Effects of Sliding Surface on the Performances of Sliding Mode Slip Ratio Controller for a HEV

Basanta Kumar Dash

Dept. of Electrical Engg., Indus College of Engineering Bhubaneswar, BPUT, Odisha, INDIA

E-mail: bkdash@indus.ac.in

Bidyadhar Subudhi, Senior Member Dept. of Electrical Engg, National Institute of Technology

Rourkela, Odisha, INDIA E-mail: bidyadhar@nitrkl.ac.in

Abstract— Slip ratio control of a ground vehicle is an

important concern for development of antilock braking system to avoid skidding when there is a transition of road surfaces. In the past, the slip ratio models of such vehicles were derived by several investigators. It is observed that the dynamics of the hybrid electric vehicle (HEV) is nonlinear, time varying and uncertain due to tire-road dynamics being a nonlinear function of road adhesion coefficient and wheel slip. Sliding mode control (SMC) is a robust control paradigm which has been widely applied successfully to develop an antilock braking system of a HEV. But the SMC performance is influenced by the choice of sliding surface. This is due to the discontinuous switching of control force arising in the vicinity of the sliding surface that produces chattering. This paper presents a detailed study on the effects of different sliding surfaces on the slip ratio control performances of a HEV.

Keywords- sliding mode control, slip ratio control, hybrid electric vehicle, sliding surface, ABS

NOMENCLATURE Jw wheel moment of inertia rw wheel radius fw viscous wheel friction force c coefficient of viscous friction nw number of driven wheels cv aerodynamic drag coefficient μ adhesive coefficient λ wheel slip ratio ωw wheel angular velocity ωv vehicle angular velocity υ linear velocity of the vehicle g acceleration due to gravity m mass of the vehicle Tb braking/traction torque fb braking force FN normal tire reaction force

I. INTRODUCTION Electric motors and electronic control system has

been widely used in recent years in passenger vehicles, resulting in a significant improvement in vehicle’s safety and stability. The power controller of the drive train of an electric vehicle regulates the motor current and voltage, and hence the output torque of the motor which in terns responsible to meet desired tractive/braking effort. Hence, electric motors are excellent actuators for motion control compared to hydraulic braking systems. Furthermore, output torque of an electric motor can be controlled faster and precisely than hydraulic

braking systems. By using electric motor regenerative braking can also be achieved in HEVs together with improved antilock performance.

A number of investigations have proposed different control techniques for achieving improved braking performance of the HEVs that employ electric vehicles such as fuzzy logic control [4–6], neural network control [1,7], feedback linearization control [8], iterative learning control [9], sliding mode control [1–3]. All these techniques try to control slip ratio accurately thereby reducing the stopping distance by preventing wheel lockup and at the same time providing directional control and stability. Among all these control techniques used to control the slip ratio, the SMC has been widely implemented [1–3, 12–14] because of its robust characteristics for handling nonlinear systems with model uncertainties.

In the development of SMC, the choice of the sliding surface influences the behavior of the overall control performance. Due to the discontinuous switching of control force in the vicinity of the sliding surface, it produces chattering which can cause system instability and damage to actuators and power plants. During the application of braking torques to actual braking systems, the performance of switching control may be degraded due to the braking torque limitation, brake actuator delay, and tire-force buildup, resulting in an oscillation near the sliding surface. Thus, a reduction of a chattering and a rapid convergence to the sliding surface in the actual brake systems are important. In order to reduce the chattering, a higher order SMC and a boundary- layer methods [10] with moderate tuning of a saturation function have been used [10]. However, the effects of choice of sliding surface on the overall performance (minimizing chattering, improving the tracking speed, error convergences etc) have not been closely investigated in slip ratio control except some limited research in [11, 16].

The determination of the maximum tractive effort of a HEV is determined by the coefficient of road adhesion and the normal load on the drive axle(s) and the characteristics of the power plant and the transmission. The smaller of these two determines the performance potential of the vehicle.

The maximum tractive/braking effort as determined by the nonlinear nature of the tire-road interaction imposes a fundamental limit on the vehicle performance characteristics, including maximum speed, acceleration/deceleration etc.

When the braking force reaches the limiting value determined by the normal load and the coefficient of road adhesion, tires are at the point of sliding. Any further increase

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in the braking force would cause the tires to lockup which is a major concern during braking on slippery or snowy roads. Wheel lock is undesirable as it prolongs the stopping distance.

When the rear tires lockup first, the vehicle will lose directional stability and the capability of the rear tires to resist lateral acceleration is reduced to zero.

The lockup of front tires will cause a loss of directional control and the driver will no longer be able to exercise effective steering. However, front tire lockup does not cause directional instability. This is because whenever lateral movement of the front tire occurs, a self-correcting movement due to the inertia force of the vehicle about the yaw center of the rear axle will be developed, which tends to bring the vehicle back to a straight line path.

When a tire is locked (i.e. 100% skid), the coefficient of road adhesion falls to its sliding value, and its ability to sustain side force is reduced to almost null. As a result, the vehicle will loose directional control and/or stability, and the stopping distance will be longer than the minimum achievable. Thus, the prime function of the braking control system is to prevent the tire from locking, and ideally to keep the skid (slip) of the tire within a desirable range. This will ensure that the tire can develop a sufficiently high braking force for stopping the vehicle in the shortest possible distance and at the same time it can retain an adequate cornering force for directional control and stability.

The maximum braking force that the tire-ground contact can support is determined by the normal load and the coefficient of road adhesion which in terns depends on the wheel slip ratio and tire/road conditions.

Slip ratio dynamics is uncertain and time varying because parameters associated with vehicle and wheel dynamics vary due to changes in the road surface, road gradient, whether condition and variation in the adhesion coefficient of the tire/road contact. The slip ratio control system design for a HEV when there is uncertainty in tire-road dynamics is a complex problem thus attracts attention of control research community. So SMC is a good choice to control slip ratio of the vehicle in the presence of model uncertainties, parameter fluctuations and disturbance. The objective of the paper is to investigate the effects of sliding surface designs on the overall performance of SMC for slip ratio control of a HEV. The design and study of slip ratio controller in this paper assumes that vehicle speed and wheel angular velocities are both available on-line by direct measurements which are the two state variables taken in our slip ratio model and both the states are observable.

II. SLIP RATIO MODEL OF THE HEV A good schematic model as well as a mathematical

model of a ground vehicle to represent its dynamic model are essentially a prerequisites for designing controllers such as slip ratio control which will be used for analysis, design of controllers and computer simulations.

Slip is defined as the difference between the vehicle and wheel speeds, normalized by the maximum of these velocity values.

Mathematically, Slip ratio: max( , )

w v

w v

ω ωλω ω−=

(1)

where ων and ωw are vehicle and wheel angular velocity respectively. The relationship between the linear and angular velocities of the vehicle is υ = rw ωv (2)

Wheel speed is greater than vehicle speed in case of acceleration and vice versa in case of braking.

To design a slip ratio controller, the system dynamic equation has to be represented in terms of the wheel slip and the state variables. Wheel speed and vehicle speed are available as state variables in the model shown in Fig.1. Choosing x1= ωv and x2= ωw as two state variables, the equation can be rewritten as

2 1

1 2max( , )x x

x xλ −= (3)

Applying a driving torque or braking torque to a pneumatic tire produces a tractive force at the tire-tread contact patch. The driving torque produces compression at the tire tread in front and within the contact patch. Consequently, the tire travels less distance than it would if it were free rolling. Similarly, when a braking torque is applied, it produces a tension at the tire tread within the contact patch and at the front. Because of this tension the tire travels more distance than it would during free rolling. This phenomenon is called the deformation slip or wheel slip.

The operation of an antilock brake system can be conceptualized as follows. It would be useful to briefly review the dynamics of the tire/wheel during braking.

Fig.1 shows the vehicle wheel model during a

braking mode. Tb is the braking/traction torque, ωw is the wheel angular velocity of the vehicle, υ is the linear velocity of the vehicle, m is the mass of the vehicle and g is the acceleration due to gravity and fb is the braking force. When a braking torque Tb is applied to the tire, a corresponding braking effort fb is developed on the tire-ground contact patch which is the primary retarding force as shown in Fig.1. This braking effort fb has a moment fbrw When the braking force is below the limit of tire-road adhesion, the braking force fb is given by

b w wb

w

T Jf

rω−

= ∑ (4)

The maximum braking force that the tire-ground contact can support is determined by the normal load and the coefficient of road adhesion.

max ( ). ( )b Nf F mgμ λ μ= = (5) where μ is the coefficient of road adhesion and is a function of slip (λ) and FN is the normal tire reaction force depends on vehicle parameters such as mass, location of center of gravity and the steering and suspension dynamics. The adhesion coefficient μ is the ratio between the tire tractive force and the normal load, depends on the tire-road contact and the value of wheel slip (λ) and can be written as

2 2

2( ) p p

p

μ λ λμ λ

λ λ=

+ (6)

( )bf mgμ λ=

υ

mg

Tb wω

wJwr

Figure 1 Wheel dynamics during braking

264

where μp and λp are the peak values. Equation (6) is compatible with experimental data in the desirable range shown in Fig. 2 [15 Wong]. For various road conditions, the curves have different peak values and slopes shown in Fig.2. The (μ- λ) characteristics also depend on operational parameters like speed and vertical load.

When the braking forces reach their corresponding values determined by equations (5), the tires are at the point of sliding. Any further increase in the braking force would cause the tires to lockup.

The braking force distribution that can ensure the maximum braking forces of the front and rear tires developed at the same time is referred to as the ideal braking force distribution. If the braking force distribution is not ideal, then either the front tire or the rear tire will lockup first.

When a tire is locked (i.e. 100% skid), the coefficient

of road adhesion falls to its sliding value, and its ability to sustain side force is reduced to almost null. As a result, the vehicle will loose directional control and/or stability, and the stopping distance will be longer than the minimum achievable. Wheel lock is undesirable as it prolongs the stopping distance.

The prime function of an antilock braking system is to prevent the tire from locking, and ideally to keep the skid (slip) of the tire within a desirable range shown in Fig 3. This will ensure that the tire can develop a sufficiently high braking force for stopping the vehicle and at the same time it can retain an adequate cornering force for directional force and stability.

Fig. 3 shows the general characteristics of the braking effort coefficient and the coefficient of cornering force at a given slip angle as a function of skid for a pneumatic tire. The rotational motion of the wheel and translational motion of the vehicle can be described as the follows. When wheel is locked, ωw=0 and λ=-1 When wheel is in free motion, ωw= ωv and λ=0

Slip ratio λ is a function of vehicle speed and wheel speed. Vehicle speed is a function of braking force fb. Wheel speed is a function of braking torque Tb and braking force fb which is unknown but related to slip ratio λ.

The dynamics of the angular motion of the wheel and the translational motion of the vehicle can be described respectively as the following equations. The total torque consists of shaft torque which is opposed by break torque and the torque components due to tire tractive force and wheel viscous friction force. The wheel viscous friction force (fw) developed on the tire-road contact surface depends on the wheel slip.

( )w w b w b wJ T r f fω = − +

( ( ) )w w b w wJ T r mg fω μ λ= − + (7)

where w wf cω= 2( )w vm n m g cυ μ λ υ= − (8)

Tire rolling resistance is neglected. Substituting (2) into (8), we have 2 2( )v w w v v wm r n m g c rω μ λ ω= −

2( )w v wv v

w

n g c rr m

ω μ λ ω= − (9)

Again from (9), we have

( )b w ww w

w w w

T r rmg fJ J J

ω μ λ= − − (10)

As the state variables are chosen as x1= ωv and x2= ωw and using (9) and (10), the state space representation of the slip ratio model of the vehicle as

1 1 1 1( ) ( )x f x c μ λ= + (11)

2 2 2 2 3( ) ( ) bx f x c c Tμ λ= + + (12) y=x2 (13)

where 21 1 1( ) v wc rf x x

m= − (14)

22 2

( )( ) w w

w

r f xf xJ

= − (15)

1

w

w

n gcr

= , 2

w

w

r mgcJ

= − and 3

1

w

cJ

= (16)

The slip ratio model given by (11), (12) and (13) is a single-input-single-output (SISO) nonlinear system where the functions f1(x1) and f2(x2) are uncertain functions due to the dependence upon the road condition parameter, μ(λ).

The actuator is a dc motor which is controlled with variable armature voltage keeping the field current constant. During the traction, the torque is considered to be positive and during braking the torque is negative. The dynamics of the actuator is described by the following equation.

bb b w

m m

dTL RV T KK d t K

ω= + + (17)

where V is the applied voltage to the armature circuit, ia is the armature current, Kb, Km are back emf constant and motor constants respectively, R, L are the resistors and inductors of the armature circuit.

Figure 3 Effect of skid on cornering force coefficientof a tire at a given slip angle

0 20 40 60 80 100% of SkidFree

Rolling

Bra

king

Eff

ort C

oeff

icie

nt

Locked Wheel

Cor

nerin

g Fo

rce

Desirable Range

Slip angle (α) = constant

Figure 2 Typical adhesion coefficients (µ) versus slip ratio (λ) curve for different road conditions [9]

Adh

esio

n C

oeff

icie

nt (µ

)

Dry Asphalt

Wet Asphalt

Icy or Snowy Road

Slip Ratio (λ) 0 0.2 0.4 0.6 0.8 1.00

0.2

0.4

0.6

0.8

1.0

265

III. PROBLEM STATEMENT In this paper longitudinal vehicle motion control has

been considered in the first quadrant of state-space spanned by x1 and x2 for the sake of simplicity i.e. (x1 >0 and x2 >0). For the case of braking, x1 > x2, from (3) we have,

2 1

1

x xx

λ −= (18)

Differentiating (18) with respect to time, we have,

1 1 2 1x x x xλ λ+ = − (19)

Substituting 1 21and x x from (11-13) in (19) gives

( , )f x buλ λ= + (20) where

2 2 1 1 2 1

1

( ) (1 ) ( ) [ (1 ) ] ( )( , ) f x f x c cf xx

λ λ μ λλ − + + − +=

(21)

1

bTux

= (22)

b=c3 (23) 1 2[ , ]Tx x x= (24)

The control gain b is unknown and assumed as constant gain with known bounds and its estimated value is b̂in the SMC design. So development of control is based on general equation (20), where f(.), b, u for braking are given in equations (21-23). The adhesion coefficient μ(λ) can be calculated from equation (6).

The (μ- λ) characteristic is not exactly known and nonlinear. So the nonlinear function f(λ, x) in (20) is assumed to be unknown. From Fig.2, Fig.3 and (6), it can be seen that by increasing slip the braking force between the tire and road surface increases because of increase in µ. For a given vehicle, this amounts to maximizing the adhesion friction coefficient at the tire road surface. However, once the peak of the characteristics is reached, any further increase in slip will reduce both breaking effort and cornering force and consequently induce locking of wheel and hence skidding.

The objective is to find a control u such that a desired slip ratio is maintained despite of uncertainties in the wheel parameters and varying road conditions. The controller must be robust with respect to uncertainties in the tire characteristics and variation in the road conditions. The slip-ratio dynamics is highly uncertain and nonlinear, mainly due to the tire friction characteristics. On the other hand, fast changing in operating conditions can appear. The objective of the proposed slip control system of a vehicle is to design robust nonlinear tracking control using the slip model of the vehicle given in (11-13) with slip as the controlled variable and the torque applied to the driven wheels as the input.

IV. ADAPTIVE SLIDING MODE CONTROLLER Schematic diagram of a slip ratio sliding mode

controller is shown in Fig.4.

The SMC design procedure for controlling slip of a

HEV involves three steps: the design of the sliding surface, the design of the controller which holds the system trajectory on the sliding surface, and the chattering free implementation.

For adaptive sliding mode control design, the equation (20) can be rewritten as

1( , ) ( , )af x h x buλ λ θ λ= + + (25) where

2 2 1 1

1

( ) (1 ) ( )( , )a

f x f xf xx

λλ − +=

(26)

2 11 2 2

1

2[ (1 ) ]( , )

( )p

p

c ch x

xλ λ λ

λλ λ

− +=

+ (27)

pθ μ= (28) The sliding mode control action is generated as follows u = ueq + ud (29) The equivalent control ueq is given by

1ˆ ˆequ b u−= (30) and the discontinuous control ud is given by

1ˆ sgn( )du b K s−= − (31) To reduce chattering effect the discontinuous control law term, sgn(s) in (31) is replaced by sat(s) function [10] around the switching surface leading to the final control law written as

1ˆ ˆ su b u K satφ

− ⎡ ⎤⎛ ⎞= −⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦

(32)

where 0φ > is the width of the boundary layer and the

function of ssat

φ⎛ ⎞⎜ ⎟⎝ ⎠

is defined as

1, 1

sg n , 1

s

ssa ts s

φφ

φ φ

⎧ ⎛ ⎞ ≤⎪ ⎜ ⎟⎝ ⎠⎛ ⎞ ⎪= ⎨⎜ ⎟

⎝ ⎠ ⎛ ⎞ ⎛ ⎞⎪ ≥⎜ ⎟ ⎜ ⎟⎪ ⎝ ⎠ ⎝ ⎠⎩

(33)

The performances of SMC are compared by defining two sliding surfaces s(λ, t) as follows:

Sliding Surface 1: 1

( , )n

eds tdt

λ ψ λ−

⎛ ⎞= +⎜ ⎟⎝ ⎠

(34)

where n is the order of the system to be controlled and is equal to 1 in this paper, ψ is a positive constant, λe is the error which is λe = λ – λd, λd is the desired slip. So, the sliding surface is

( , ) ds tλ λ λ= − (35) Sliding Surface 2: By adding an integral term in (34)

( )0

( , )t

d ds t dtλ λ λ λ λ= − + −∫ (36)

In the case of sliding surface 1: ds λ λ= − (37)

On setting 0s = , equation (25) gives

1ˆ ˆˆˆ ( , ) ( , )a du f x h xλ θ λ λ= − − + (38)

The final control law, u can be written as

11

ˆ ˆ ˆˆ( , ) ( , )a dsu b f x h x Ksatλ θ λ λφ

− ⎡ ⎤⎛ ⎞= − − + −⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦

(39)

The precise knowledge of fa(λ,x) and b are unknown. The nonlinear function fa(λ,x) is estimated as ˆ ( , )af xλ and the estimation error is assumed to be bounded by some known Figure 4 Structure of Slip Ratio Sliding Mode Controller

Sliding Mode

Controller

Vehicle Dynamics

V

max( , )w

w

υ

υ

ω ωω ω−

Actuator Dynamics

Tb y= ων

λ

λd

ωw ,ων

266

continuous function Fa(λ,x), so thatˆ( , ) ( , ) ( , )a a af x f x F xλ λ λ− ≤ .

The nonlinear function (λ,x1) is estimated as 1ˆ( , )h xλ

and the estimation error is assumed to be bounded by some known continuous function H(λ,x1), so that

1 1 1ˆ( , ) ( , ) ( , )h x h x H xλ λ λ− ≤ . So we have

1 1 1ˆ( , ) ( , ) ( , )h x h x H xλ λ λ≤ + (40)

1 1 1ˆ( , ) ( , ) ( , )h x h x H xλ λ λ≤ +

Fa and H can be calculated using (26) and (27). Considering the uncertainty limits of the constant c1, c2 and c3 to be ±20%. Fa(λ,x) can be thus calculated as ˆ

a a aF f f≤ − .

Assuming the unknown control gain b is of constant sign and known bounds (a 20% bound in the estimation error can be considered), which is estimated by b, i.e.

min max min maxˆ0 , 0b b b b b b< ≤ ≤ < ≤ ≤ (41)

where min maxˆ ( )b b b= and 1 b̂

bβ β− ≤ ≤ (42)

where m ax

m in

bb

β = (43)

It should be noted that K should be chosen to satisfy the sliding condition

212

d s sdt

η≤ (44)

where η is a positive constant. Thus the value of K can be found as

m ax ˆ( ) ( 1)aK F H uβ η θ β≥ + + + − (45) where θmax is the maximum value of θ. Using (36), K can be written as

1 1 1

1 11 m ax 1

ˆ ˆ ˆ ˆ1 ( , )

ˆ ˆ ˆˆ1 ( , ) ( ( , ))

a a dK b b F b b b b f x

b b h x b b H x

η λ λ

θ λ θ λ

− − −

− −

≥ + + − − +

− +

(46)

The tuning rule is chosen [1] as

1ˆ( , )ˆ h x sλθ

γ= (47)

In the case of sliding surface 2:

d ds λ λ λ λ= − + − (48) The final control law, u can be written as

11

ˆ ˆ ˆˆ( , ) ( , )a d dsu b f x h x Ksatλ θ λ λ λ λφ

− ⎡ ⎤⎛ ⎞= − − + − + −⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦

(49)

K can be written as 1 1 1

m a x 1

1

11

ˆ ˆ ˆ ( ( , ) )ˆ ˆ1 ( , )

ˆ ˆˆ1 ( , )

a

a d d

K b b F b b b b H x

b b f x

b b h x

η θ λ

λ λ λ λ

θ λ

− − −

−

−

≥ + + +

− − − + +

−

(50)

V. RESULTS AND DISCUSSIONS Two sliding surfaces have been selected as given in

(35) and (36) for comparison of SMC performances for slip ratio control of a HEV and hence a simulation set up is prepared using MATLAB. The initial velocities of the vehicle and wheel are taken as 90 km/h and 89.2 km/h respectively. The breaking torque is limited to 2000 Nm. The simulation parameters of vehicle and actuator are given in Table.1.

For the choice of both the sliding surfaces, it can be seen from Fig.5(a) and Fig.5(b) that the wheel and vehicle speeds are decelerating and the controller avoids locking.

TABLE 1: HEV and Actuator Simulation Parameters

Torque Constant (Kt) 0.2073 m-kg/amp

m 1400 kg Back emf constant (Kb) 2.2 V/rad/sec J 0.65 kg m2 Inductance (L) 0.0028 H rw 0.31 m Resistance (R) 0.125 ohm fn 3560 N nw 4 cv 0.595 N/m2/s2

Velocities

Figure 5(a) Velocities vs Time for sliding surface (1)

Figure 5(b) Velocities vs Time for sliding surface (2)

Braking Torque

Figure 6(a) Braking torque vs Time for sliding surface (1)

Figure 6(b) Braking torque vs Time for sliding surface (2)

Slip

Figure 7(a) Slip vs Time for sliding surface (1)

0 0.5 1 1.5 2 2.5 30

20

40

60

80

100

Time (sec)

Vel

ociti

es (k

m/h

)

x1x2

0 0.5 1 1.5 2 2.5 30

20

40

60

80

100

Time (sec)

Vel

ociti

es (k

m/h

)x

1

x2

0 0.5 1 1.5 2 2.5 3-2000

-1000

0

1000

Time (sec)

Bra

king

torq

ue (N

m)

0 0.5 1 1.5 2 2.5 3-2000

-1000

0

1000

Time (sec)

Bra

king

torq

ue (N

m)

0 0.5 1 1.5 2 2.5 3-0.8

-0.6

-0.4

-0.2

0

Time (sec)

Slip

267

Figure 7(b) Slip vs Time for sliding surface (2)

Required Voltage

Figure 8(a) Required voltage vs Time for sliding surface (1)

Figure 8(b) Required voltage vs Time for sliding surface (2)

The response of the braking torques for both the

sliding surfaces given in (35) and (36) are compared in Fig.6(a) and Fig.6(b). The break torque control force is the summation of the equivalent and discontinuous braking torques with the actuator limitation. After application of brake, the wheel speed suddenly falls and where as the vehicle speed slowly decreases, resulting in higher slip ratio and hence the wheel is in the verge of locking. But to avoid wheel locking, the wheel speed is suddenly increases by the controller at 2.3 seconds to a desirable range to maintain the slip ratio in a desired range shown in Fig.5(a) and Fig.5(b). This is done by increasing braking torque demand at 2.3 seconds shown in Fig.6(a), Fig.6(b), by increasing the applied voltage to the actuator shown in Fig.8(a) and Fig.8(b). This phenomena relating braking torque, velocities and voltage requirement are true for choice of both the sliding surface in the design of adaptive sliding mode slip ratio controller. In both the cases, the controller maintains the desired slip ratio shown in Fig.7(a) and Fig.7(b). The controller tracks the desired slip ratio (0.2) excellently for a slippery road as seen from Fig.7(a) and Fig.7(b). It has been noted here that while maintaining desired slip ratio, the control activities of braking torque and required voltage have less chattering in case of sliding surface (2) shown in Fig.6(a) Fig.6(b) for braking torque and in Fig.8(a) Fig.8(a) for voltage requirement to the actuator. The sliding variable converges to zero which indicates that for both the cases of sliding surfaces, system states are always on the sliding plane.

VI. CONCLUSIONS In the application of adaptive sliding mode slip ratio

control for a HEV, the performance of controllers with different two sliding surfaces has been compared through MATLAB simulation. The controller’s parameters were tuned off line during simulation to track desired slip ratio of a HEV. It can be concluded from the simulation result that the choice of sliding surface having integral term in (45) for the design of adaptive sliding mode slip ratio controller for a HEV results chattering reduction in the actuator action and estimation error convergence. It has also been concluded that by increasing reaching time for sliding variables in the choice of 2nd sliding surface also improves in chattering reduction in the actuator action and estimation error convergence.

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[15] J. Y. Wong, “Theory of Ground Vehicles”, Wiley, New York 4th Edison [16] T. Shim, S. Chang, and S. Lee, “Investigation of Sliding- Surface Design

of Sliding Mode Controller in Antilock Braking Systems” IEEE Trans on Vehicular Technology, Vol.57, No.2, March 2008.

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