# Traction Control of Electric Vehicles Using Sliding-Mode Controller with Tractive Force Observer

Authors: Suwat Kuntanapreeda
Journal: International Journal of Vehicular Technology
 (2014)

---

## Abstract

Traction control is an important element in modern vehicles to enhance drive efficiency, safety, and stability. Traction is produced by friction between tire and road, which is a nonlinear function of wheel slip. In this paper, a sliding-mode control approach is used to design a robust traction controller. The control objective is to operate vehicles such that a desired wheel slip ratio is achieved. A nonlinearity observer is employed to estimate tire tractive forces, which are used in the control law. Simulation and experimental results have illustrated the success of the proposed observer-based controller.


---

## Body

## 1. Introduction

Electric vehicles (EVs) have become very attractive in replacing conventional internal combustion engine vehicles because of environmental and energy issues. They have received a great attention from the research community. Control methodologies have been actively developed and applied to EVs to improve the EVs performances [1–8].Traction control plays an important role in vehicle motion control because it can directly enhance drive efficiency, safety, and stability [9, 10]. Traction is the vehicular propulsive force produced by friction between tire and road. The characteristics of the friction are nonlinear and uncertain, which make traction control difficult. The friction depends on many factors such as tire type, road surface, road condition, and wheel slip. Accordingly, an objective of the traction control is to operate vehicles such that a desired wheel slip ratio is obtained. The slip ratio yielding the maximum friction coefficient is usually desired because it yields the maximum torque from the propulsion system to drive the vehicle forward.Traction control of electric vehicles has drawn extensive attention since electric motors can produce very quick and precise torques compared to conventional internal combustion engines. In [1], traction control based on a maximum transmission torque estimation (MTTE) approach was proposed. The estimation was carried out by an open-loop disturbance observer, which required only the input torque and the wheel speed. The estimated maximum transmission torque was used in the control law as a constraint to prevent the slip. Experimental results illustrated the effectiveness and practicality of the proposed control design. The MTTE approach was extended by replacing the open-loop observer with a closed-loop observer in [2]. By doing this, the robustness of the control system was markedly enhanced. In [3], traction control of electric vehicles using a sliding-mode observer to estimate the maximum friction was presented. The observer was based on the LuGre friction model. The controller used the estimated maximum friction to determine the suited maximum torque for the wheels.Sliding-mode control has been extensively used for control of uncertain nonlinear systems because of its robustness property. The essence of the sliding-mode control is to use a switching control command to drive the controlled system’s state trajectory onto a specified sliding surface in the state space and then to keep the state trajectory moving along this surface [11, 12].As far as sliding-mode control of vehicles is concerned, wheel slip control of electric vehicles based on a sliding-mode framework was proposed in [4]. A conditional integrator approach was employed to overcome the chattering, enabling a smooth transition to a PI control law when the slip is close to the set point. Experimental results demonstrated a good slip regulation and robustness to disturbances. In [13], a sliding-mode approach to the design of an active braking controller was proposed. The controlled variable was a convex combination of wheel deceleration and wheel slip. The approach offered advantages with respect to pure slip and deceleration control. In [14], a second-order sliding-mode traction force controller for vehicles was proposed. The traction control was achieved by maintaining the wheel slip at a desired value.This paper presents a robust control scheme for traction control of electric vehicles. The control objective is to operate the vehicles at a desired wheel slip ratio. The paper proposes a simple approach to design a traction controller based on a sliding-mode control framework. The main motivation of the design is the robustness to uncertainties. The implementation of the control design requires tractive forces for feedback, but they are not usually available in practices. To overcome this problem, a PI observer developed in [15, 16] is used to estimate the tractive forces. The PI observer has an attractive zero-steady-state feature similar to the well-known PI controllers. This synthesis of the sliding-mode traction controller and the PI observer makes the implementation practical. At the end, the resulting observer-based controller is experimentally validated in a single-wheel test rig.The rest of the paper is organized as follows. In the next section some preliminaries are provided. The longitudinal dynamic model of the vehicles used in the paper is presented in Section3, followed by the controller and observer design in Section 4. Simulation results are given in Section 5. In Section 6 an experimental study is presented. The conclusions of this paper are drawn in Section 7.


## 2. Preliminaries

### 2.1. Sliding-Mode Control

Sliding-mode control based on the equivalent control method is summarized in this subsection. The reader is referred to [11, 12] for more details of the method.The controlled system is expressed as(1)
x
˙
=
f
x
+
g
x
u
,
where x
∈
R
n is the state variable vector, u
∈
R
m is the input vector, and f
(
·
) and g
(
·
) are nonlinear functions. Let S
(
x
) be a desired sliding surface, which is usually chosen according to the control objective. Based on the equivalent control method, the control input vector is written as
(2)
u
=
u
eq
+
u
sw
,
where u
eq and u
sw are called the equivalent control and the switching control, respectively. The equivalent control u
eq is determined based on the assumption that the system trajectory is staying on the sliding surface. Thus, it is simply obtained by setting
(3)
S
˙
=
0
.
The switching control u
sw is designed to guarantee that the system trajectory moves towards the sliding surface and stays on it. It is determined such that the reachability condition
(4)
S
S
˙
=
-
η
S
,
η
>
0
is satisfied.


### 2.2. Nonlinearity Observer

The nonlinearity observer developed in [15, 16] is provided in this subsection. Consider the following nonlinear system:
(5)
x
˙
=
A
x
+
N
α
t
+
M
σ
t
+
B
u
,
y
=
C
x
,
where x
∈
R
n is the state variable vector, u
∈
R
m is the control vector, y
∈
R
p is the output vector, A is the system matrix, B is the control input matrix, C is the output matrix, and M and N are the constant matrices. Here, α
(
t
) is an unknown nonlinear function whereas σ
(
t
) is a known function. The observer is designed to estimate α
(
t
).The fundamental idea of the observer is to approximateα
(
t
) by a fictitious system:
(6)
α
t
≈
H
v
t
,
v
˙
(
t
)
=
V
v
(
t
)
.
By substituting (6) into (5), the system can be expressed as
(7)
x
˙
v
˙
=
A
N
H
0
V
x
v
+
B
M
0
0
u
σ
.
Thus, the observer is chosen as
(8)
x
^
˙
v
^
˙
=
A
N
H
0
V
x
^
v
^
+
B
M
0
0
u
σ
+
L
x
L
v
y
-
C
x
^
,
where L
x and L
v are the observer gain matrices that must be chosen such that the observer is asymptotically stable.In a special case whenH
=
I and V
=
0 are chosen, the observer is reduced to the proportional-integral (PI) observer:
(9)
x
^
˙
=
A
x
^
+
B
u
+
M
σ
+
L
x
y
-
C
x
^
+
N
L
v
∫
y
-
C
x
^
d
t
,
v
^
=
L
v
∫
y
-
C
x
^
d
t
and the estimated nonlinearity is given by
(10)
α
^
t
=
v
^
t
.
The PI observer has been successfully applied to control problems [17, 18]. The reader is referred to [16] for a proof of the estimation convergence and an analysis of the estimation errors.


## 2.1. Sliding-Mode Control

Sliding-mode control based on the equivalent control method is summarized in this subsection. The reader is referred to [11, 12] for more details of the method.The controlled system is expressed as(1)
x
˙
=
f
x
+
g
x
u
,
where x
∈
R
n is the state variable vector, u
∈
R
m is the input vector, and f
(
·
) and g
(
·
) are nonlinear functions. Let S
(
x
) be a desired sliding surface, which is usually chosen according to the control objective. Based on the equivalent control method, the control input vector is written as
(2)
u
=
u
eq
+
u
sw
,
where u
eq and u
sw are called the equivalent control and the switching control, respectively. The equivalent control u
eq is determined based on the assumption that the system trajectory is staying on the sliding surface. Thus, it is simply obtained by setting
(3)
S
˙
=
0
.
The switching control u
sw is designed to guarantee that the system trajectory moves towards the sliding surface and stays on it. It is determined such that the reachability condition
(4)
S
S
˙
=
-
η
S
,
η
>
0
is satisfied.


## 2.2. Nonlinearity Observer

The nonlinearity observer developed in [15, 16] is provided in this subsection. Consider the following nonlinear system:
(5)
x
˙
=
A
x
+
N
α
t
+
M
σ
t
+
B
u
,
y
=
C
x
,
where x
∈
R
n is the state variable vector, u
∈
R
m is the control vector, y
∈
R
p is the output vector, A is the system matrix, B is the control input matrix, C is the output matrix, and M and N are the constant matrices. Here, α
(
t
) is an unknown nonlinear function whereas σ
(
t
) is a known function. The observer is designed to estimate α
(
t
).The fundamental idea of the observer is to approximateα
(
t
) by a fictitious system:
(6)
α
t
≈
H
v
t
,
v
˙
(
t
)
=
V
v
(
t
)
.
By substituting (6) into (5), the system can be expressed as
(7)
x
˙
v
˙
=
A
N
H
0
V
x
v
+
B
M
0
0
u
σ
.
Thus, the observer is chosen as
(8)
x
^
˙
v
^
˙
=
A
N
H
0
V
x
^
v
^
+
B
M
0
0
u
σ
+
L
x
L
v
y
-
C
x
^
,
where L
x and L
v are the observer gain matrices that must be chosen such that the observer is asymptotically stable.In a special case whenH
=
I and V
=
0 are chosen, the observer is reduced to the proportional-integral (PI) observer:
(9)
x
^
˙
=
A
x
^
+
B
u
+
M
σ
+
L
x
y
-
C
x
^
+
N
L
v
∫
y
-
C
x
^
d
t
,
v
^
=
L
v
∫
y
-
C
x
^
d
t
and the estimated nonlinearity is given by
(10)
α
^
t
=
v
^
t
.
The PI observer has been successfully applied to control problems [17, 18]. The reader is referred to [16] for a proof of the estimation convergence and an analysis of the estimation errors.


## 3. Longitudinal Dynamic Model

A longitudinal dynamic model of two-axle vehicles is presented. The model is known as a bicycle model, which has three degrees of freedom. The model can be found in [14] and is given as
(11)
Vehicle
body

:
F
f
+
F
r
-
F
loss
=
m
V
˙
x
,
Front
axle

:
T
f
-
r
f
F
f
=
I
f
ω
˙
f
,
Rear
axle

:
T
r
-
r
r
F
r
=
I
r
ω
˙
r
,
where V
x is the longitudinal velocity of the vehicle center of gravity, ω
f
,
ω
r are the tire rotational speeds (the subscripts f and r stand for the front and rear axle, resp.), m is the vehicle mass, I
f
,
I
r are the moments of inertia, r
f
,
r
r are the tire effective rolling radius, T
f
,
T
r are the input torques, F
f
,
F
r are the tractive forces, F
loss
=
c
x
V
x
2
·
sgn
⁡
(
V
x
)
+
f
roll
m
g combining the aerodynamic drag and the rolling resistance, and c
x and f
roll are the aerodynamic drag and rolling resistance coefficients, respectively.The tractive forces are given by(12)
F
i
=
μ
λ
i
N
i
,
i
=
f
,
r
,
where μ
(
λ
i
) is the friction coefficient function, λ
i is the wheel slip ratio defined as
(13)
λ
i
=
r
i
ω
i
-
V
x
r
i
ω
i
,
for
driving
,
V
x
-
r
i
ω
i
V
x
,
for
braking
and N
f
,
N
r are the normal loads given as
(14)
N
f
=
l
r
m
g
-
l
h
m
V
˙
x
l
f
+
l
r
,
N
r
=
l
f
m
g
+
l
h
m
V
˙
x
l
f
+
l
r
,
where l
f is the distance from the front axle to the center of gravity, l
r is the distance from the rear axle to the center of gravity, and l
h is the height of the center of gravity (see Figure 1).Figure 1
Longitudinal model of vehicles.The friction coefficient depends on many factors: tire type, road surface, road condition, wheel slip, and so forth. This makes behaviors of the tractive forces complicated. The friction coefficient usually has to be measured experimentally. The typical friction coefficient in dependency of the slip ratio is shown in Figure2. It illustrates that some amount of slip is necessary to produce the tractive force; on the other hand, an excessive slip leads to a loss of the force.Figure 2
Typical trends of longitudinal friction coefficient.


## 4. Control Design

### 4.1. Sliding-Mode Controller Design

A sliding-mode controller is designed for the system described by (11)–(14). The control objective is to drive the vehicle such that the desired slip ratio λ
* is achieved. First, define the sliding surface as
(15)
S
i
λ
i
,
ω
i
=
λ
i
-
λ
*
ω
i
,
i
=
f
,
r
,
ω
i
≠
0
.
Taking derivative of the sliding surface yields
(16)
S
˙
i
λ
i
,
V
x
=
F
loss
r
i
m
-
1
r
i
m
F
f
-
1
r
i
m
F
r
-
1
-
λ
*
r
i
I
i
F
i
+
1
-
λ
*
1
I
i
u
i
,
where u
i
=
T
i, i
=
{
f
,
r
}. Assuming S
˙
i
=
0 in (16) results in
(17)
u
i
,
eq
=
I
i
1
-
λ
*
×
-
F
loss
r
i
m
+
1
r
i
m
F
f
+
1
r
i
m
F
r
+
(
1
-
λ
*
)
r
i
I
i
F
i
.
Next, letting u
i
=
u
i
,
eq
+
u
i
,
sw and utilizing the reachability condition (4) result in
(18)
u
i
,
sw
=
-
η
i
I
i
1
-
λ
*
sgn
⁡
(
S
i
)
,
η
i
>
0
,
where
(19)
sgn
⁡
(
S
i
)
=
S
i
S
i
=
1
,
S
i
>
0
,
0
,
S
i
=
0
,
-
1
,
S
i
<
0
.
From (17) and (18), the sliding-mode control law can be concluded as
(20)
u
i
=
I
i
1
-
λ
*
-
F
loss
r
i
m
+
1
r
i
m
F
f
+
1
r
i
m
F
r
hhhhhhhhhhh
+
(
1
-
λ
*
)
r
i
I
i
F
i
-
η
i
sgn
⁡
(
S
i
)
.
Note that the control law requires F
f
,
F
r for feedback, but they are usually not available. Thus, an observer is needed to estimate F
f
,
F
r.


### 4.2. Nonlinearity Observer Design

A nonlinearity observer is designed to estimateF
f
,
F
r. First, express (11) in the form (5) as
(21)
x
˙
=
0
0
0
0
0
0
0
0
0
x
+
1
m
1
m
-
r
f
I
f
0
0
-
r
r
I
r
α
(
t
)
x
˙
=
+
1
m
0
0
σ
(
t
)
+
0
0
1
I
f
0
0
1
I
r
u
,
y
=
1
0
0
0
1
0
0
0
1
x
,
where x
T
=
[
V
x
ω
f
ω
r
], u
T
=
[
T
f
T
r
], α
(
t
)
T
=
[
F
f
F
r
], and σ
(
t
)
=
F
loss. Consequently, the observer (8) with H
=
I and V
=
0 can be expressed as
(22)
X
^
˙
=
0
0
0
1
m
1
m
0
0
0
-
r
f
I
f
0
0
0
0
0
-
r
r
I
r
0
0
0
0
0
0
0
0
0
0
X
^
+
0
0
1
m
1
I
f
0
0
0
1
I
r
0
0
0
0
0
0
0
U
+
L
y
-
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
X
^
,
where X
^
T
=
[
V
^
x
ω
^
f
ω
^
r
F
^
f
F
^
r
] and U
T
=
[
T
f
T
r
F
loss
]. Here, L is the observer gain matrix that must be chosen such that X
^
→
X, resulting in F
^
f
→
F
f and F
^
r
→
F
r as desired.


## 4.1. Sliding-Mode Controller Design

A sliding-mode controller is designed for the system described by (11)–(14). The control objective is to drive the vehicle such that the desired slip ratio λ
* is achieved. First, define the sliding surface as
(15)
S
i
λ
i
,
ω
i
=
λ
i
-
λ
*
ω
i
,
i
=
f
,
r
,
ω
i
≠
0
.
Taking derivative of the sliding surface yields
(16)
S
˙
i
λ
i
,
V
x
=
F
loss
r
i
m
-
1
r
i
m
F
f
-
1
r
i
m
F
r
-
1
-
λ
*
r
i
I
i
F
i
+
1
-
λ
*
1
I
i
u
i
,
where u
i
=
T
i, i
=
{
f
,
r
}. Assuming S
˙
i
=
0 in (16) results in
(17)
u
i
,
eq
=
I
i
1
-
λ
*
×
-
F
loss
r
i
m
+
1
r
i
m
F
f
+
1
r
i
m
F
r
+
(
1
-
λ
*
)
r
i
I
i
F
i
.
Next, letting u
i
=
u
i
,
eq
+
u
i
,
sw and utilizing the reachability condition (4) result in
(18)
u
i
,
sw
=
-
η
i
I
i
1
-
λ
*
sgn
⁡
(
S
i
)
,
η
i
>
0
,
where
(19)
sgn
⁡
(
S
i
)
=
S
i
S
i
=
1
,
S
i
>
0
,
0
,
S
i
=
0
,
-
1
,
S
i
<
0
.
From (17) and (18), the sliding-mode control law can be concluded as
(20)
u
i
=
I
i
1
-
λ
*
-
F
loss
r
i
m
+
1
r
i
m
F
f
+
1
r
i
m
F
r
hhhhhhhhhhh
+
(
1
-
λ
*
)
r
i
I
i
F
i
-
η
i
sgn
⁡
(
S
i
)
.
Note that the control law requires F
f
,
F
r for feedback, but they are usually not available. Thus, an observer is needed to estimate F
f
,
F
r.


## 4.2. Nonlinearity Observer Design

A nonlinearity observer is designed to estimateF
f
,
F
r. First, express (11) in the form (5) as
(21)
x
˙
=
0
0
0
0
0
0
0
0
0
x
+
1
m
1
m
-
r
f
I
f
0
0
-
r
r
I
r
α
(
t
)
x
˙
=
+
1
m
0
0
σ
(
t
)
+
0
0
1
I
f
0
0
1
I
r
u
,
y
=
1
0
0
0
1
0
0
0
1
x
,
where x
T
=
[
V
x
ω
f
ω
r
], u
T
=
[
T
f
T
r
], α
(
t
)
T
=
[
F
f
F
r
], and σ
(
t
)
=
F
loss. Consequently, the observer (8) with H
=
I and V
=
0 can be expressed as
(22)
X
^
˙
=
0
0
0
1
m
1
m
0
0
0
-
r
f
I
f
0
0
0
0
0
-
r
r
I
r
0
0
0
0
0
0
0
0
0
0
X
^
+
0
0
1
m
1
I
f
0
0
0
1
I
r
0
0
0
0
0
0
0
U
+
L
y
-
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
X
^
,
where X
^
T
=
[
V
^
x
ω
^
f
ω
^
r
F
^
f
F
^
r
] and U
T
=
[
T
f
T
r
F
loss
]. Here, L is the observer gain matrix that must be chosen such that X
^
→
X, resulting in F
^
f
→
F
f and F
^
r
→
F
r as desired.


## 5. Numerical Studies and Results

The vehicle model in (11)–(14) is used as the control plant. The vehicle parameters are adopted from [14] and are summarized in Table 1. The friction coefficient function used in this paper is the Burckhardt friction model [9]:
(23)
μ
λ
=
C
1
1
-
e
-
C
2
λ
-
C
3
λ
,
where C
1, C
2, and C
3 are model parameters. The values of the friction model are also listed in Table 1. This friction model yields the maximum friction coefficient at λ
≈
0.2.Table 1
Parameters of the vehicle and the friction model.
Parameter
Value
m
1202 kg
I
f
,
I
r
1.07 kgm2
l
f
1.15 m
l
r
1.45 m
l
h
0.53 m
c
x
0.4
f
roll
0.013
r
f
,
r
r
0.32 m
C
1
1.05
C
2
20.02
C
3
0.4646


### 5.1. Sliding-Mode Controller

The control law (20) is used. Here, it is assumed that F
f
,
F
r are available for feedback. This assumption will be removed later in Section 5.3. The tuning parameters η
f
=
η
r
=
120 are chosen. To achieve the maximum tractive forces, the desired slip ratio λ
* is set to 0.2. The results are shown in Figures 3–5. Note that, at time = 20 sec, a ten-time decrease of the friction coefficient was introduced to simulate a change in road condition. Figure 3 shows that the sliding-mode controller successfully drove the system to operate at the desired wheel slip ratio.Figure 3
Wheel slip ratios and tire tractive forces.Figure 4
Input torques.Figure 5
Velocity and rotational speeds.Time histories of the input torques are presented in Figure4. It is seen from this figure that the controller effectively reduced the input torques to keep the wheel slip ratios constant when the friction coefficient was decreased. Figure 5 displays time histories of the vehicle velocity and the tire rotational speeds. It shows that the velocity and the rotational speeds decreased corresponding to the input torques.


### 5.2. Nonlinearity Observer

The observer (22) is used. The observer gain L is determined through the pole-placement technique. The desired poles are simply chosen to be −1, −2, −3, −4, and −5 and the following gain matrix is obtained:
(24)
L
=
3.041
-
0.079
0.128
-
0.239
6.545
0.911
0.241
0.664
5.414
1.853
-
30.11
-
8.293
-
0.715
-
5.375
-
16.21
.
The results are shown in Figures 6–8. The input torques T
f
,
T
r were initially set to 500 Nm and then stepped up to 1500 Nm at time = 200 sec. The results show that the observer correctly estimated all state variables including the tractive forces.Figure 6
Comparison of true and estimated velocity and rotational speeds.Figure 7
Wheel slip ratios and comparison of true and estimated tractive forces.Figure 8
Comparison of true and estimated tractive forces around time = 200 sec.Comparisons of true and estimated velocity and rotational speeds are presented in Figure6. It is seen from the figure that there are overshoots in the estimated rotational speeds when the step change of the input torques occurred. However, they were quickly converged back to the true values. The estimations of the tractive forces are presented in Figures 7 and 8. In Figure 8, it is seen that the estimations converged to the true values within 3 and 6 sec for the front and rear wheels, respectively.Moreover, to investigate the measurement noises effects on the accuracy of the estimations, we added zero-mean normal distribution noises with the variance of 50 into all measurements. We found that the observer still worked well. The results similar to those in Figures6–8 were obtained. Figure 9 displays an example of the results. Note that there are some small oscillations in the estimations due to the noises.Figure 9
Comparison of true and estimated tractive forces when the measurements degraded by noises.


### 5.3. Sliding-Mode Controller with Nonlinearity Observer

In this subsection, the scenario in Section5.1 is resimulated without the availability assumption of F
f
,
F
r. The observer presented in Section 5.2 is used to estimate F
f
,
F
r. The results are shown in Figures 10–12.Figure 10
Wheel slip ratios and tire tractive forces using the observer-based controller.Figure 11
Input torques using the observer-based controller.Figure 12
Velocity and rotational speeds using the observer-based controller.Figure10 shows that the observer-based controller was able to drive the system to operate at the desired wheel slip ratio. Note that there are jumps of the slip ratios occurring when the friction coefficient was decreased, but they asymptotically converged back to the desired value. The jumps are due to the dynamics of the observer.Time histories of the input torques are presented in Figure11. Similar to the results in Figure 4, the controller effectively reduced the input torques to keep the wheel slip ratios constant. Figure 12 displays time histories of the vehicle velocity and the tire rotational speeds. It is observed that there were large overshoots in the rotational speeds when the friction was decreased. These overshoots are due to the dynamics of the observer.


## 5.1. Sliding-Mode Controller

The control law (20) is used. Here, it is assumed that F
f
,
F
r are available for feedback. This assumption will be removed later in Section 5.3. The tuning parameters η
f
=
η
r
=
120 are chosen. To achieve the maximum tractive forces, the desired slip ratio λ
* is set to 0.2. The results are shown in Figures 3–5. Note that, at time = 20 sec, a ten-time decrease of the friction coefficient was introduced to simulate a change in road condition. Figure 3 shows that the sliding-mode controller successfully drove the system to operate at the desired wheel slip ratio.Figure 3
Wheel slip ratios and tire tractive forces.Figure 4
Input torques.Figure 5
Velocity and rotational speeds.Time histories of the input torques are presented in Figure4. It is seen from this figure that the controller effectively reduced the input torques to keep the wheel slip ratios constant when the friction coefficient was decreased. Figure 5 displays time histories of the vehicle velocity and the tire rotational speeds. It shows that the velocity and the rotational speeds decreased corresponding to the input torques.


## 5.2. Nonlinearity Observer

The observer (22) is used. The observer gain L is determined through the pole-placement technique. The desired poles are simply chosen to be −1, −2, −3, −4, and −5 and the following gain matrix is obtained:
(24)
L
=
3.041
-
0.079
0.128
-
0.239
6.545
0.911
0.241
0.664
5.414
1.853
-
30.11
-
8.293
-
0.715
-
5.375
-
16.21
.
The results are shown in Figures 6–8. The input torques T
f
,
T
r were initially set to 500 Nm and then stepped up to 1500 Nm at time = 200 sec. The results show that the observer correctly estimated all state variables including the tractive forces.Figure 6
Comparison of true and estimated velocity and rotational speeds.Figure 7
Wheel slip ratios and comparison of true and estimated tractive forces.Figure 8
Comparison of true and estimated tractive forces around time = 200 sec.Comparisons of true and estimated velocity and rotational speeds are presented in Figure6. It is seen from the figure that there are overshoots in the estimated rotational speeds when the step change of the input torques occurred. However, they were quickly converged back to the true values. The estimations of the tractive forces are presented in Figures 7 and 8. In Figure 8, it is seen that the estimations converged to the true values within 3 and 6 sec for the front and rear wheels, respectively.Moreover, to investigate the measurement noises effects on the accuracy of the estimations, we added zero-mean normal distribution noises with the variance of 50 into all measurements. We found that the observer still worked well. The results similar to those in Figures6–8 were obtained. Figure 9 displays an example of the results. Note that there are some small oscillations in the estimations due to the noises.Figure 9
Comparison of true and estimated tractive forces when the measurements degraded by noises.


## 5.3. Sliding-Mode Controller with Nonlinearity Observer

In this subsection, the scenario in Section5.1 is resimulated without the availability assumption of F
f
,
F
r. The observer presented in Section 5.2 is used to estimate F
f
,
F
r. The results are shown in Figures 10–12.Figure 10
Wheel slip ratios and tire tractive forces using the observer-based controller.Figure 11
Input torques using the observer-based controller.Figure 12
Velocity and rotational speeds using the observer-based controller.Figure10 shows that the observer-based controller was able to drive the system to operate at the desired wheel slip ratio. Note that there are jumps of the slip ratios occurring when the friction coefficient was decreased, but they asymptotically converged back to the desired value. The jumps are due to the dynamics of the observer.Time histories of the input torques are presented in Figure11. Similar to the results in Figure 4, the controller effectively reduced the input torques to keep the wheel slip ratios constant. Figure 12 displays time histories of the vehicle velocity and the tire rotational speeds. It is observed that there were large overshoots in the rotational speeds when the friction was decreased. These overshoots are due to the dynamics of the observer.


## 6. Experimental Studies and Results

The experimental studies are carried on a single-wheel test rig (see Figure13). The test rig simulates the longitudinal dynamic of electric vehicles. It consists of a drum set, a wheel set, and a measurement/control unit. The role of the drum set is to let the wheel behave like rolling on a road. The diameter and width of the drum are approximately 1.0 and 0.3 meters, respectively. The drum set is also equipped with a rotational speed sensor. The wheel set consists of a tire, a 4-inch-rim wheel, a brushed DC motor, and loading masses. The wheel is directly driven by the motor through a rigid shaft. The wheel set is also equipped with another rotational speed sensor. The measurement/control unit consists of a signal condition circuit and a PC computer. The computer is installed with a 12-bit analog/digital interface board. The two rotational speed signals are fed back to the computer through the interface board. The control sampling period is 0.5 sec.Figure 13
Photograph of the single-wheel test rig.A mathematical model of the test rig can be summarized:(25)
Drum

:
r
d
2
F
-
c
d
V
x
=
I
d
V
˙
x
,
Wheel

:
b
V
in
-
r
w
F
-
a
ω
=
I
w
ω
˙
,
where V
x
=
Ω
r
d is the simulated longitudinal velocity, Ω is the drum rotational speed, r
d is the drum radius, ω is the wheel rotational speed, V
in is the motor voltage input, I
d
,
I
w are the moments of inertia (the subscripts d and w stand for the drum and wheel, resp.), r
w is the wheel radius, a
=
k
T
k
b
/
R
+
c
w and b
=
k
T
/
R
K
0 are constants, k
T is the motor torque constant, k
b is the motor back-EMF constant, R is the motor resistance, c
d
,
c
w are the damping constants, K
0 is the gain of the motor-drive unit, and F is the tractive force. The parameters of the test rig are summarized in Table 2.Table 2
Parameters of the experimental test rig.
Parameter
Value
I
d
24.95 kgm2
I
w
0.0098 kgm2
r
d
0.5 m
r
w
0.13125 m
a
0.0392
b
0.5096
c
d
0.05 N/ms−1


### 6.1. Control Design

First, a sliding-mode controller is designed. Define the sliding surfaces as(26)
s
λ
,
ω
i
=
λ
-
λ
*
ω
i
,
ω
i
≠
0
.
Taking derivative of the sliding surface yields
(27)
s
˙
λ
,
V
x
=
c
d
I
d
r
w
V
x
-
r
d
2
I
d
r
w
F
μ
-
1
-
λ
*
a
I
w
ω
-
1
-
λ
*
r
w
I
w
F
μ
+
1
-
λ
*
b
I
w
u
,
where u
=
V
in. By following the procedure in Section 4.1, the sliding-mode control law can be found as
(28)
u
=
I
w
(
1
-
λ
*
)
b
-
c
d
I
d
r
w
V
x
+
r
d
2
I
d
r
w
F
μ
+
(
1
-
λ
*
)
a
I
w
ω
h
h
h
h
h
h
h
h
h
h
h
h
h
h
+
1
-
λ
*
r
w
I
w
F
μ
-
η
sgn
⁡
s

-
c
d
I
d
r
w
V
x
+
r
d
2
I
d
r
w
F
μ
+
(
1
-
λ
*
)
a
I
w
ω
.Next, a tractive force observer is designed. Rewrite (25) as
(29)
x
˙
=
-
c
d
I
d
0
0
-
a
I
w
x
+
r
d
2
I
d
-
r
w
I
w
f
(
t
)
+
0
b
I
w
u
,
y
=
1
0
0
1
x
,
where x
T
=
[
V
x
ω
], u
=
V
in, and f
(
t
)
=
F
μ. By following the procedure in Section 4.2, the observer can be concluded as
(30)
X
^
˙
=
-
c
d
I
d
0
r
d
2
I
d
0
-
a
I
w
-
r
w
I
w
0
0
0
X
^
+
0
b
I
w
0
u
+
L
y
-
1
0
0
0
1
0
X
^
,
where X
^
T
=
[
V
^
x
ω
^
F
^
μ
] and u
=
V
in. The gain matrix L must be chosen such that X
^
→
X.


### 6.2. Experimental Results

The controller comprises the control law (28) and the observer (30). The parameters in the control law (28) are chosen as λ
* = 0.2 and η = 10. The gain matrix L in the observer (30) is designed by simply placing the observer poles at −1, −2, and −3. Since the observer is implemented in a digital computer with the sampling period 0.5 seconds, the corresponding discrete-time desired poles become 0.6065, 0.3679, and 0.2231. The following gain matrix is obtained:
(31)
L
=
0.6306
0.0006
0.2680
0.3062
-
0.0591
-
0.1058
.Experimental results are shown in Figure14. Here, V
in was set to 6 volt for the first few seconds prior the activation of controller. The results show that the observer-based controller was able to drive the system to operate at λ
=
0.2 successfully.Figure 14
Experimental results using the observer-based controller.


## 6.1. Control Design

First, a sliding-mode controller is designed. Define the sliding surfaces as(26)
s
λ
,
ω
i
=
λ
-
λ
*
ω
i
,
ω
i
≠
0
.
Taking derivative of the sliding surface yields
(27)
s
˙
λ
,
V
x
=
c
d
I
d
r
w
V
x
-
r
d
2
I
d
r
w
F
μ
-
1
-
λ
*
a
I
w
ω
-
1
-
λ
*
r
w
I
w
F
μ
+
1
-
λ
*
b
I
w
u
,
where u
=
V
in. By following the procedure in Section 4.1, the sliding-mode control law can be found as
(28)
u
=
I
w
(
1
-
λ
*
)
b
-
c
d
I
d
r
w
V
x
+
r
d
2
I
d
r
w
F
μ
+
(
1
-
λ
*
)
a
I
w
ω
h
h
h
h
h
h
h
h
h
h
h
h
h
h
+
1
-
λ
*
r
w
I
w
F
μ
-
η
sgn
⁡
s

-
c
d
I
d
r
w
V
x
+
r
d
2
I
d
r
w
F
μ
+
(
1
-
λ
*
)
a
I
w
ω
.Next, a tractive force observer is designed. Rewrite (25) as
(29)
x
˙
=
-
c
d
I
d
0
0
-
a
I
w
x
+
r
d
2
I
d
-
r
w
I
w
f
(
t
)
+
0
b
I
w
u
,
y
=
1
0
0
1
x
,
where x
T
=
[
V
x
ω
], u
=
V
in, and f
(
t
)
=
F
μ. By following the procedure in Section 4.2, the observer can be concluded as
(30)
X
^
˙
=
-
c
d
I
d
0
r
d
2
I
d
0
-
a
I
w
-
r
w
I
w
0
0
0
X
^
+
0
b
I
w
0
u
+
L
y
-
1
0
0
0
1
0
X
^
,
where X
^
T
=
[
V
^
x
ω
^
F
^
μ
] and u
=
V
in. The gain matrix L must be chosen such that X
^
→
X.


## 6.2. Experimental Results

The controller comprises the control law (28) and the observer (30). The parameters in the control law (28) are chosen as λ
* = 0.2 and η = 10. The gain matrix L in the observer (30) is designed by simply placing the observer poles at −1, −2, and −3. Since the observer is implemented in a digital computer with the sampling period 0.5 seconds, the corresponding discrete-time desired poles become 0.6065, 0.3679, and 0.2231. The following gain matrix is obtained:
(31)
L
=
0.6306
0.0006
0.2680
0.3062
-
0.0591
-
0.1058
.Experimental results are shown in Figure14. Here, V
in was set to 6 volt for the first few seconds prior the activation of controller. The results show that the observer-based controller was able to drive the system to operate at λ
=
0.2 successfully.Figure 14
Experimental results using the observer-based controller.


## 7. Conclusions

This paper has proposed a robust observer-based sliding-mode control scheme for a vehicle traction control problem. The sliding-mode controller is designed using an equivalent control method. The control objective is to operate the vehicles at a desired wheel slip ratio. Based on a PI observer design, the observer estimates the tractive forces to be used in the control law. Only a vehicle longitudinal velocity measurement and the tire rotational speed measurements are required in the control loop. However, since the vehicle longitudinal velocity measurement might not be commonly found in vehicles, this requirement is considered as the limitation of the proposed scheme. The simulation and experimental results illustrated the effectiveness of the proposed observer-based controller. Last but not least, although the derivation of the sliding-mode control law guarantees the convergence of the desired slip, the coupling of the observer and the controller still does not theoretically confirm the convergence. This theoretical work may be considered as a needed future research.



---

*Source: 829097-2014-12-21.xml*