Control of Vehicle Driving ModelBy Non-linear Controller

M. Mubin*, S. Ouchi**, N. Kodani**, N. Mokhtar*, H. Arofi

*Department of Electrical Engineering, Faculty of Engineering,University of Malaya, 50603 Kuala Lumpur, Malaysia.

E-mail: marizanL&uimedumy* *Department of Applied Computer Engineering, Tokai University,

1117 Kitakaname, Hiratsuka-shi, Kanagawa 259-1292, Japan.E-mail: ouchis(fkeyaki.cc.u-tokai.ac

Abstract A new drive control system which has an effect on = a1v + bmi1pcontrolling the vehicle slip during accelerating and braking is cb = a2co+ bm2jU + b2rproposed in this paper. This drive control system uses a non- = atr + b ulinear controller designed by following the Lyapunov theorem. |:= -v/max(c,v)The controller is designed in order that it can work at both lu = f(A)conditions that is the slippery and non-slippery road. Theeffectiveness of this control system is proved by a basic where all the coefficients used are given as follows:experiments.

a, -B2/J2 bn1 := W(r2jr1)/J2I. INTRODUCTION a2 b=-B 2 :=W(r1jr2)/J1 b2 :=1/J1

Traction control system is one of the safety developmentsthat have reached the automobiles during this period. It works Wheelto ensure maximum contact between the tire and the road |Wsurface so that the wheels are prevented from losing gripespecially when accelerating, decelerating or braking on the B1low friction road condition such as wet or icy road. As to date,many researches have been developed addressing this / B 0problem. L3],[4],[5],[6],[7],[8],L9] However, the traction control r21(0Vproblems are still far from reaching the final solution and a lot F dof works must be done.

This paper proposes the control of vehicle driving model bya non-linear controller. This controller is designed by Fig. 1 Two-inertia system modelfollowing the Lyapunov theorem, in order that it can work at All the parameters of the controlled object are given inboth conditions: the slippery and non-slippery road. Then, the Table 1.application of the designed controller on the drive-control of a Table 1 Parameter of controlled objectvehicle is discussed, including the structure of the control r Driving torque j1 Moment of inertia of 1stwheelsystem and it is followed by the implementation of this [Nm] 14.55x0-2[k 2]controller into a specially modified device. The discussion on Moment of inertia of 2d wheelthe estimation of the car-body speed is also given in this paper. Road friction coefficient 3.87179 [kgm2J]Finally, the feasibility of this controller has been proven bytheprmnareut.W Weight of car-body r1 Radius of P t wheelthe experimental results. w ri

15.4872 [N] 0.137[m]

II. THE CONTROLLED OBJECT F Driving power r Radius of 2nd wheel[N] 0.25 [m]

A two-inertia system, which consists of two rotating wheels B, Friction of 1st wheel v Speed of 1st wheelwhich represent the wheel and the car-body as shown in Fig. 1 4.5 x 10-2[Nmlradls] [radls]is considered as a controlled object. The equation of motion B2 Friction of2d wheel v Speed of2d wheelofthis system is expressed in Eq. (1). 2.5x10-2[Nm/rad/s] L[radis]

Tf Torque time constant

O.Ols]

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From Eq. (1), friction coefficient p is a nonlinear function A. r -2 Controland it is different for the various road conditions, depends on In order to stabilize the controlled object expressed in Eq.the slip ratio 2 . The ,u - A characteristics, generated by (1), the following performance index is considered:Pacejka model for the normal and icy road is shown in Fig. 2. 3Meanwhile, the relationship between the driving torque and S = -a()the reference torque is given as where

V=VTr ±+u(2) {a:=4-o < v)

Stabi In order to obtain S 0 (at t -c), the following Lyapunovsl|Stat function candidate is considered:

Unstable State V=S (S0) (4)

I Here, the time derivation for V can be obtained as__ V 2S{bm,UnstableState(u+ )}

dv: MyU=O-9 b =b -ab(5)9 eumyu-O2 (5)

where the control law is considered as

-1 -akft. 6j 4 -I2X a#4 6 u3i U := (a1i - aa2c + k1 S| SI + k2S)/ab2 (6)Fig. 2 p - A characteristic To satisfy the Lyapunov stability theorem, VX < 0 must be

obtained. Therefore, VX must satisfy the following condition:III. ESTIMATION OF CAR-BODY SPEED USING DISTURBANCE

OBSERVER V < 2{(lbm,p-ab2rr -k1)|S|-k21S2} (7)

Initially, the car-body speed is estimated by using the Here, in order to obtain S -> O (at t -> oo), a reference valuedisturbance observer. The minimal order observer is applied for torque z-r is defined as given in Eq. (8), where Pr is ato the system; instead of full order one, because the system r f

ed as givenindEq (8) wer is aalso has measurable states. The result of the car-body speed reference value for friction coefficient and bmr(b 2 b) s athat estimated using the observer is shown in Fig. 3. standard value for the change of car-body's weight.

4r (bbmr/ab2)pr (8)

20 Therefore, Eq. (7) can be written as

0 ovh 1 . I V<2t(bmr P11rIk1)ISK2I } (9)O 5 10 15 20 25 30

Imo Isco When the following conditions are satisfied(a) Speed response IU-/rI <7y , lbmrl.<kl , O<k2

the control law is obtained as expressed in Eq. (11).

,u =FoF+Fv+KS/|SK2S (10)___________ where

10 20 30 rF, :=-a lb, F2 :a,l/abIlime Isec I2l(b) ,u response [K :=klab2 , K:2 k2 ab2

Fig. 3 Results of estimation of car-body speedThe block diagram of r - A control system is shown in Fig. 4.

However, from Fig. 3, the accuracy of the results obtainedis not promising. It is because of many factors such as thetemperature characteristic change, the running resistance andthe modeling error margin of accurate system identification.

IV. CONTROL SYSTEM DESIGN

Since the results obtained from the observer is notpromising, the encoder is used to measure the car-body speed.To achieve the objective that is to control the slip ratio, thetheory of the control system is restructured.

2008 International Symposium on Communications and Information Technologies (ISCIT 2008) 395

I ~~~~~Vehicle ob{sterver

K1K WheCotrl Fi. otrlsse

syste B u SIpi

sg _ On the oThe had fo th cae ofS 0 I c Is otiedS Ca atV.v,wee,.i h ee enicau fslprto

Fig.4 - oto ytmMawie t~< v,, sotaie+hr fl oto

a |#' l~~v Ra id Swfi6o + +-<js

B. Sc of l m cn be d s ca contro s+ I s

Next, the structure of the controlsystem is discussed. From The block diagramig.5rt controlsystemlissowei

Eq.a(11),the equation can bediide B.2 During Slippingf tem an L k On the other hand, for the case of Sr e, A A is obtained

term u~~~~1 ,as shown in the followingAequation:rncevalu ofsl'prat

u~~~~1:~~~~~~K1S/S~ ~ ~ ~ ~ ~ ~ ~11±0

Fig. 4 - control system Meanwhile, at co < v, A = -AO is obtained where full control

B. StructureofControl System ~can be done. Th's control system'is called A control system.B.~~~~ ~ ~ ~ ~ ~ ~ ~ ~~~ ~hStrutur

diaraConrrSyste IcotoI ytmssoniNext, the structure of the control system iS discussed. From TebokdarmfrteAcnrlsse ssoni

Eq. (I11), the equation can be divided into two terms that is the Fig. 6.linear feedback termn u, and also the non-linear feedback Ditrac

terrn u,j , as shown in the following equation:CO

u, Ectl)+Fv+ 2S 11++Unl :K1S|S ( )(f+ 'b Vehicle

For the non-linear feedback term unl, ± + co

KS + K, (S > °) (12) K

At S # 0, the driving torque v can be expressed as sgn F v

T =FlcO+F2V±Ki +K2S+Tr (13) S syte

B. ] During Non-slipping Fig. 6 A control system

At S .0 that is when slip does not occur, the time variation V. EXPERIMENTAL RESULTSfor the estimated value of car-body speed v~and the vehicle

In order to check the feasibility of the designed controller,speed c are small. Therefore, it can be considered that i i' ~~~~~~~itis implemented into a specially modified experimental

v0) 0. device as shown in Fig. 7. The device consists of two rotatingp= (lbmr /bm)(pUr ±y), pU <Y (14) wheels, which represent the vehicle wheel and the car-body as

labeled.From Eq. (14), at bmr = b , the friction coefficient p is

equivalent to the reference value of friction coefficient PIwith the addition of limit value y.

The block diagram of the control system for S. 0 case isshown in Fig. 5. This system is known as r control system.

Fig.7 Experimental device of 2 inertia system

396 2008 International Symposium on Communications and Information Technologies (ISCIT 2008)

The controller gains that are used in this control system aregiven in Table 2. Different values of gains are used duringacceleration, constant speed and also deceleration. t [l. T [sl

5STime [sec] io iSf Tie[sc* ~~~~~~~~~~~~~~~(d)S ResponseTable 2 Controller gain _

Acceleration Constant Deceleration102 2 2F 4.5'xl 4.5xl 20 4.5x10d

F2 -36724x 104 -2.9409 x 10i4 -23503 x 10-4 0 Time [sec]i 1.(e) Switching Response

K 3.6x10 3.42xI-2 2.1x 4 Fig. 8 Non-control case Fig. 9 Control caseK1 T.x1 r4x0 1

K2 3.3 2.85 2.1

AO1 0.2 0.001 -0.20001 -0--2 _._..VI. CONCLUSIONS7 0.17 0.001 0.18

The proposed nonlinear controller for the vehicle drivingThe expenrmental results for non-control and control cases model gives satisfactory experimental results. The works

are shown in Fig. (8) and Fig. (9), respectively. Experimental presented in this paper will be continued in the future wheretime is set to 15 sec, where the vehicle accelerates for the first an amendment regarding the temperature characteristic will3 sec, travels at a constant speed for the next 9 sec, and be made to increase the accuracy of the disturbance observerdecelerates for the last 3 sec. Here, it is assumed that the as an estimation system for the car-body speed.vehicle is moving on the low friction road condition.

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+g~~~~~ ji.........

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