-
linear backlash was neglected in favour of a linear
model. Nevertheless, the backlash and the flexibility do
exist in the real powertrain, and their effects on the
vehicle driveability during regenerative braking have
rarely been reported.
In this study, an active control algorithm of an elec-
tric powertrain for compensation for the backlash and
flexibility of the driveline is investigated. System mod-
els, including a non-linear electric powertrain model
and a hydraulic brake system model, are developed.
The impacts of the powertrain’s non-linear backlash
and flexibility on the blended braking and the vehicle
driveability are analysed. To improve further an electric
vehicle’s driveability during regenerative deceleration, a
hierarchical mode-switching-based active control algo-
rithm is developed to compensate for the powertrain’s
backlash and flexibility. Simulations of the proposed
control algorithms are carried out during regenerative
brake processes. Some of the simulation results are pre-
sented in this article.
System modelling
Figure 1 shows the overall structure of the regenerative
and hydraulic blended braking system considered in
this study. A central electric motor is installed at the
front axle of the vehicle. During deceleration, a regen-
erative braking torque, which is transmitted by the dri-
veline, is exerted on the axle. In the meantime, the
friction braking torque is modulated by the hydraulic
modulator. The blended braking torque produces the
overall braking operation.
Electric powertrain system
The electric powertrain consists of an electric motor, a
gearbox, a final drive, a differential and half-shafts.
Figure 2 shows a simplified powertrain model, whereas
a two-inertia model is used in this study. One inertia
indicates the electric motor, and the other corresponds
to the contribution of the wheel. The gearbox, consist-
ing of the transmission, the final drive, the differential
and inner and outer constant-velocity joints, is located
close to the motor inertia. The backlash contributions
throughout the powertrain are lumped together into
one single backlash angle 2a. The main flexibility of the
driveline is assumed to be in the half-shafts, represented
by the stiffness and the damping properties. Assuming
that the half-shafts are of the same length, the motor’s
output torque is considered to be equally distributed to
the left half-shaft and the right half-shaft.
1430 Proc IMechE Part D: J Automobile Engineering 229(11)
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Considering the effect of the electrical system
dynamics, the motor torque is modelled as a first-order
reaction with a small time constant tmbeing taken into
account,
2
as shown by
tm
_
Tm+Tm=Tm,ref ð1Þ
where Tmis the real value of motor torque and Tm,ref is
the reference value.
The dynamic equation for the transmitted torque
from the motor output shaft to the half-shafts is
Jm
um+bm
_
um=Tm1
i0ig
2Ths ð2Þ
where Jmis the motor inertia, bmis the viscous friction
of the motor, i0is the final drive ratio, igis the transmis-
sion ratio and Ths is the half-shaft torque.
A flexible half-shaft with non-linear backlash con-
nects the gearbox and the wheel inertia. The non-linear
model for the half-shaft torque is given by
21
Ths =khsus+chs
_
usð3Þ
us=udubð4Þ
ud=u1u3,ub=u2u3ð5Þ
where khs and chs are the stiffness coefficient and the
damping coefficient respectively of the half-shaft, udis
the shaft twist angle, ubis the position in the backlash
and u1,u2and u3are the angles at the indicated posi-
tions on the shaft, as shown in Figure 2, where
u1=um=i0igand u3=uw.
The non-linear model for the backlash position is
described by
16
Figure 1. Overall structure of the regenerative and hydraulic blended braking system.
CV: constant velocity.
Figure 2. Simplified model of the electric powertrain system.
Lv et al. 1431
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_
ub=
max 0, _
ud+khs
chs udub
ðÞ

,ub=a
_
ud+khs
chs udub
ðÞ,ub
jj
\a
min 0, _
ud+khs
chs udub
ðÞ

,ub=a
8
>
>
<
>
>
:
ð6Þ
where 2ais the backlash gap size.
The above equation (6) indicates that ubcan change
only within the backlash gap and not beyond the
boundaries. When stuck at a boundary, the shaft dis-
placement rate _
udmust be sufficiently large in relation
to the shaft twist in order for ubto start to move into
the gap.
The dynamic equation for a driven wheel is
Jw
uw+bw
_
uw=Ths Thb Tbx ð7Þ
where Jwis the wheel inertia and the road load is
divided into a friction term bwand an exogenous longi-
tudinal force Tbx on the tyre. The friction braking tor-
que Thb generated by mechanical hydraulic brake
devices can be considered as a disturbance to the wheel.
Hydraulic brake system
In order to simulate and analyse the brake blending
performance during regeneration deceleration, the
hydraulic brake system, including a master brake cylin-
der, wheel cylinders and inlet and outlet valves, was
modelled. The schematic diagram of the hydraulic
brake system is shown in Figure 3.
The structure of the wheel cylinder is simplified to be
a piston and a spring. Based on the hydraulic fluid flow
and valve dynamics, the wheel cylinder pressure is given
by
_
pFW =kFW
p2r4
FW
CdAvffiffiffiffiffiffiffiffiffi
2Dp
rfluid
sð8Þ
where kFW is the spring stiffness of the wheel brake
cylinder, rFW is the radius of the piston of the wheel
cylinder, Cdand Avare the flow coefficient and cross-
sectional area respectively of the valve opening, rfluid is
the density of the hydraulic fluid and Dpis the pressure
difference across the valve. Detailed models have been
given by Lv et al.
8
The tyre and vehicle
In some studies in powertrain control, the wheel slip is
neglected, and only the vehicle inertia is considered as a
load.
16,26
However, since the tyre behaviour is of great
importance for research on braking, to achieve more
accurate results, the well-known Pacejka magic formula
tyre model is adopted in this study. The tyre behaviour
can be accurately described under a combined longitu-
dinal slip and lateral slip condition. Also, models of
vehicle dynamics with eight degrees of freedom have
also been built in MATLAB/Simulink. Detailed models
of the vehicle and the tyre have been reported by Zhang
et al.
2,27
and Lv et al.
8
Verification of simulation models
In order to guarantee the feasibility and effectiveness of
the established system models, verification needs to be
carried out before simulations.
The electric powertrain model built in MATLAB/
Simulink has been fed with real experimental data com-
ing from the vehicle tests for calibration. As the upper
left plot in Figure 4 shows, the real test data of motor
output torque are utilized as the input of the simulation
models, and the output results of the state variables,
such as the motor speed, the wheel speed and the vehi-
cle speed, are compared with the real experimental
data. Based on Figure 4, the simulation results of those
Figure 3. Schematic diagram of the hydraulic braking system.
1432 Proc IMechE Part D: J Automobile Engineering 229(11)
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state variables match the real experimental results well,
demonstrating the feasibility and the correctness of the
powertrain model.
The detailed experimental validations of the models
for the hydraulic brake, the vehicle and the tyre, have
been previously accomplished and reported by Zhang
et al.
2,28
and Lv et al.
8
Some key parameters are listed in Table 1.
Effects of the backlash and the flexibility of
the powertrain on the electric vehicle’s
driveability and the regenerative braking
performance
In existing studies on blended braking, the powertrain
is usually modelled as a rigid system, with the non-
linear backlash and flexibility neglected, to simplify
control design modelling. Thus, the half-shaft torque is
regarded as an amplification of the motor’s output tor-
que, multiplied by the gear ratio,
1–4,29
whereas the
motor torque is open-loop controlled. However, it is
obvious that backlash non-linearity and flexibility exist
in the real world. Their impacts on the vehicle drive-
ability and the regenerative braking performance are
discussed below.
Effect of the powertrain backlash on the vehicle
driveability during regenerative deceleration
For an electric vehicle, if we define the contact as being
positive when a half-shaft is transmitting a driving tor-
que, then the contact during regenerative deceleration
is negative. Once the electric vehicle goes from driving
mode to regenerative braking, the backlash is traversed,
exciting the non-linearity. In the backlash (non-contact)
mode, the system, which is described by equations (1)
to (7) in the second section, can be implemented in a
state-space formulation, according to
_
Tm
_
us
um
uw
_
ub
0
B
B
B
B
B
B
@
1
C
C
C
C
C
C
A
=
1
tm0000
0khs
chs 000
1
Jm0bm
Jm00
000bw
Jw0
0khs
chs
1
i0ig10
0
B
B
B
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
C
C
C
A
Tm
us
_
um
_
uw
ub
0
B
B
B
B
B
B
@
1
C
C
C
C
C
C
A
+
1
tm000
00 00
00 00
01
Jw1
Jw0
00 00
0
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
A
Tm,ref
Thb
Tbx
Tl
0
B
B
B
@
1
C
C
C
A
ð9Þ
This indicates that no torque is transmitted through
the half-shaft in the backlash gap and that the motor
and the load are decoupled.
Figure 5 illustrates regenerative braking of the simu-
lated electric car. During the first second, the vehicle is
Figure 4. Verification of the simulation models using experimental data.
Table 1. Parameters of the case-study electric vehicle.
Parameter Value Units
Total mass m1360 kg
Wheelbase L2.50 m
Frontal area A2.40 m
2
Coefficient C
D
of air resistance 0.32
Nominal radius rof a tyre 0.295 m
Lv et al. 1433
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operating in the drive mode, with the backlash in posi-
tive contact. At 1s, the driver depresses the brake pedal,
requesting a deceleration operation, so that a transition
from a positive motor torque to a negative motor tor-
que occurs. At approximately 1 s, unexpected torque
oscillations occur on the half-shaft during the transi-
tion, resulting in an uncomfortable jerk in the vehicle’s
deceleration.
Focusing on the torque transition procedure shown
in Figure 6, the backlash traverse happens at approxi-
mately 1.1 s, and the contact is changed from the posi-
tive side (CO + ) to the negative side (CO–) for 40 ms.
Because the motor is decoupled from the load, i.e. the
half-shaft torque Ths is zero, based on equation (2),
within the backlash gap, all the motor output torque is
applied to its own inertia, Jm
um=Tm, which acceler-
ates the motor greatly. Therefore, when the negative
contact occurs, the speed difference between the motor
and the load exceeds –2rad/s. This speed difference
results in a great impact (shunt), which causes torque
oscillations (shuffle) on the half-shaft and an
unexpected jerk in vehicle deceleration, which indicates
the driveability, as the plots in Figure 6 show. Thus, a
controller that does not take traversing of the backlash
into account will have a very difficult task in damping
the torque oscillations.
Effect of the powertrain flexibility on the regenerative
braking performance
The flexibility is mainly contributed by the damping
and elastic characteristics of the driveline, especially the
half-shaft. Unlike backlash traverse, which occurs only
during transitions between the driving mode and the
braking mode, flexibility exists throughout the overall
operating process, including traction and deceleration.
As Figure 5 shows, during blended braking, after nega-
tive contact is established (i.e. in the contact mode),
because of the drivetrain flexibility, the half-shaft
Figure 6. Transition procedure of vehicle going from the
driving mode to regenerative braking.
Regen: regenerative.
Figure 5. Simulation results of regenerative braking under
non-active powertrain control.
Regen: regenerative.
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torque oscillations last for approximately 1 s and then
gradually decrease. Also, the torque is consumed by
the flexibility characteristics of the driveline during its
transmission. Compared with the target value, the tor-
que consumed in the drivetrain reaches 80 N m on the
half-shaft in the static state.
Motivated by observation of the above phenomena
in the time domain, the effects of the half-shaft stiffness
and the damping on the powertrain dynamics in the
contact mode are analysed in the frequency domain. In
the contact mode, the system state-space formulation is
reformulated according to
_
Tm
_
us
um
uw
_
ub
0
B
B
B
B
B
B
@
1
C
C
C
C
C
C
A
=
1
tm
0000
0khs
chs
000
1
Jm
0bm
Jm
00
000bw
Jw
0
00000
0
B
B
B
B
B
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
C
C
C
C
C
A
Tm
us
_
um
_
uw
ub
0
B
B
B
B
B
B
@
1
C
C
C
C
C
C
A
+
1
tm
000
01
chs
01
chs
02
i0igJm
02
i0igJm
001
Jw
1
Jw
0000
0
B
B
B
B
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
C
C
C
C
A
Tm,ref
Thb
Tbx
Tl
0
B
B
B
@
1
C
C
C
A
ð10Þ
The state vectors indicate that the torque transmis-
sion is recovered on the shaft and the connection
between the motor and the load is re-established.
Based on the above state-space formulation, the
transfer function from the motor torque to the half-
shaft torque can be expressed as
Ths
Tm
=Jws2+(Jwchs +bwkhs)s+bwchs
a1s3+a2s2+a3s+a4
ð11Þ
where
a1=1
i0ig
JmJw
a2=1
i0ig
Jm(bw+khs)+ 1
i0ig
Jwbm+2Jwkhs
a3=1
i0ig
Jmchs +1
i0ig
bm(bw+khs)+2(Jwchs +bwkhs )
a4=chs
i0ig
(bm+2bw)
Figure 7 shows the Bode plots of the half-shaft tor-
que to an input motor torque, based on the above
transfer function. In the low-frequency range, the dri-
veline can be regarded as rigid. When the frequency
exceeds 30 rad/s, the response of the load torque is
characterized by a resonance peak. When the damping
coefficient chs decreases (c0\c1\c2\c3\+),
both the magnitude and the phase responses are subject
to amplitude growth at approximately the resonance
point. As the frequency increases beyond 50 rad/s, the
gradient of the magnitude response gradually decreases
from 0 dB to –20 dB, whereas the phase value converges
to –90°, reflecting the elastic and damping characteris-
tics of the system.
Therefore, because of its elastic and damping prop-
erties, the electrified powertrain cannot be regarded as
simply rigid. Moreover, the open-loop dynamic
response of the electric powertrain is not satisfied for
advanced control of electric vehicles.
Active powertrain control algorithm
design
To enhance further the driveability of the electric vehi-
cle during blended braking, an active control algorithm
that considers compensation for the powertrain’s back-
lash non-linearity and flexibility is developed, as
described in this section.
Hierarchical control architecture
Based on the analysis presented above, the obvious
goal of backlash control is to reduce the impact force
of the motor on the load when contact is re-established,
realizing a ‘soft landing’ which avoids chatter. This can
be accomplished by requiring the speed difference
_
uw(t)_
um(t)=i0igto be small. We also desire a fast tra-
verse with a short time delay, because the waiting time
limits the torque tracking performance. Thus, the back-
lash compensation can be seen as a speed tracking
problem. The control objective is to track the reference
speed _
uref(t), which is the wheel speed _
uw(t) in this
study, with the motor speed _
um(t)=i0ig.
In the contact mode, active control for flexibility
compensation can be seen as a torque tracking prob-
lem, rather than speed tracking as in backlash control.
The control objective is therefore to track the target
torque Tm,tgt(t) with the half-shaft torque 2Ths(t)=i0ig.
Based on the considerations discussed above, an
overall control protocol is developed. The control pro-
tocol has a hierarchical architecture consisting of a
high-level mode-switching supervisor and a low-level
active controller, as shown in Figure 8.
Sliding-mode-based control to compensate for the
powertrain backlash
Because of its ability to address non-linearity and to
achieve a good performance and a fast response, a
sliding-mode control (SMC) scheme is adopted. As dis-
cussed in the section on hierarchical control architec-
ture, the objective in the backlash mode is to track the
reference speed with the motor speed, which is set by
Lv et al. 1435
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the high-level supervisory controller. Thus, the error
term is defined as
eBL =
_
um
i0ig
_
uwð12Þ
The first step in designing the sliding-mode control-
ler is to define the sliding surface. To guarantee zero
steady error, an integral-type sliding surface is chosen,
30
as given by
S=d
dt+l

nðeBL dt=0 ð13Þ
where nis the order of the system and lis the positive
gain. Based on equation (2), the order of the system is
one. Therefore, the sliding surface can be defined as
S=eBL +lðeBL dt=0 ð14Þ
One method for designing a control law that derives
the system trajectories to the sliding surface is the
Lyapunov direct method. The Lyapunov function used
for the single-input single-output system is
V=1
2SS ð15Þ
To ensure the stability of the system, the derivative
of the Lyapunov function should satisfy the condition
_
V=S_
S40ð16Þ
If S_
S=SKS, where Kis a positive control gain,
the above inequality can be satisfied. Hence,
S_
S=SKS )S(_
S+KS)=0 ð17Þ
Based on equation (14), the derivative of Scan be
expressed as
_
S=
um
i0ig
uw+leð18Þ
To guarantee the stability and reachability of the
SMC system, the positive control gain Kcan be chosen
as
30,31
K4
uw
um
i0ig
+l_
uw
_
um
i0ig
 ð19Þ
Figure 7. Bode plot of the half-shaft torque responses to an input motor torque.
1436 Proc IMechE Part D: J Automobile Engineering 229(11)
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Combining equations (2) and (18), neglecting the
electric motor’s dynamics and substituting _
Sinto
_
S+KS = 0, the expression obtained is
uw1
i0igJm
Tm,ref +leBL +KS =0 ð20Þ
Thus, the control input in the backlash mode can be
written as
Tm,ref =Tref,BL =i0igJm
uw+leBL +KS

ð21Þ
A block diagram of the backlash-mode active con-
troller is shown in Figure 9.
In fact, the standard SMC law for this system is
defined as
T0
ref,BL =i0igJm
uw+leBL +Ksgn(S)

ð22Þ
sgn(S)=
1, S.0
0, S=0
1, S\0
8
<
:
ð23Þ
However, it is well known that, in standard SMC,
the discontinuous signum function, sgn(S), may cause
chatter when the state trajectories are closing to the slid-
ing surfaces. To avoid this phenomenon, the above dis-
continuous term is replaced by the continuous function
KS in equation (21), which removes the chatter from
the control input.
During the blended braking process, the hydraulic
brake, whose behaviour exerts a strong influence on
the backlash control performance, also needs to be
considered.
Because the half-shaft torque in the backlash phase
is zero, based on equation (7), given a stable operating
point (Tbx,0,_
uw,0), the wheel dynamics can be repre-
sented by
uw(t)= 1
Jw
Thb(t)1
Jw
Tbx,0 +bw
_
uw,0

ð24Þ
According to equation (24), if _
Thb(t)
\0, i.e. if the
hydraulic braking torque is decreased, the wheel decel-
eration will increase, which will make the speed
Figure 8. The mode-switching-based hierarchical control protocol.
Lv et al. 1437
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difference become larger. Therefore, a greater control
effort will be required for the motor to traverse the
gap. If _
Thb(t)
.0 and thus, if the friction braking tor-
que is increased, even though the relative speed will
become smaller, the regeneration efficiency will be wea-
kened. Considering a worst case, if the frictional brake
is over-applied, the contact might be re-established on
the opposite side. To simplify the implementation, the
friction braking torque, i.e. the wheel cylinder pressure,
can be maintained during the backlash phase, accord-
ing to
_
pFW,ref(t)=0 ð25Þ
Torque tracking control to compensate for the
powertrain flexibility
As described in the section on hierarchical control
architecture, in the contact mode, compensation for the
powertrain flexibility can be seen as a torque tracking
problem. The control objective is to track the target
torque Tm,tgt with 2Ths=i0ig. A combined feedforward
and feedback control structure is adopted
32
according
to
Tm,ref =Tff +Tfb ð26Þ
where Tff is the feedforward input term required for
tracking and Tfb is the feedback component designed
to reduce the control error.
Based on the control objective, the feedforward term
can be determined by
Tff =Tm,tgt ð27Þ
Tm,tgt
=min Tb,f
,1
i0ig
Tm, lim
jj

ð28Þ
where Tm,tgt is the target value of the motor torque,
Tb,fis the total brake demand of the front wheels and
Tm, lim is the torque limit of the electric motor, calcu-
lated on the basis of the battery’s state of charge and
the motor speed.
Since the value of the half-shaft torque is unable to
be measured by a sensor in vehicle implementation,
estimation techniques have been studied by some
researchers.
19,20
Assuming that the value of the half-
shaft torque is available, the error term between the
target and the real value of the half-shaft torque can be
represented by
eCO =Tm,tgt 2Ths
i0ig
ð29Þ
For the feedback term, a linear proportional–inte-
gral–derivative control law is adopted according to
Tfb =KPeCO +KIðeCO dt+KD
d
dteCO ð30Þ
where the feedback gains KP,KIand KDare tuning
parameters.
Therefore, in the contact mode, the control input
can be written as
Tm,ref =Tref,CO
=Tm,tgt +KPeCO +KIÐeCO dt+KDd
dteCO
ð31Þ
A block diagram of the torque tracking controller
for flexibility compensation in the contact mode is
shown in Figure 10.
To meet the overall braking demand of the vehicle
in the contact mode, the reference value of the hydrau-
lic brake pressure is calculated on the basis of the total
brake demand Tb,fof the front wheels and the value of
the half-shaft torque Ths as given by
pFW,ref =k0(Tb,f2Ths)ð32Þ
where k0is the conversion coefficient of the wheel cylin-
der pressure and braking torque, determined by the
parameters of the mechanical brake devices.
Some key parameters of the active controllers pro-
posed are shown in Table 2.
Simulations
To evaluate the control performance of the proposed
algorithm during normal deceleration processes, simu-
lations are carried out in MATLAB/Simulink using the
models described in the second section.
Figure 9. Block diagram of the backlash-mode controller.
1438 Proc IMechE Part D: J Automobile Engineering 229(11)
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In the simulations, the initial braking speed is set to
40 km/h. The vehicle is powered during the first second,
and then the brake torque is requested by the driver at
1 s. The transition of contact occurs from the CO +
mode to the CO– mode, and the backlash is traversed.
The master cylinder pressure is taken as a ramp input
which stabilizes at 3MPa. The road is assumed to have
no slope and to have a dry surface with a high adhesion
coefficient of 0.8.
Conventional open-loop non-active control is taken
as a baseline. The simulation results for this baseline
control are shown in Figures 5 and 6 in the third sec-
tion. To demonstrate the importance of the active con-
trol system developed and its effectiveness in improving
the control performance, contact-mode active control
(‘contact active’) for flexibility compensation alone and
active control in combined contact and backlash modes
(‘combined active’) for both flexibility compensation
and backlash compensation are simulated. The value of
the longitudinal acceleration is selected as a parameter
to evaluate the vehicle’s driveability. Some results are
described below.
Simulation results of contact-mode active control
Figure 11 shows the simulation results of contact-mode
active control for flexibility compensation, neglecting
the effect of the backlash gap. According to Figure 11,
to compensate for the power loss during regenerative
brake torque transmission in the driveline, the motor
torque is increased and ensures that the actual half-
shaft torque reaches the target quickly, beginning at
1.4 s. However, because backlash compensation is not
involved in this control, the speed difference between
the motor and the wheel is greatly increased, to almost
–3 rad/s, during the backlash traverse. Although the
waiting time for the gap traverse is shortened to 38 ms,
this effort results in drastic oscillations in the half-shaft
torque, leading to the occurrence of unexpected jerks at
the beginning of the re-occurrence of the negative
Figure 11. Simulation results of contact-mode active control.
Figure 10. Block diagram of the contact-mode controller.
Table 2. Key parameters of the proposed active controllers.
Controller Parameter Value
SMC K3
l0.1
PID KP1.1
KI2.5
KD0.1
SMC: sliding-mode control; PID: proportional–integral–derivative.
Lv et al. 1439
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contact. Because the hydraulic brake pressure is regu-
lated on the basis of the motor’s brake torque, it also
experiences an undesired frequency modulation during
the period between 1.1 s and 1.4 s. These results indicate
that active control for backlash compensation is worth
implementing in an advanced electric powertrain con-
trol system.
Simulation results of active control in combined
contact and backlash modes
Figure 12 shows the simulation results of active control
for both flexibility compensation and backlash
compensation. As with the contact-mode active control
discussed above, this strategy also ensures that the half-
shaft torque remains consistent with the target value to
compensate for the torque loss in the driveline.
Furthermore, during the backlash mode, the active
sliding-mode controller reduces the motor torque
effort, which reduces the speed difference between the
motor and the wheel. A hydraulic brake pressure is
maintained during backlash control, as seen in the third
plot from the top in Figure 12. Although the waiting
time for the backlash traverse increases slightly to
46 ms, the half-shaft torque oscillations are significantly
reduced. The motor’s regenerative brake torque is
smoothly applied during the transition, ensuring com-
fortable deceleration.
Comparisons of three control algorithms
Figure 13 shows that both of the two active control
algorithms ensure a good torque-tracking performance
while compensating for the powertrain flexibility. The
combined active control strategy is more advantageous
than the other two with respect to the backlash com-
pensation achieved during the transition process, which
further improves the vehicle driveability, as shown in
Figure 14.
The performance and regeneration efficiencies of the
three control strategies are also compared
Figure 12. Simulation results of combined contact-mode
active control and backlash-mode active control.
Figure 13. Simulation results of the half-shaft torque under
three different control algorithms.
Figure 14. Simulation results of the vehicle driveability under
three different control algorithms.
1440 Proc IMechE Part D: J Automobile Engineering 229(11)
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quantitatively below. To compare the control perfor-
mance of each strategy, the tracking errors etof the
half-shaft torque from t=1s to t= 2.5 s are examined
by various means. The average tracking error
et
jj
and
the standard deviation of the errors setare selected as
the evaluation parameters. As Table 3 shows, the com-
bined active control strategy yields the best tracking
performance of the three strategies, as indicated by the
values
et
jj
=21:05 N m and set=55:31 N m
respectively.
The control performances of the three control strate-
gies are also compared at the vehicle level, as shown in
Table 4. The two active control algorithms developed
are more advantageous than the conventional algo-
rithm with respect to the deceleration and the regenera-
tion efficiency. Although the two improved control
algorithms perform almost the same in deceleration,
the combined active control algorithm performs better
with respect to the variation in the vehicle jerk, which
confirms the validity and effectiveness of the active con-
trol algorithm for backlash compensation.
Conclusions
During the blended braking process, the backlash and
the flexibility of the powertrain may excite driveline
oscillations which cause the vehicle driveability to dete-
riorate and adversely affect the control performance.
This article discussed the active control of the power-
train for backlash compensation and flexibility com-
pensation for an electric vehicle during regenerative
deceleration. System models, including the non-linear
powertrain model and the hydraulic brake model, were
developed. The electric powertrain model was cali-
brated and verified using experimental data from the
road tests on the vehicle, guaranteeing its feasibility
and correctness. The effects of the non-linear backlash
and the powertrain flexibility on the vehicle driveability
during regenerative deceleration were analysed. To
improve the driveability and the blended braking
performance further, a mode-switching-based active
control algorithm was developed and simulated under
normal deceleration processes. The simulation results
showed that, under combined active control, the aver-
age tracking error of the half-shaft torque and the
r.m.s. vehicle jerk were 21.05 N m and 3.38 m/s
3
respec-
tively, demonstrating the feasibility and the effective-
ness of the proposed algorithm.
Further work will be carried out in the following
areas: algorithms for backlash and half-shaft torque
online estimation; road tests on the different active con-
trol strategies.
Declaration of conflict of interest
The authors declare that there is no conflict of interest.
Funding
This work was supported by the Natural Science
Foundation of China (grant number 51475253).
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... However, simulative investigations with consideration of the torsional oscillations of the drivetrain during regenerative braking as well as during brake blending maneuvers are described in [19,20]. For the damping of these torsional oscillations, corresponding control algorithms are explained. ...
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Sudden torque change caused by the TM (traction motor) of HEV (hybrid electric vehicle) can lead to severe driveline oscillation, furthermore the backlash between engaging components in the driveline could increase the vibration amplitude. This paper focuses on the driveline oscillation of a series-parallel hybrid vehicle on launch condition. An analytic model of driveline torsional vibration considering disturbance brought by nonlinear backlash is established and simplified for the convenience of controller design. Based on the simplified model, a feedforward control strategy is proposed to suppress HEV launch vibration. However, the control effect will deteriorate sharply with the increase of equivalent backlash in the driveline system. To improve vibration attenuation performance, a feedback compensator is established and incorporated into the control system, with a robust sliding mode observer which is designed to estimate unknown disturbances as feedback signal. Numerical simulation is applied to validate vibration attenuation performance of the feedforward-feedback hybrid control strategy. Simulation results demonstrate that the feedforward-feedback control strategy has better robustness to cancel the unwanted effects of disturbance brought by nonlinear backlash compared with other control methods, which can be regarded as an effective approach to attenuate driveline oscillation and thus improve ride comfort on launch condition.
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The influence of the electromagnetic parameters on the torsional dynamics of the electric vehicle powertrain is studied by considering the electromechanical coupling effect. By adding the electromagnetic torque on the drive side, the powertrain is simplified as nonlinear drive-shaft model. The number, stability, and bifurcation conditions of the equilibrium points of the nonlinear drive-shaft model are deduced. Based on the averaged equations and the amplitude-frequency response equation, the stability and bifurcation conditions, such as fold bifurcation and Hopf bifurcation, of the resonance curve are discussed. The influence of electromagnetic parameters on the torsional dynamics is studied by simulation. It is shown that with the change of the parameters, the number as well as the stability of the equilibrium points may be changed which is affected by fold bifurcation. It is also shown that the resonance curve may lose its stability when fold bifurcation happens. By limiting the parameters in the region without fold bifurcation, the unstable dynamics of the resonance curve can be controlled.
Conference Paper
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As one of the key technologies of electrified vehicles, regenerative braking offers the capability of fuel saving by converting the kinetic energy of the moving vehicle into electric energy during deceleration. To coordinate the regenerative brake and friction brake, improving regeneration efficiency and guaranteeing brake performance and brake safety, development of special brake systems for electrified vehicles is needed. This paper presents a new type of electrically-controlled regenerative braking system (EABS) that has been developed for electrified passenger vehicles, which has the potential to be brought into production in China. By utilizing as much as possible mature components, integrating cooperative regeneration with ABS/TCS functions, EABS can achieve high regeneration efficiency and brake safety while providing system reliability, low development cost and development risk. This article describes the layout of the newly developed regenerative braking system. The operation modes and control methods of the system are introduced. Road test data from a commercialized electric vehicle prove the good performance of this system. The energy consumption of vehicle reduced by EABS developed is over 25% under ECE driving cycle.
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In this paper a novel adaptive regenerative braking control concept for electric vehicles with an electric motor at the front axle is presented. It is well known that the “phased” type regenerative braking systems of category B maximize the amount of regenerative energy during braking. However, there is an increased risk of maneuvering capability loss especially during cornering. An integrated braking controller which determines - in a single step - the desired yaw moment and allocates the braking demand between hydraulic brakes and electric motor during cornering is designed using the State Dependent Riccati Equation (SDRE) method. A unique method for deriving the State Dependent Coefficient (SDC) formulation of the system dynamics is proposed. Soft constraints are included in the state dynamics while an augmented penalty approach is followed to handle hard constraints. The performance of the controller has been evaluated for different combined cornering-braking scenarios using simulations in a Matlab/Simulink environment. For this an eight degrees of freedom (DOF) nonlinear vehicle model has been utilized. The numerical results show that the controller is able to optimize (locally) the amount of regenerative braking energy while respecting system’s constraints such as tire force saturation, vehicle yaw rate and slip angle errors.
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This paper addresses the problem of adaptive compensation of actuator nonlinearities with unknown parameters, for state feedback control of unknown multi-input multi-output (multivariable) linear time-invariant dynamic systems. A new controller parametrization is derived to deal with bilinear parameters that resulted from the two sets of the actuator nonlinearity parameters and the dynamic system parameters. To overcome the difficulty caused by a nondiagonal high-frequency gain matrix and a special actuator nonlinearity parameter structure, the new controller structure is parametrized on the basis of the high-frequency gain matrix LDU or SDU decomposition, leading to a linear parametrization coordinating the actuator nonlinearity parameters. The developed control scheme is a model reference adaptive control based design, employing an adaptive state feedback control law combined with an adaptive inverse, to deal with uncertain parameters in the system dynamics and actuator nonlinearities. To ensure the robustness with respect to actuator nonlinearity parametrization errors and the nonsingularity of the adaptive inverse, the parameter estimates are updated with an adaptive law using a combined switching-� and parameter projection modification. An application is studied for adaptive aircraft flight control with synthetic jet actuators, which have nonlinear characteristics to be compensated. Simulation results show the desired adaptive control system performance.
Conference Paper
This paper extends an LQR anti-jerk torque compensator for an automotive driveline with an optimization based handling of the backlash. The time derivative of the drive shaft torque, which is closely related to the vehicle jerk, is used as a virtual system output and regulated to zero. Thereby the controller does not need a reference model for generation of reference trajectories for the control law evaluation. The properties of the controller are discussed and the behavior is illustrated by simulation examples and verified by experiments.