A Model Predictive Sliding Mode Control with Integral Action for

Slip Suppression of Electric Vehicles

Tohru Kawabe

Faculty of Engineering, Information and Systems, University of Tsukuba, Tsukuba 305-8573, Japan

Keywords:

Electric Vehicle, Slip Ratio, Robustness, Sliding Mode Control, Model Predictive Control.

Abstract:

This paper proposes a new SMC (Sliding Mode Control) method with MPC (Model Predictive Control) al-

gorithm for the slip suppression of EVs (Electric Vehicles). This method introducing the integral term with

standard SMC gain, where the integral gain is optimized for each control period by solving an optimization

problem based on the MPC algorithm to improve the acceleration performance and the energy consumption

of EVs. Numerical simulation results are also included to demonstrate the effectiveness of the method.

1 INTRODUCTION

Over the past decades, the automobile population has

been increasing rapidly in the developing countries,

such as BRICs (Brazil, Russia, India and China) (Dar-

gay et al., 2007). With the wide spread of automobiles

all over the world, especially internal-combustion en-

gine vehicles (ICEVs), the environment and energy

problems: air pollution, global warming, oil resource

exhaustion and so on, are going severely (Mamalis

et al., 2013). As a countermeasure to these prob-

lems, the development of next-generation vehicles

have been focused. EVs run on electricity only and

they are zero emission and eco-friendly. So EVs have

attracted great interests as a powerful solution against

the problems mentioned above (Brown et al., 2010;

Hirota et al., 2011; Tseng et al., 2013).

EVs are propelled by electric motors, using elec-

trical energy stored in batteries or another energy stor-

age devices. Electric motors have several advantages

over ICEs (Internal-Combustion Engines):

• The input/output response is faster than for gaso-

line/diesel engines.

• The torque generated in the wheels can be de-

tected relatively accurately

• Vehicles can be made smaller by using multiple

motors placed closer to the wheels.

The travel distance per charge for EV has been in-

creased through battery improvements and using re-

generation brakes, and attention has been focused on

improving motor performance. The above-mentioned

facts are viewed as relatively easy ways to improve

maneuverability and stability of EVs.

It’s, therefore, important to research and develop-

ment to achieve high-performance EV traction con-

trol. Several methods have been proposed for the

traction control (Fujii and Fujimoto, 2007) by us-

ing slip ratio of EVs, such as the method based

on MFC (Model Following Control) in (Hori, 2000)

We have been proposed MP-PID (Model Predictive

Proportional-Integral-Derivative) method in (Kawabe

et al., 2011) and MP-2DOF-PID method in (Kawabe,

2014).

These methods show good performances under

the nominal conditions where the situations, for ex-

ample, mass of vehicle, road condition, and so on,

are not changed. To meet the high performance even

variation happened in such conditions, it is signiﬁ-

cant to construct the robust control systems against

the changing of situation. About this point, SMC per-

forms good robustness against the uncertainties and

nonlinearities of the systems.

However, for slip suppression with the conven-

tional SMC (Slotine and Li, 1991; Eker and Aki-

nal, 2008) , the control performance will get degrada-

tion due to the chattering which always occurs when

switching the control inputs due to the structure of

SMC. To overcome such disadvantages, the SMC

method introducing the integral action with gain to

design the sliding surface (SMC-I) has been proposed

in (Li and Kawabe, 2013), where the integral gain

is derived by trial and error. In order to get better

control performance and save more energy for slip

suppression of EVs with changing the mass of vehi-

151

Kawabe T..

A Model Predictive Sliding Mode Control with Integral Action for Slip Suppression of Electric Vehicles.

DOI: 10.5220/0005500401510158

In Proceedings of the 12th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2015), pages 151-158

ISBN: 978-989-758-123-6

Copyright

c

2015 SCITEPRESS (Science and Technology Publications, Lda.)

cle and road condition, the optimal gain derived on-

line is expected. Therefore, we have developed the

Model Predictive Sliding Mode Control with Integral

action (MP-SMC-I) (Li and Kawabe, 2014), which

determines the integral gain adaptively at each step by

MPC algorithm (Maciejowski, 2005). However, there

are some room to improve the robust performance of

this method. This paper, therefore, proposes the im-

proved MP-SMC-I method. The simulation results

are shown to verify the effectiveness of the proposed

method.

2 ELECTRIC VEHICLE

DYNAMICS

As a ﬁrst step toward practical application, this paper

restricts the vehicle motion to the longitudinal direc-

tion and uses direct motors for each wheel to simplify

the one-wheel model to which the drive force is ap-

plied. In addition, braking was not considered this

time with the subject of the study being limited to

only when driving.

From ﬁg. 1, the vehicle dynamical equations are

expressed as eqs. (1) to (4).

M

dV

dt

= F

d

(λ) − F

a

−

T

r

r

(1)

J

dω

dt

= T

m

− rF

d

(λ) − T

r

(2)

F

m

=

T

m

r

(3)

F

d

= µ(c,λ)N (4)

Where M is the vehicle weight, V is the vehicle body

velocity, F

d

is the driving force, J is the wheel inertial

moment, F

a

is the resisting force from air resistance

and other factors on the vehicle body, T

r

is the fric-

tional force against the tire rotation, ω is the wheel

angular velocity, T

m

is the motor torque, F

m

is the

motor torque force conversion value, r is the wheel

radius, and λ is the slip ratio. N is the normal tire

force deﬁned as N = Mg where g is the acceleration

of gravity. The slip ratio is deﬁned by eq. (5) from the

wheel velocity (V

ω

) and vehicle body velocity (V ).

λ =

V

ω

−V

V

ω

(accelerating)

V −V

ω

V

(braking)

(5)

λ during accelerating can be shown by eq. (6) from

ﬁg. 1.

λ =

rω −V

rω

(6)

Figure 1: One-wheel car model.

The frictional forces that are generated between

the road surface and the tires are the force generated

in the longitudinal direction of the tires and the lateral

force acting perpendicularly to the vehicle direction

of travel, and both of these are expressed as a func-

tion of λ. The frictional force generated in the tire

longitudinal direction is expressed as µ, and the rela-

tionship between µ and λ is shown by eq. (7) below,

which is a formula called the Magic-Formula(Pacejka

and Bakker, 1991) and which was approximated from

the data obtained from testing.

µ(λ) = −c

road

× 1.1 × (e

−35λ

− e

−0.35λ

) (7)

Where c

road

is the coefﬁcient used to determine the

road condition and was found from testing to be ap-

proximately c

road

= 0.8 for general asphalt roads, ap-

proximately c

road

= 0.5 for general wet asphalt, and

approximately c

road

= 0.12 for icy roads. For the var-

ious road conditions (0 < c < 1), the µ − λ surface is

shown in ﬁg. 2.

It shows how the friction coefﬁcient µ increases

with slip ratio λ (0.1 < λ < 0.2) where it attains the

maximum value of the friction coefﬁcient. As deﬁned

in eq. (4), the driving force also reaches the maximum

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.2

0.4

0.6

0.8

1

Slip Ratio λ

Friction coecient µ

Icy road

Wet asphalt road

Dry asphalt road

Figure 2: µ − λ curve for road conditions.

ICINCO2015-12thInternationalConferenceonInformaticsinControl,AutomationandRobotics

152

value corresponding to the friction coefﬁcient. How-

ever, the friction coefﬁcient decreases to the minimum

value where the wheel is completely skidding. There-

fore, to attain the maximum value of driving force for

slip suppression, it should be controlled the optimal

value of slip ratio. the optimal value of λ is derived as

follows. Choose the function µ

c

(λ) deﬁned as

µ

c

(λ) = −1.1 × (e

−35λ

− e

−0.35λ

). (8)

By using eqs. (7) and (8), it can be rewritten as

µ(c,λ) = c

road

· µ

c

(λ). (9)

Evaluating the values of λ which maximize µ(c,λ)

for different c(c > 0), means to seek the value of λ

where the maximum value of the function µ

c

(λ) can

be obtained. Then let

d

dλ

µ

c

(λ) = 0 (10)

and solving equation (10) gives

λ =

log100

35 − 0.35

≈ 0.13. (11)

Thus, for the different road conditions, when λ ≈ 0.13

is satisﬁed, the maximum driving force can be gained.

Namely, from eq. (7) and ﬁg. 2, we ﬁnd that regard-

less of the road condition (value of c), the λ − µ sur-

face attains the largest value of µ when λ is the opti-

mal value 0 .13.

3 MP-SMC WITH INTEGRAL

ACTION DESIGN FOR SLIP

SUPPRESSION

3.1 SMC with Integral Action (SMC-I)

Method

In this section, the previous proposed control strategy

based on SMC with integral action (SMC-I) (Li and

Kawabe, 2013) is explained. Without loss of gener-

ality, one wheel car model in ﬁg. 1 is used for the

design of the control law. The nonlinear system dy-

namics can be presented by a differential equation as

˙

λ = f + bT

m

(12)

where λ ∈ R is the state of the system representing the

slip ratio of the driving wheel which is deﬁned as eq.

(5) for the case of acceleration, T

m

is the control in-

put. f describes the nonlinearity of system and b is

the input gain, and they are all time-varying. Differ-

entiating eq. (5) with respect to time gives

˙

λ =

−

˙

V + (1 − λ)

˙

V

w

V

w

(13)

and substituting eqs. (1), (2) and (4) into eq. (13), the

following equations can be attained,

f = −

g

V

w

1 + (1 − λ)

r

2

M

J

w

µ(c,λ), (14)

b =

(1 − λ)r

J

w

V

w

. (15)

The sliding mode controller is described to main-

tain the value of slip ratio λ at the desired value λ

∗

.

Referring to (Li and Kawabe, 2013), in order to

reduce the undesired chattering effect for which it is

possible to excite high frequency modes, and guaran-

tee zero steady-state error, an integral action with gain

has been introduced to the design of sliding surface.

By adding an integral item to the difference between

the actual and desired values of the slip ratio, the slid-

ing surface function s is given by

s = λ

e

+ K

in

∫

t

0

λ

e

(τ)dτ, (16)

where λ

e

is deﬁned as λ

e

= λ− λ

∗

and K

in

is the inte-

gral gain, K

in

> 0.

The sliding mode occurs when the state reaches

the sliding surface deﬁned by s = 0. The dynamics of

sliding mode is governed by

˙s = 0. (17)

By using eqs. (12) to (17), the sliding mode con-

trol law is derived by adding a switching control input

T

msw

to the nominal equivalent control input T

meq n

as

in (Li and Kawabe, 2013)

T

m

= T

meq n

+ T

msw

, (18)

T

meq n

=

1

b

[− f

n

− K

in

λ

e

], (19)

T

msw

=

1

b

−Ksat(

s

Φ

)

, (20)

sat

s

Φ

=

−1 s < −Φ

s

Φ

−Φ ≤ s ≤ Φ,

1 s > Φ

(21)

where “

n

” is used to indicate the estimated model

parameters. f

n

is the estimation of f calculated by us-

ing the nominal values of vehicle mass M

n

and road

surface condition coefﬁcient c

n

. Φ > 0 is a design pa-

rameter which deﬁnes a small boundary layer around

the sliding surface. The sliding gain K > 0 is selected

as

K = F + η (22)

by deﬁning Lyapunov candidate function in (Li and

Kawabe, 2013), where F = | f − f

n

| and η is a design

parameter.

By using eqs. (18), (19), (20) and (22), the control

law of SMC-I can be represented as

T

m

=

1

b

− f

n

− K

in

λ

e

− (F + η)sat

s

Φ

. (23)

AModelPredictiveSlidingModeControlwithIntegralActionforSlipSuppressionofElectricVehicles

153

3.2 Improved Model Predictive SMC

with Integrl Action (MP-SMC-I)

Method

In this section, we show the improved MP-SMC-I

method. Although the MP-SMC-I method has been

developed by us (Li and Kawabe, 2014), it has disad-

vantage to take time to calculate the predicted control

input (

ˆ

T

m

). Then we improved this point to propose

new calculation method.

Generally, MPC algorithm is used to predict the

future state behavior base on the discrete-time state

space model (Maciejowski, 2005) . The continuous

time state space model for the slip ratio control repre-

sented by eq. (12) can not be dealt with in the same

way. It is transformed to the discrete time state space

model at sampling time t = kT , T is the sampling pe-

riod. The torque input is deﬁned by

T

m

(t) = T

m

(kT ), kT ≤ t < (k + 1)T. (24)

For convenience, we will omit T in the following

equations.

The controlled object of vehicle dynamics can be

described as follows.

λ(k + 1) = f

d

k, λ(k)

+ b

d

k, λ(k)

T

m

(k) (25)

where λ(k) is the state variable representing the slip

ratio at time k. f

d

k, λ(k)

describes the nonlinearity

of the discrete time system, b

d

k, λ(k)

is the input

gain, and they are given by

f

d

k, λ(k)

= −

g

V

w

(k)

1 +

1 − λ(k)

r

2

M

J

w

×µ

c,λ(k)

(26)

b

d

k, λ(k)

=

1 − λ(k)

r

J

w

V

w

(k)

. (27)

The control input T

m

(t) given by eq. (23) can be

rewritten as

T

m

(k) =

1

b

d

k, λ(k)

f

dn

k, λ(k)

− K

in

(k)

λ(k) − λ

∗

−

F

d

k, λ(k)

+ η

sat

s

k, λ(k),K

in

(k)

Φ

where λ

∗

is the reference slip ratio, η is the design pa-

rameter, and both of them are constants. f

dn

k, λ(k)

is the estimation of f

d

k, λ(k)

and is deﬁned as

f

dn

k, λ(k)

=

−gµ

c

n

,λ(k)

V

w

(k)

1 +

1 − λ(k)

r

2

M

n

J

w

.

(28)

Now, we set current time to k. For a prediction

horizon H

p

, the predicted slip ratios

ˆ

λ(k + i) for i =

1,··· ,H

p

depend on the known values of current slip

ratios, current torque input and future torque inputs.

By using eq. (25), the predicted slip ratios can be

represented as

ˆ

λ(k + H

p

) = f

d

k + H

p

− 1,

ˆ

λ(k + H

p

− 1)

+ b

d

k + H

p

− 1,

ˆ

λ(k + H

p

− 1)

×

ˆ

T

m

(k + H

p

− 1) (29)

where

ˆ

T

m

(k + i), i = 0, · · · , H

p

− 1 are predicted con-

trol inputs.

ˆ

T

m

(k + i) is given by

ˆ

T

m

(k + i) =

f

dn

k + i,

ˆ

λ(k + i)

− K

in

ˆ

λ(k + i) − λ

∗

b

d

k + i,

ˆ

λ(k + i)

−

F

d

k + i,

ˆ

λ(k + i)

+ η

sat

s

k+i,

ˆ

λ(k+i),K

in

Φ

b

d

k + i,

ˆ

λ(k + i)

.

Where

F

d

k, λ(k)

=

g

V

w

k, λ(k)

µ

c

max

,λ(k)

− µ

c

n

,λ(k)

+

g

1 − λ(k)

r

2

J

w

V

w

k, λ(k)

M

max

µ

c

max

,λ(k)

− M

n

µ

c

n

,λ(k)

and where M

n

is the estimated value of vehicle mass

M and c

n

is estimated for the viscous friction coefﬁ-

cient c. This calculation method of

ˆ

T

m

is improved

method of previous our method (Li and Kawabe,

2014).

Here, we deﬁne the estimated values of these pa-

rameters respectively as the arithmetic mean of the

value of the bounds.

c

n

=

c

min

+ c

max

2

(30)

M

n

=

M

min

+ M

max

2

. (31)

Actually, the mass of the car often changes with the

number of passengers and the weight of luggage. Be-

sides, the car has to always travel on various road sur-

faces. Then the ranges of variation in parameter c and

parameter M are assumed to be deﬁned as

c

min

≤ c ≤ c

max

(32)

M

min

≤ M ≤ M

max

. (33)

Here, the objective function J for deciding the

value of K

in

can be written as

ICINCO2015-12thInternationalConferenceonInformaticsinControl,AutomationandRobotics

154

J =

H

p

−1

∑

i=0

q|

ˆ

λ(k + i + 1) − λ

∗

| + r|

ˆ

T

m

(k + i)|

(34)

where q, r are the positive weights. By using eqs. (29)

and (30), both

ˆ

λ and

ˆ

T

m

can be expressed by K

in

, thus

J can be represented by a function of K

in

. Our aim

is to ﬁnd the parameter K

in

that minimizes this objec-

tive function J(K

in

). In a nutshell, the optimization

problem is given by

min

K

in

J(K

in

)

s. t. [ Some constraint conditions

with input or output (if exist)] (35)

At time k, the optimal K

in

(k) can be found by solving

eq. (35) with some optimization method (here, a grid

search method is made to the discretized K

in

) by MPC

Algorithm. Once the optimal K

in

(k) is determined, it

is used as the continuous K

in

(t) for kT ≤ t < (k +

1)T , then the continuous control input T

m

(t) can be

calculated by eq. (23). At the next sampling time k +

1, the optimal K

in

(k + 1) is calculated as the previous

step. At each sampling period, the same operation

is repeated. Therefore, using the MP-SMC-I method

could determine the optimal parameter K

in

by solving

the optimization problem.

4 NUMERICAL EXAMPLES

4.1 Simulation Settings

This section shows the numerical simulation results to

demonstrate the effectiveness of the proposed method

as shown in previous section. The performance of the

proposed MP-SMC-I method is compared with that

of no-control, SMC and SMC-I methods. In the sim-

ulation examples, the vehicle starts from rest and ac-

celerates on icy road, wet asphalt road and dry as-

phalt road respectively with the parameters of dynam-

ics shown in table 1. The maximum simulation time is

set to 20[s] and the maximum velocity of the vehicle

is 180[km/h].

Table 1: Parameters used in the simulations.

J

w

:Inertia of wheel 21.1[kg/m

2

]

r:Radius of wheel 0.26[m]

λ

∗

:Desired slip ratio 0.13

g:Acceleration of gravity 9.81[m/s

2

]

The SMC controller parameters Φ,η as well as the

integral gain K

in

are listed in table 2 which are deter-

mined by trial and error.

Table 2: Controller settings

SMC Φ = 1,η = 5

SMC-I Φ = 1,η = 5

K

in

= 10

MP-SMC-I Φ = 1,η = 5

0 ≤ K

in

≤ 200, ∆K

in

= 1

q = 1.0 × 10

8

r = 1.0

In order to evaluate the energy consumption by

the electric motor, we estimate the energy consumed

for driving the wheel based on the following assump-

tions. Firstly, the electric power is all used to drive the

wheel. Secondly, the power consumed by the vehicle

is in proportional with the rotational energy due to the

rotation of driven wheel. The rotational energy E

rot

is

deﬁned by the rotational inertia of wheel J

w

and the

angular velocity w is given by

E

rot

=

1

2

J

w

w

2

. (36)

In the simulations, the total distance traveled is calcu-

lated by integrating the vehicle velocity from 0 to the

simulation time t

int

and is deﬁned as

D

dis

=

∫

t

int

0

V dt. (37)

To learn how much the energy consumed with respect

to the distance, the cost of the energy per distance (en-

ergy consumption rate) are also calculated.

4.2 Simulation Results

In order to verify the robustness of proposed MP-

SMC-I with variation both in the mass of vehicle and

road condition, the range of the uncertainties in the

mass of vehicle M is [1000,1400][kg] and the range of

road condition coefﬁcient c is [0.1,0.9]. The nominal

values used in the controller design taking the arith-

metic mean of the edge values are M

n

= 1200[kg],

c

n

= 0.5.

Furthermore, three road surfaces switch in the

simulation by time as: an icy road (c = 0.12) dur-

ing [0,0.45)[s], another icy road (c = 0.20) dur-

ing [0.45, 8)[s], a wet asphalt road(c = 0.50) dur-

ing [8,9)[s] and a dry asphalt road(c = 0.80)) during

[9,10][s].

As shown in ﬁg. 3, the slip ratio can be suppressed

to reference value 0.13 represented by black dot

line accurately and rapidly regardless of the masses

changing by 100[kg] from 1000[kg] to 1400[kg].

This implies that MP-SMC-I acts robustly to the vari-

ation in vehicle mass. It also makes a good transient

performance at the switching spots on the road condi-

tion.

AModelPredictiveSlidingModeControlwithIntegralActionforSlipSuppressionofElectricVehicles

155

0 1 2 3 4 5 6 7 8 9 10

0

0.2

0.4

0.6

0.8

1

Time [s]

Slip ratio

b

b

M=1000[kg]

M=1100[kg]

M=1200[kg]

M=1300[kg]

M=1400[kg]

Reference value

Figure 3: Time response of slip ratio with MP-SMC-I for

different vehicle masses.

Next, we compared MP-SMC-I with the conven-

tional SMC, SMC-I and no control. For saving of

space, only results with the case of vehicle mass

M assigned to 1000[kg] and 1400[kg] are limited to

shown below. From ﬁgs. 4 and 5 show the responses

of slip ratio and motor torque under three different

road conditions. The slip ratio by MP-SMC-I can

be suppressed to the reference value more accurately

and rapidly than SMC-I. When the road condition

changes, the slip ratio by MP-SMC-I also keep an

suitable transient performance, reducing the steady

0 1 2 3 4 5 6 7 8 9 10

0

0.2

0.4

0.6

0.8

1

Time [s]

Slip ratio

No control

SMC

SMC−I

MP−SMC−I

Reference value

(a) Slip ratio

0 1 2 3 4 5 6 7 8 9 10

0

200

400

600

800

1000

Time [s]

Torque [N m]

b

b

No control

SMC

SMC−I

MP−SMC−I

(b) Motor torque

Figure 4: Simulation results with No control, SMC, SMC-I

and MP-SMC-I (M = 1000[kg]).

0 1 2 3 4 5 6 7 8 9 10

0

0.2

0.4

0.6

0.8

1

Time [s]

Slip ratio

b

b

No control

SMC

SMC−I

MP−SMC−I

Reference value

(a) Slip ratio

0 1 2 3 4 5 6 7 8 9 10

0

200

400

600

800

1000

Time [s]

Torque [N m]

b

b

No control

SMC

SMC−I

MP−SMC−I

(b) Motor torque

Figure 5: Simulation results with No control, SMC, SMC-I

and MP-SMC-I (M = 1400[kg]).

state error and rising time. The motor torque utilized

by MP-SMC-I sufﬁciently to drive the wheel. The

ﬂuctuation in torque occurs at the time of road condi-

tion switching, which leads to K

in

setting based on the

prediction in the set prediction interval by the MPC

algorithm. K

in

is adjusted on-line for the optimum

value from the objective function during the predic-

tion interval when the road condition switches.

In ﬁg. 6, the response of K

in

varies sharply at

the road condition switching spot because the tire

grip margin changes, which leading to adjust K

in

to

achieve the sufﬁcient driving force. Once the slip ra-

tio deviates from the reference value, K

in

is adjusted

much based on MPC algorithm to get the appropriate

force to drive the wheel to reach the reference value

ﬁnally.

Then, to conﬁrm the acceleration performance, we

compare MP-SMC-I with SMC-I, SMC and No con-

trol. As shown in ﬁg. 7, the curve of the vehicle ve-

locity with MP-SMC-I coincides with the one with

SMC-I almost. But we can see that the vehicle with

MP-SMC-I represented by the red curve achieves the

maximum driving force for the best acceleration dur-

ing the whole simulation time.

Finally, table 3 show the results of the energy con-

sumed E

r

, the total distance D

d

and the energy con-

sumption rate E

p

for different mass of vehicle. We

can see that the energy consumption rate with MP-

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0 1 2 3 4 5 6 7 8 9 10

0

50

100

150

200

Time [s]

K

in

(i) M = 1000[kg]

0 1 2 3 4 5 6 7 8 9 10

0

50

100

150

200

Time [s]

K

in

(ii) M = 1400[kg]

Figure 6: Time response of K

in

for MP-SMC-I).

0 1 2 3 4 5 6 7 8 9 10

0

5

10

15

20

25

Time [s]

Body velocity [m/s]

b

b

No control

SMC

SMC−I

MP−SMC−I

(i) M = 1000[kg]

0 1 2 3 4 5 6 7 8 9 10

0

5

10

15

20

25

Time [s]

Body velocity [m/s]

b

b

No control

SMC

SMC−I

MP−SMC−I

(ii) M = 1400[kg]

Figure 7: Time response of body velocity with No control,

SMC, SMC-I and MP-SMC-I.

SMC-I is nearly as much as the one with SMC-I, but

the total distance is longer than SMC-I. This indicates

that the vehicle with MP-SMC-I achieve a better ac-

celeration performance. As the same result as the pre-

Table 3: Results of energy consumption rate with No con-

trol, SMC, SMC-I and MP-SMC-I.

(i) ( M = 1000[kg])

E

r

D

d

E

p

[Wh] [m] [Wh/km]

No control 80.34 55.52 1447

SMC 28.30 64.82 437

SMC-I 28.11 69.58 404

MP-SMC-I 28.64 70.03 409

(ii) (M = 1400[kg])

E

r

D

d

E

p

[Wh] [m] [Wh/km]

No control 62.75 56.33 1114

SMC 35.10 64.98 540

SMC-I 36.22 69.54 521

MP-SMC-I 36.90 70.02 527

vious chapter described, due to the mass increasing

caused much energy cost that the EV should be made

more light without detriment to performance.

5 CONCLUSIONS

In this paper, the improved MP-SMC-I method has

been proposed for robust slip suppression problem of

EVs. We can veriﬁed that the the proposed method

shows good robust performance against the changing

of road condition and vehicle mass by numerical sim-

ulations. Also this method shows the good energy

consumption performance. At the present stage, ef-

fectiveness of the proposed method is only conﬁrmed

by numerical simulations. Next step, therefore, is to

realize the real system using the proposed method

and to proof the advantage of such system by real

experiments. Furthermore, in future work, the suit-

ability of the proposed method must be studied not

only for the slip suppression addressed by this paper

but also for overall driving including during braking.

Even for this issue, however, the basic framework of

the method can be expanded relatively easily to the

foundation for making practical EVs with high per-

formance and safety traction control systems.

ACKNOWLEDGEMENTS

This research was partially supported by Grant-

in-Aid for Scientiﬁc Research (C) (Grant number:

24560538; Tohru Kawabe; 2012-2014) from the Min-

istry of Education, Culture, Sports, Science and Tech-

nology of Japan.

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157

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