Using of Fuzzy PID Controller to Improve Vehicle Stability ...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]

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



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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.



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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)




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.



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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



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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.



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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.



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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.



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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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Documents

Using of Fuzzy PID Controller to Improve Vehicle Stability ...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]