International Journal of Applied Engineering Research ISSN 0973-4562 Volume 12, Number 5 (2017) pp. 671-680
© Research India Publications. http://www.ripublication.com
671
Using of Fuzzy PID Controller to Improve Vehicle Stability for Planar
Model and Full Vehicle Models
Abdusslam Ali Ahmed1
Institute of Natural Sciences, Okan university/Istanbul, Turkey.
E-mail: [email protected]
1ORCID: 0000-0002-9221-2902
Başar Özkan2
Mechanical Engineering Department, Okan university/Istanbul, Turkey.
E-mail: [email protected]
Abstract
Stability control system plays an important role in vehicle
dynamics in order to improve the vehicle handling and
stability performances. In this paper, vehicle stability control
system is studied by using of two vehicle models which are a
planar vehicle model and full vehicle model, design of a
complete vehicle dynamic control system is required to
achieve high performance of vehicle stability and handling,
this control system structure includes three parts which are a
reference model (2 DOF vehicle model) used for yaw stability
control analysis and controller design, actual vehicle model,
and the controller. In this work a fuzzy PID controller was
used and designed to improve the stability of the vehicle.
Many simulation results show the structures of control
systems for vehicle models used in this work were successful
and can improve the handling and stability of the vehicle.
Keywords: vehicle model, Fuzzy PID controller, vehicle
stability, yaw rate, Planar vehicle model, Full vehicle model.
INTRODUCTION
Thanks to the development of electric motors and batteries,
the performance of EV is greatly improved in the past few
years. The most distinct advantage of an EV is the quick and
precise torque response of the electric motors. A further merit
of a 4 in-wheel-motor drived electric vehicle (4WD EV) is
that, the driving/braking torque of each wheel is
independently adjustable due to small but powerful motors,
which can be housed in vehicle wheel assemblies. Besides,
important information including wheel angular velocity and
torque can be achieved much easier by measuring the electric
current passing through the motor. Based on these remarkable
advantages, a couple of advanced motion controllers are
developed, in order to improve the handling and stability of a
4WD EV [17].
Modern motor vehicles are increasingly using active chassis
control systems to replace traditional mechanical systems in
order to improve vehicle handling, stability, and comfort.
These chassis control systems can be classified into the three
categories, according to their motion control of vehicle
dynamics in the three directions, i.e. vertical, lateral, and
longitudinal directions: 1) suspension, e.g. active suspension
system (ASS) and active body control (ABC); 2) steering, e.g.
electric power steering system (EPS) and active front steering
(AFS), and active four-wheel steering control (4WS); 3)
traction/braking, e.g. anti-lock brake system (ABS), electronic
stability program (ESP), and traction control (TRC).[1].
Based on our survey of research trends in this topic, many
researchers have worked about vehicle dynamics control,
stability, handling, and passengers comfort. All works give
very practical results for specific estimation parameters like
lateral acceleration, yaw rate and sideslip angle but most of
researchers used just vehicle planar models in their studies
with many control methods that means they didn’t take into
consideration heave, pitch, and roll motions of sprung mass.
Some studies[2,3,4] discuss the stability control for electric
vehicle in general and used planar vehicle model and sliding
mode control method theory in their studies, while another
works also used planar model to study handling and stability
of vehicles but with different control strategies such as using
of H∞ control theory [5], designing of controller dynamically
allocates the drive torque in terms of the vertical load and slip
rate of the four wheels[6],robust control method [7] to
preserve vehicle stability and improve vehicle handling
performance, and Fuzzy PID Method[8] to improve yaw
stability control for in-Wheel-Motored electric vehicle.
Other researches[9,10,11,12,13,14,15] in vehicle dynamic
field discussed the handling and stability in cars with
many control techniques and they took into account the full
model of vehicle which includes the suspension system and
rotational motions: yaw motion, pitch motion, and roll motion.
The main different between this work and the other is using of
Fuzzy PID controller with both of planar vehicle model and
full vehicle model to improve stability and handling. The
proposed control architecture in this paper similar control
architecture was used in [16] for the simulation of vehicle
stability control system in addition of rolling, pitching, and
bouncing of the sprung mass in the case of full vehicle model.
VEHICLE MODELS
In this paper, three types of vehicle dynamic model are
established: a non-linear vehicle planar dynamic model, full
vehicle model developed for simulating the vehicle dynamics,
and a linear 2-DOF reference model used for designing
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 12, Number 5 (2017) pp. 671-680
© Research India Publications. http://www.ripublication.com
672
controllers and calculating the desired responses( desired
vehicle yaw rate and desired sideslip angle) to driver’s
steering input. This work includes comparing of vehicle
stability performance in the case of using of planar vehicle
dynamic model and the other case which is full active vehicle
model.
Planar Vehicle Model
This vehicle model is a four-wheel vehicle, only considering
the planar motion: longitudinal, lateral, and yaw. And the
vehicle is modeled as a rigid body with
three-degree-of-freedom. The bob, pitch and roll motions are
ignored. Figure 1 shows the vehicle diagram with planar
motion.
Figure 1: Planar Vehicle Model.
The differential equation of vehicle motion is shown as
follows:
For yaw movement:
)]sin()(2
)(2
)cos()(2
)()cos()()sin()([
fR
y
fL
y
rL
x
rR
x
fL
x
fR
x
rR
y
rL
y
fL
y
fR
y
fL
x
fR
xz
FFd
FFd
FFd
FFbFFaFFawI
(1)
For longitudinal movement:
])sin()()cos()[(1 rR
x
rL
x
fL
y
fR
y
fL
x
fR
xzyx FFFFFFm
wvv
(2)
For lateral movement:
])sin()()cos()[(1 rR
y
rL
y
fR
x
fL
x
fR
y
fL
yzxy FFFFFFm
wvv
(3)
Whrer fR
xF , fL
xF , rR
xF , rL
xF , fR
yF , fL
yF , rR
yF ,
rL
yF are force components for front right, front left, rear right
and rear left tire along x, y coordinates respectively; a ,b the
distance of the center of gravity of the vehicle to front and
rear axle; d distance between leaf and right wheels; vx, vy
longitudinal and lateral velocity , wz yaw rate, δ is the front
wheel steering angle, m is the total vehicle mass, I is the
moment of inertia of the vehicle about its yaw.
Full Vehicle Model
A vehicle dynamic model is established and the three typical
vehicle rotational motions, including yaw rate motion, the
pitch motion, and the roll motion, are considered. They are
illustrated in Figure. 2(a), Figure. 2(b), and Figure. 2(c),
respectively.
The equations of motion can be derived as:
For yaw motion of sprung mass shown in Figure. 1(a)
)())cos()sin()cos()sin(( rR
y
rL
y
fL
y
fL
x
fR
y
fR
xxzzz FFbFFFFaIIw
(4) Where wz is the vehicle yaw rate, Iz is yaw moment of inertia
of sprung mass ,a and b is horizontal distance between the
C.G. of the vehicle and the front, rear axle, and ϕ is the roll
angle of sprung mass.
The equations of motion in the longitudinal direction and the
lateral direction can be written as:
mgfFFF
FFFwhmwvvm
r
rR
x
rL
x
fL
y
fL
x
fR
y
fR
xzszyx
])sin(
)cos()sin()cos([)(
(5)
Where m is the mass of the vehicle , vy is the vehicle speed in
the lateral direction, vx is the vehicle speed in the longitudinal
direction is the , h is the vertical distance between the C.G. of
sprung mass and the roll center , and fr is the rolling resistance
coefficient.
])cos(
)sin()cos()sin([()(
rR
y
rL
y
fL
y
fL
x
fR
y
fR
xszxy
FFF
FFFhmwvvm
(6)
For pitch motion of the sprung mass:
)()( 2143 zzzzy FFaFFbI (7)
Figure 2: Three typical vehicle rotational motions: (a) yaw
motion; (b) pitch motion; (c) roll motion.
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 12, Number 5 (2017) pp. 671-680
© Research India Publications. http://www.ripublication.com
673
4321 ,,, zzzz FFFF in equation (7) are the total force of the
suspension acting on the front and rear sprung masses, Iy is the
pitch moment of inertia of sprung mass , and θ is the pitch
angle of sprung mass.
And for roll motion od sprung mass:
)()( 4132 zzzzszxzzxysx FFFFdghmwIhwvvmI
(8)
Where Ix is the roll moment of inertia of sprung mass and Ixz is
the product of inertia of sprung mass about the roll and yaw
axes.
We also have the equations for the vertical motions of sprung
mass and unsprung mass.
4321 zzzzss FFFFZm (9)
Where sZ is the vertical displacement of sprung mass.
111111 )( zugtuu FZZkZm (10)
222222 )( zugtuu FZZkZm (11)
333333 )( zugtuu FZZkZm (12)
444444 )( zugtuu FZZkZm (13)
Where mui is the mass of the unsprung mass at wheel i., Zui is
the vertical displacement of unsprung mass, kti is the stiffness
of tire at wheel i, and Zgi is the road excitation.
The total force of the suspension acting on the front and rear
sprung masses can be calculated as:
1
12
11111112
)(
2)()( f
d
ZZ
d
kZZcZZkF uuaf
sususz
(14)
2
12
22222222
)(
2)()( f
d
ZZ
d
kZZcZZkF uuaf
sususz
(15)
3
43
33333332
)(
2)()( f
d
ZZ
d
kZZcZZkF uuar
sususz
(16)
4
43
44444442
)(
2)()( f
d
ZZ
d
kZZcZZkF uuar
sususz
(17)
Where ksi is the stiffness of the suspension at wheel i , ci is the
damping coefficient of the suspension at wheel i, kaf is the
stiffness of the anti-roll bars for the front suspension, kar is the
stiffness of the anti-roll bars for the rear suspension , and fi is
the control force of front and rear active suspension controller.
When the pitch angle of sprung mass θ and the roll angle of
sprung mass ϕ are small, the following approximation can be
reached.
daZZ ss 1 (18)
daZZ ss 2 (19)
dbZZ ss 3 (20)
dbZZ ss 4 (21)
Vehicle Reference Model
In vehicle dynamic studies, the reference vehicle model as
shown in Figure 3 is commonly used for yaw stability control
analysis and controller design. This model is linearized from
the nonlinear vehicle model based on the some assumptions:
Tires forces operate in the linear region, the vehicle moves on
plane surface/flat road (planar motion), and Left and right
wheels at the front and rear axle are lumped in single wheel at
the center line of the vehicle.
Figure 3: Vehicle refernce model(bicycle model).
The driver tries to control the vehicle’s stability during normal
and moderate cornering from the steer ability point of view.
Therefore, the reference model reflects the desired
relationship between the driver performance and the vehicle
stability factors. Hence, the model is designed to generate the
desired values of the yaw rate and the sideslip angle at each
instance, according to the driver’s steering wheel angle input
and the vehicle velocity, while considering a constant forward
velocity. The desired sideslip angle of the vehicle is tried to be
maintained as closest as possible to zero, since a vehicle
slipping to the sides is not a desired behavior.
0d On the other hand, while cornering, the yaw rate value cannot
be assumed as zero. Instead, it has to have a value that
depends on the front wheel inclination angle, the forward
velocity and the vehicle dimensions, and could be calculated
as follows [37]:
21)( xus
xzd
vkL
vw
(22)
arafaffarrus CLCCLCLmk /)( (23)
Where wzd and βd are the desired yaw rate and desired sideslip
angle, kus is the understeer parameter, and Caf , Car are
longitudinal and lateral stiffness of front and rear tire.
The simulation result for the vehicle reference model which
represents the desired yaw rate is shown in Figure 4.
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 12, Number 5 (2017) pp. 671-680
© Research India Publications. http://www.ripublication.com
674
Figure 4: The desired yaw rate of the vehicle reference
model.
MODEL OF WHEEL DYNAMIC
A schematic of a modeled wheel is shown in Figure 5. The
wheel has a moment of inertia Iw and an effective radius Rw.
Torque T can be applied to the wheel and longitudinal tire
force Fx is generated at the bottom of the wheel. The wheel
rotates with angular velocity ω and moves with a longitudinal
velocity vx. A summation of the moments about the axis of
rotation of the wheel generates the dynamical equation shown
in following equation:
111 . wwx IRFT (24)
222 . wwx IRFT (25)
333 . wwx IRFT (26)
444 . wwx IRFT (27)
Figure 5: Wheel schematic diagram.
TIRE MODEL
At extreme driving condition, the tire may run at non-linear
religion. This paper uses Dugoff’s tire model which provides
for calculation of forces under combined lateral and
longitudinal tire force generation.. The longitudinal and lateral
force of tire were expressed as [11]:
)(1
1
fCF
x
xx
(28)
)(1
tan2
fCF
x
y
(29)
Where C1 and C2 are the longitudinal and cornering
stiffness of the tire, x is the longitudinal slip ratio of
the tire and defined in equation 30 and equation 31,and
i is the tire slip angle at each tire.
x
iw
xv
R 1 in the case of deceleration (30)
iw
x
xR
v
1 in the case of acceleration (31)
The individual tire slip angles i are calculated using
vehicle geometry and wheel vehicle velocity vectors. If
the velocities at wheel ground contact points are known
the tire slip angles can be easily derived geometrically
and are given by:
zx
zy
fR
wd
v
wav
.2
.tan 1 (32)
zx
zy
fL
wd
v
wav
.2
.tan 1 (33)
zx
zy
rL
wd
v
wbv
.2
.tan 1 (34)
zx
zy
rL
wd
v
wbv
.2
.tan 1 (35)
The variable and the function )(f are given by
}))tan((){(2
)1(
2
2
2
1
CC
F
x
xz
(36)
And
11)(
1)2()(
iff
iff (37)
Where is the tire-road friction coefficient and zF is the
vertical tire force for each wheel.
FUZZY PID CONTROLLER
Fuzzy PID controller is composed of adjustable PID controller
and fuzzy controller. The core is fuzzy controller. It contains
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 12, Number 5 (2017) pp. 671-680
© Research India Publications. http://www.ripublication.com
675
fuzzification, repository, fuzzy inference, defuzzification, and
input/output quantification and so on. Fuzzy logic is a rule
which can map a space-input to another space-output. In
engineering application, fuzzy logic has the following
characters: 1) Fuzzy logic is flexible; 2) Fuzzy logic is based
on natural language, and the requirement for intensive reading
of data is not very high; 3) Fuzzy logic can take full advantage
of expert information; 4) Fuzzy logic is easy to combine with
traditional control technique.
Fuzzy PID controller takes E (the error between feedback
value and desired yaw rate value of controlled station as input.
Using fuzzy reasoning method it adjusts the PID parameters
(Kp, Ki, Kd). Change the PID parameters on line by using the
fuzzy rules; these functions form the self-setting fuzzy PID
controller. Its control system architecture is shown in Figure
6.
Figure 6: The structure of fuzzy PID controller.
CONTROL STRATEGY
In this paper the Fuzzy PID Controller is designed to improve
the vehicle yaw stability in the case of vehicle planar model
and the full vehicle model. The fuzzy PID controller can be
decomposed into the equivalent proportional control, integral
control and the derivative control components. we design a
control system which includes three parts: the reference
vehicle model(Bicycle model), full vehicle model( Actual
vehicle), and the fuzzy PID controller. The structure of the
two vehicle models with controller are given in Figure 7 and
Figure 8.
Depending on 𝛿 which can be gained by the driver action and
vx, the expected vehicle’s yaw rate wzd can be calculated
through the reference model. The PID controller is applied
widely, because it possesses the virtues such as the simple
structure , better adaptability and faster response time.
Regulating the parameters of the PID is the fussy task by
manual work, while regulating parameter by fuzzy controller
is very convenient. The sideslip angle 𝛽 is compared with
desired sideslip 𝛽𝑑 and the yaw rate wz is compared with
desired yaw rate wzd, and then the yaw rate is calculated. The
fuzzy control is implemented as follow: the input variable,
i.e., vehicle yaw rate error, is fuzzificated firstly, and then the
output variables, the direct yaw moment Mz. The fuzzification
is the process of transforming fact variables to corresponding
fuzzy variables according to selected member functions.
Figure 7: The structure of vehicle model with controller for planar model.
Figure 8: The structure of vehicle model with controller for full vehicle model.
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 12, Number 5 (2017) pp. 671-680
© Research India Publications. http://www.ripublication.com
676
SIMULATION RESULTS AND DISCUSSION
In this paper, the simulation models are a planar vehicle
model which includes longitudinal, lateral, and yaw motions
only and a full vehicle model that includes active suspension
dynamics, roll, yaw and pitch motion. Using the vehicle
speed, yaw rate, longitudinal and lateral accelerations of the
vehicle at c.g. As the measurement vectors, the steering angle
as input signal. The control system is analyzed using
Matlab/Simulink. We assume that the vehicle travels at a
constant speed vx = 20m/s, and is subject to a steering input
from steering wheel. The steering input is set as a step signal
which sown in the figure 9.
The values of the vehicle physical parameters used in the
simulations are listed in Table 1.
Table 1. vehicle physical parameters.
Parameter Unit Value Parameter Unit Value
ms Kg 810 c1 KN.s/m 1570
m Kg 1030 c2 KN.s/m 1570
a m 0.968 c3 KN.s/m 1760
b m 1.392 c4 KN.s/m 1760
d m 0.64 ks1 KN/m 20.6
Rw m 0.303 ks2 KN/m 20.6
Iw Kg.m2 4.07 ks3 KN/m 15.2
μ --- 0.4 ks4 KN/m 15.2
C1 KN/rad 52.526 kaf (N.m/ rad) 6695
C2 KN/rad 29000 kar (N.m/ rad) 6695
Caf KN/rad 95.117 Ix Kg.m2 300
Car KN/rad 97.556 Iy Kg.m2 1058.4
g m/s2 9.81 Iz Kg.m2 1087.8
mu1 Kg 26.5 v0 m/s 20
mu2 Kg 26.5 fr --- 0.015
mu3 Kg 24.4 h m 0.505
mu4 Kg 24.4
Figure 9: The angle of front steering wheel on step maneuver.
In this paper, the mathematical model for vehicle dynamic
model has been presented above for planar vehicle model and
full vehicle model. Figure 10-Figure 14 present the selected
results of the planar vehicle model with and without control.
Figure 10 and Figure 11 present the longitudinal velocity and
lateral acceleration of the vehicle while, Figure 12 shows
Comparison between the vehicle yaw rate with and without
control and (Figure 13) presents the Comparison between
Sideslip angle with and without control. Fuzzy rule viewer
shown in Figure 14 gives the effect of control signal for
change in rules. Finally it's obvious that using of Fuzzy PID
controller improved the vehicle stability performance.
Figure 10: Comparison between longitudinal velocity with
and without control.
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 12, Number 5 (2017) pp. 671-680
© Research India Publications. http://www.ripublication.com
677
Figure 11: Comparison between Lateral acceleration with and
without control.
Figure 12: Comparison between yaw rate with and without
control.
Figure 13: Comparison between Sideslip angle with and
without control.
Figure 14: Rule viewer of fuzzy PID controller in the case of
planar vehicle model
The simulation results of the full vehicle model are shown in
Figure 15-Figure 18, these simulation results illustrate
comparison of the full vehicle model with and without fuzzy
PID controller. Also, in this model the vehicle stability
performance with fuzzy PID controller is improved. Fuzzy
rule viewer shown in Figure 19 gives the effect of control
signal for change in rules.
Figure 15: Comparison between longitudinal velocity with
and without control.
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 12, Number 5 (2017) pp. 671-680
© Research India Publications. http://www.ripublication.com
678
Figure 16: Comparison between Lateral acceleration with and
without control.
Figure 17: Comparison between yaw rate with and without
control.
Figure 18: Comparison between Side slip angle with and
without control.
Figure 19: Rule viewer of fuzzy PID controller in the case of
planar vehicle model
CONCLUSION
Aiming at improving vehicle stability,the simulink model of
the vehicle is constructed with Fuzzy PID controller to
ensure and improve the vehicle stability in this paper. The
results and plots show a significant difference between the
vehicle performance in the case of without control and the
vehicle stability and performance in the case of using Fuzzy
PID controller. Also, it is found that, using Fuzzy PID
controller with both of planar and full vehicle models was
successful in improve the vehicle performance. The lateral
acceleration, yaw rate , and sideslip angle improved
significantly. Therefore vehicle stability control system using
fuzzy PID controller can enhance the performance and
stability of vehicle.
ACKNOWLEDGEMENT
I express my deep thanks to the Asst.Prof.Başar Özkan for his
support and guidance in this work.
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