Content uploaded by Chouki Sentouh
Author content
All content in this area was uploaded by Chouki Sentouh on Apr 25, 2019
Content may be subject to copyright.
1
A New Control Strategy of an Electric Power
Assisted Steering System
Alaa Marouf, Mohamed Djema¨
ı, Chouki Sentouh and Philippe Pudlo
Abstract—The control of Electric Power Assisted Steering
(EPAS) systems is a challenging problem due to the multiple
objectives and the need of several pieces of information to
implement the control. The control objectives are to generate
assist torque with fast responses to driver’s torque commands,
insure system stability, attenuate vibrations, transmit the road
information to the driver, and improve the steering wheel
returnability and free control performance. The control must also
be robust against modeling errors and parameter uncertainties.
To achieve these objectives, a new control strategy is introduced
in this paper. A reference model is used to generate ideal motor
angle that can guarantee the desired performance, and then a
sliding-mode control is used to track the desired motor angle.
This reference model is built using dynamic mechanical EPAS
model, driven by driver torque, road reaction torque and the
desired assist torque. To implement the reference model with
a minimum of sensors, a sliding-mode observer with unknown
inputs and robust differentiators are employed to estimate driver
torque, road reaction torque and the system’s states. With
the proposed control strategy, there is no need for different
algorithms, rules to switch between these algorithms, and fine-
tuning several parameters. In addition, our strategy improves the
system’s performance and robustness and reduces the cost. The
simulation results show that the proposed control structure can
satisfy the desired performance.
Index Terms—Electric power assisted steering (EPAS), sliding
mode, observer with unknown inputs, robust differentiators,
reference model.
I. INTRODUCTION
ELECTRIC Power Assisted Steering (EPAS) Systems are
replacing hydraulic power steering in many new vehicles
today. They have many advantages over traditional hydraulic
power steering systems [1]: engine independence/fuel econ-
omy, steering feel tenability, modularity and quick assembly,
compact size, and environmental compatibility. Figure. 1 ex-
plains the operation of a EPAS system. When a driver steers,
the steering torque is detected by a torque sensor (i.e., torsion
bar), placed between the steering wheel and the motor. The
measured torque used as an approximation of the torque that
the driver has applied to the steering wheel to determine
the amount of assist torque to be provided by the electric
motor. The amount of assist torque is typically calculated
from tunable torque boost based on the vehicle’s speed and
the steering torque applied to the steering wheel. The output
Alaa Marouf, M. Djema¨
ı, Chouki Sentouh and Philippe Pudlo are with
Univ. Lille Nord de France, F-59000 Lille, France. UVHC, LAMIH, CNRS,
UMR 8201, Campus du Mont Houy F-59313 Valenciennes, France
Alaa.Marouf@meletu.univ-valenciennes.fr,
Mohamed.Djemai@univ-valenciennes.fr,
Chouki.Sentouh@univ-valenciennes.fr,
Philippe.Pudlo@univ-valenciennes.fr
torque of the motor is multiplied by the gear arrangement. This
assisted torque, combined with the driver’s torque, provides the
total steering torque.
There are several research and development on EPAS sys-
tem. In Badawy et al. [1], the industrial concern, Delphi Auto-
motive, presented the bock diagram of an EPAS controller. The
controller consists of the three algorithms: the assist torque al-
gorithm, the damping algorithm and the return algorithm. The
objectives of these algorithms are illustrated in this paper [1].
Similarly, Mitsubishi Electric Corporation [2] presented a
block diagram of their controller. The control of EPAS must
generate the desired assist torque and insure the stability of the
system with a high assist gain in the boost curve. Zaremba et
al. [3] proposed a fixed-structure compensator to stabilize the
system and minimize the torque vibration. The controller com-
bines traditional lead-lag compensator networks. The proposed
control can reduce the vibrations for given assist gain but does
not make it possible to arbitrarily impose the system’s dynamic
behavior for each assist gain and slows the system response.
Chabaan and Wang [4] employed H-infinity control to generate
assist torque, improve the driver’s steering feel, and enhance
the closed-loop robustness. A driver torque estimator was
introduced. This estimator uses the measurement of the pinion
torque, the control signal and the transfer functions of the
EPAS model. The estimator uses improper transfer functions,
which are approximated by stable, proper transfer functions.
The proposed control improves system robustness but can’t
eliminate the vibrations. In addition, it requires several pieces
of information, such as steering wheel position, torque sensor
readings, motor velocity and road reaction force. In [5], a
robust integral sliding mode controller is proposed to generate
the assist torque, stabilize the system, and improve the EPAS
damping characteristics.
The driver receives the road information through the steering
system, which the driver uses to determine his/her inputs to
control the vehicle’s direction. The EPAS performance requires
that the control must deal with the transmissibility from the
tire/road to the driver to achieve a good steering feel. Sugitani
et al. [6] proposed H-infinity control to transmit the useful road
information (i.e., low frequency dynamics) of tire/road contact
to the driver in order to improve the steering feel. Chabaan and
Wang [5] defined the desired steering torque (i.e., the torque
measured by the torque sensor) so that it is proportional to road
reaction force in a specified low frequency range (K= 0.15).
Chen et al. [7] proposed a two-controller structure to attenuate
road disturbances. The structure consists of two controllers:
an H-infinity controller, used to address the driver’s feelings
and regulate the motion response, and a proportional-integral
2
(PI) controller, used to produce the assist torque according to
the command from H-infinity controller. Parmar and Hung [8]
proposed an optimal LQR controller for dual pinion EPAS.
The controller was designed for two inputs - driver torque and
motor terminal voltage - to insure stability and reduce torsion
vibrations. Increasing assist gain to reduce the steering torque
produces undesirable steering vibration [9]. To eliminate the
vibration, Kurishige et al. [9] proposed control algorithm based
on the motor velocity, which was estimated.
Friction torque prevents the steering wheel from coming
back to the exact center position. Return control is used to
accomplish the best centering quality. Damping control is
needed to avoid free control oscillations, which are usually
present at high speeds. In Kim and Song [10], a proportional-
integral-derivative (PID) controller is proposed to improve the
return-to-center position. To improve steering maneuverability
and steering wheel returnability, Kurishige et al. [11] proposed
a steering angle feedback control algorithm based on a road
reaction torque estimation. The steering feedback control gain
is adjusted according to the estimated reaction torque. Chen et
al. [12] proposed sliding-mode return control. The controller
requires a road reaction torque estimation and triggers rules
to switch between the assist control and return control. Li et
al. [13] proposed an estimator of road reaction torque to design
a fault-tolerant control in cases of faulty torque sensor.
The EPAS system control is a challenging control design
problem. The control must insure several objectives: gener-
ate assist torque, assure system stability, attenuate vibrations
caused by each of the system inputs, transmit the road informa-
tion to the driver, and improve the steering wheel returnability
and free control performance. The control must also be robust
against modeling errors and parameter uncertainties. In addi-
tion, several pieces of information are required to implement
the control, such as steering wheel angle, motor velocity, driver
torque and road reaction torque.
A new control strategy is proposed in this paper to satisfy
these several objectives without the need for different control
algorithms and rules for switching between these algorithms.
This strategy avoids the need for fine-tuning several parameters
and reduces the number of sensors. A reference model is
employed to generate ideal motor angle, which can guarantee
the desired performance; then, a sliding-mode control is used
to track the desired motor angle. The reference model is
built using dynamic mechanical EPAS model, driven by driver
torque, road reaction torque and the desired assist torque. The
model parameters are fine-tuned to attenuate the vibrations
caused by each of the system inputs. To implement the
reference model, a sliding-mode observer with unknown inputs
and robust differentiators is employed [14].
The main contribution of this paper is the adoption of
reference model to generate the desired motor angle that can
guarantee several objectives. The proposed reference model
can serve also as a reference model to control the steering
motor in the Steer-By-Wire systems [15][16][17]. The paper
also provides a robust estimator for estimating the main infor-
mation (e.g., driver torque, road reaction torque, motor speed,
motor acceleration, steering wheel speed, steering wheel ac-
celeration) for EPAS system, which is generally required to
Fig. 1. EPAS dynamic model
achieve good performance.
The remainder of this paper is organized as follows. Sec-
tion II presents the EPAS system model. Section III presents
the control objectives. Driver torque estimation is discussed
in section IV. Section V describes the control design. First,
the reference model is introduced, then the reference model
parameter choices are discussed, and finally the second-order
sliding-mode control to track the desired motor angle is
presented. Section VI explains the sliding-mode observer with
unknown inputs and robust differentiators, which is used to
implement the controller. Section VII reports the simulation
results in order to validate the performance of control strategy,
while section VIII presents our conclusions.
II. SYSTEM MODELING
This section introduces the EPAS dynamic model and
vehicle model for control design and simulation.
A. Dynamic Model of the EPAS System
The EPAS dynamic model establishes a relationship be-
tween the steering mechanism, the motor’s electrical dynam-
ics, and the tire/road contact forces. Figure. 1 shows the model
of steering mechanism equipped with brushed DC motor.
According to Newton’s laws of motion, the motion equa-
tions can be written as follows:
Jc¨
θc=−Kcθc−Bc˙
θc+Kc
θm
N−Fcsign ˙
θc+Td(1)
Jeq ¨
θm=Kcθc
N−Kc
N2+KrR2
p
N2θm−Beq ˙
θm+KtIm
−Fmsign ˙
θm−Rp
NFr(2)
The motor’s electrical dynamics is given by:
Lm˙
Im=−RmIm−Kt˙
θm+U(3)
where Jeq =Jm+R2
p
N2Mr,Beq =Bm+R2
p
N2Brand θm=
Nxr
Rp.
3
Tdis the driver torque; Fris the road reaction force; θc,θm
,xrare, respectively, the steering wheel angle, the motor’s
angular position, and the rack position. Table I defines and
quantifies the other EPAS model parameters.
TABLE I
NOMENCLATURE DEFINITIONS
Symbol Description Value [units]
Jcsteering column moment of inertia 0.04Kg.m2
Bcsteering column viscous damping 0.072N.m./(rad/s)
Kcsteering column stiffness 115N.m/rad
Fcsteering column friction 0.027N.m
Mrmass of the rack 32kg
Brviscous damping of the rack 3820N/(m/s)
Rpsteering column pinion radius 0.007m
Krtire spring rate 43000N.m/m
Jmmotor moment of inertia 0.0004Kg.m2
Bmmotor shaft viscous damping 0.0032N.m./(rad/s)
Fmmotor friction 0.056N.m
Ktmotor torque, voltage constant 0.05N.m/A
Lmmotor inductance 0.0056H
Rmmotor resistance 0.37Ω
Immotor current A
Nmotor gear ratio 13.65
The steering torque acted on the steering column Tc, the
road reaction torque Trand the assist torque Tacan be written
as:
Tc=Kcθc−θm
N(4)
Tr=RpFr(5)
Ta=NKtIm(6)
The linear model of the EPAS system can be expressed in
the following state-space form:
˙x=Ax +B1u1+B2u2
z=Czx
y=Cx =C1C2Tx=y1y2T(7)
where x=θc˙
θcθm˙
θmImTis the state vector
of the EPAS system; u1=Uis the voltage of the DC motor;
u2=TdTrTrepresents the vector of unknown inputs,
which is composed with driver torque and road reaction torque;
z=Tcis the steering torque acting on the steering column,
which is used as indicator of the steering feel because it acts
directly on the driver’s hands via the steering wheel [1][6];
and y=θcθmTis the measurement signal.
A=
0 1 0 0 0
−Kc
Jc
−Bc
Jc
Kc
Jc0 0
0 0 0 1 0
Kc
JeqN0−R2
pKr+Kc
JeqN2−Beq
Jeq
Kt
Jeq
0 0 0 −Kt
Lm
−Rm
Lm
B1=
0
0
0
0
1
Lm
, B2=
0 0
1
Jc0
0 0
0−1
NJeq
0 0
C=10000
00100, Cz=Kc0−Kc
N0 0
B. Vehicle Model
The four-wheel vehicle model is used to generate the road
reaction force Fr, which acts on the rack. This vehicle model
is used for the simulation (Figure. 2). The vehicle model input
is the front wheel angle (δ=θm/Nv), and the vehicle model
output is the road reaction force. The non-linear vehicle model
is described by the following equations [18][19]:
m˙vx= (Fx1+Fx2) cos δ−(Fy1+Fy2) sin δ+
+ (Fx3+Fx4) + mvyψ
m˙vy= (Fx1+Fx2) sin δ+ (Fy1+Fy2) cos δ+
+ (Fy3+Fy4)−mvxψ
Iz¨
ψ=lf((Fx1+Fx2) sin δ+ (Fy1+Fy2) cos δ)−
−lr(Fy3+Fy4)−Sb
2(Fy2−Fy1) sin δ+
+Sb
2(Fx4−Fx3) + Sb
2(Fx2−Fx1) cos δ
Iw˙wi=−RwFxi +Ti(8)
where vy, vxare respectively the vehicle’s lateral and
longitudinal velocity, ˙
ψis the vehicle yaw rate, βis the vehicle
slip angle, δis the front steer angle, mis the vehicle mass,
Izis the moment of inertia with respect to the vertical axle,
Iwis the wheels inertia, Rwis the wheels radius, wiis the
angular velocity of the wheel i,Tiis the torque applied at the
wheel i, and Fyi, Fxi represents respectively the lateral and
longitudinal forces.
Fig. 2. Four-wheel vehicle model
These forces are calculated using Pacejka tire model [20]:
Y(x) = Dsin [Carctan {(1 −E)Bx +Earctan (Bx)}]
(9)
where Y(X) = y(x) + Svand x=X+Sh.Shis the
horizontal shift and Svis the vertical shift. In this case Y(x)
is either the lateral force Fy, the aligning moment Mzor the
longitudinal force Fxand Xis either the slip angle αor
the longitudinal slip ratio λ. The main coefficients are: the
peak value, D; the shape factor, C; the stiffness factor, B; the
curvature factor, E. The parameters in the Pacejka formula
depend on the type of the tire and road conditions.
Once the vehicle dynamics have been determined, we can
calculate the road reaction force, which acts on the rack by
using the geometry of the front axle that links the front tires to
the rack (Figure. 3). The steering pivot axis shown in Figure. 3
4
Fig. 3. Front axle
has a geometrical configuration specific to each vehicle [21].
The steering pivot axis is characterized by two angles: camber
angleγpcaster angle γc. The camber angle γpis the angle
between the steering pivot axis and the vertical axis of the
vehicle when viewed from the front or the rear. The caster
angle γcis the angle towards which the steering pivot axis is
tilted, forward or rearward from the vertical, as viewed from
the side. The interaction between the steering pivot axis and
the ground defines the scrub radius dsand the geometric trail
cg.The road reaction force Frapplied on the rack due to
longitudinal forces, lateral forces and aligning torque is given
by [21]:
Fr=cos (γp) cos (γc)
ln
Ms(10)
where lnis the length of knuckle arm and Msis the reaction
torque around the steering axis, given by:
Ms=−dscos (γp) cos (γc) (Fx1−Fx2)
−cgcos (γp) cos (γc) (Fy1+Fy2)
−cos (γp) cos (γc) (Mz1+Mz2)
The block diagram shown in Figure. 4 illustrates the system
inputs and outputs.
Fig. 4. System block diagram
III. CONTROL OBJECTIVES
The objectives of the EPAS control are to generate assist
torque, attenuate vibration, supply road information, improve
steering wheel returnability and provide damping compensa-
tion.
−6 −4 −2 0 2 4 6
−40
−30
−20
−10
0
10
20
30
40
Driver Torque (NM)
Assist Torque (NM)
high
speed
0 mpH
Ka
Fig. 5. Assist boost curve characteristics
A. Generate assist torque
The main objective of EPAS control is to reduce the driver’s
steering effort by generating assist torque using electric motor.
Figure. 5 shows the assist boost curve characteristics, obtained
from the car manufacturer. The torque required to be produced
by the motor (i.e., desired assist torque) depends on the driver’s
torque and vehicle speed. The torque assistance is high at low
vehicle speeds; as the vehicle speed increases, the amount
of assistance decreases. The EPAS control must insure the
generation of the desired assist torque, a stable system with
large amount of assistance, and the motor’s rapid response to
the driver torque command.
B. Attenuate vibration
The EPAS system is a resonance system, with the torsion as
spring and the motor as mass. This resonance can produce vi-
brations transmitted to steering wheel and worsen the steering
feel. Figure. 6 shows the resonance characteristics in the open
−80
−60
−40
−20
0
20
Magnitude (dB)
100101102
−180
−90
0
90
180
270
360
Phase (deg)
Bode Diagram
Frequency (rad/sec)
To driver torque
To reaction torque
To control
Fig. 6. Open-loop frequency response of the steering torque to driver torque,
reaction force and control input
5
loop, with maximum peak at 63 rad/sec (10 Hz). As Figure. 6,
the resonance is a function of driver torque, reaction force
and control input. It is necessary to compensate the vibrations
caused by each of these system inputs in order to ensure good
performance in all operating conditions (e.g., variety of assist
torques, the driver’s steering inputs and the road conditions).
C. Supply road information
The forces generated between the tire and the ground
surface (i.e., road information) mainly determine the vehicle’s
motion and the driver’s maneuvers. To achieve a good steering
feel, the driver must receive appropriate amount of road
information [6][22]. The driver must ”feel” the road and must
not feel the fluctuations.
D. Improve steering wheel returnability and provide damping
compensation
At low vehicle speeds, the friction torque prevents the
steering wheel from coming back to the exact center position.
At high vehicle speeds, the self-aligning torque increases and
makes the steering wheel return to center with excessive
overshooting and oscillations [1][2][10][12]. This character-
istic leads to generating an unexpected yaw motion of the
vehicle. For these reasons, return control is needed to insure
that the steering wheel comes back to the center position and
damping control is needed at high vehicle speeds to avoid the
oscillations.
IV. DRIVER TORQUE ESTIMATION
Traditionally, in EPAS systems, the driver torque is not
measured directly. Thus, the measured torque Tcis used as
approximation of the driver torque to determine the desired
assist torque. To show the effects of using measured torque
instead of driver torque to determine the desired assist torque
(Tar =KaTc, Tar =KaTd), the transfer functions (G=
θm
Td, H =Tc
Td) are calculated, using equations (1) and (2)
and neglecting the electronic dynamic and the motor’s internal
control. This implies that the motor’s electromagnetic torque is
equal to the motor’s voltage V=Tar /N. Here, the function H
indicates the steering feel and the difference between the driver
torque and the measured torque. The function G indicates the
speed response of the motor’s assistance to the driver torque
command.
•Case 1 : Tar =KaTc
G1=θm
Td
=
Kc
N(1 + Ka)
Fm+K2
c
N2KaFc−K2
c
N2(1 + Ka)(11)
H1=Tc
Td
=Kc
Fc1 + (Kc−Fc)
NG1(12)
•Case 2: Tar =KaTd
G2=θm
Td
=
1
N(FcKa+Kc)
FcFm−K2
c
N2
(13)
H2=Tc
Td
=Kc
Fc1 + (Kc−Fc)
NG2(14)
0 0.5 1 1.5 2 2.5 3
0
5
10
15
Motor angle responce
Time (sec) (sec)
Angle (rad)
Ka=1
Ka=2
Ka=3
Ka=4
Ka=5
−− with driver torque amplification
.. with measured torque amplification
Fig. 7. Motor angle with the variation of assist gain, with driver torque
amplification and measured torque amplification
where Fc=Jcs2+Bcs+Kcand Fm=Jeqs2+Beq s+
Kc+R2
pKr
N2.
When the measured torque is used (case 1) to determine the
desired assist torque, the value of the poles varies accordingly
to the assist gain (Ka), so the system speed and its oscillatory
behavior is related to the assist gain. This control strategy
increases the control requirements and the complexity of the
design. The control must insure system stability and attenuate
the vibrations with the variation of the assist gain, without
slowing the system response resulting from the compensa-
tion phase. In addition, it must attenuate the road reaction
torque transmitted to the steering torque, while simultaneously
making the driver recognize the road reaction torque and the
changes in the road conditions [3][9].
When the driver torque is used to determine the desired
assist torque (case 2), the value of the poles of the system
does not depend on the assist gain. Thus, the control design is
simpler, and the maximum assist torque that can be generated
depends on the motor and gearbox characteristics. Figure. 7
shows the time response of the system in both cases (1 & 2)
with the variation of the assist gain. As shown in Figure. 7,
with the increased assist gain, the system response with driver
torque amplification is considerably faster than the system with
measured torque amplification. Chabaan and Wang [5] report
other important reasons for driver torque estimation, such as
the difference between driver torque and measured torque
in the transition dynamics and the difficulty of determining
exactly the values of steering column stiffness and steering
column inertia. So, it is highly desirable to estimate driver
torque.
V. CONTROL DESIGN
Figure. 8 illustrates the EPAS system’s block diagram and
the location of current (torque) controller C. As can be seen in
this figure, the current controller is inside the armature’s loop
and the load torque disturbances are outside the control loop.
Consequently, the torque control, without the compensation
loops, can’t provide robust control of the motor angle. The
changes in the load torque will be reflected in the changes
6
of motor velocity and acceleration, thus in the changes of
steering wheel angle, speed and acceleration, which results
in a worse steering feel. Therefore, Compensation algorithms
are needed, which leads to several control loops. On the other
hand, the assist torque generated by the motor and the front
wheel steering angle is proportional to the assist torque motor
angle. By controlling the motor angle, the assist torque is
generated and the vehicle’s directional control is insured. This
led us to choose the motor angle as a reference signal.
Fig. 8. Block diagram of the EPAS system with conventional current
controller
A. Reference Model
Using the equations (1) and (2) and neglecting the non
linearities, which represent friction, the actual angular position
of the motor is given by:
θm=Kθm −Jeq ¨
θm−Beq ˙
θm−Jc
N¨
θc−Bc
N˙
θc
+Kθm Ta
N+Td
N−Tr
N(15)
where Kθm =KrR2
pN2,Ta=NKtImis the assist torque,
and Tr=RpFris the road reaction torque. Using equation
(15), it can be shown that the motor angle depends on the
inputs: driver torque, road reaction torque and assist torque.
Thus, the control in position can respond to the changes in
each of these inputs.
In addition, the motor angle depends on the design parame-
ters (i.e., Jeq, Beq, Jc, Bc) of the electro-mechanical system.
These parameters can be chosen to impose ideal performance.
We define the reference motor angle θmr as follows:
θmr =Kθm −Jeqr ¨
θm−Beqr ˙
θm−Jcr
N¨
θc−Bcr
N˙
θc
+Kθm Tar
N+Td
N−Trr
N(16)
where:
-Tar is the desired assist torque (i.e., the output of the
boost curve).
-Trr =KfTr:Kfis a programmable gain chosen
to improve the steering maneuverability and return-
to-center performance. In this way, the road reaction
torque is used to add a steering torque through the
electric motor. This allows the driver to feel the road
reaction torque clearly through the steering wheel
and improves the steering feel.
-Jeqr , Beqr , Jcr and Bcr are programmable gains
chosen to force the desired performance of the EPAS
system. The choice of these parameters can be con-
sidered as the ideal steering system design or as the
fine-tuning of the control gains.
-˙
θc,¨
θc,˙
θmand ¨
θmare, respectively, the real steering
wheel speed, real steering wheel acceleration, real
motor speed and real motor acceleration.
The model has as inputs driver torque, desired assist torque,
road reaction torque, steering wheel speed, steering wheel
acceleration, motor speed and motor acceleration. These inputs
give a good evaluation of the steering column dynamics,
as well as the interaction between the road surface and the
vehicle’s tires. The model output is the reference motor angle.
The proposed reference model can serve also as a reference
model to control the steering motor in the Steer-By-Wire
systems [15] [16].
Fig. 9. Structure of our control strategy
The reference model can’t be implemented if the system
inputs are not available. These inputs are very important
information in the design of the EPAS control system. In
the literature, it is known that the driver torque is required to
determine the amount of assist torque, while the road reaction
torque is used to improve the steering wheel returnability and a
good steering feel [23][11]. The speed is used to add damping
compensation [1][10][2], and the acceleration is used to add
inertia compensation [1][2]. The estimation of these inputs
is necessary to reach a good performance and reduce system
costs.
Figure. 9 shows the structure of our control strategy. The
control strategy comprises the following steps: 1) fine-tuning
the reference model parameters to eliminate the vibrations;
2) estimating the inputs of the reference model; 3) generating
the desired motor angle, using equation (16); and 4) designing
control law to track the desired motor angle.
B. Fine-tuning the Reference Model Parameters
Now, it is time to choose the parameters:
Jeqr , Beqr , Jcr , Bcr and Kf .
1) Choosing Jeqr , Beqr , Jcr and Bcr :
The objective is to choose the parameters of the reference
model (i.e., Jeqr , Beqr , Jcr , Bcr ) in order to eliminate or
reduce the vibration caused in response to the driver torque,
reaction torque and control input. The effect of each parameter
on the steering torque (z=Tc) is studied using the EPAS
dynamic model (7). z=Tcis used as indicator of the steering
7
feel because it acts directly on the driver’s hands via the
steering wheel [1][6]. The other possible choice is the steering
wheel velocity: ˙
θc.
−50
−40
−30
−20
−10
0
10
20
Magnitude (dB)
100102
−180
−135
−90
−45
0
Phase (deg)
Tc/Td
Frequency (rad/sec)
−60
−40
−20
0
20
Magnitude (dB)
100102
−180
−90
0
90
180
Phase (deg)
Tc/Tr
Frequency (rad/sec)
−80
−60
−40
−20
0
20
Magnitude (dB)
100102
−90
0
90
180
270
360
Phase (deg)
Tc/u1
Frequency (rad/sec)
Jeq1 = Jeq
Jeq2 < Jeq1
Jeq3 < Jeq2
Fig. 10. Influence of the equivalent inertia parameter on frequency response
•The influence of the equivalent inertia, Jeq:
Decreasing the value of the equivalent inertia results in grad-
ually moving the poles of the system towards a negative
direction in order to reduce the vibrations caused in response
to driver torque, road reaction torque and control input and to
increase the amplitude of the gain response to road reaction
torque and control inputs at high frequencies. Figure 10 shows
these effects; the other parameters are fixed to their original
value (table I).
−80
−60
−40
−20
0
20
Magnitude (dB)
100102
−180
−135
−90
−45
0
Phase (deg)
Tc/Td
Frequency (rad/sec)
−60
−40
−20
0
20
Magnitude (dB)
100102
−180
−90
0
90
180
Phase (deg)
Tc/Tr
Frequency (rad/sec)
−80
−60
−40
−20
0
20
Magnitude (dB)
100102
−90
0
90
180
270
360
Phase (deg)
Tc/u1
Frequency (rad/sec)
Jc1 = Jc
Jc2 > Jc1
Jc3 > Jc2
Fig. 11. Influence of the steering column inertia parameter on frequency
response
•The influence of the steering wheel inertia, Jc:
Increasing the value of the steering column inertia results in
a slight change in the poles’ position in order to increase the
gain in lower frequencies in response to road reaction torque
and control inputs, to decrease the magnitude of resonance
peak, and to decrease the amplitude of gain response to driver
torque at high frequencies. Figure. 11 shows these effects; the
other parameters are fixed to their original value (table I).
•The influence of the damping parameters, Bc&Beq :
Figures 12 and 13 show the effects of increasing the steering
column damping parameter Bcand equivalent damping
Beq; the other parameters are fixed to their original value
(table I). It can be shown that increasing the steering column
−50
−40
−30
−20
−10
0
10
20
Magnitude (dB)
100102
−180
−135
−90
−45
0
Phase (deg)
Tc/Td
Frequency (rad/sec)
−60
−40
−20
0
20
Magnitude (dB)
100102
−180
−90
0
90
180
Phase (deg)
Tc/Tr
Frequency (rad/sec)
−80
−60
−40
−20
0
20
Magnitude (dB)
100102
−90
0
90
180
270
360
Phase (deg)
Tc/u1
Frequency (rad/sec)
Bc1 = Bc
Bc2 > Bc1
Bc3 > Bc2
Fig. 12. Influence of the steering column damping parameter on frequency
response
−50
−40
−30
−20
−10
0
10
20
Magnitude (dB)
100102
−180
−135
−90
−45
0
Phase (deg)
Tc/Td
Frequency (rad/sec)
−60
−40
−20
0
20
Magnitude (dB)
100102
−180
−90
0
90
180
Phase (deg)
Tc/Tr
Frequency (rad/sec)
−80
−60
−40
−20
0
20
Magnitude (dB)
100102
−90
0
90
180
270
360
Phase (deg)
Tc/u1
Frequency (rad/sec)
Beq1 = Beq
Beq2 > Beq1
Beq3 > Beq2
Fig. 13. Influence of the equivalent damping parameter on frequency response
damping parameter Bcresults in decreasing the magnitude
of resonance peak. In addition, the gain in lower frequencies
in response to road reaction torque and control inputs is
increased and the response of steering torque to the driver
torque is slowed. Increasing the value of the equivalent
damping parameter Beq results in decreasing the magnitude
of resonance peak, and slowing the system’s response.
−50
−40
−30
−20
−10
0
10
20
Magnitude (dB)
100102
−180
−135
−90
−45
0
Phase (deg)
Tc/Td
Frequency (rad/sec)
−60
−50
−40
−30
−20
−10
0
10
20
Magnitude (dB)
100102
−180
−90
0
90
180
Phase (deg)
Tc/Tr
Frequency (rad/sec)
−80
−70
−60
−50
−40
−30
−20
−10
0
10
20
Magnitude (dB)
100102
−90
0
90
180
270
360
Phase (deg)
Tc/u1
Frequency (rad/sec)
EPAS SYS
MOD−REF SYS
Fig. 14. Frequency response of the EPAS system vs the reference system
model
8
The parametric analysis shows that, with a judicious choice
of these parameters, the vibrations can be attenuated, thus
obtaining good steering feel. One of the most important
parameters is the equivalent inertia (Jeq) and the damping
value (Beq and Bc) to obtain proper damping. We chose the
parameters as follows:
Jeqr = 4.52e−5Kg.m2, Jcr = 0.04 K g.m2
Beqr = 2e−3N.m(rad/s), Bcr = 0.225 N.m(rad/s)
Figure 14 shows the frequency responses of steering torque
and steering wheel velocity of the original EPAS system and
the reference system model. As this figure shows, the refer-
ence system model doesn’t have a resonance characteristics
compared to the original EPAS system.
2) choosing the gain Kf:
The objective is to choose the parameter Kfin order to
improve the steering maneuverability and the return-to-center
performance. The gain Kfis defined as (Kf=Kd·KV). The
gain Kdcan be fine-tuned to make the steering wheel come
back to center without oscillation at high speeds. The gain
Kdis set to one when the absolute driver torque is larger than
threshold ∆Tdand set to zero, otherwise. With this selection,
the desired motor angle is damped by the first term in equation
(16) and is forced to return to zero when the driver releases the
steering wheel. The gain KVis set to increase as the vehicle
speed increases so that it takes more turns of the steering
wheel to move the wheels, thus increasing vehicle stability.
This selection also makes the steering wheel comes back to
the center position faster (figure. 15).
3 4 5 6
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
Time (sec)
Angle (rad)
Steering wheel angle with RM for Kv=1
V=16 m/s
V=18 m/s
V=20 m/s
V=22 m/s
3 3.5 4 4.5 5
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Time (sec)
Angle (rad)
Steering wheel angle with RM and varying Kv value at V=22 m/s
Increasing Kv
Without control
With RM Control
Fig. 15. Steering wheel angle returnability
The road reaction torque decreases on slippery roads com-
pared to high friction roads, and the delay time between
driver torque and road reaction torque also increases. The
amplitude of the steering wheel angle increases, and the delay
time between driver toque and steering wheel angle decreases.
Consequently, the driver’s steering experience would worsen,
thus decreasing vehicle safety.
Furthermore, assisting the driver in this case of low road ad-
herence and generating the desired assist torque will increase
the wheel steering angle and decrease vehicle stability. In this
case, the gain Kvcan be fine-tuned to improve the driver’s
0 2 4 6 8 10 12
−3
−2
−1
0
1
2
3
Time (sec)
(rad)
Steering wheel angle
0 2 4 6 8 10 12
−0.4
−0.2
0
0.2
0.4
Time (sec)
(rad/s)
Yaw rate
µ=1, Kv=1
µ=0.25, Trr=Trd
µ=0.25, Kv=2.5
µ=0.25, Kv=1
Fig. 16. Steering wheel angle and yaw rate response at vehicle speed V=
10m/s, when µ= 1 and µ= 0.25
maneuvers and the vehicle stability. By defining desired road
reaction torque as a function of the steering angle and vehicle
speed based on the experimental data on a high friction road,
then comparing the estimated road reaction torque to the
desired road reaction torque, it can be decided whether or
not the road friction is low. If the absolute difference between
the desired road reaction torque Trd and the estimated road
reaction torque Tris larger than threshold ∆Trthat means the
road friction is low, and the value of Kvis set to Kv=|Trd|
|Tr|,
or set the value of Trr in equation (16) to Trd. This choice
makes the driver feel he/she steers his/her vehicle like he/she
is driving on the high road friction.
Another solution is to build a look-up table for Kvbased
on the vehicle speed and the difference between the desired
reaction torque and estimated reaction torque. In this case, the
value of Kvis chosen large enough to create solid steering
experience for safe driving. Figure. 16 shows the performance
of these solutions with road friction coefficient µ= 0.25
and vehicle speed fixed at 10m/s. As this figure shows, the
two solutions improve the steering maneuvers and the vehicle
stability, compared to the case Kv= 1. The first solution
(Trr =Trd ) is best for the steering feel: the driver steers
his/her vehicle like if he/she is driving on the high road
friction. The second solution (Kv= 2.5) is best for vehicle
stability: there is smaller time delay and smaller amplitude in
yaw rate response.
C. Second Order Sliding Mode Controller
The sliding-mode control method is used because of its
robustness, fast dynamic response and the relative ease of
implementation [24][25][26][27]. The synthesis of a sliding-
mode control is done in two steps: first, we determine a sliding
surface Son which the control objectives are developed.
Next, we derive a control law in order to bring the state
trajectory to this output and maintain this trajectory there at
all time. The major disadvantage of first-order sliding-mode
control lies in the generation of oscillations, called chattering.
The high-order sliding-mode control algorithms have been
developed to reduce this chattering phenomenon, while at
9
the same time keeping the main features of the first-order
sliding mode: precision, robustness, simplicity, and finite-time
convergence [27][28][29].
The main characteristics of higher-order sliding-mode con-
trol is that the control acts on the high-order derivative of the
sliding surface. These algorithms consider the system input u
as a new state variable, while using the derivative of uas an
actual control. In the second-order sliding modes (2-SM), the
control acts on the second derivative of the sliding variable,
¨
S= 0, and the sliding set is defined as S=˙
S= 0. Second-
order sliding-mode control has been successfully applied to
DC motor drives [30][31].
The objective of the control is to track the desired angle
generated by the reference model. The tracking errors eand
the sliding surface are defined as:
e=θmr −θm, S =ce + ˙e(17)
Differentiating equation (17), and considering equations (1),
(2) and (3), the time derivative of the sliding surface can be
expressed as:
˙
S=g(t) + cKt
RmJeq
U(18)
with: g(t) = C1θm+C2˙
θm+C3˙
Im+C4θc+C5Fr
¨
S= ˙g(t) + cKt
RmJeq
˙
U(19)
Taking driver torque, road reaction force, and current and
voltage limitations into account, it is possible to evaluate
a positive constant Φ, such that (|˙g(t)| ≤ Φ). Under this
condition it has been proved in, [27][28], that the application
of the controller (20) allows us to steer both S, ˙
Sto zero in
finite time.
U=−λ|S|0.5sign (S) + U1
˙
U1=−W sign (S)(20)
with λand Wsatisfying the following inequality:
λ > Φ, W 2≥4Φλ+ Φ
λ−Φ(21)
The control algorithm reported in equation (20) is referred to
as the ”Super-Twisting ” algorithm (see [15] for more details).
VI. OBSERVER DESIGN
In order to implement the reference model, we need to
estimate the inputs of the reference model. The sliding-mode
observers have been used for their robustness, minimization of
error dynamics, finite time convergence and reconstruction of
unknown inputs [26][32][33][34]. In the literature, it is well
known that the observer matching condition is required to
construct the observer. Recently, sliding-mode observers have
been developed for systems that don’t satisfy the observer
matching condition. According to Floquet et al. [33], addi-
tional independent output signals from the available measure-
ments are generated used sliding-mode differentiator so that
conventional sliding-mode observers with unknown inputs can
be developed for systems without observer matching condition.
Since our system doesn’t satisfy the observer matching
condition, additional outputs are generated using high-order
sliding-mode exact differentiator and then employed in the
design of the sliding-mode observer with unknown inputs.
The advantage of this method is to optimize the number of
the sensors and parameters so we don’t need to use a vehicle
model or add new sensors.
A. The Sliding-Mode Observer with Unknown Inputs
The necessary and sufficient conditions for the existence of
the sliding-mode observer with unknown inputs for the system
(7) are [32]:
•The observer matching condition, expressed in equation
(22), is satisfied:
rank (B2) = rank (C B2) = 2 (22)
•The system zeros of the triple (A, B2, C)are in the open
left-hand complex plane.
The system (7) is observable, but doesn’t satisfy the ob-
server matching condition (rank (B2)6=rank (CB2)). There-
fore, we can’t construct the sliding-mode observer directly. To
extend the existing of the observer when the observer matching
condition is not satisfied, Floquet et al. [33] proposed the idea
of generating additional outputs.
- Introduce the notion of relative degree µj∈N,
1≤j≤pof the system with respect to the output
yj(i.e., the number of times the output yjmust be
differentiated so that the unknown inputs u2appears
explicitly), thus defining formule as follows:
CjAkB2= 0 for all k < µj−1
CjAηj−1B26= 0 (23)
- Choose the integers (1 ≤µαi ≤µi)such that
rank(CaB2) = rank(B2)and Pp
i=1 µαi is mini-
mal.
Ca=C1··· C1Aα1−1··· Cp··· CpAαp−1T
Floquet et al. [33] proved that invariant zeros of triples
(A, B2, C),(A, B2, Ca)are identical. Consequently, if the
system satisfies the condition (rank(CaB2) = rank(B2), we
can design the sliding-mode observer with unknown inputs for
the following system model:
˙x=Ax +B1u1+B2u2
ya=Cax=C1··· CpAαp−1Tx(24)
Applying (23), we check that C1B2= 0 and C2B2= 0,
C1AB26= 0 and C2AB26= 0, and we choose µα1=µ1= 2
and µα2=µ2= 2, so that:
Ca=
C1
C1A
C2
C2A
=
1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0 1 0
is a full row rank with rank (B2) = rank (CaB2). Thus, the
additional outputs needed to construct the observer (y12, y22)
are the angular speed of the steering wheel and the mo-
tor speed. However, these additional outputs, which are the
derivative of the measured signal, are not available. In order
10
to obtain these additional outputs, the second-order sliding-
mode differentiator is used. The main advantages of this
differentiator are that it is exact and robust with respect
to measurement errors, input noises, and finite convergence
process time [35][36][37].
In order to estimate the state and the unknown inputs of the
system (7), the following sliding-mode observer, Walcott and
Zak [32], is proposed:
˙
ˆx=Aˆx+B1u1+L(ya−ˆya)−B2E(ˆya, ya, η)(25)
where ˆya=Caˆxand ya=y1y12 y2y22 T, and
E(ˆya, ya, η)is the discontinuous output injection,
E( ˆya, ya, η) = (ηF( ˆya−ya)
kF( ˆya−ya)k2for F ( ˆya−ya)6= 0
0for F ( ˆya−ya) = 0
where ηis the positive constant larger than the upper bound
of u2and the pair of matrices (P, F ) satisfying (26) and (27)
for some L and Q , where P and Q are symmetric positive
definite matrices (see [32] for more details).
(A−LCa)TP+P(A−LCa) = −Q(26)
F Ca=BT
2P(27)
u2=−E(ˆya, ya)(28)
B. Second-order sliding-mode differentiator
Now, we employ two second-order sliding-mode differentia-
tors to estimate the additional outputs (i.e., steering wheel and
motor speeds) and the accelerations (i.e., steering wheel and
motor accelerations). Let us consider a signal y(t)measured
in real time to be differentiated, whose third derivative has a
known Lipschitz Y. The second-order sliding-mode differen-
tiator proposed by Levant [36] is given by:
˙z0=v0
v0=−λ0Y1/3 |z0−y(t)|2/3 sign (z0−y(t)) + z1
˙z1=v1
v1=−λ1Y1/2 |z1−v0|1/2 sign (z1−v0) + z2
˙z2=−λ2Y sig n (z2−v1)
(29)
where: λ2, λ1, λ0are positive design parameters. After finite
time, the following equalities are verified [36]:
z0=y(t), z1= ˙y(t), z2= ¨y(t)(30)
The accuracy for the first and second derivative (z1, z2)
are proportional, respectively, to ε2/3 and ε1/3, where ε
is the magnitude of measurement noises [36]. The differ-
entiator design requires computing four positive constants
(λ0, λ1, λ2, Y ) for each differentiator. One possible choice
of the differentiator parameters is (λ2= 1.1, λ1= 1.5, λ0=
2) is given by Levant [36][37]. Taking into account driver
torque, reaction force, current and voltage limitations, it is pos-
sible to evaluate two constants such that |...
θc|< Yc,|...
θm|<
Ym.
VII. SIMULATION RESULTS
The purpose of this section is to validate the effective-
ness and the robustness of our control strategy. Figure. 9
shows the simulation block diagram implemented using Mat-
lab/Simulink. White noise was added to the measurement
signals and the system inputs, using Simulink uniform noise
generator. The magnitude of noise added to the measurement
signals is 5% of the signal magnitude with a sampling time
of 0.001s. The magnitude of noise added to the system inputs
(i.e., driver torque and road reaction force) is 10% of the signal
magnitude with a sampling time of 0.1s. In the following
simulations, the driver steers the steering wheel left from 0to
4Nm, right 4N m back to 3N m and then releases the steering
wheel (Figure. 17). The vehicle speed is set to 10m/s when
the amount of assistance is high.
0 2 4 6 8 10 12 14 16
−5
0
5
Time (sec)
(Nm)
Driver torque
simulated
estimated
0 2 4 6 8 10 12 14 16
−10
−5
0
5
10
Time (sec)
(Nm)
Road reaction torque
plant
estimated
Fig. 17. Driver torque and road reaction torque estimation
0 2 4 6 8 10 12 14 16
−3
−2
−1
0
1
2
Time (sec)
Steering column speed
(rad/s)
plant
estimated
0 2 4 6 8 10 12 14 16
−30
−20
−10
0
10
20
Time (sec)
Motor speed
(rad/s)
plant
estimated
Fig. 18. Steering column speed and motor speed estimation
Figures 17, 18 and 19 show the performance of the observer
and the differentiators. As shown in these figures, the unknown
inputs (i.e., driver torque and road reaction torque), additional
outputs (i.e., steering column velocity and motor speed) and
the accelerations (i.e., steering wheel and motor accelerations)
can be estimated with good accuracy. As shown in Figures 18
11
0 2 4 6 8 10 12 14 16
−6
−4
−2
0
2
4
Time (sec)
Steering column angular acceleration
(rad/s2)
plant
estimated
0 2 4 6 8 10 12 14 16
−60
−40
−20
0
20
40
Time (sec)
Motor angular acceleration
(rad/s2)
plant
estimated
Fig. 19. Steering column angular acceleration and motor angular acceleration
estimation
and 19, there is a time delay between simulated and estimated
values caused by the necessary filtered noise. In fact, we
found in our simulations that high Lipschitz Yvalue reduces
the time delay, but with the cost of increasing the sensitivity
to measurement noise. The curves in these figures show the
robustness of the observer and the second-order differentiators
versus measurement noise.
Figure. 20 shows the motor angle with respect to its
reference value. The tracking of the desired motor angle is
of good quality, even with the measurement noise. Figure. 21
shows the comparison between the desired assist torque and
generated assist torque. It can be seen that the control strategy
provides a good response to the reference assist torque. In
0 2 4 6 8 10 12 14 16
−15
−10
−5
0
5
10
15
Time (sec)
angle (rad)
Refrence and Generated motor angle
Reference
Generated
Fig. 20. Reference and generated motor angle
0 2 4 6 8 10 12 14 16
−10
−8
−6
−4
−2
0
2
4
6
8
10
Time (sec)
Assist Torque (Nm)
Refrence and Generated assist Torque
Reference
Generated
Fig. 21. Reference and generated assist torque
addition, we can see that there is a delay between the generated
assist torque and reference assist torque. This delay is due to
the difference between the real and estimated driver torque
and the big difference between the damping values in the
original EPAS system and the reference model. However, this
delay is not significant and thus will not be felt by the driver.
Figure. 22 shows the relationship between the driver torque
and the desired assist torque produced by our strategy. It can
be seen that the shape is close to the desired boost curve (see
Figure. 5). Figure. 23 shows the response of the steering wheel.
The steering wheel angle returns to its center position rapidly
without overshooting, after the driver releases the steering
wheel.
−5 0 5
−10
−5
0
5
10
Driver Torque (Nm)
Assist Torque (Nm)
Driver torque and generated assist Torque
Fig. 22. Response of the generated assist torque with respect to driver torque
0 2 4 6 8 10 12 14 16
−1.5
−1
−0.5
0
0.5
1
1.5
Time (sec)
Steering wheel angle (rad)
Steering wheel angle
Fig. 23. Steering wheel angle
In order to test the robustness to parameters uncertainty,
we performed another simulation with a 10% variation of
the parameters values (Jc,Bc,Jeq,Beq ,Kt,Rm,Lm). The
simulation results depicted in Figures 24 and 25 show the good
performance of our strategy, even with parameter uncertainty.
VIII. CONCLUSION
In this paper, a new control strategy for the EPAS system
is proposed. A reference model is employed to generate ideal
motor angle that can guarantee the desired performance, then a
second-order sliding-mode control is used to track the desired
motor angle. This model is built using the dynamic EPAS
system model, whose inputs are driver torque, road reaction
torque, and desired assist torque. The model is fine-tuned
to attenuate the vibrations, improve the response time and
improve the steering feel. To implement the reference model
without adding new sensors, a sliding-mode observer with
12
0 5 10 15
−5
0
5
Time (sec)
(Nm)
Driver torque
0 5 10 15
−10
−5
0
5
10
Time (sec)
(Nm)
Road reaction torque
0 5 10 15
−1.5
−1
−0.5
0
0.5
1
1.5
Time (sec)
Steering column speed
(rad/s)
0 5 10 15
−20
−10
0
10
20
Time (sec)
Motor speed
(rad/s)
0 5 10 15
−4
−2
0
2
4
Time (sec)
Steering column angular acceleration
(rad/s2)
0 5 10 15
−50
0
50
Time (sec)
Motor angular acceleration
(rad/s2)
simulated
estimated plant
estimated
plant
estimated plant
estimated
plant
estimated plant
estimated
Fig. 24. Estimator performance, with 10% variation of the parameters
0 5 10 15
−15
−10
−5
0
5
10
15
Time (sec)
angle (rad)
Refrence and Generated motor angle
Reference
Generated
0 5 10 15
−10
−5
0
5
10
Time (sec)
Assist Torque (Nm)
Refrence and Generated assist Torque
Reference
Generated
0 5 10 15
−2
−1
0
1
2
Time (sec)
Tar−Ta
(Refrence − Genereated) assist torque
−5 0 5
−10
−5
0
5
10
Driver Torque (Nm)
Assist Torque (Nm)
Driver torque and generated assist Torque
Fig. 25. Control performance, with 10% variation of the parameters
unknown inputs and second-order sliding-mode differentiators
were developed. Simulation results show the effectiveness and
the robustness of our strategy. Our control strategy reduces
design time, improves system performance and robustness, and
reduces system cost. It is also can be applied to the Steer-By-
Wire systems.
IX. ACKNOWLEDGMENTS
This research was supported by the French National Re-
search Agency (ANR) (VOLHAND project ”ANR-09-VTT-
14-01/06), the International Campus on Safety and Intermodal-
ity in Transportation, the European Community, the Regional
Delegation for Research and Technology, the Ministry of
Higher Education and Research, the Nord/Pas-de-Calais Re-
gion and the National Center for Scientific Research.
X. NOTATION
Symbol Description
Tddriver torque
Tcsteering torque
Trroad reaction torque
Taassist torque
Tar desired assist torque
Trd desired road reaction torque
θcsteering wheel angle
θmmotor angle
θmr reference motor angle
Jcsteering column moment of inertia
Bcsteering column viscous damping
Kcsteering column stiffness
Jmmotor moment of inertia
Bmmotor shaft viscous damping
Mrmass of the rack
Brviscous damping of the rack
Jeq equivalent inertia
Beq equivalent damping
Rpsteering column pinion radius
Nmotor gear ratio
Ktmotor torque, voltage constant
Kaassist gain
Lmmotor inductance
Rmmotor resistance
Immotor current
Umotor terminal voltage
Fcsteering column friction
Fmmotor friction
lnlength of knuckle arm
Frroad reaction force
mvehicle mass
Izyaw moment inertia
Krtire spring rate
Rwwheels radius
βvehicle slip angle
˙
ψyaw rate
δfront steer angle
Fyi lateral force
Fxi longitudinal force
REFERENCES
[1] A. Badawy, J. Zuraski, F. Bolourchi, and A. Chandy, “Modeling and
analysis of an electric power steering system,” SAE Technical Paper
1999-01-0399, 1999, doi:10.4271/1999-01-0399, 1999.
[2] T. Satake, M. Kurishige, N. Inoue, K. Ikemoto, and T. Kifuku, “Eval-
uation of eps control strategy using driving simulator for eps,” SAE
Technical Paper 2003-01-0582, 2003, doi:10.4271/2003-01-0582, 2003.
[3] A. T. Zaremba, M. K. Liubakka, and R. M. Stuntz, “Control and steering
feel issues in the design of an electric power steering system,” Proc. of
the ACC, pp. 36–40, 1998.
[4] R. C. Chabaan and L. Y. Wang, “Control of electrical power assist
systems: H∞design, torque estimation and structural stabilitys,” JSAE
Review, vol. 22, no. 4, pp. 435–444, 2001.
[5] A. Marouf, C. Sentouh, M. Djemai, and P. Pudlo, “Control of electric
power assisted steering system using sliding mode control,” IEEE
Conference on Intelligent Transportation Systems, pp. 107–112, 2011.
[6] N. Sugitani, Y. Fujuwara, K. Uchida, and M. Fujita, “Electric power
steering with h-infinty control designed to obtain road information,”
Proc. of the ACC, pp. 2935–2939, 1997.
[7] X. Chen, X. Chen, and K. Li, “Robust control of electric power-assisted
steering,” IEEE Vehicle Power and Propulsion Conference, pp. 97 – 102,
2005.
[8] M. Parmar and J. Y. Hung, “A sensorless optimal control system for an
automotive electric power assist steering system,” IEEE Transactions on
Industrial Electronics, vol. 51, no. 2, pp. 290–298, 2004.
[9] M. Kurishige, T. Kifuku, N. Inoue, S. Zeniya, and S. Otagaki, “A control
strategy to reduce steering torque for stationary vehicles equipped with
eps,” SAE Technical Paper 1999-01-0403, 1999, doi:10.4271/1999-01-
0403, 1999.
13
[10] J. Kim and J. Song, “Control logic for an electric power steering system
using assist motor,” Mechatronics, pp. 447 – 459, 2002.
[11] M. Kurishige, N. m. H. Tanaka, K. Tsutsumi, and T. Kifuku, “An eps
control strategy to improve steering maneuverability on slippery roads,”
SAE Technical Paper 2002-01-0618, 2002, doi:10.4271/2002-01-0618,
2002.
[12] B. Chen, W. Hsu, and S. Huang, “Sliding-mode return control of
electric power steering,” SAE Technical Paper 2008-01-0499, 2008,
doi:10.4271/2008-01-0499, pp. 59–64, 2008.
[13] X. Li, X. P. Zhao, J. Chen, and J. L. Men, “Controller design for soft-
disability remedy of the electric power steering system,” International
Journal of Automotive Technology, vol. 10, pp. 497–503, 2009.
[14] A. Marouf, M. Djemai, C. Sentouh, and P. Pudlo, “Driver torque and
road reaction force estimation of an electric power steering using sliding
mode observer with unknown inputs,” IEEE Conference on Intelligent
Transportation Systems, pp. 354 – 359, 2010.
[15] A. Cetin, M. Adli, D. Barkana, and H. Kucuk, “Implementation and
development of an adaptive steering-control system,” IEEE Transactions
on Vehicular Technology, vol. 59, N1, pp. 75–83, 2010.
[16] P. Setlur, J. Wagner, D. Dawson, and D. Braganza, “A trajectory tracking
steer-by-wire control system for ground vehicles,” IEEE Transactions on
Vehicular Technology, vol. 55, N1, pp. 76–85, 2006.
[17] A. Baviskar, J. Wagner, D. Dawson, D. Braganza, and P. Setlur, “An
adjustable steer-by-wire haptic-interface tracking controller for ground
vehicles,” IEEE Transactions on Vehicular Technology, vol. 58, N2, pp.
546–554, 2009.
[18] S. Glaser, S. Mammar, and C. Sentouh, “Integrated driver vehicle infras-
tructure road departure warning unit,” IEEE Transactions on Vehicular
Technology, vol. 59, N6, pp. 2757–2771, 2010.
[19] H. Imine, L. Fridman, H. Shraim, and M. Djemai, Sliding Mode Based
Analysis and Identification of Vehicle Dynamics. Springer Verlag 414:
Lecture Notes in Control and Information Sciences, 2011.
[20] H. B. Pacejka, Tyre and vehicle dynamics. Elsevier, Butterworth
Heinemann, ISBN 10:0-7506-6918-7, 2005.
[21] L. Nehaoua, A. Marouf, J. Santin, P. Pudlo, and M. Djemai, “To-
wards an electrical power-assisted steering simulator: Modelisation
specifications,” 5th IFAC Symposium on Mechatronic Systems, no.
10.3182/20100913-3-US-2015.00084, 2010.
[22] H. Imine, Y. Delanne, and N. K. M’Sirdi, “Road profiles inputs estima-
tion in vehicle dynamics simulation,” IAVSD, International Journal of
Vehicle System Dynamics, vol. 44, N 4, pp. 285–303, 2006.
[23] H. Tanaka and M. Kurishige, “The torque controlled active steer for
eps,” AVEC’04, pp. 501 – 506, 2004.
[24] V. I. Utkin, Sliding Modes in Control and Optimization. Berlin,
Germany: Springer, 1992.
[25] V. I. Utkin, J. Gulder, and J. Shi, Sliding Mode Control in Electrome-
chanical Systems. London, UK: Taylor and Francis, 1999.
[26] C. Edwards and S. Spurgeon, Sliding mode control: theory and appli-
cations. Taylor and Francis, 1998.
[27] W. Perruquetti and J. P. Barbot, Sliding mode control in engineering.
Control engineering. New York.Basel, USA: Marcel Dekker, 2002.
[28] L. Fridman and A. Levant, “Higher-order sliding modes,” in In W.
Perruquetti and J. P. Barbot, Sliding mode control in engineering.
Control engineering, New York.Basel, USA, 2002, pp. 53–101.
[29] M. Djemai, J. P. Barbot, and K. Busawon, “Designing r-sliding mode
control using smooth iterative manifolds,” In Med. J. of Measurement
and Control, vol. 4, N 2, pp. 86–93, 2008.
[30] A. Damiano, G. Gatto, I.Marongiu, and A. Pisano, “Second-order sliding
mode control of dc drives,” IEEE Trans. Ind. Electron., vol. 51, N 2,
pp. 364–373, 2004.
[31] M. Djemai, K. Busawon, K. Benmansour, and A. Marouf, “High order
sliding mode control of a dc motor drive via a switched controlled multi-
cellular converter,” Int. Journal of System Science, vol. 42, Issue 11, pp.
86–93, 2011.
[32] B. L. Walcott and S. Zak, “State observation of nonlinear uncertain
dynamical systems,” IEEE Trans. Automat. Contr, vol. AC-32, No. 2,
pp. 166–170, 1987.
[33] T. Floquet, C. Edwards, and S. Spurgeon, “On sliding mode observers
for systems with unknown inputs,” International Journal of Adaptive
Control and Signal Processing, vol. 21, pp. 638–656, 2007.
[34] K. Klasi, J. Lian, S. Hui, and S. Zak, “Sliding mode observers with
unknown inputs: A high gain approach,” Automatica, vol. 46, pp. 347–
353, 2010.
[35] H. Saadaoui, N. Manamanni, M. Djemai, J. Barbot, and T. Floquet, “Ex-
act differentiation and sliding mode observers for switched lagrangian
systems,” Nonlinear Analysis, vol. 65, pp. 1050–1069, 2006.
[36] A. Levant, “Higher-order sliding mode differentiation and otput-
feedback control,” International journal of Control, vol. 76, Issue 9-10,
pp. 924–941, 2003.
[37] ——, “Exact differentiation of signals with unbounded higher deriva-
tives,” 45th IEEE Conference on Decision and Control, pp. 5585–5590,
2006.
Alaa MAROUF received the B.Sc degree in Con-
trol and Electronic System Engineering from the
Higher Institute of Applied Science and Technology,
(HIAST), Syria, and the M.Sc degree in Electronic
of Autonomous Systems from the University of
Cergy-Pontoise and Ecole Nationale Suprieur de
l’Electronique et de Ses Applications (ENSEA),
France. He is currently PhD Student at the LAMIH
(CNRS UMR 8201) laboratory at the University of
Valenciennes, France. His main research interests are
in automotive control and driver assistance systems.
Mohamed DJEMAI is actually full professor at
University of Valenciennes and Hainaut-Cambrsis
France. He obtained his Ph.D. degree in Automatic
Control in 1995 from the University Paris Sud-
Orsay, France. From 2000 to 2008 he was asso-
ciate professor in automatic control in Ecole Na-
tionale Suprieur de l’Electronique et de Ses Appli-
cations (ENSEA), Cergy, France. He obtained a full
Professorship at Valenciennes University in 2008.
He is currently working in the laboratory LAMIH
(CNRS UMR 8201) (http://www.univvalenciennes.
fr/LAMIH/). Actually, his main research activities deal with hybrid systems,
sliding mode control and observers, fault detection and residual generation.
With application to power systems and automotive control.
Chouki SENTOUH received the B.S. degree in
engineering on control and automation systems sci-
ence from the University of Bab Ezzouar (USTHB),
Algeria, in 2000, the M.S. degree in automatic
control from the University of Versailles, France, in
2003, and the Ph.D. degree in automatic control from
the University of Evry, France, in 2007. Since 2009,
he has been an Associate Professor of electrical
engineering with the University of Valenciennes,
France. He works with the Automatic Control and
Human-Machine Systems research team of the In-
dustrial and Human Informatics, Mechanics and Control Laboratory. His
research interests include automotive control, risk analysis, and cooperation
in intelligent transportation systems.
Philippe PUDLO is actually full professor at
University of Valenciennes and Hainaut-Cambrsis,
France. He received his Ph.D. in Industrial and
Human Automation from the University of Va-
lenciennes, France in 1999. From 1999 to 2011
he was associate professor in automatic control
in University Institute of Technology of Univer-
sity of Valenciennes, France. He obtained a full
Professorship at Valenciennes University in 2011.
He is currently working in the laboratory LAMIH
(CNRS UMR 8201) (http://www.univvalenciennes.
fr/LAMIH/). Actually, his current interest fields are biomechanics, modeling
and simulation, and ergonomics, with application to compensation system of
the handicap for the mobility.
