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applied
sciences
Article
Adaptive Global Fast Sliding Mode Control for
Steer-by-Wire System Road Vehicles
Junaid Iqbal 1ID , Khalil Muhammad Zuhaib 1, Changsoo Han 2, *, Abdul Manan Khan 2
and Mian Ashfaq Ali 3
1Department of Mechatronics Engineering, Hanyang University ERICA Campus, Ansan 15588, Korea;
jibssp@gmail.com (J.I.); kmzuhaib@gmail.com (K.M.Z.)
2Department of Robot Engineering, Hanyang University ERICA Campus, Ansan 15588, Korea;
kam@hanyang.ac.kr
3School of Mechanical and Manufacturing Engineering (SMME), National University of Science and
Technology (NUST), Islamabad 44000, Pakistan; ishfaqaries@gamil.com
*Correspondence: cshan@hanyang.ac.kr; Tel.: +82-31-400-4062
Academic Editor: Felipe Jimenez
Received: 21 June 2017; Accepted: 13 July 2017; Published: 19 July 2017
Abstract:
A steer-by-wire (SbW) system, also known as a next-generation steering system, is one
of the core elements of autonomous driving technology. Navigating a SbW system road vehicle in
varying driving conditions requires an adaptive and robust control scheme to effectively compensate
for the uncertain parameter variations and external disturbances. Therefore, this article proposed
an adaptive global fast sliding mode control (AGFSMC) for SbW system vehicles with unknown
steering parameters. First, the cooperative adaptive sliding mode observer (ASMO) and Kalman filter
(KF) are established to simultaneously estimate the vehicle states and cornering stiffness coefficients.
Second, based on the best set of estimated dynamics, the AGFSMC is designed to stabilize the impact
of nonlinear tire-road disturbance forces and at the same time to estimate the uncertain SbW system
parameters. Due to the robust nature of the proposed scheme, it can not only handle the tire–road
variation, but also intelligently adapts to the different driving conditions and ensures that the tracking
error and the sliding surface converge asymptotically to zero in a finite time. Finally, simulation
results and comparative study with other control techniques validate the excellent performance of
the proposed scheme.
Keywords:
adaptive global fast sliding mode (AGFSM); adaptive sliding mode observer (ASMO);
Kalman filter (KF); Steer-by-Wire (SbW)
1. Introduction
The automobile industry is immensely working to transform conventional road vehicles into
partial/full autonomous vehicles. SAE International and NHTSA have classified six levels of driving
autonomy from “no automation” to “full automation” [
1
,
2
]. In particular, from the lane-keeping
assistance system [
3
] to fully automated maneuvering [
4
–
6
], Steer-by-Wire (SbW) technology is
playing a fundamental role in advanced driving assistance systems [
7
]. Nissan introduced the first
commercialized SbW system in 2013 with the Infiniti Q50 vehicle [
8
,
9
]. The SbW system delivers better
overall steering performance with comfort, reduces power consumption, provides active steering
control, and significantly improves the passenger safety. Compared with a conventional steering
system, the SbW system has replaced the mechanical shaft between the steering wheel and front
wheels with two actuators, controllers, and sensors. The first actuator steers the front wheels and the
second actuator provides steering feel feedback to the driver, obtained from the road and tire dynamics.
Appl. Sci. 2017,7, 738; doi:10.3390/app7070738 www.mdpi.com/journal/applsci
Appl. Sci. 2017,7, 738 2 of 26
Over the last decade, many researchers have proposed a number of control techniques to
compensate for the system parameter variation, change in road conditions, and external disturbances
for obtaining the robust performance of the SbW system. In [
10
,
11
], sliding mode based control
schemes are proposed for a partially known SbW system with unknown lumped uncertainties to track
the reference signal. However, it is hard to classify the wide range of nominal parameters under the
sideslip, and the robust performance may not be guaranteed over different road conditions. In [
12
–
16
],
the upper bound sliding mode control (SMC) technique is proposed for the bounded unknown SbW
system parameters and uncertain dynamics. However, the process of obtaining these proper bounds
is not evident. In [
17
–
20
] proportional-derivative (PD) control is proposed to follow the driver’s
steering wheel signal closely. However, under uncertain dynamics, it is difficult to achieve satisfactory
performance with a conventional control scheme. In [
21
] cornering stiffness and chassis side slip
angle are estimated to calculate the self-aligning torque. The authors used the proportion of estimated
torque as a feedback to the driver for artificial steering feel. In [
22
] three suboptimal sliding mode
techniques are evaluated for yaw-rate tracking problem in over-actuated vehicles. In [
23
,
24
], adaptive
control is implemented for path tracking via SbW system and the authors estimated the sliding gains
by considering the known steering parameters and cornering coefficients. However, they did not
use any mechanism to stop the estimation. Consequently, the controller could lead to saturation by
estimating too large a sliding gain. In [
25
,
26
] the authors proposed a hyperbolic tangent function
with adaptive SMC based schemes to counter the effect of self-aligning torque. In [
27
] the frictional
torque and self-aligning torque are replaced by a second-order polynomial function that acts as an
external disturbance over the SbW system. The authors proposed an adaptive terminal SMC (ATSMC)
to estimate the upper bounds of parameters and disturbance.
Apart from the control design, a robust estimation methodology is also needed for the SbW
system to estimate the vehicle states, uncertain parameters, and tire–road conditions for eliminating
the effect of external disturbances from the controller. For instance, in recent years Kalman filter (KF)
and nonlinear observers have gained much more attention from researchers; for example, in [
28
] a
dual extended KF is used to estimate vehicle states and road friction. In [
29
] the authors estimated five
DOF vehicle states and inertial parameters, such as overloaded vehicle’s additional mass, respective
yaw moment of inertia, and its longitudinal position using the dual unscented KF by considering the
constant road–tire friction over a flat road. In [
30
–
32
] a fixed gain based full-state nonlinear observer is
designed to estimate the longitudinal, lateral, and yaw velocities of the vehicle. However, for good
estimation performance the observer gains must be tuned to a wide range of driving conditions.
In order to reduce the burden of gain tuning from a-nonlinear observer [
33
], employed linear matrix
inequality based convex optimization to obtain the gains of reduced order observer for estimating
vehicle velocities. In [
34
] the authors implemented an adaptive gain based sliding mode observer to
estimate the battery’s charging level and health in electric vehicles.
In this paper first, we have established the adaptive sliding mode observer (ASMO) and the
Kalman filter (KF) to simultaneously estimate the vehicle states and cornering stiffness coefficients
by using the yaw rate and the strap down [
35
] lateral acceleration signals. Then, based on the
simultaneously estimated dynamics, the two-fold adaptive global fast sliding mode control (AGFSMC)
is designed for SbW system vehicles, considering that the steering parameters are unknown. In the
first fold, estimated dynamics-based control (EDC) is utilized to stabilize the impact of self-aligning
torque and frictional torque. In the second fold, the AGFSMC is developed to estimate the uncertain
SbW parameters and eliminate the effect of residual disturbance left out from the EDC. The adaptation
capability of the proposed scheme not only intelligently handles the tire–road environmental changes,
but also adapts the system parameters and sliding gains according to the different driving conditions.
Finally, for avoiding overestimations of parameters and gains, discontinuous projection
mapping [
36
] is incorporated to stop the estimation and adaptive mechanism as the tracking error
converges to the designed dead zone bounds [
37
]. In the simulated results section, the comparative
Appl. Sci. 2017,7, 738 3 of 26
study will show the effectiveness of the proposed AGFSMC scheme, which ensures that the tracking
error and sliding surface converge asymptotically to zero in a finite time.
The rest of the paper is structured as follows: In Sections 2and 3, vehicle dynamics modeling
and SbW system modeling with external disturbance are discussed. In Section 4, the ASMO and KF
are established to estimate the vehicle states and parameters. In Section 5, the AGFSMC scheme is
developed for the SbW system and the convergence analysis with bounded conditions is discussed
in detail. Section 6describes the simulation results and findings to validate the proposed scheme,
followed by the last section that concludes the paper.
2. Vehicle Dynamics Modeling
Figure 1illustrates the simplest bicycle model of a vehicle, which has a central front wheel and
a central rear wheel, in place of two front and two rear wheels. The vehicle has two degrees of freedom,
represented by the lateral motion
y
and the yaw angle
ψ
. According to Figure 1, the dynamics along
the yaxis and yaw axis are described as [38,39]:
m..
y+Vx
.
ψ=Fy f cos δf w +Fx f sin δf w +Fyr (1)
Iz
..
ψ=lfFy f cos δf w +Fx f sin δf w −lrFyr, (2)
where
..
y
,
.
ψ
, and
..
ψ
. are the acceleration with respect to the
y
axis motion, yaw rate, and yaw acceleration,
respectively.
lf
and
lr
represent the distance of front and rear axles from the center of gravity,
respectively.
m
and
Iz
are the mass of vehicle and the moment of inertia along the yaw axis, respectively.
Vx
denotes the longitudinal vehicle velocity at the center of gravity.
Fxf
and
Fxr
are the longitudinal
forces of the front and rear wheels, respectively.
Fy f
and
Fyr
are the lateral frictional forces of front and
rear wheels, respectively, as shown in Figure 1.
Appl. Sci. 2017, 7, 738 3 of 27
study will show the effectiveness of the proposed AGFSMC scheme, which ensures that the tracking
error and sliding surface converge asymptotically to zero in a finite time.
The rest of the paper is structured as follows: In Sections 2 and 3, vehicle dynamics modeling
and SbW system modeling with external disturbance are discussed. In Section 4, the ASMO and KF
are established to estimate the vehicle states and parameters. In Section 5, the AGFSMC scheme is
developed for the SbW system and the convergence analysis with bounded conditions is discussed
in detail. Section 6 describes the simulation results and findings to validate the proposed scheme,
followed by the last section that concludes the paper.
2. Vehicle Dynamics Modeling
Figure 1 illustrates the simplest bicycle model of a vehicle, which has a central front wheel and
a central rear wheel, in place of two front and two rear wheels. The vehicle has two degrees of
freedom, represented by the lateral motion and the yaw angle . According to Figure 1, the
dynamics along the axis and yaw axis are described as [38,39]:
+=cos +sin+ (1)
=cos +sin−,(2)
where ,, and are the acceleration with respect to the axis motion, yaw rate, and yaw
acceleration, respectively. and represent the distance of front and rear axles from the center of
gravity, respectively. and are the mass of vehicle and the moment of inertia along the yaw
axis, respectively. denotes the longitudinal vehicle velocity at the center of gravity. and
are the longitudinal forces of the front and rear wheels, respectively. and are the lateral
frictional forces of front and rear wheels, respectively, as shown in Figure 1.
Figure 1. Bicycle model of vehicle.
In order to simplify the model, it is assumed that longitudinal forces , are equal to zero
and by using the small angle approximation, i.e., cos ≈1, the simplified dynamics can be
modeled as follows:
+=+ (3)
=−.(4)
For small slip angles, the lateral frictional forces are proportional to slip-angle and at the
front and rear wheels, respectively. Therefore, lateral forces are defined as:
=2. (5)
=2.
,
(6)
where
=−+
(7)
=−−
, (8)
Figure 1. Bicycle model of vehicle.
In order to simplify the model, it is assumed that longitudinal forces
Fx f
,
Fxr
are equal to zero
and by using the small angle approximation, i.e.,
cos δf w ≈
1, the simplified dynamics can be modeled
as follows:
m..
y+Vx
.
ψ=Fy f +Fyr (3)
Iz
..
ψ=lfFy f −lrFyr. (4)
For small slip angles, the lateral frictional forces are proportional to slip-angle
αf
and
αr
at the
front and rear wheels, respectively. Therefore, lateral forces are defined as:
Fy f =2Cf.αf(5)
Fyr =2Cr.αr, (6)
where
αf=δf w −Vy+lf
.
ψ
Vx(7)
Appl. Sci. 2017,7, 738 4 of 26
αr=−Vy−lf
.
ψ
Vx, (8)
where
Cf
and
Cr
are the front and rear tires’ cornering stiffness coefficients.
δf w
denotes the steering
angle of front wheels, which is considered the same for both front wheels, and factor 2 accounts for
two front and two rear wheels, respectively.
By using the small angle approximation Vy=.
y[38], Equations (7) and (8) can be written as:
αf=δf w −
.
y+lf
.
ψ
Vx(9)
αr=−
.
y−lf
.
ψ
Vx. (10)
Substituting Equations (5), (6), (9) and (10) into Equations (3) and (4), the state space model is
represented as: .
x=Ax +Bδf w , (11)
where
x=h.
y.
ψiT
A=
−2Cf+Cr
mVx−Vx+2lfCf−lrCr
mVx
−2lfCf−lrCr
IzVx−2l2
fCf+l2
rCr
IzVx
,B=
2Cf
m
2lfCf
Iz
.
(12)
3. Steer-by-Wire System Modeling
Figure 2depicts the standard model of SbW system for road vehicles. As shown, the steering
wheel angle sensor is used to detect the driver’s reference angle and the feedback motor is used to
provide the artificial steering feel.
Appl. Sci. 2017, 7, 738 4 of 27
where and are the front and rear tires’ cornering stiffness coefficients. denotes the
steering angle of front wheels, which is considered the same for both front wheels, and factor 2
accounts for two front and two rear wheels, respectively.
By using the small angle approximation = [38], Equations (7) and (8) can be written as:
=−+
(9)
=−−
. (10)
Substituting Equations (5), (6), (9) and (10) into Equations (3) and (4), the state space model is
represented as: =
+, (11)
where =
=
−2+
−
+2−
−2−
−2
+
,=
2
2
.
(12)
3. Steer-by-Wire System Modeling
Figure 2 depicts the standard model of SbW system for road vehicles. As shown, the steering
wheel angle sensor is used to detect the driver’s reference angle and the feedback motor is used to
provide the artificial steering feel.
Figure 2. Steer-by-wire model.
Similarly, the front wheel angle is detected by the pinion angle sensor. Based on the error
between the reference angle and the front wheel angle, the control signal is provided to the front
wheel steering motor to closely steer the front wheels according to the driver’s reference angle.
The equivalent second-order dynamics of the front wheels’ steering motor is expressed as
follows [10,15]:
+++=, (13)
Steering wheel
angle sensor
Steering wheel
Steering feel
Feedback motor
Pinion angle
sensor
Front wheel
steering motor Controller
Figure 2. Steer-by-wire model.
Similarly, the front wheel angle is detected by the pinion angle sensor. Based on the error between
the reference angle and the front wheel angle, the control signal is provided to the front wheel steering
motor to closely steer the front wheels according to the driver’s reference angle.
Appl. Sci. 2017,7, 738 5 of 26
The equivalent second-order dynamics of the front wheels’ steering motor is expressed as
follows [10,15]:
Jeq
..
δf w +Beq
.
δf w +τF+τa=ku, (13)
where
Jeq
and
Beq
are the equivalent moment of inertia and the equivalent damping of the SbW system,
respectively.
u
is the front wheels’ steering motor control input and
k
is the steering ratio between the
steering wheel angle and the front wheels’ angle, given by δf w =δsw/k.
It is known that many modern road vehicles use the variable steering ratio. Therefore, dividing
both sides of Equation (13) by
k
eliminates the impact of the variable steering ratio from the proposed
control scheme without compromising the steering performance. Thus, the dynamics of SbW system
can be written as:
Jek
..
δf w +Bek
.
δf w +τFk +τak =u, (14)
where
Jek =Jeq
k=Jf w
k+Jf mk(15)
Bek =Beq
k=Bf w
k+Bf mk(16)
τFk =τF
k(17)
τak =τa
k, (18)
where
Jf w
and
Jf m
are the moment of inertia of the front wheels and the front wheel steering motor,
respectively.
Bf w
and
Bsm
are the damping factors of the front wheels and the front wheel steering
motor, respectively.
When the vehicle is turning, the steering system experiences torque that tends to resist the
attempted turn, known as self-aligning torque
τa
. It can be seen from Figure 3that the resultant lateral
force developed by the tire manifests the self-aligning torque. The lateral force is acting behind the
tire center on the ground plane and tries to align the wheel plane with the direction of wheel travel.
Therefore, the total self-aligning torque is given by [14]:
τa=Fy f tp+tm, (19)
where
tp
is the pneumatic trail (the distance between the application point of lateral force
Fy f
to the
center of tire),
tm
is the mechanical trail, also known as the caster offset, which is the distance between
the tire center and the point where the steering axis intersects with the ground plane.
Appl. Sci. 2017, 7, 738 5 of 27
where and are the equivalent moment of inertia and the equivalent damping of the SbW
system, respectively. is the front wheels’ steering motor control input and is the steering ratio
between the steering wheel angle and the front wheels’ angle, given by =
⁄.
It is known that many modern road vehicles use the variable steering ratio. Therefore, dividing
both sides of Equation (13) by eliminates the impact of the variable steering ratio from the
proposed control scheme without compromising the steering performance. Thus, the dynamics of
SbW system can be written as:
++ + =, (14)
where
=
=
+
(15)
=
=
+ (16)
=
(17)
=
, (18)
where and are the moment of inertia of the front wheels and the front wheel steering motor,
respectively. and are the damping factors of the front wheels and the front wheel steering
motor, respectively.
When the vehicle is turning, the steering system experiences torque that tends to resist the
attempted turn, known as self-aligning torque . It can be seen from Figure 3 that the resultant
lateral force developed by the tire manifests the self-aligning torque. The lateral force is acting behind
the tire center on the ground plane and tries to align the wheel plane with the direction of wheel
travel. Therefore, the total self-aligning torque is given by [14]:
=+, (19)
where is the pneumatic trail (the distance between the application point of lateral force to
the center of tire), is the mechanical trail, also known as the caster offset, which is the distance
between the tire center and the point where the steering axis intersects with the ground plane.
Figure 3. Self-aligning torque on front wheel.
By substituting Equations (5) and (9) into Equation (19), can be written as:
=2−+
+. (20)
Moreover, is the coulomb frictional torque acting on SbW system, expressed as [23]:
Side view
Top view
Figure 3. Self-aligning torque on front wheel.
Appl. Sci. 2017,7, 738 6 of 26
By substituting Equations (5) and (9) into Equation (19), τacan be written as:
τa=2Cf δfw −
.
y+lf
.
ψ
Vx!tp+tm. (20)
Moreover, τFis the coulomb frictional torque acting on SbW system, expressed as [23]:
τF=Fz f µtpsign.
δf w(21)
Fz f =mglr
lf+lr, (22)
where
Fz f
is the normal load on front axle,
µ
is the coeffient of friction,
mg
is the vehicle’s weight without
any external load, and
sign()
signum function is used to identify the direction of the frictional torque.
The stability of the SbW system mainly depends on the road and environmental conditions.
The uncertain road surface such as dry, wet, or icy can produce considerable variations in the tire
cornering stiffness coefficients
Cf
,
Cr
, which can adversely affect the controller performance. Therefore,
to estimate the vehicle states and the uncertain parameter variation, the cooperative ASMO and KF are
designed in the next section.
4. ASMO and KF
In this section, we will first design the ASMO for yaw rate
.
ψ
and lateral velocity
.
y
, and then
use the KF parameter estimator to estimate the tire cornering stiffness coefficients under varying
road conditions.
In order to design the observer, a few assumptions are made, such as: the yaw rate
.
ψ
is directly
measurable from yaw rate sensor; and the vehicle’s longitudinal velocity is obtained from
Vx=reω
,
where
re
is the effective tire radius and
ω
is the averaged free wheels angular speed measured from
the wheel encoders. Moreover, it is considered that there is no effect of gravitation acceleration
g
on
the lateral acceleration ay, such that the aymeasurement model is defined as [29]:
ay,sensor =..
y+Vx
.
ψ. (23)
Therefore, the lateral velocity can be obtained from a strapdown algorithm [35] as follows:
.
y(t)=.
y(t−1)+Z(ay,sensor −Vx
.
ψ)dt, (24)
where .
y(t−1)is the prior lateral velocity.
The conventional sliding mode observer (SMO) for vehicle states (Equation (11)) can be
designed as: ..
ˆ
y=−A11
.
ˆ
y−A12
.
ˆ
ψ+B1δf w +L1sign.
y−.
ˆ
y(25)
..
ˆ
ψ=−A21
.
ˆ
y−A22
.
ˆ
ψ+B2δf w +L2sign.
ψ−
.
ˆ
ψ, (26)
where
A11
,
A12
,
B1
, and
B2
are the elements of Equation (12) with nominal
m0
and
Iz0
;
L1
and
L2
are
the observer gains, which must satisfy the following conditions, such that:
L1>max(|A11e1|+|A12 e2|)(27)
L2>max(|A12e1|+|A22 e2|), (28)
where e1=.
y−.
ˆ
yand e2=.
ψ−
.
ˆ
ψ.
Appl. Sci. 2017,7, 738 7 of 26
The road surface variation is a critical factor for tuning the observer gains
L1
and
L2
during
the design process. Any inappropriate selection of
L1
and
L2
will significantly reduce the SMO
performance, resulting in a possible deviation of state estimation from the original trajectory.
Due to the aforementioned fact, an adaptive gain based sliding mode observer [
34
] is proposed,
which improves the estimation performance by adapting the observer gains according to tire road
conditions. Therefore, Equations (25) and (26) are changed to new forms, as follows:
..
ˆ
y=−A11
.
ˆ
y−A12
.
ˆ
ψ+B1δf w +ˆ
L1(t)sign(e1)(29)
..
ˆ
ψ=−A21
.
ˆ
y−A22
.
ˆ
ψ+B2δf w +ˆ
L2(t)sign(e2), (30)
where the ASMO gain adaptation law for i=1, 2 is expressed as:
.
ˆ
Li(t)=(ρi|ei|,|ei|>εi
0, otherwise, (31)
where
ˆ
Li(t)>
0, is strictly positive time varying adaptive ASMO gain.
ρi
is a positive scalar used
to adjust the adaption speed.
εi
1 are small positive constants used to activate the adaptation
mechanism with the condition defined in Equation (31); therefore, as the error converges to the bound
|ei|≤εiin finite time, ˆ
Li(t)will stop increasing.
For convergence proof, the Lyapunov function of ASMO for lateral velocity is defined as:
V1=1
2e2
1+1
2ρ1e
L1, (32)
where e
L1=ˆ
L1−L1is the adaptive gain convergence error.
The derivative of V1, with the consideration that .
L1=0, is obtained as follows:
.
V1=e1
.
e1+1
ρ1e
L1
.
ˆ
L1
=e1[−A11e1−A12 e2−ˆ
L1sign(e1)] + 1
ρ1e
L1
.
ˆ
L1
≤e1[−A11e1−A12 e2]−ˆ
L1|e1|+ ( ˆ
L1−L1)|e1|
≤e1[−A11e1−A12 e2]−L1|e1|.
(33)
Thus, by considering Equation (27): .
V1≤0. (34)
Similarly, the Lyapunov function
V2
for yaw rate convergence error
e2
and adaptive gain
convergence error e
L2=ˆ
L2−L2can be written as:
V2=1
2e2
2+1
2ρ2e
L2. (35)
The time derivative of Equation (35) will asymptotically converge to zero,
.
V2≤
0, by considering
.
L2=0 and L2>max(|a12 e1|+|a22e2|).
Remark 1.
In the practical implementation, the direct strapdown of lateral acceleration may incorporate the
small continuous noise to the lateral velocity that can diverge the ASMO estimation over time. Therefore, to deal
with the issue, a lateral velocity-based damping term [
40
] is added to cancel out the incremental noise. Now
Equation (24) can be written as:
.
y(t)=.
y(t−1)(1−σ)+Z(ay,sensor −Vx
.
ψ)dt, (36)
Appl. Sci. 2017,7, 738 8 of 26
where σ>0is an adjustable small damping parameter.
Remark 2.
The designed ASMO may encounter high-frequency chattering due to the discontinuous signum
function
sign()
; therefore, it is replaced by the continuous function
ei/(|ei|+εi)
, such that Equations (29) and
(30) are rewritten as: ..
ˆ
y=−A11
.
ˆ
y−A12
.
ˆ
ψ+B1δf w +ˆ
L1(t)e1
|e1|+ε1
(37)
..
ˆ
ψ=−A21
.
ˆ
y−A22
.
ˆ
ψ+B2δf w +ˆ
L2(t)e2
|e2|+ε2
. (38)
The estimation performance of the ASMO for lateral velocity and yaw rate primarily depends
upon the knowledge of tire cornering stiffness coefficients Cfand Cr, which are unknown in practice
and cannot be measured directly from the onboard vehicle sensors. Therefore, a Kalman filter (KF) [
41
]
is proposed in cooperation with ASMO to estimate these stiffness coefficients under different tire–road
conditions. Once the KF estimates a sufficient set of tire cornering stiffness coefficients, the parameter
estimation can be switched off.
The KF algorithm [41] for tire cornering stiffness estimation is given in Table 1.
Table 1. Kalman filter algorithm.
1. Initialize ˆw0,P0:
ˆ
w0=E[w(0)]
P0=E[(w(0)−ˆ
w0)(w(0)−ˆ
w0)T]
2. Time Update:
ˆ
w−
t=ˆ
wt−1
P−
t=Pt−1+Q
3. Measurement Update:
Kt=P−
tHTHP−
tHT+R−1
ˆ
wt=ˆ
w−
t+Ktzt−Hˆ
w−
t
Pt=(I−KtH)P−
t
P
denotes the estimate error covariance,
Q
is the process noise covariance, and
R=r2
s
is the
measurement noise covariance, whereas rsrepresents the sensor’s zero-mean white noise.
The tire cornering coefficients vector
w
and the measurement
z
, consisting of the lateral
acceleration ay, are defined as:
w=hCfCriT,z=Hw, (39)
where
z=ay
H="−2
m0 .
ˆ
y+lf
.
ˆ
ψ
Vx−δfw !−2
m0 .
ˆ
y−lf
.
ˆ
ψ
Vx!#.(40)
It is to be noted that the tire cornering stiffness coefficient’s vector
w
is considered as constant,
therefore, the time derivative of
w
is zero, (
.
w=
0
)
. Then,
w
and
z
can be written in Euler’s discretized
form as:
w(k)=w(k−1)+v(k)(41)
z(k)=Hw(k)+r(k), (42)
where vand rare the zero mean process noise and measurement noise, respectively.
In order to improve the estimation performance and the convergence accuracy of KF, the difference
e3=zt−Hˆ
w−
t
, known as residual, is utilized to switch off the KF estimator. Therefore, on the basis of
Appl. Sci. 2017,7, 738 9 of 26
e3
, a bounded condition is selected, such that when
e3
reaches the specified bound
|e3|≤ε3
, the KF
will stop the estimation process and thereafter the estimated parameters will become constant until
e3
exceeds the specified condition. ε3(ε3>0)is the small positive constant.
Thus, the estimated tire cornering stiffness-based ASMO for Equations (37) and (38) is revised as:
..
ˆ
y=−A11(ˆ
wt−1).
ˆ
y−A12(ˆ
wt−1)
.
ˆ
ψ+B1(ˆ
wt−1)δf w +ˆ
L1(t)e1
|e1|+ε1
(43)
..
ˆ
ψ=−A21(ˆ
wt−1).
ˆ
y−A22(ˆ
wt−1)
.
ˆ
ψ+B2(ˆ
wt−1)δf w +ˆ
L2(t)e2
|e2|+ε2
. (44)
5. AGFSMC Control Design
In this section, the estimated dynamics-based adaptive global fast sliding mode control (AGFSMC)
is designed in two steps to estimate the uncertain steering parameters and eliminate the effect of
varying tire–road disturbance forces, so that the front wheels asymptotically track the driver’s reference
command in finite time.
The tracking error
eθ
between the front wheel angle
δf w
and the scaled reference hand wheel
angle δdis defined as:
eθ(t)=δf w(t)−δsw(t)
k=δf w(t)−δd(t). (45)
The combination of linear sliding surface and the terminal sliding surface is known as the global
fast terminal sliding surface, s, which is defined as [42]:
s=.
eθ+λ1(eθ)q/p+λ2eθ, (46)
where
λ1
and
λ2(λ1
,
λ2>
0), are strictly positive constants, and
q
and
p
, are positive odd numbers,
such that q<p.
Thus, the time derivative of sis obtained as:
.
s=..
eθ+λ1
q
p(eθ)(q
p−1).
eθ+λ2
.
eθ. (47)
.
scan be written as: .
s=..
δf w −..
δr, (48)
where ..
δris expressed as:
..
δr=..
δd−λ1
q
p(eθ)(q
p−1)+λ2.
eθ. (49)
Thus, for stabilizing the SbW system (Equation (14)) and exponentially converging the tracking
error (Equation (45)) to zero, the two-step closed loop control law
u
for the SbW system is designed as:
u=uE+uA, (50)
where, in the first step, the estimated dynamics based control (EDC)
uE
, is designed to counter the
tire–road disturbance acting on the SbW system as follows:
uE=−sign(s)(|ξτak|+|ξτFk|), (51)
where
ξτak
, is the estimated self-aligning torque, which is computed from the best set of estimated
vehicle states and front wheel cornering stiffness provided by the ASMO and KF.
ξτFk
is the nominal
frictional torque obtained from the nominal set of vehicle parameters, such as mass, nominal coefficient
of friction, and the geometry of the vehicle.
Appl. Sci. 2017,7, 738 10 of 26
Both |ξτak |and |ξτFk|are expressed as:
|ξτak |=2ˆ
Cf
kotpo+tm
δf w −
.
ˆ
y+lf
.
ˆ
ψ
Vx
(52)
|ξτFk |=m0glr
lf+lrk0
µotposign.
δf w, (53)
where
.
ˆ
y
,
.
ˆ
ψ
, and
ˆ
Cf
are the observed vehicle states and the front wheel’s estimated cornering stiffness, as
worked out in the previous section, respectively.
µo
,
tpo
,
m0
, and
ko
are the nominal system parameters.
Second, to tackle the residual disturbance left by the EDC and estimate the uncertain steering
parameters, the adaptive global fast sliding mode control (AGFSMC) uAis designed as follows:
uA=−sign(s)
| |ˆ
a+ˆ
T
.
ˆ
y+lf
.
ˆ
ψ
Vx+ˆ
β1|u(t−1)|
−β2s, (54)
where
ˆ
a(t)
is the estimated parameter’s vector and
| |
is the signal feedback vector; they are defined
as follows:
ˆ
a=ˆ
Jek ˆ
Bek ˆ
Fˆ
TT(55)
| | = [|..
δr|| .
δf w||sign(.
δf w)||δf w |]. (56)
Moreover,
ˆ
β1
and
β2
,
(ˆ
β1
,
β2>
0) are the fixed and adaptive gains used to control the convergence
speed of AGFSMC, respectively, and
|u(t−1)|
is the prior control input obtained at the time step
t−
1.
Therefore, the adaptation laws for updating the ˆ
a(t)and ˆ
β1are designed as:
.
ˆ
a=Γ|T||s|(57)
.
ˆ
β1=|s||u(t−1)|, (58)
where
Γ(Γ>
0
)
is the diagonal positive definite gain matrix used to tune the parameter adaptation
speed. Figure 4shows the framework of the proposed AGFSMC scheme.
Appl. Sci. 2017, 7, 738 10 of 27
Second, to tackle the residual disturbance left by the EDC and estimate the uncertain steering
parameters, the adaptive global fast sliding mode control (AGFSMC) is designed as follows:
=−sign()||
+
+
+
|(−1)|−, (54)
where () is the estimated parameter’s vector and || is the signal feedback vector; they are
defined as follows:
=
ℱ
(55)
||=
sign. (56)
Moreover, and , (,>0) are the fixed and adaptive gains used to control the
convergence speed of AGFSMC, respectively, and |(−1)| is the prior control input obtained at
the time step −1.
Therefore, the adaptation laws for updating the () and are designed as:
=Γ|||| (57)
=|||(−1)|, (58)
where Γ(Γ>0) is the diagonal positive definite gain matrix used to tune the parameter adaptation
speed. Figure 4 shows the framework of the proposed AGFSMC scheme.
Figure 4. AGFSMC scheme framework.
Convergence Proof
The Lyapunov function candidate is defined as:
=1
2
+1
2
Γ
+1
2
, (59)
where ()=()− is the parameter estimation error, ()=
()− is the adaptive gain
convergence error, and Γ is the inverse of gain matrix.
ASMO & KF
Estimator
Parameter & Gain
Adaption
Law
EDC –
AGFSM
Controller
Steer-by-Wire
System
Vehicle
Dynamics
Sensor
−
+
()
()
1
Global
Terminal
Sliding
Surface
,
,,
(−1)
Figure 4. AGFSMC scheme framework.
Convergence Proof
The Lyapunov function candidate is defined as:
Appl. Sci. 2017,7, 738 11 of 26
V3=1
2Jeks2+1
2e
aTΓ−1e
a+1
2e
β12, (59)
where
e
a(t)=ˆ
a(t)−a
is the parameter estimation error,
e
β1(t)=ˆ
β1(t)−β1
is the adaptive gain
convergence error, and Γ−1is the inverse of gain matrix.
The time derivative of Lyapunov function
V3
in terms of the SbW system (Equation (14)) and the
control input (Equation (50)), with the considerations that .
a=0, .
β1=0, are obtained as follows:
.
V3=sJek
.
s+.
ˆ
aTΓ−1e
a+e
β1
.
ˆ
β1
=s[−Jek
..
δr−Bek
.
δf w −τFk −τak +u] + .
ˆ
aTΓ−1e
a+e
β1
.
ˆ
β1
=s[−Jek
..
δr−Bek
.
δf w −τFk −τak −sign(s)(|ξτFk|
+|ξτak |) + uA] + .
ˆ
aTΓ−1e
a+e
β1
.
ˆ
β1
=s[−Jek
..
δr−Bek
.
δf w +uA]−(sτFk +|s||ξ τFk|)
−(sτFk +|s||ξτak|) + .
ˆ
aTΓ−1e
a+e
β1
.
ˆ
β1
≤s[−Jek
..
δr−Bek
.
δf w +uA]− |s|(|ξτFk|−|τFk|)
−|s|(|ξ τak|−|τak |) + .
ˆ
aTΓ−1e
a+e
β1
.
ˆ
β1.
(60)
It is considered that |ξτFk |<τFk and |ξτaFk|<τak , such that:
|ξτFk |−|τFk|=−F |sign(.
δf w)|(61)
|ξτak |−|τak|=−T δf w +
.
y+lf
.
ψ
Vx!, (62)
where
F
and
T
are the uncertain residual parameters of frictional torque and self-aligning
torque, respectively.
Substituting Equations (61), (63) and AGFSMC
uA
(Equation (54)) into (Equation (60)), then the
inequality is written as:
.
V3≤s[−Jek
..
δr−Bek
.
δf w −sign(s){ˆ
Jek|..
δr|+ˆ
Bek|.
δf w|
+ˆ
T |δf w |+ˆ
T
.
ˆ
y+lf
.
ˆ
ψ
Vx+ˆ
β1|u(t−1)|} − β2s]
+|s|F |sign(.
δf w)|+|s|T |δf w |+|s|T
.
y+lf
.
ψ
Vx
+.
ˆ
aTΓ−1e
a+e
β1
.
ˆ
β1
=−(|s|ˆ
Jek|..
δr|+sJek
..
δr)−(|s|ˆ
Bek|.
δf w|+sBek
.
δf w)
−(|s|ˆ
F|sign(.
δf w)| − s|F|sign(.
δf w)|)
−(|s|ˆ
T |δf w |−|s|T |δf w |) + |s|T
.
y+lf
.
ψ
Vx− |s|ˆ
T
.
ˆ
y+lf
.
ˆ
ψ
Vx
−|s|ˆ
β1|u(t−1)| − β2s2+.
ˆ
aTΓ−1e
a+e
β1
.
ˆ
β1
=−|s|| ..
δr|e
Jek − |s|| .
δf w|e
Bek − |s||sign(.
δf w)|e
F − |s||δf w |e
T
−|s| ˆ
T
.
ˆ
y+lf
.
ˆ
ψ
Vx− T
.
y+lf
.
ψ
Vx!− |s|ˆ
β1|u(t−1)| − β2s2
+.
ˆ
aTΓ−1e
a+e
β1
.
ˆ
β1
=−|s|| |e
a+.
ˆ
aTΓ−1e
a− |s| ˆ
T
.
ˆ
y+lf
.
ˆ
ψ
Vx− T
.
y+lf
.
ψ
Vx!
−|s|ˆ
β1|u(t−1)|+ ( ˆ
β1−β1)
.
ˆ
β1−β2s2.
(63)
Appl. Sci. 2017,7, 738 12 of 26
With the adaptation laws of
.
ˆ
a
(Equation (57)) and
.
ˆ
β1
(Equation (58)), substituting into Equation
(63) satisfies:
.
V3≤ −|s|
ˆ
T
.
ˆ
y+lf
.
ˆ
ψ
Vx− T
.
y+lf
.
ψ
Vx
−|s|β1|u(t−1)|−β2s2(64)
The convergence proof shows that the proposed AGFSMC is stable and the inequality
(Equation (64)) ensures that the global fast terminal sliding surface variable exponentially converges to
zero (s=0) in the finite time.
Remark 3.
The signum function
sign(s)
incorporates the chattering and discontinuity in the proposed
controller. Therefore, to eliminate the chattering phenomenon the signum function is replaced by the boundary
layer saturation function sat(·)such that Equations (51) and (53) are re-written as:
uE=−sat(s)(|ξτak|+|ξτFk|)(65)
uA=−sat(s)
| |ˆ
a+ˆ
T
.
ˆ
y+lf
.
ˆ
ψ
Vx+ˆ
β1|u(t−1)|
−β2s. (66)
The boundary layer saturation function is defined as:
sat(s)=(s
φ|s|<φ
sign(s)otherwise , (67)
where
φ>
0represents the boundary layer thickness. Due to the boundary layer, the closed-loop error cannot
converge to zero. However, a carefully selected value of
φ
would lead the error to a user-specified bounded region.
Remark 4.
In order to avoid overestimation of
ˆ
a
and
ˆ
β1
, which can lead the control input
u(t)
to saturation,
Equations (57) and (58) can be re-written for the permissible bounds of
eθ
using the discontinuous projection
mapping [36] as follows:
.
ˆ
a=(0 if |eθ|≤ε4
Γ| |T|s|otherwise (68)
.
ˆ
β1=(0 if |eθ|≤ε5
|s||u(t−1)|otherwise , (69)
where
ε4
and
ε5
are defined as dead zone bounds [
27
]in terms of tracking error. Therefore, when the tracking
error converges to the respective dead zone bound, the adaption mechanism will be switched off and after that ˆ
a
and ˆ
β1become constant.
6. Simulation Results
In this section, the estimation accuracy of vehicle states and cornering stiffness coefficients, and
the control input performance of the proposed AGFSMC scheme for SbW system road vehicles, are
validated over the three different maneuvering tests, in compression with adaptive sliding mode
control (ASMC) and adaptive fast sliding mode control (ATSMC).
The first test (test 1) is sinusoidal maneuvering with varying tire–road conditions—snowy for
the first 30 s and a dry asphalt road for the next 30 s—with the selected coefficient of friction as
µt<30 =
0.45,
µt≥30 =
0.85 and the tire cornering stiffness coefficients for the front and rear wheels
as
Cf(t<30)=
4000,
Cf(t≥30)=
8000,
Cr(t<30)=
5000,
Cr(t≥30)=
10, 000, respectively. The second test
(test 2) is known as circular maneuvering, conducted over a dry asphalt road. Moreover, a high speed
cornering test (test 3) is also introduced to further evaluate the robustness of the proposed scheme.
Appl. Sci. 2017,7, 738 13 of 26
It is worth noting that the first two tests are carried out at longitudinal speed
Vx=
10
m/s
and the
third test at
Vx=
20
m/s
with the same sampling rate of
∆T=
0.001
s
. Furthermore, the vehicle and
SbW system parameters are listed in Table 2.
Table 2. Vehicle and SbW system parameters.
Parameter Value (s)
m(kg)1270
Izkg·m21537
lf,lr(m)1.015, 1.895
Jek 0.28
Bek 0.88
k18
tm,tp(m)0.023, 0.016
The parameters for the proposed cooperative ASMO and KF estimator with the termination
bounds are selected as:
ˆ
L1(0)=ˆ
L2(0)=
8,
ρ1=ρ2=
10,
σ=
0.001,
ε1=ε2=
0.005,
ε3=
0.01,
m0=
1150
kg
,
Iz0=
1430
kg·m2
,
ˆ
w0=[100 100]T
,
P0=
10000
×I2x2
,
Q=1×10−6I2×2
,
and rs=0.001.
In addition, the parameters for the designed AGFSMC scheme with dead zone bounds are chosen
as:
λ1=λ2=
12,
p=
7,
q=
5,
φ=
0.8,
β2=
4,
tpo=tm=
0.016
m
,
µo=
0.6,
ko=
16,
Γ=I4×4
,
ε4=ε5=0.002, and the initial conditions are considered as ˆ
a(0)=ˆ
β1(0)=0.
To compare the performance of the proposed AGFSMC scheme with the adaptive sliding mode
control (ASMC), as designed in [25], we used the following equations:
u=1
k(Je0(λ.
e+..
δd) + Be0
.
δf w +ξf0sign(.
δf w) + vs+Ksat(s) + ˆ
ρτtanh(δf w ))
K=0.1(Je0(λ|.
e|+|..
δd|) + Be0|.
δf w|+ξf0)
.
ˆ
ρτ=µv
Je0+µ.
stanh(δf w ),
(70)
where the tracking error
e=δd−δf w
and the sliding surface
s=.
e+λe
with
.
s=(sk−sk−1)/∆t
are defined in Equation (71). The saturation function
sat(·)
is also taken to be the same as Equation
(67) with boundary layer thickness
φ=
0.8. Moreover, the nominal SbW system parameters
Je0=
3,
Be0k=
12,
ξf0=
100,
k=
18, and the control parameters
λ=
12,
v=
72,
µ=
450 are selected
according to the methodology defined in [25].
For performance comparison with ATSMC, as designed as [
27
], the calculations are given
as follows:
u=−sat(s)ˆ
a1..
δd+ˆ
b1
.
δf w+ˆ
c0+ˆ
c1δf w +ˆ
c2
.
δf w+λˆ
aq
p(e)(q
p−1).
e−ˆ
ρ
2s
−k1sign(s)−k2s
.
ˆ
c0=η1|s|(1−σˆ
c0)
.
ˆ
c1=η2|s|δf w (1−σˆ
c1)
.
ˆ
c2=η3|s|
.
δf w(1−σˆ
c2)
.
ˆ
a1=η4|s|..
δd+η4|s|λq
p(e)(q
p−1).
e(1−σˆ
a1)
.
ˆ
b1=η5|s|
.
δf w1−σˆ
b1,.
ˆ
ρ=η6s2
2(1−σˆ
ρ),
(71)
where
λ
,
p
,
q
,
sat(s)
, and
φ
have the same values as those defined in AGFSMC. Moreover, the control
parameters and adjustable parameters for adaptive laws are selected according to [
27
] as follows:
η1=4, η2=η3=η4=η5=η6=2, k1=0.001, k2=4, ..
δd=2 and ρ=0.001, resptectively.
Appl. Sci. 2017,7, 738 14 of 26
6.1. Sinusoidal Maneuvering Test (Test 1)
The reference steering wheel angle is generated by:
δd=0.4 sin(0.5πt)rad. (72)
Figure 5shows the simultaneously estimated lateral velocity, yaw rate, and cornering stiffness
coefficients. It is observed that the cooperative ASMO and KF scheme intelligently cope with the
tire–road variations and estimate the vehicle states and cornering stiffness coefficients by self-tuning
the gains according to the driving environment. Figure 5c shows that the estimated
ˆ
Cf
,
ˆ
Cr
have not
only converged to the neighborhood of the actual values in both dry and snowy conditions, but also
become constant after the condition e3reached a specified termination bound.
Appl. Sci. 2017, 7, 738 14 of 27
Figure 6 represents the tracking response and the control input performance of the AGFSMC
scheme against the varying tire–road disturbance forces. We can see from Figure 6b that the proposed
methodology effectively eliminates the impact of self-aligning torque (Equation (20)) and Coulomb
frictional torque (Equation (21)) from the SbW system and ensures that the front wheels are precisely
tracking the reference steering angle with a steady state tracking error of 0.002rad .
It is noted that at the beginning of sinusoidal maneuvering, after 3 s, the tracking error reached the
peak value of 0.01rad. This is because we started all the parameter estimations from very low values,
such as =[100100], (0)=
(0)=0. Therefore, right after the peak error, all the estimated
parameters converged to the sufficient estimation set. As a result, the peak tracking error also
converged to the steady-state dead zone region.
(a) (b)
(c)
Figure 5. Estimation results of vehicle states and cornering stiffness coefficients in test 1: (a) Estimated
lateral velocity; (b) Estimated yaw rate; (c) Estimated cornering stiffness coefficients.
(a) (b)
Lateral velocity (m/s)
Yaw rate (rad/s)
Cornering Stiffness (N/rad)
0 102030405060
Time
(
s
)
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0 102030405060
Time
(
s
)
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
0.06
Figure 5.
Estimation results of vehicle states and cornering stiffness coefficients in test 1: (
a
) Estimated
lateral velocity; (b) Estimated yaw rate; (c) Estimated cornering stiffness coefficients.
Figure 6represents the tracking response and the control input performance of the AGFSMC
scheme against the varying tire–road disturbance forces. We can see from Figure 6b that the proposed
methodology effectively eliminates the impact of self-aligning torque (Equation (20)) and Coulomb
frictional torque (Equation (21)) from the SbW system and ensures that the front wheels are precisely
tracking the reference steering angle with a steady state tracking error of 0.002
rad
. It is noted that
at the beginning of sinusoidal maneuvering, after 3 s, the tracking error reached the peak value of
0.01
rad
. This is because we started all the parameter estimations from very low values, such as
ˆ
w0=[100 100]T
,
ˆ
a(0)=ˆ
β1(0)=
0. Therefore, right after the peak error, all the estimated parameters
converged to the sufficient estimation set. As a result, the peak tracking error also converged to the
steady-state dead zone region.
Moreover, Figure 7shows the estimated SbW system parameters and the sliding gain adaptation
profile. It is observed that the estimated SbW system parameters did not converge to the listed actual
constants, but due to the adaptive capability of the proposed control scheme, all parameters as well
as the sliding gain are adaptively adjusted in time for both driving conditions, which ensure the
closed-loop stability of the SbW system. Hence, the outstanding steering performance of the SbW
Appl. Sci. 2017,7, 738 15 of 26
system vehicle is achieved against the nonlinear tire–road disturbance forces and the uncertain SbW
system parameters.
Appl. Sci. 2017, 7, 738 14 of 27
Figure 6 represents the tracking response and the control input performance of the AGFSMC
scheme against the varying tire–road disturbance forces. We can see from Figure 6b that the proposed
methodology effectively eliminates the impact of self-aligning torque (Equation (20)) and Coulomb
frictional torque (Equation (21)) from the SbW system and ensures that the front wheels are precisely
tracking the reference steering angle with a steady state tracking error of 0.002rad .
It is noted that at the beginning of sinusoidal maneuvering, after 3 s, the tracking error reached the
peak value of 0.01rad. This is because we started all the parameter estimations from very low values,
such as =[100100], (0)=
(0)=0. Therefore, right after the peak error, all the estimated
parameters converged to the sufficient estimation set. As a result, the peak tracking error also
converged to the steady-state dead zone region.
(a) (b)
(c)
Figure 5. Estimation results of vehicle states and cornering stiffness coefficients in test 1: (a) Estimated
lateral velocity; (b) Estimated yaw rate; (c) Estimated cornering stiffness coefficients.
(a) (b)
Lateral velocity (m/s)
Yaw rate (rad/s)
Cornering Stiffness (N/rad)
0 102030405060
Time
(
s
)
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0 102030405060
Time
(
s
)
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
0.06
Appl. Sci. 2017, 7, 738 15 of 27
(c)
Figure 6. Control performance of the proposed AGFSMC scheme in test 1: (a) Tracking performance; (b)
Tracking error; (c) Control input torque.
Moreover, Figure 7 shows the estimated SbW system parameters and the sliding gain adaptation
profile. It is observed that the estimated SbW system parameters did not converge to the listed actual
constants, but due to the adaptive capability of the proposed control scheme, all parameters as well
as the sliding gain are adaptively adjusted in time for both driving conditions, which ensure the
closed-loop stability of the SbW system. Hence, the outstanding steering performance of the SbW
system vehicle is achieved against the nonlinear tire–road disturbance forces and the uncertain SbW
system parameters.
(a) (b)
(c) (d)
(e)
Figure 7. Estimated SbW system parameters and sliding gain with AGFSMC scheme in test 1:
(a–d) Estimated SbW system parameters; (e) Estimated sliding gain.
Control input (Nm)
0 102030405060
Time
(
s
)
-0.2
0
0.2
0.4
0.6
0 102030405060
Time
(
s
)
-0.2
0
0.2
0.4
Figure 6.
Control performance of the proposed AGFSMC scheme in test 1: (
a
) Tracking performance;
(b) Tracking error; (c) Control input torque.
Appl. Sci. 2017, 7, 738 15 of 27
(c)
Figure 6. Control performance of the proposed AGFSMC scheme in test 1: (a) Tracking performance; (b)
Tracking error; (c) Control input torque.
Moreover, Figure 7 shows the estimated SbW system parameters and the sliding gain adaptation
profile. It is observed that the estimated SbW system parameters did not converge to the listed actual
constants, but due to the adaptive capability of the proposed control scheme, all parameters as well
as the sliding gain are adaptively adjusted in time for both driving conditions, which ensure the
closed-loop stability of the SbW system. Hence, the outstanding steering performance of the SbW
system vehicle is achieved against the nonlinear tire–road disturbance forces and the uncertain SbW
system parameters.
(a) (b)
(c) (d)
(e)
Figure 7. Estimated SbW system parameters and sliding gain with AGFSMC scheme in test 1:
(a–d) Estimated SbW system parameters; (e) Estimated sliding gain.
Control input (Nm)
0 102030405060
Time
(
s
)
-0.2
0
0.2
0.4
0.6
0 102030405060
Time
(
s
)
-0.2
0
0.2
0.4
Figure 7.
Estimated SbW system parameters and sliding gain with AGFSMC scheme in test 1: (
a
–
d
)
Estimated SbW system parameters; (e) Estimated sliding gain.
Figure 8demonstrates that the steering performance of the ASMC scheme is not as good as that
of the proposed AGFSMC scheme. This is because the hyperbolic tangent function used in ASMC is
unable to replicate the actual self-aligning torque acting on the SbW system. Also, the adaptation law
Appl. Sci. 2017,7, 738 16 of 26
cannot estimate the appropriate equivalent coefficient of self-aligning torque to compensate for the
varying tire–road conditions. Consequently, the overall tracking error is much higher, particularly in
the dry asphalt road condition: the tracking error peaks to the steady-state value of 0.06 rad, which is
almost 30 times higher than in the proposed scheme. Although the ASMC scheme has the information
of nominal parameters and utilized the saturation function, it incorporates high-frequency chattering
during the first 3 s of the simulation, where the reference angle is set to zero.
Appl. Sci. 2017, 7, 738 16 of 27
Figure 8 demonstrates that the steering performance of the ASMC scheme is not as good as that
of the proposed AGFSMC scheme. This is because the hyperbolic tangent function used in ASMC is
unable to replicate the actual self-aligning torque acting on the SbW system. Also, the adaptation law
cannot estimate the appropriate equivalent coefficient of self-aligning torque to compensate for the
varying tire–road conditions. Consequently, the overall tracking error is much higher, particularly in
the dry asphalt road condition: the tracking error peaks to the steady-state value of 0.06 rad, which
is almost 30 times higher than in the proposed scheme. Although the ASMC scheme has the
information of nominal parameters and utilized the saturation function, it incorporates high-
frequency chattering during the first 3 s of the simulation, where the reference angle is set to zero.
Figure 9 shows that the overall tracking response of ATSMC is better than the ASMC under the
varying driving conditions, while both schemes cannot outperform the proposed AGFSMC. It can be
seen that the control input overshoots the allowable control limit, which causes irregular spikes in
the tracking error. We noticed two reasons for that: (1) The designed adaptation law for estimating
the control parameter does not include a provision to maintain the positive estimation; and (2)
the ATSMC does not possess any mechanism to bound or stop the parameter adaptation process for
avoiding overestimations, as compared to the one proposed in AGFSMC. Therefore, the tracking
error is consistently converging to a smaller region with spikes due to the large and continuous
parameter estimation, which may lead the controller to saturation state.
(a)
(b)
(c)
Steering angle (rad)Tracking error (rad)Control input (Nm)
Appl. Sci. 2017, 7, 738 17 of 27
(d)
Figure 8. Control performance of adaptive sliding mode controller in test 1: (a) Tracking performance;
(b) Tracking error; (c) Control input torque; (d) Estimated equivalent coefficient of self-aligning
torque.
(a)
(b)
(c) (d)
Figure 9. Control performance of adaptive terminal sliding mode control in test 1: (a) Tracking
performance; (b) Tracking error; (c) Control input torque; (d) Estimated parameters.
Steering angle (rad)Tracking error (rad)Control input (Nm)
Figure 8.
Control performance of adaptive sliding mode controller in test 1: (
a
) Tracking performance;
(
b
) Tracking error; (
c
) Control input torque; (
d
) Estimated equivalent coefficient of self-aligning torque.
Appl. Sci. 2017,7, 738 17 of 26
Figure 9shows that the overall tracking response of ATSMC is better than the ASMC under the
varying driving conditions, while both schemes cannot outperform the proposed AGFSMC. It can be
seen that the control input overshoots the allowable control limit, which causes irregular spikes in
the tracking error. We noticed two reasons for that: (1) The designed adaptation law for estimating
the control parameter
ˆ
a1
does not include a provision to maintain the positive estimation; and (2)
the ATSMC does not possess any mechanism to bound or stop the parameter adaptation process for
avoiding overestimations, as compared to the one proposed in AGFSMC. Therefore, the tracking error
is consistently converging to a smaller region with spikes due to the large and continuous parameter
estimation, which may lead the controller to saturation state.
Appl. Sci. 2017, 7, 738 17 of 27
(d)
Figure 8. Control performance of adaptive sliding mode controller in test 1: (a) Tracking performance;
(b) Tracking error; (c) Control input torque; (d) Estimated equivalent coefficient of self-aligning
torque.
(a)
(b)
(c) (d)
Figure 9. Control performance of adaptive terminal sliding mode control in test 1: (a) Tracking
performance; (b) Tracking error; (c) Control input torque; (d) Estimated parameters.
Steering angle (rad)Tracking error (rad)Control input (Nm)
Figure 9.
Control performance of adaptive terminal sliding mode control in test 1: (
a
) Tracking
performance; (b) Tracking error; (c) Control input torque; (d) Estimated parameters.
6.2. Circular Maneuvering Test (Test 2)
The circular maneuvering test is carried out over the dry asphalt road for 25 s with these selected
tire–road parameters: Cf=8000, Cr=10000, and u=0.85.
Figures 10–12 portray the promising results of the proposed AGFSMC scheme in all aspects
during test 2. We can see the fine estimation of vehicle states and cornering coefficients in Figure 10.
The estimated cornering coefficients takes less than a second to converge to the sufficient estimation
set over the dry asphalt, such as,
ˆ
Cf∼
=
7250,
ˆ
Cr∼
=
9050, and becomes constant after
e3
satisfies the
Appl. Sci. 2017,7, 738 18 of 26
selected
ε3
bound. Thus, Figure 11 exhibits the excellent tracking response of the front wheels with an
observed peak tracking error of 0.008
rad
, which eventually converged to the
ε4
bound after the rapid
adjustment of all adaptive parameters
ˆ
Jek
,
ˆ
Bek
,
ˆ
F
,
ˆ
T
, and
ˆ
β1
to certain constants, as shown in Figure 12.
Appl. Sci. 2017, 7, 738 18 of 27
6.2. Circular Maneuvering Test (Test 2)
The circular maneuvering test is carried out over the dry asphalt road for 25 s with these selected
tire–road parameters: =8000,=10000, and =0.85.
Figures 10–12 portray the promising results of the proposed AGFSMC scheme in all aspects
during test 2. We can see the fine estimation of vehicle states and cornering coefficients in Figure 10.
The estimated cornering coefficients takes less than a second to converge to the sufficient estimation
set over the dry asphalt, such as, ≅7250, ≅9050, and becomes constant after satisfies the
selected bound. Thus, Figure 11 exhibits the excellent tracking response of the front wheels with
an observed peak tracking error of 0.008rad, which eventually converged to the bound after the
rapid adjustment of all adaptive parameters ,
,ℱ
,
, and to certain constants, as shown in
Figure 12.
(a) (b)
(c)
Figure 10. Estimation results of vehicle states and cornering stiffness coefficients in test 2:
(a) Estimated lateral velocity; (b) Estimated yaw rate; (c) Estimated cornering stiffness coefficients.
Lateral velocity (m/s)
Yaw rate (rad/s)
Cornering Stiffness (N/rad)
Figure 10.
Estimation results of vehicle states and cornering stiffness coefficients in test 2: (
a
) Estimated
lateral velocity; (b) Estimated yaw rate; (c) Estimated cornering stiffness coefficients.
Appl. Sci. 2017, 7, 738 19 of 27
(a) (b)
(c)
Figure 11. Control performance of the proposed AGFSMC scheme in test 2: (a) Tracking performance;
(b) Tracking error; (c) Control input torque.
(a) (b)
(c) (d)
(e)
Figure 12. Estimated SbW system parameters and sliding gain with AGFSMC scheme in test 2:
(a–d) Estimated SbW system parameters; (e) Estimated sliding gain.
Steering angle (rad)
Tracking error (rad)
Control input (Nm)
Figure 11.
Control performance of the proposed AGFSMC scheme in test 2: (
a
) Tracking performance;
(b) Tracking error; (c) Control input torque.
Appl. Sci. 2017,7, 738 19 of 26
Appl. Sci. 2017, 7, 738 19 of 27
(a) (b)
(c)
Figure 11. Control performance of the proposed AGFSMC scheme in test 2: (a) Tracking performance;
(b) Tracking error; (c) Control input torque.
(a) (b)
(c) (d)
(e)
Figure 12. Estimated SbW system parameters and sliding gain with AGFSMC scheme in test 2:
(a–d) Estimated SbW system parameters; (e) Estimated sliding gain.
Steering angle (rad)
Tracking error (rad)
Control input (Nm)
Figure 12.
Estimated SbW system parameters and sliding gain with AGFSMC scheme in test 2: (
a
–
d
)
Estimated SbW system parameters; (e) Estimated sliding gain.
In contrast to the proposed scheme, the ASMC shows the worst tracking performance throughout
test 2. It can be seen from Figure 13 that the tracking error is unable to obtain any steady state bound
and reached a peak value of 0.076
rad
, which is almost 9.5 times higher than in the proposed AGFSMC
scheme. Moreover, the adaptation law also shows inconsistent behavior in the last 7 s of this test,
where the estimated coefficient of self-aligning torque rapidly drops to a highly negative value. As a
result, neither tracking error nor sliding surface converged to the steady state boundary at a finite time
in the Lyapunov’s sense.
Appl. Sci. 2017, 7, 738 20 of 27
In contrast to the proposed scheme, the ASMC shows the worst tracking performance
throughout test 2. It can be seen from Figure 13 that the tracking error is unable to obtain any steady
state bound and reached a peak value of 0.076rad, which is almost 9.5 times higher than in the
proposed AGFSMC scheme. Moreover, the adaptation law also shows inconsistent behavior in the
last 7 s of this test, where the estimated coefficient of self-aligning torque rapidly drops to a highly
negative value. As a result, neither tracking error nor sliding surface converged to the steady state
boundary at a finite time in the Lyapunov’s sense.
(a) (b)
(c) (d)
Figure 13. Control performance of adaptive sliding mode controller in test 2: (a) Tracking performance; (b)
Tracking error; (c) Control input torque; (d) Estimated equivalent coefficient of self-aligning torque.
On the other hand, the ATSMC performed slightly better than the ASMC in terms of tracking
response and also managed to converge the tracking error to the steady state bound during test 2.
The peak tracking error observed under the ATSMC scheme is 0.067rad as shown in Figure 14,
which is marginally less than the ASMC but almost 8.35 times higher than the proposed scheme.
Moreover, the abrupt shift in parameter estimation and the multiple control input overshoots are
again observed in this test.
Steering angle (rad)
Tracking error (rad)
Control input (Nm)
Figure 13.
Control performance of adaptive sliding mode controller in test 2: (
a
) Tracking performance;
(
b
) Tracking error; (
c
) Control input torque; (
d
) Estimated equivalent coefficient of self-aligning torque.
Appl. Sci. 2017,7, 738 20 of 26
On the other hand, the ATSMC performed slightly better than the ASMC in terms of tracking
response and also managed to converge the tracking error to the steady state bound during test 2.
The peak tracking error observed under the ATSMC scheme is 0.067
rad
as shown in Figure 14, which is
marginally less than the ASMC but almost 8.35 times higher than the proposed scheme. Moreover, the
abrupt shift in
ˆ
a1
parameter estimation and the multiple control input overshoots are again observed
in this test.
Appl. Sci. 2017, 7, 738 21 of 27
(a)
(b)
(c) (d)
Figure 14. Control performance of adaptive terminal sliding mode control in test 2: (a) Tracking
performance; (b) Tracking error; (c) Control input torque; (d) Estimated parameters.
6.3. High Speed Cornering Test (Test 3)
In order to further evaluate the estimation accuracy, tracking response, and control input
performance of the proposed AGFSMC scheme, a high-speed cornering test is performed on a dry
asphalt road for 45 s.
As expected, Figures 15–17 clearly indicate the remarkable performance of the proposed scheme
against the parametric uncertainties and tire–road disturbance. The cooperative ASMO and KF also
maintain the robustness and provide adequate estimated dynamics to stabilize the effect of self-
aligning torque and frictional torque at high speed. The peak tracking error observed during test 3
under the AGFSMC scheme is 0.0095rad, which is almost eight times lower than ASMC (0.076rad),
and four times lower than ATSMC (0.04rad). Compared to other control schemes, Figure 18 shows
that the tracking error under ASMC was again unable to attain any steady state bound and also
incorporates high-frequency chattering at constant steering angle inputs. The ATSMC shows a decent
performance regarding the tracking error convergence as compared to ASMC. However, the sudden
parameter estimation shift with control overshoot still exists in this test, as shown in Figure 19.
Steering angle (rad)
0 5 10 15 20 25
-20
0
20
0 5 10 15 20 25
0
2
4
0 5 10 15 20 25
0
20
40
0 5 10 15 20 25
0
5
10
0 5 10 15 20 25
0
2
4
0 5 10 15 20 25
Time (s)
0
5
10
0 5 10 15 20 25
Time
(
s
)
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
0.06
Control input (Nm)
Figure 14.
Control performance of adaptive terminal sliding mode control in test 2: (
a
) Tracking
performance; (b) Tracking error; (c) Control input torque; (d) Estimated parameters.
6.3. High Speed Cornering Test (Test 3)
In order to further evaluate the estimation accuracy, tracking response, and control input
performance of the proposed AGFSMC scheme, a high-speed cornering test is performed on a dry
asphalt road for 45 s.
As expected, Figures 15–17 clearly indicate the remarkable performance of the proposed scheme
against the parametric uncertainties and tire–road disturbance. The cooperative ASMO and KF also
maintain the robustness and provide adequate estimated dynamics to stabilize the effect of self-aligning
torque and frictional torque at high speed. The peak tracking error observed during test 3 under the
AGFSMC scheme is 0.0095
rad
, which is almost eight times lower than ASMC
(0.076 rad)
, and four
times lower than ATSMC
(
0.04
rad
). Compared to other control schemes, Figure 18 shows that the