Content uploaded by Junaid Iqbal

Author content

All content in this area was uploaded by Junaid Iqbal on Jul 22, 2017

Content may be subject to copyright.

Content uploaded by Junaid Iqbal

Author content

All content in this area was uploaded by Junaid Iqbal on Jul 22, 2017

Content may be subject to copyright.

Content uploaded by Junaid Iqbal

Author content

All content in this area was uploaded by Junaid Iqbal on Jul 22, 2017

Content may be subject to copyright.

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