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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 66, NO. 2, FEBRUARY 2017 981

Distributed Adaptive Sliding Mode Control Strategy

for Vehicle-Following Systems With Nonlinear

Acceleration Uncertainties

Xianggui Guo, Jianliang Wang, Fang Liao, and Rodney Swee Huat Teo

Abstract—This paper investigates the distributed adaptive con-

trol problems for nonlinear vehicle-following systems subject to

nonlinear acceleration uncertainties involving vehicle acceleration

disturbances, wind gusts, and parameter uncertainties. It is worth

mentioning that the acceleration of the leader in most existing

studies is always assumed to be zero or constant; the evolution of

the leader in this paper may be subject to some unknown bounded

input. Distributed adaptive control strategies based on an integral

sliding mode control (ISMC) technique are proposed to maintain

a rigid formation for a string of vehicle platoon in one dimension.

First, distributed adaptive control based on traditional constant

time headway (TCTH) policy under the assumption that the initial

spacing and velocity errors are zero is developed to guarantee that

all spacing errors are uniformly ultimately bounded and that the

string stability of the whole vehicle platoon is also satisﬁed. Then, a

modiﬁed constant time headway (MCTH) policy is proposed to re-

move the assumption of zero initial spacing and velocity errors and

simultaneously effectively decrease the intervehicle spacing (i.e.,

increase the trafﬁc density), making them nearly equal to those

by using the constant spacing (CS) policy. Adaptive compensation

terms without requiring apriorknowledge of upper bounds of the

uncertainties are constructed to compensate for the time-variant

effects caused by nonlinear acceleration uncertainties. Finally,

numerical simulation results show the validity and advantages of

the proposed policy up to a signiﬁcant higher trafﬁc density.

Index Terms—Distributed control, integrated sliding mode con-

trol (ISMC), modiﬁed constant time headway (MCTH) policy,

nonlinear acceleration uncertainties, string stability.

Manuscript received November 24, 2015; revised March 22, 2016; accepted

April 14, 2016. Date of publication April 20, 2016; date of current version

February 10, 2017. This work was supported in part by the National Natural

Science Foundation of China under Grant 61403279, by the Tianjin Natural

Science Foundation under Grant 13JCQNJC04000, and by the National Uni-

versity of Singapore under Grant RCA-14/123. The review of this paper was

coordinated by Dr. D. Cao.

X. Guo is with the School of Automation and Electrical Engineering, Univer-

sity of Science and Technology Beijing, Beijing 100083, China, and also with

the School of Electrical and Electronic Engineering, Nanyang Technological

Uni versity, Singapore 639798 (e-mail: guoxianggui@163.com).

J. Wang is with the School of Electrical and Electronic Engineering, Nanyang

Technological University, Singapore 639798 (e-mail: ejlwang@ntu.edu.sg).

F. Liao is with the Temasek Laboratories, National University of Singapore,

Singapore 117508 (e-mail: tsllf@nus.edu.sg).

R. S. H. Teo is with the T emasek Laboratories, National University of

Singapore, Singapore 117508 (e-mail: tsltshr@nus.edu.sg).

Color versions of one or more of the ﬁgures in this paper are available online

at http://ieeexplore.ieee.org.

Digital Object Identiﬁer 10.1109/TVT.2016.2556938

I. INTRODUCTION

N recent years, the problems of contro lling vehicle forma-

tions have become hot topics due to the variety of applica-

tions in various areas, such as satellite formation ﬂying [1], car

platoons [2]–[11], or unmanned aerial vehicles [12], [13]. The

simple case of a 1-D vehicle platoon has been studied exten-

sively, and the main goals of platoon control are to maintain a

desired and safety distance between intervehicles in the platoon

[14]–[18] and increase the capacity of trafﬁc ﬂow by reducing

the intervehicle spacing [2], [7], [18]. It is worth mentioning that

a key property of platoon control is string stability [19]. String

stability indicates that spacing errors do not amplify as they

propagate upstream from one vehicle to another vehicle [14].

The relationship b etween string stability and spacin g policies

has been an important issue [5]. There are two major spacing

policies for the control o f vehicle platoon: constant spacing

(CS) policy and traditional constant time headway (TCTH)

policy. CS policy is usually used, such as [2]–[5], [20]–[22],

due to the very high achievable trafﬁc capacity. However, the

aforementioned results are established under the assumption

that the initial spacing and the initial velocity erro rs are zero

simultaneously. This is not a realistic assumption, and the initial

spacing and the initial velocity errors are known to cause large

transient engine and brake torques [14]. On the other hand, the

string stability is guaranteed by using CS policy at the cost

of intervehicle communication by requiring the preceding ve-

hicle’s acceleration information [20]. Therefore, constant time

headway (CTH) policy is used to replace CS policy in many ex-

isting results such as [6], [11], [14], [17], [18], and [23] because

it can ensure string stability just by using onboard information

[14], [18]. Nevertheless, the steady-state intervehicle spacings

of CTH policy are very large at high speed, and thus, the trafﬁc

density is low. Recently, to reduce the intervehicle spacing, a

modiﬁed CTH (MCTH) policy by introducing a virtual truck

was p roposed in [18], but it will increase the risks of collisions.

Thus, in this paper, some efforts will be made in the directio n of

reducing the intervehicle spacing and simultaneously removing

the assumption of zero initial spacing and velocity errors, which

is important and challenging in both theory and practice.

As the vehicle platoon is an interconnected coupled system,

disturbances acting on one vehicle may affect other vehicles

and even amplify spacing errors along the string, i.e.„ strin g

instability [4], [24], [25]. Generally speaking, distur bances and

parameter uncertainties in the system are two main underly-

ing causes of the str ing instability problem [2]. The primary

See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

982 IEEE TR ANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 66, NO. 2, FEBRUARY 2017

disturbances existing in a vehicle platoon include th e vehicle

acceleration disturbances caused by environmental circum-

stances such as gust, friction on grounds, and rolling resistance,

whereas the uncertainties in a vehicle platoon consist of the

aerodynamic drag, the mass of passenger vehicles [2], and inter-

mediate uncertainties induced by networks [24]. Thus, how to

reduce the effect of disturbances and parameter uncertainties on

string stability has attracted considerable interest (see [2]–[4] ,

[17], [21], [24], and [26] and the references therein). Note that,

in the aforementioned studies ( except for [2 4]) on string stabil-

ity, the research is all based on a common assumption that the

acceleration of the leader is always assumed to be zero or con-

stant, which has clear shortcomings in practice due to the com-

plexity of the environment. Furthermore, in practice, it is almost

impossible to get an exact mathematical model of a dynamic

system due to sensor measurement noise or environmental dis-

turbances and so on [27]. Then, it is natural to build our analysis

and design on full account of nonlinear vehicle dynamics

directly, which can potentially lead to a more accurate platoon

control model, although highly nonlinear platoon dynamics in-

deed make the controller design more complex and challenging.

Motivated by the aforementioned points, based on TCTH and

MCTH policies, novel distributed control strategies combining

integral sliding mode control (ISMC) with adaptive control

technique are proposed to solve the string stability of vehicle-

following systems subject to unknown bounded nonlinear ac-

celeration uncertainties in both the leader and the followers. A

new MCTH policy is proposed to increase the trafﬁc density

and remove the u sual assumption in many existing results

[2]–[6], [10], [11], [18] , [ 20]–[23] , [26] that the initial spacing

and velocity errors are zer o. In addition to the new MCTH

policy a nd based on it, effective platoon control strategies

are proposed to guarantee the bounded stability of individual

vehicle and the string stability of the platoon. In addition, the

bound on the spacing errors can be rendered arbitrarily small

by adjusting the design parameters. The main contributions of

this paper are as follows.

1) Distribu ted adaptive ISMC strategies: Based on TCTH

and MCTH policies, new distributed adaptive control

strategies combining the ISMC method with adaptive

control technique are proposed to guarantee the bounded

stability of the spacing er rors and the string stability of

the whole vehicle platoon.

2) New MCTH policy: This policy is effective to guarantee

smooth transient response for nonzero initial spacing

and velocity deviations and simultaneously increase the

trafﬁc density, which is nearly equal to that of CS policy.

Unfortunately, increasing the trafﬁc density is at the cost

of increasing the communication required since the infor-

mation o f the leader needs be broad casted to all followers

via wireless communication.

3) Nonlinear acceleration uncertainties: Unlike in most ex-

isting studies, which assume that the acceleration of the

leader is zero or constant, both the leader and the fol-

lowers in this paper may be subject to some unknown

bounded nonlinear uncertainties. Moreover, adaptive

compensation terms without requiring apriorknowledge

of the upper bound of the uncertainties are constructed

to effectively reduce the effect of the unknown bounded

nonlinear acceleration uncertainties.

This paper is organized as follows. In Section II, the vehicle

platoon and the problem considered in this paper are described.

In Section III, based on the TCTH and MCTH policies, two

different distributed adaptive ISMC control strategies are pro-

posed. The bounded stability and the string stability are also

proved in this section. Simulation results are provided to show

the effectiveness and advantages in Section I V. Concluding

remarks are presented in Section V.

Throughout this paper, the following notations are used:

·stands for the Euclidean norm of a vector, and the symbol

|·|represents the absolute value of real numbers.

II. V

EHICLE PLATOON AND PROBLEM FORMULATION

A. Vehicle Platoon

Consider N follower vehicles and a leader that are organized

into a platoon running in a straight line. Let x

(t), v

(t),and

(t)(i = 0, 1,...,N) denote the position, the velocity, and

the acceleration of the ith vehicle in the platoon, respectively,

with i = 0 standing for the lead vehicle and the others being the

followers. For brevity, we let V

= {1, 2,...,N}; V

/{N}

denotes the relative complement of {N} in V

,andV

∪{0}

denotes the union of V

and {0}. A nonlinear vehicle model

is utilized to describe the dynamic behavior of the vehicles

as realistically as possible. Let the ith follower vehicle in the

platoon be represented by the following nonlinear differential

equations:

˙x

(t)=v

(t)

˙v

(t)=u

(t)+f

(t),v

(t),t) ,i∈V

(1)

and the leader dynamic be described as

˙x

(t)=v

(t)

˙v

(t)=a

(t)+f

(t),v

(t),t)

(2)

where u

(t) denotes the control input, and f

(t),v

(t),t)

for vehicle i is an unknown bounded nonlinear time-varying

uncertainty involving vehicle acceleration disturbances, wind

gust, and parameter uncertainties and inter mediate uncertainties

induced by networks. Since f

(t),v

(t),t) is introduced in

the leader and follower acceleration dynamics, the model is

thus more general than the one studied in [10], [11], [17], [28],

and [29]. The acceleration a

(t) is a known function of time.

According to the practical case, it is reasonable to assume

v

(t)≤¯v

since the desired acceleration a

(t) and the non-

linear various uncertainties f

(t),v

(t),t) actingonthe

leader vehicle are bounded.

Throughout this paper, the following assumption is made for

the development of control strategies.

Assumption 1: Nonlinearities f

(t),v

(t),t)(i ∈V

∪{0})

satisfy the following bounded condition:

f

i−1

(t),v

i−1

(t),t) − f

(t),v

(t),t)

≤ μ

x

i−1

(t) − x

(t) + η

v

i−1

(t) − v

(t) + θ

(3)

where μ

> 0, η

> 0, and θ

≥ 0 are unknown nonnega-

tive constants. In addition, f

(t),v

(t),t) is bounded as

f

(t),v

(t),t)≤

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