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Wang, H.; Kong, H.; Man, Z.; Tuan, D. M.; Cao, Z.; Shen, W. (2014). Sliding

mode control for steer-by-wire systems with AC motors in road vehicles.

Originally published in IEEE Transactions on Industrial Electronics, 61(3), 1596–1611.

Available from: http://dx.doi.org/10.1109/TIE.2013.2258296

Copyright © 2013 IEEE. This is the author’s version of the work, posted here with the permission of the publisher for your personal use. No further distribution is permitted. You may also be able to access the published version from your library. The definitive version is available at http://ieeexplore.ieee.org/.

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Abstract—In this paper, the modelling of Steer-by-Wire (SbW)

systems is further studied and a sliding mode control scheme for the SbW systems with uncertain dynamics is developed. It is shown that an SbW system, from the steering motor to the steered front wheels, is equivalent to a second-order system. A sliding mode controller can then be designed based on the bound information of uncertain system parameters, uncertain self-aligning torque and uncertain torque pulsation disturbances, in the sense that not only the strong robustness with respect to large and nonlinear system uncertainties can be obtained, but also the front wheel steering angle can converge to the hand-wheel reference angle asymptotically. Both the simulation and experimental results are presented in support of the excellent performance and effectiveness of the proposed scheme.

Index Terms— Robustness, self-aligning torque, sliding mode control, steer-by-wire.

I. INTRODUCTION

ANY RESEARCHERS and engineers in automotive industry are currently working on the Steer-by-Wire

(SbW) systems that are known as the next generation of steering systems. The advantages of using SbW systems in the road vehicles are to improve the overall steering performance, lower the power consumption, and enhance the safety and comfort of the passengers. The modern SbW systems have the following distinct characteristics: (i) The mechanical link in conventional road vehicles used to connect the hand-wheel to the steered front wheels, through the rack and pinion gearbox, is removed; (ii) The hand-wheel angle sensor is installed on the steering column to provide the reference signal for the front wheel steering angle to follow; (iii) The steering motor, coupled to the rack and pinion gearbox, is adopted to steer the front wheels based on the reference information provided by the hand-wheel angle sensor. In addition, a feedback motor is employed on the hand-wheel side to provide drivers with the feeling of the effects of self-aligning torque between the front wheels and the road surface, based on the error information

H. Wang is with the Faculty of Engineering and Industrial Sciences,

Swinburne University of Technology, Hawthorn, Melbourne VIC 3122, Australia (Tel: +61-3-92144610. Email: haiwang@swin.edu.au)

Z. Man, D. M. Tuan, Z. Cao, and W. Shen are with the Faculty of Engineering and Industrial Sciences, Swinburne University of Technology, Hawthorn, Melbourne VIC 3122, Australia (Email: zman@swin.edu.au; mdo@swin.edu.au; zcao@swin.edu.au; wshen@swin.edu.au).

H. Kong is with the school of Electrical Engineering and Automation, Hefei University of Technology, Hefei 230009, China (E-mail: konghuifang@163.com).

between the reference angle and the actual steering angle measured indirectly by the pinion angle sensor.

Recently, many studies on the mathematical modelling of SbW systems have been carried out. In [1], the dynamics of the test vehicle’s SbW system was described with a simple second-order model based on the observation of the experimental results by ignoring tire forces and considering tire-to-road contact. In [2] and [3], two second-order models considering the effect of tire forces and vehicle dynamics were utilized in both the steered-wheel side and the hand-wheel side, respectively. However, the dynamics of the two motors are not included in the SbW system modelling. In [4] and [5], the hand-wheel, the front wheels, and two motors are all represented by the second-order differential equations, two independent closed loops are then designed for the hand-wheel with the feedback motor and the front wheels with the steering motor, respectively. In [6] and [7], the hand-wheel with the feedback motor and the front wheel directional assembly described by the rack motion were represented by two second-order models. The whole closed-loop SbW system is developed based on the relationship between the rack displacement and the hand-wheel rotational angle. However, the disadvantages of this SbW modelling structure are that the tire dynamics, especially the tire self-aligning torque, were not considered, and the effect of the self-aligning torque on the steering performance cannot be compensated effectively in the SbW controller design.

Although the mathematical models of SbW systems have been extensively explored as briefly discussed in the above, the detailed modelling, viewing from the front wheel steering motor to the front wheels, has not been fully studied yet. Considering the fact that a complete SbW system consists of three main components: the front wheel steering motor, the rack and pinion gearbox, and the steered front wheels, it is essential to develop a full mathematical model for SbW systems in order to understand the interactions of these components in practical operation, and design robust controllers for achieving excellent steering performance against uncertain system parameters and the tire self-aligning torque that is treated as the most significant disturbance on the SbW systems.

In most existing SbW control systems, several control methods have been used to realize perfect steering characteristics. In [1], [2], [4]-[7], the conventional proportional-derivative (PD) control technique was popularly used with the aim of enabling front wheels to closely follow the driver’s command. In [8] and [9], a state feedback controller using the linear quadratic control technique was developed, aiming at driving the rolling angle of the SbW

Sliding Mode Control for Steer-by-Wire Systems with AC Motors in Road Vehicles

Hai Wang, Student Member, IEEE, Huifang Kong, Zhihong Man, Member, IEEE, Do Manh Tuan, Student Member, IEEE, Zhenwei Cao, and Weixiang Shen, Member, IEEE

M

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motorcycle to track the reference angle. In [12], in order to realize the virtual steering characteristics, an adaptive control method was applied for controlling the front-wheel actuators through the estimation of the front tire cornering stiffness. In [14], the adaptive online estimation method was used to identify the uncertain parameters of the vehicle directional-control and driver-interaction units.

However, because the controllers are designed based on the partial mathematical models in these schemes, good steering performance may not be guaranteed when the road conditions are varying. Particularly, the poles of the closed-loop SbW systems with the PD control may change their locations on the complex plane with the variable road conditions. Such a change may result in the instability of SbW systems.

In this paper we will further study the modelling of SbW systems by taking into account the dynamics from the steering motor to the front wheels. It is worth noting that, in addition to the inertia, the damping, and the friction, the tire self-aligning moment and torque pulsation disturbances are also considered in the system modelling. We then propose a sliding mode control scheme for SbW systems in order to achieve good steering performance.

It is well known that for both linear and nonlinear systems, sliding mode control technique is widely used for tracking control and stabilization with bounded uncertainty information [16], [17], [22]-[30].With the proper choice of sliding mode surface, the stability of the closed-loop system can be obtained asymptotically. It will be shown that the sliding mode controller (SMC) to be designed in this paper is capable of driving the steering angle to closely follow the hand-wheel command with a strong robustness against uncertainties. The merit of this control scheme, from the viewpoint of design, is that only the bound information of the unknown system parameters, self-aligning torque and torque pulsation disturbances is required for designing SMC. It will be confirmed from the simulation and experiment sections that, the designed SMC will drive the sliding variable to reach the sliding surface, and then the tracking error between the steering angle and its reference signal can asymptotically converge to zero on the sliding mode surface.

The rest of the paper is organized as follows. In Section II, the mathematical modelling of SbW systems is formulated, and the self-aligning torque as well as the total torque pulsation disturbances are briefly analysed. In Section III, an SMC is proposed and the convergence of tracking error dynamics and robustness with respect to system uncertainties are discussed in detail. In Section IV and Section V, the numerical simulations as well as the experimental studies are carried out, respectively, to show the good performance of the proposed SMC. Section VI gives conclusions.

Fig. 1. Steer-by-Wire system with rotary motors.

II. PROBLEM FORMULATION

The basic principle of an SbW system in road vehicles is shown in Fig. 1 [1], [2]. It is seen that the SbW system can be divided into two parts: the upper part includes the hand-wheel, the hand-wheel angle sensor, and the feedback motor, respectively; and the lower part is composed of the steering motor, the pinion angle sensor, the rack and pinion gearbox, and the steered front wheels.

The hand-wheel feedback motor is controlled in the sense that it can provide the driver to feel the interactions between the front wheels and road surfaces during driving. The front wheel steering motor generates the actual torque for steering the two front wheels through the rack and pinion gearbox and the steering arm. The control of the steering motor aims at driving the front wheel steering angle to closely follow the hand-wheel reference command.

In this paper, we model the steering system, from the steering actuator to the steered front wheels, as a motor driving a load (the steered wheels) through the rack and pinion gearbox. First, the dynamic equation of the front wheel steering motor is described by the following second-order differential equation [13], [21]:

𝐽𝑠𝑚�̈�𝑠𝑚 + 𝐵𝑠𝑚�̇�𝑠𝑚 + 𝜏12 = 𝜏𝑠𝑚∗ + 𝜏𝑑𝑖𝑠 (1) where 𝐽𝑠𝑚 and 𝐵𝑠𝑚 are the moments of the inertia and the viscous friction of the steering motor, respectively, 𝜃𝑠𝑚 is the shaft angle of the steering motor, 𝜏12 is the torque exerted on the motor shaft by the two steered wheels through the rack and pinion gearbox, 𝜏𝑑𝑖𝑠 represents the motor torque pulsation disturbances that will be described below, and 𝜏𝑠𝑚∗ is the torque control input for the steering motor.

For the further analysis, the road vehicle equipped with the SbW system in Fig. 1 is represented by the following linear bicycle model as shown in Fig. 2 [1]. As shown in Fig. 2, the two front wheels and two rear wheels of the vehicle are represented by a central front wheel and a central rear wheel, respectively, VCG is the vehicle velocity at the centre of gravity (CG), 𝛿𝑓 is the steering angle of the central front wheel, VX and VY are the longitudinal and lateral components of the CG velocity, Vf and Vr are the velocities of the central front wheel and central rear wheel, 𝑙𝑓 and 𝑙𝑟 are the distances of the central front wheel and central rear wheel from the CG of the vehicle, 𝐹𝑓𝑦 and 𝐹𝑟

𝑦 are the lateral forces of the central front wheel and central rear wheel, respectively, 𝛼𝑓 and 𝛼𝑟 are the tire slip an-

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Fig. 2. The bicycle model of vehicle.

gles of the central front wheel and central rear wheel, and 𝛽 is the vehicle body slip angle at CG.

In this paper, the steered central front wheel in Fig. 2 can be treated as the load of the steering motor and rotates about the vertical axis crossing the wheel centre. Therefore, the rotation of the central front wheel satisfies the following dynamic equation [19]-[21]:

𝐽𝑓𝑤𝛿�̈� + 𝐵𝑓𝑤�̇�𝑓 + 𝐹s 𝑠𝑖𝑔𝑛��̇�𝑓� + 𝜏𝑒 = 𝜏𝑠 (2) where 𝐽𝑓𝑤 , and 𝐵𝑓𝑤 are the moments of the inertia and the viscous friction of the front wheels, respectively, 𝜏𝑠 is the torque applied on the steering arm by the steering motor through the rack and pinion gearbox, 𝜏𝑒 is the self-aligning torque which reflects the interaction between the road surface and the front wheels while the vehicle is turning, 𝐹𝑠 𝑠𝑖𝑔𝑛��̇�𝑓� is the Coulomb friction in the steering system with 𝐹s defined as the Coulomb friction constant, and 𝑠𝑖𝑔𝑛(�̇�𝑓) is the sign function with

𝑠𝑖𝑔𝑛��̇�𝑓� = �1 for �̇�𝑓 > 00 for �̇�𝑓 = 0 −1 for �̇�𝑓 < 0

(3)

Assuming that there is no backlash between the rack and gear teeth, we have the following relationships held [21]:

𝛿𝑓𝜃𝑠𝑚

= 𝑁1𝑟𝑁2

= 𝜏12𝜏𝑠

(4a)

where 𝑁1 and 𝑁2 are the tooth numbers of rack and pinion gearbox, respectively, r is a scale factor to account for the conversion from the linear motion of the rack to the rotation at the steering arm or the steering angle of the steered front wheels.

Further, differentiating 4(a) twice, we obtain the following relationships about the motor shaft angle𝜃𝑠𝑚 , the steering angle 𝛿𝑓, and their derivatives [21]:

𝛿𝑓𝜃𝑠𝑚

= �̇�𝑓�̇�𝑠𝑚

= �̈�𝑓�̈�𝑠𝑚

= 𝑁1𝑟𝑁2

= 𝜏12𝜏𝑠

(4b)

Then using (4b) in (2), we have

𝐽𝑒𝑞𝛿�̈� + 𝐵𝑒𝑞𝛿�̇� −𝑟𝑁2𝑁1𝜏𝑑𝑖𝑠 + 𝐹s 𝑠𝑖𝑔𝑛��̇�𝑓� + 𝜏𝑒 = 𝜏𝑒𝑞 (5)

where

𝐽𝑒𝑞 = 𝐽𝑓𝑤 + �𝑟𝑁2𝑁1�2𝐽𝑠𝑚 (6)

𝐵𝑒𝑞 = 𝐵𝑓𝑤 + �𝑟𝑁2𝑁1�2𝐵𝑠𝑚 (7)

and the equivalent drive torque signal

𝜏𝑒𝑞 = 𝑟𝑁2𝑁1𝜏𝑠𝑚∗ (8)

Remark 2.1: It is seen from (5) that the SbW system, from the steering motor to the steered front wheels, is equivalent to a second-order direct drive system. Although many researchers in [1]-[7] have extensively considered the modelling issue of SbW systems, this is the first time to systematically derive a complete mathematical model for the SbW system in Fig. 1. Remark 2.2: Although the moments of the inertias 𝐽𝑓𝑤 and 𝐽𝑠𝑚 , the effective viscous friction coefficients 𝐵𝑓𝑤 and 𝐵𝑠𝑚 , the conversion parameter r in (6)-(7), and the Coulomb friction constant 𝐹s are all unknown in practice, the following bounded conditions can be assumed:

𝐽𝑓𝑤0 ≤ 𝐽𝑓𝑤 ≤ 𝐽𝑓𝑤1 (9)

𝐵𝑓𝑤 ≤ 𝐵𝑓𝑤1 (10)

𝐽𝑠𝑚0 ≤ 𝐽𝑠𝑚 ≤ 𝐽𝑠𝑚1 (11)

𝐵𝑠𝑚 ≤ 𝐵𝑠𝑚1 (12)

𝑟0 ≤ 𝑟 ≤ 𝑟1 (13)

𝐹s0 ≤ 𝐹s ≤ 𝐹s1 (14)

where 𝐽𝑓𝑤0 , 𝐽𝑓𝑤1 , 𝐽𝑠𝑚0 , 𝐽𝑠𝑚1 , 𝐵𝑓𝑤1 ,𝐵𝑠𝑚1 ,𝑟0 , 𝑟1 , 𝐹s0 , and 𝐹s1 are positive constants. Thus, considering (6) and (7), we can express the upper and lower bounds of 𝐽𝑒𝑞 and 𝐵𝑒𝑞 as follows:

𝐽𝑒𝑞1 = 𝐽𝑓𝑤1 + �𝑟1𝑁2𝑁1�2𝐽𝑠𝑚1 (15)

𝐽𝑒𝑞0 = 𝐽𝑓𝑤0 + �𝑟0𝑁2𝑁1�2𝐽𝑠𝑚0 (16)

𝐵𝑒𝑞1 = 𝐵𝑓𝑤1 + �𝑟1𝑁2𝑁1�2𝐵𝑠𝑚1 (17)

Then, 𝐽𝑒𝑞 and 𝐵𝑒𝑞 satisfy the following bounded conditions: 𝐽𝑒𝑞0 ≤ 𝐽𝑒𝑞 ≤ 𝐽𝑒𝑞1 (18)

0 < 𝐵𝑒𝑞 ≤ 𝐵𝑒𝑞1 (19)

Remark 2.3: In addition to the uncertainties in 𝐽𝑓𝑤, 𝐽𝑠𝑚, 𝐵𝑓𝑤 , 𝐵𝑠𝑚, r and 𝐹s , another significant uncertainty exists in the self-aligning torque 𝜏𝑒 that is the total aligning moment of the tire. Fig. 3(a) and Fig. 3(b) show the tire forces and the self- aligning torque at the steered central front wheel of the bicycle model, respectively, and the parameters of the tire dynamics are listed in Table I.

As shown in Fig. 3 (b), the aligning torque generated by the tire lateral force is given by [18], [34]:

𝜏𝑒 = (𝑙𝑐 + 𝑙𝑝)𝐹𝑓𝑦 (20)

where 𝑙𝑐 is the mechanical trail, describing the distance between the tire centre and the point on the ground about the tire pivots as a result of the wheel caster angle, and 𝑙𝑝 is the pneumatic trail, the distance from the tire centre to the application of the lateral force 𝐹𝑓

𝑦.

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Fig. 3. Tire force at front central wheel. (a) Tire forces. (b) Self-aligning torque.

TABLE I

PARAMETERS OF VEHICLE DYNAMICS

Parameter Definition 𝛿𝑓 Front wheel steering angle 𝛼𝑓 Front tire slip angle Vf Front wheel velocity 𝐼𝑧 Vehicle inertia around CG M Vehicle mass 𝐹𝑓𝑥 Front wheel longitudinal driving force 𝐹𝑓𝑦 Front wheel lateral force

𝐶𝐹𝑓 Front wheel cornering force 𝐶𝑓 Front wheel cornering stiffness 𝜏𝑒 Front wheel self-aligning torque 𝑙𝑐 Front wheel mechanical trail 𝑙𝑝 Front wheel pneumatic trail

Remark 2.4: At a small slip angle, such as 4 degrees or less, the lateral force is linearly related to the tire slip angle 𝛼𝑓 and can be modelled as follows:

𝐹𝑓𝑦 = −𝐶𝑓 .𝛼𝑓 (21)

where 𝐶𝑓 is the front tire cornering stiffness coefficient, a parameter closely related to the tire-road friction. Assuming that the vehicle body slip angle is close to zero, we can approximate the front wheel side slip angle 𝛼𝑓 as follows [11], [12]:

𝛼𝑓 ≈ 𝛽 + 𝛾.𝑙𝑓𝑉𝐶𝐺

− 𝛿𝑓 (22)

Thus, the front tire lateral force 𝐹𝑓𝑦 and the self-aligning

torque 𝜏𝑒 can be expressed as: 𝐹𝑓𝑦 = −𝐶𝑓 . �𝛽 + 𝛾.𝑙𝑓

𝑉𝐶𝐺− 𝛿𝑓� (23)

and

𝜏𝑒 = −𝐶𝑓�𝑙𝑐 + 𝑙𝑝� �𝛽 + 𝛾.𝑙𝑓𝑉𝐶𝐺

− 𝛿𝑓� (24)

For the design of the sliding mode controller in the next section, the upper bound of the self-aligning torque 𝜏𝑒 is estimated as follows:

|𝜏𝑒| ≤ 𝜏�̅� (25) where

𝜏�̅� = 𝐶�̅��𝑙�̅� + 𝑙�̅�� ��𝛽 + 𝛾.𝑙𝑓𝑉𝐶𝐺

− 𝛿𝑓�� (26)

where 𝑙�̅�, 𝑙�̅� are the upper bounds of the mechanical trail and the pneumatic trail, respectively, and 𝐶�̅� is the upper bound of the front tire cornering stiffness coefficient.

Remark 2.5: In (24), the steering angle of the steered front wheels 𝛿𝑓 and the yaw rate of the vehicle 𝛾 are measured by using the pinion angle sensor indirectly and the yaw rate sensor, respectively. However, the vehicle body slip angle 𝛽 can be estimated from the bicycle model of the road vehicle as [1], [12]:

𝛽 = 𝑡𝑎𝑛−1 �𝑙𝑟 𝑡𝑎𝑛 𝛿𝑓𝑙𝑟+𝑙𝑓

� (27)

Remark 2.6: As for the SbW systems, in the final analysis, it is to control the front wheel steering motor that plays an essential role to handle the uncertain dynamics and disturbances. In this research, the steering motor is a three phase permanent magnet (PM) AC motor. Then, the mathematical model of the PM AC motor and the sources of torque pulsation disturbances will be briefly discussed as follows.

Considering the rotor rotating coordinates (d-q axes) of the motor as the reference coordinates, the d-q axes stator voltages of the three phase PM AC motor can be modelled as follows [39]-[44]:

𝑣𝑑 = 𝐿𝑑𝑑𝑖𝑑𝑑𝑡

+ 𝑅𝑠𝑖𝑑 − 𝜔𝑒𝐿𝑞𝑖𝑞 (28)

𝑣𝑞 = 𝑅𝑠𝑖𝑞 + 𝐿𝑞𝑑𝑖𝑞𝑑𝑡

+ 𝜔𝑒(𝐿𝑑𝑖𝑑 + 𝜓𝑑𝑚) (29)

where 𝑣𝑑 and 𝑣𝑞 are the d-q axes stator voltages, respectively, 𝑖𝑑 and 𝑖𝑞 are the d-q axes stator currents, respectively, 𝐿𝑑 and 𝐿𝑞 are the d-q axes stator synchronous inductances, respectively, 𝜓𝑑𝑚 is the d-axis flux linkage due to the permanent magnet, Rs is the stator resistance, and 𝜔𝑒 is the rotor electrical speed, which is related to the rotor mechanical speed as 𝜔𝑒 = 𝑝

2𝜔𝑚 with 𝑝 as the number of poles (even

number). Generally, the current 𝑖𝑑 and 𝑖𝑞 can be calculated from 𝑖𝑎 and 𝑖𝑏 (obtained from current measurements) by using Clarke and Park transformations [42], [46]. The corresponding electromagnetic torque equation is expressed as follows:

𝜏𝑠𝑚 = 32𝑝2�𝜓𝑑𝑚𝑖𝑞 + �𝐿𝑑 − 𝐿𝑞�𝑖𝑑𝑖𝑞� (30)

In industrial applications, field-oriented control principle is widely adopted to control a PM AC motor. Thus, in order to simplify the model and reduce the costs, the desired current component 𝑖𝑑∗ in the direct axis is set to zero, that is, 𝑖𝑑∗ = 0. Usually, with the current loop controller, 𝑖𝑑 can be easily regulated to zero. In this case, the torque expression in (30) can be rewritten as:

𝜏𝑠𝑚 = 32𝑝2𝜓𝑑𝑚𝑖𝑞 (31)

Remark 2.7: In the ideal case, the q-axis reference current 𝑖𝑞∗ can be directly achieved from (31) due to the constant d-axis flux linkage 𝜓𝑑𝑚 = 𝜓𝑑0. However, the torque disturbance 𝜏𝑑𝑖𝑠 in (1) always exists in motor torque generation. Therefore, in designing the torque control signal for the steering motor, these perturbations need to be taken into account and compensated effectively in the meantime. In terms of the flux harmonics, the actual flux linkage 𝜓𝑑𝑚 is described as follows [43]: 𝜓𝑑𝑚 = 𝜓𝑑0 + ∑ 𝜓𝑑6𝑖 𝑐𝑜𝑠(𝑖6𝜃𝑒)∞

𝑖=1

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= 𝜓𝑑0 + 𝜓𝑑6 𝑐𝑜𝑠(6𝜃𝑒) + 𝜓𝑑12 𝑐𝑜𝑠(12𝜃𝑒) + ⋯

= 𝜓𝑑0 + 𝜓𝑑6𝜃 + 𝜓𝑑12𝜃 + ⋯ (32) where 𝜓𝑑0, 𝜓𝑑6,and 𝜓𝑑12 are the known constant average dc, the 6th , and the 12th harmonics amplitude of the d-axis flux linkage, respectively, 𝜓𝑑6𝜃 and 𝜓𝑑12𝜃 are the 6th , and the 12th harmonic terms, respectively, and 𝜃𝑒 is the electrical angle of the rotor. For the purpose of simplicity, only the 6th and 12th harmonics are considered as the principal source of torque pulsations in this paper. Remark 2.8: The dccurrent offsets always exist in the motor terminals in virtue of the digital-to-analog converter offsets of the motion controller as well as the current offset error of the current tracking amplifier. Thus, sinusoidal torque disturbances and the corresponding velocity ripples at the system output are inevitably generated, which will severely affect the high-precision tracking performance in practice.

Let 𝑖𝑎∗(𝑡), 𝑖𝑏∗(𝑡), and 𝑖𝑐∗(𝑡) be the desired currents at the motor terminals and the dc current offsets in the measured currents of phases a and b be ∆𝑖𝑎 and ∆𝑖𝑏 , respectively. Then, the third phase current offset is calculated as ∆𝑖𝑐 = −(∆𝑖𝑎 +∆𝑖𝑏), and the actual currents are of the forms 𝑖𝑎 = 𝑖𝑎∗ + ∆𝑖𝑎, 𝑖𝑏 = 𝑖𝑏∗ + ∆𝑖𝑏 , and 𝑖𝑐 = 𝑖𝑐∗ + ∆𝑖𝑐 , respectively. The currents 𝑖𝑑 , 𝑖𝑞 , and 𝑖0 can be calculated by using Clarke and Park transformations based on the three phase currents with the offsets (abc frame to dq0 frame). Therefore, the actual current 𝑖𝑞 can be expressed as [42], [45]-[47]:

𝑖𝑞 = 𝑖𝑞∗(𝑡) + ∆𝑖𝑞𝑜𝑓𝑓(𝑡) (33) where the desired d-axis current 𝑖𝑞∗ is given by

𝑖𝑞∗(𝑡) = 23�𝑖𝑎∗(𝑡) cos �𝜃𝑒 + 𝜋

2�

+ 𝑖𝑏∗(𝑡) cos �𝜃𝑒 −𝜋6� + 𝑖𝑐∗(𝑡) cos �𝜃𝑒 + 7𝜋

6�� (34)

and the current disturbance ∆𝑖𝑞𝑜𝑓𝑓 is of the form:

∆𝑖𝑞𝑜𝑓𝑓 = 23�∆𝑖𝑎 �cos �𝜃𝑒 + 𝜋

2� − cos �𝜃𝑒 + 7𝜋

6��

+∆𝑖𝑏 �cos �𝜃𝑒 −𝜋6� − cos �𝜃𝑒 + 7𝜋

6���

= 2√3

sin(𝜃𝑒 + 𝜑)��∆𝑖𝑎2� + ∆𝑖𝑎∆𝑖𝑏 + �∆𝑖𝑏2� (35)

where 𝜑 is the phase related to ∆𝑖𝑎 and ∆𝑖𝑏 . Accordingly, the upper bound of ∆𝑖𝑞𝑜𝑓𝑓 can be estimated as

∆𝑖𝑞𝑜𝑓𝑓 ≤ �∆𝑖𝑞𝑜𝑓𝑓� ≤ ∆�𝑖𝑞𝑜𝑓𝑓 (36) and ∆�𝑖𝑞𝑜𝑓𝑓 is defined as

∆�𝑖𝑞𝑜𝑓𝑓 = 2√3𝜉𝑐 (37)

where the upper bound 𝜉𝑐 is defined as

��∆𝑖𝑎2� + ∆𝑖𝑎∆𝑖𝑏 + �∆𝑖𝑏2� ≤ 𝜉𝑐 (38)

Then, based on the above analysis, we re-write (31) as follows:

𝜏𝑠𝑚 = 32𝑝2

(𝜓𝑑0 + 𝜓𝑑6𝜃 + 𝜓𝑑12𝜃)𝑖𝑞

= 32𝑝2

(𝜓𝑑0 + 𝜓𝑑6𝜃 + 𝜓𝑑12𝜃)�𝑖𝑞∗(𝑡) + ∆𝑖𝑞𝑜𝑓𝑓(𝑡)�

= 32𝑝2𝜓𝑑0𝑖𝑞∗(𝑡) + 𝜏𝑑𝑖𝑠 = 𝜏𝑠𝑚∗ + 𝜏𝑑𝑖𝑠 (39)

where 𝜏𝑠𝑚∗ is the desired torque signal for the motor, and 𝜏𝑑𝑖𝑠 represents the total pulsation disturbances in the motor torque generation, which satisfies as

𝜏𝑑𝑖𝑠 = 32𝑝2�(𝜓𝑑6𝜃 + 𝜓𝑑12𝜃)�𝑖𝑞∗(𝑡) + ∆𝑖𝑞𝑜𝑓𝑓(𝑡)�

+𝜓𝑑0∆𝑖𝑞𝑜𝑓𝑓(𝑡)�

= 32𝑝2

[(𝜓𝑑6𝜃 + 𝜓𝑑12𝜃)]𝑖𝑞 + 32𝑝2𝜓𝑑0∆𝑖𝑞𝑜𝑓𝑓(𝑡) (40)

Then, the first part in (40) can be described as the sum of the 6th and the 12th torque harmonics as:

32𝑝2

[(𝜓𝑑6𝜃 + 𝜓𝑑12𝜃)]𝑖𝑞 = 𝜏𝑠𝑚6 𝑐𝑜𝑠(6𝜃𝑒)

+𝜏𝑠𝑚12 𝑐𝑜𝑠(12𝜃𝑒) (41) where 𝜏𝑠𝑚6 and 𝜏𝑠𝑚12 are the 6th and the 12th harmonic torque amplitudes, respectively, and the bound information is given by 𝜏𝑠𝑚6 𝑐𝑜𝑠(6𝜃𝑒) + 𝜏𝑠𝑚12 𝑐𝑜𝑠(12𝜃𝑒) ≤ |𝜏𝑠𝑚6| + |𝜏𝑠𝑚12|

≤ 𝜏�̅�𝑚6 + 𝜏�̅�𝑚12 (42)

where 𝜏�̅�𝑚6and 𝜏�̅�𝑚12 are the upper bounds of the 6th and the 12th harmonic torque amplitudes, respectively. For the second part in (40), the bound information is given as 32𝑝2𝜓𝑑0∆𝑖𝑞𝑜𝑓𝑓(𝑡) ≤ 3

2𝑝2𝜓𝑑0�∆𝑖𝑞𝑜𝑓𝑓� ≤

32𝑝2𝜓𝑑0∆�𝑖𝑞𝑜𝑓𝑓

= √32𝜓𝑑0𝜉𝑐 (43)

Therefore, the upper bound of 𝜏𝑑𝑖𝑠 can be estimated as follows:

𝜏𝑑𝑖𝑠 ≤ 𝜏�̅�𝑖𝑠 (44) where

𝜏�̅�𝑖𝑠 = 𝜏�̅�𝑚6 + 𝜏�̅�𝑚12 + √32𝜓𝑑0𝜉𝑐 (45)

III. THE DESIGN OF THE SLIDING MODE CONTROLLER

In this section, we will develop a robust SMC for the SbW system in (5) with uncertain dynamics, under the condition that the information of the upper and the lower bounds of the unknown system parameters, the self-aligning torque, and the total torque pulsation disturbances in (9)-(14), (26), and (45) are known, respectively. The controller design and stability analysis are presented in the following theorem: Theorem 1: Consider the SbW system in (5) with the uncertainty bounds in (9)-(14), (26), and (45), respectively. The tracking error 𝜀𝜃 asymptotically converges to zero if the motor control torque 𝜏𝑠𝑚∗ is designed as:

𝜏𝑠𝑚∗ = − 𝑁1𝑟0𝑁2

𝑠𝑖𝑔𝑛(𝑠) �𝐽𝑒𝑞1 ��̅̈�ℎ𝑟 + 𝜆|𝜀�̇�|� + 𝐵𝑒𝑞1��̇�𝑓�

+ 𝑟1𝑁2𝑁1

𝜏�̅�𝑖𝑠 + 𝐹s1 + 𝜏�̅�� (46)

where 𝜀𝜃 = 𝛿𝑓 − 𝜃ℎ𝑟 is the tracking error between the front wheel steering angle and the hand-wheel reference angle 𝜃ℎ𝑟,

6

𝑠𝑖𝑔𝑛(𝑠) is the sign function defined in (3), �̅̈�ℎ𝑟 is the upper bound of the second-order derivative of 𝜃ℎ𝑟, which satisfies as

��̈�ℎ𝑟� ≤ �̅̈�ℎ𝑟 (47) the sliding variable s is defined as:

𝑠 = 𝜀�̇� + 𝜆𝜀𝜃 (48) with 𝜆 the designed positive parameter.

Proof: Choosing a Lyapunov function candidate 𝑉 = 𝑠2

2 and

differentiating V with respect to time, we have �̇� = 𝑠�̇�

= 𝑠[𝜀�̈� + 𝜆𝜀�̇�]

= 𝑠��̈�𝑓 − �̈�ℎ𝑟 + 𝜆𝜀�̇��

= 𝑠 �� 1𝐽𝑒𝑞�𝑟𝑁2𝑁1

𝜏𝑠𝑚∗ − 𝐵𝑒𝑞�̇�𝑓 + 𝑟𝑁2𝑁1𝜏𝑑𝑖𝑠 − 𝜏𝑒 − 𝐹s 𝑠𝑖𝑔𝑛��̇�𝑓����

+𝑠�−�̈�ℎ𝑟 + 𝜆𝜀�̇��

= 𝑠 �𝑟𝑁2𝑁1

𝜏𝑠𝑚∗

𝐽𝑒𝑞− 𝐵𝑒𝑞

𝐽𝑒𝑞�̇�𝑓 + 𝑟𝑁2

𝑁1

𝜏𝑑𝑖𝑠𝐽𝑒𝑞

− 𝜏𝑒𝐽𝑒𝑞

− 𝐹s 𝑠𝑖𝑔𝑛��̇�𝑓�

𝐽𝑒𝑞�

+𝑠�−�̈�ℎ𝑟 + 𝜆𝜀�̇��

= 𝑠 𝑟𝑁2𝑁1

𝜏𝑠𝑚∗

𝐽𝑒𝑞− 𝑠 𝐵𝑒𝑞

𝐽𝑒𝑞�̇�𝑓 + 𝑠 𝑟𝑁2

𝑁1

𝜏𝑑𝑖𝑠𝐽𝑒𝑞

− 𝑠 𝜏𝑒𝐽𝑒𝑞

− 𝑠 𝐹s 𝑠𝑖𝑔𝑛��̇�𝑓�

𝐽𝑒𝑞

−𝑠�̈�ℎ𝑟 + 𝑠𝜆𝜀�̇�

= −|𝑠|𝑟𝑟0𝐽𝑒𝑞

�𝐽𝑒𝑞1 ��̅̈�ℎ𝑟 + 𝜆|𝜀�̇�|� + 𝐵𝑒𝑞1��̇�𝑓� + 𝐹s1

+ 𝑟1𝑁2𝑁1

𝜏�̅�𝑖𝑠 + 𝜏�̅�� − 𝑠 𝐵𝑒𝑞𝐽𝑒𝑞

�̇�𝑓 + 𝑠 𝑟𝑁2𝑁1

𝜏𝑑𝑖𝑠𝐽𝑒𝑞

− 𝑠 𝜏𝑒𝐽𝑒𝑞

−𝑠 𝐹s 𝑠𝑖𝑔𝑛��̇�𝑓�

𝐽𝑒𝑞− 𝑠�̈�ℎ𝑟 + 𝑠𝜆𝜀�̇�

= −�|𝑠| 𝐽𝑒𝑞1𝐽𝑒𝑞

𝑟𝑟0�̅̈�ℎ𝑟 + 𝑠𝜃ℎ𝑟̈ � − �|𝑠| 𝐵𝑒𝑞1

𝐽𝑒𝑞

𝑟𝑟0�𝛿�̇�� + 𝑠 𝐵𝑒𝑞

𝐽𝑒𝑞𝛿�̇��

−�|𝑠| 𝑟𝑟0

𝑟1𝑁2𝑁1

𝜏�𝑑𝑖𝑠𝐽𝑒𝑞

− 𝑠 𝑟𝑁2𝑁1

𝜏𝑑𝑖𝑠𝐽𝑒𝑞� − �|𝑠| 𝑟

𝑟0

𝜏�𝑒𝐽𝑒𝑞

+ 𝑠 𝜏𝑒𝐽𝑒𝑞�

−�|𝑠| 𝑟𝑟0

𝐹s1𝐽𝑒𝑞

+ 𝑠 𝐹s 𝑠𝑖𝑔𝑛��̇�𝑓�

𝐽𝑒𝑞� − �𝜆 𝑟

𝑟0

𝐽𝑒𝑞1𝐽𝑒𝑞

|𝑠||𝜀�̇�| − 𝜆𝑠𝜀�̇��

≤ −|𝑠| � 𝑟𝑟0

𝐽𝑒𝑞1𝐽𝑒𝑞

− 1� �̅̈�ℎ𝑟 − |𝑠| 1𝐽𝑒𝑞� 𝑟𝑟0𝐵𝑒𝑞1 − 𝐵𝑒𝑞� �𝛿�̇��

−|𝑠| 𝑟𝐽𝑒𝑞

𝑁2𝑁1�𝑟1𝑟0𝜏�̅�𝑖𝑠 − |𝜏𝑑𝑖𝑠|� − |𝑠| 1

𝐽𝑒𝑞( 𝑟𝑟0𝜏�̅� − |𝜏𝑒|)

−|𝑠| 1𝐽𝑒𝑞� 𝑟𝑟0𝐹s1 − |𝐹𝑠|� − 𝜆|𝑠| � 𝑟

𝑟0

𝐽𝑒𝑞1𝐽𝑒𝑞

− 1� |𝜀�̇�|

≤ −|𝑠| 1𝐽𝑒𝑞� 𝑟𝑟0𝐹s1 − |𝐹𝑠|� < 0 for |𝑠| ≠ 0 (49)

(49) indicates that the sliding variable s is asymptotically stable. However, if 𝐹s1, the upper bound of 𝐹𝑠, is chosen such that

𝑟𝑟0𝐹s1 − 𝑚𝑎𝑥{|𝐹𝑠|} ≥ 𝜎𝐹 (50)

where 𝜎𝐹 is a positive constant number. (49) can then be written as:

�̇� < − 𝜎𝐹𝐽𝑒𝑞

|𝑠| (51)

(51) ensures that the sliding variable s converges to the sliding mode surface in a finite time [33]. The sliding mode controller in (46) can constrain the closed-loop error dynamics on the sliding mode surface, and the tracking error between the steering angle and the reference signal can then exponentially converge to zero. Remark 3.1: As the sign function sign(s) is involved in the sliding mode control signal in (46), the chattering may occur in the control input. Using the boundary layer control technique in [31]-[33], we can modify the control law in (46) as follows:

𝜏𝑠𝑚∗ = − 𝑁1𝑟0𝑁2

𝑠𝑎𝑡(𝑠) �𝐽𝑒𝑞1 ��̅̈�ℎ𝑟 + 𝜆|𝜀�̇�|� + 𝐵𝑒𝑞1��̇�𝑓�

+ 𝑟1𝑁2𝑁1

𝜏�̅�𝑖𝑠 + 𝐹𝑠1 + 𝜏�̅�� (52)

where

𝑠𝑎𝑡(𝑠) = �𝑠𝛿

for |𝑠| < 𝛿𝑠𝑖𝑔𝑛(𝑠) for |𝑠| ≥ 𝛿

(53)

and the constant 𝛿 > 0. (52) is called the boundary layer sliding mode controller (BL-SMC). As shown in [31]-[33], the output tracking error cannot converge to zero as the sign function is replaced by the sigmoid function. However, by properly choosing the value of the positive constant 𝛿 in (53), the tracking error can be small enough to satisfy the tracking precision requirement in practice. Remark 3.2: In this paper, the hand-wheel dynamics can be described by the following equation:

��̇�ℎ�̈�ℎ� = �

0 1−𝐶ℎ

𝐽ℎ− 𝐵ℎ

𝐽ℎ� �𝜃ℎ�̇�ℎ� + �01�

(𝜏ℎ − 𝜏𝑟) (54a)

where 𝐽ℎ , 𝐵ℎ and 𝐶ℎ are the moment of inertia, the viscous friction coefficient and the torsional stiffness of the hand-wheel shaft, respectively, 𝜃ℎ is the hand-wheel rotational angle, 𝜏ℎ is the input torque provided by the driver, and 𝜏𝑟 is the feedback torque generated by the hand-wheel feedback motor. It should be noted that the hand-wheel feedback motor is controlled by a PD regulator of the tracking error between the reference angle from the hand-wheel, and the steering angle, providing the driver with the true feeling of the steering effort. Moreover, the control parameters of the PD regulator must be chosen in the sense that the closed-loop in the hand-wheel side is stable [2].

In this design, the desired reference angle for the front wheel to follow can be expressed as:

𝜃ℎ𝑟 = 1𝑁𝜃𝜃ℎ (54b)

where 𝑁𝜃 is the ratio between the hand-wheel rotational angle and the front wheel steering angle. Remark 3.3: It is well known that the variable gear ratio steering (VGRS) has been widely used in advanced road vehicles recently. Here we would like to address that the proposed sliding mode control scheme in this paper is also applicable for the SbW systems with VGRS. The variable gear

7

ratio 𝑁2𝑁1

is actually embedded in the steering ratio 𝑟𝑁2𝑁1

, which has been involved in all of the parameters of the equivalent second-order model in (5). By properly choosing the upper and lower bounds of all parameters of the SbW system model in (5) with VGRS, the proposed SMC can ensure the good steering performance.

IV. NUMERICAL SIMULATION

In order to show the good performance of the proposed SMC, a simulation is carried out in comparison with the PD controller with feed-forward torque and the 𝐻∞ controller for SbW systems.

A. Parameters of SbW System and Vehicle Dynamics In this simulation, both the front wheel steering motor (PM

AC motor) and the feedback motor are chosen as the same in order to agree with the SbW platform used in the experimental section, where the steering motor is connected to a gearhead with ratio 𝑟𝑔. The nominal parameters of the SbW system are listed in Table II. The nominal parameters of the three-phase PM AC motors are given as follows: rated speed 𝜔 =2000𝑟𝑝𝑚 , rated torque 𝑇𝑚 = 4.77 𝑁𝑚 , peak instantaneous torque 𝑇𝑚𝑎𝑥 = 14.3 𝑁𝑚 , rated power 𝑃 = 1 𝑘𝑊 , rated voltage 𝑈𝑚 = 200 𝑉 (phase to phase), rated current 𝐼𝑚 =5.3 𝐴, number of poles 𝑝 = 6, constant magnet flux 𝜓𝑑0 =0.2 𝑊𝑏. Then we assume that the dc current offsets in phase a and phase b are ∆𝑖𝑎 = −0 .08 𝐴 and ∆𝑖𝑏 = 0.06 𝐴 , respectively. In addition, the parameters of the vehicle dynamics and motor harmonic torque used in the simulation are given in Table III [10], [35].

In this simulation, we assume that the central front wheel parameters 𝐽𝑓𝑤 and 𝐵𝑓𝑤 , the steering motor parameters 𝐽𝑠𝑚 and 𝐵𝑠𝑚, the conversion parameter 𝑟, the actual gear ratio 𝜂, the tire parameters 𝐶𝑓, 𝐶𝑟 , 𝑙𝑐 and 𝑙𝑝, the 6th and 12th harmonic torque amplitudes 𝜏𝑠𝑚6 and 𝜏𝑠𝑚12 , the dc current offsets ∆𝑖𝑎 and ∆𝑖𝑏 are all unknown, but the following uncertainty bounds are known:

0.015 𝑘𝑔.𝑚2 ≤ 𝐽𝑠𝑚 ≤ 0.028 𝑘𝑔.𝑚2 (55a)

𝐵𝑠𝑚 ≤ 0.045 𝑁𝑚𝑠/𝑟𝑎𝑑 (55b)

2.1 𝑘𝑔.𝑚2 ≤ 𝐽𝑓𝑤 ≤ 3 𝑘𝑔.𝑚2 (56a)

𝐵𝑓𝑤 ≤ 14 𝑁𝑚𝑠/𝑟𝑎𝑑 (56b)

5.6 ≤ 𝑟 ≤ 6.3 (57)

8 = 𝑟𝑔0 ≤ 𝑟𝑔 ≤ 𝑟𝑔1 = 9 (58)

2.2 𝑁𝑚 ≤ 𝐹s ≤ 3.2 𝑁𝑚 (59)

𝑙�̅� = 0.02 𝑚 and 𝑙�̅� = 0.03 𝑚 (60)

𝐶�̅� = �47000 𝑁/𝑟𝑎𝑑 for wet asphalt road 14000 𝑁/𝑟𝑎𝑑 for snowy road 82000 𝑁/𝑟𝑎𝑑 for dry asphalt road

(61)

𝜏�̅�𝑚6 = 0.05 𝑁𝑚, 𝜏�̅�𝑚12 = 0.01 𝑁𝑚 (62)

TABLE II

NOMINAL PARAMETERS OF THE SBW SYSTEM

Parameter Value

𝐽𝑓𝑤 (𝑘𝑔.𝑚2) 2.6

𝐵𝑓𝑤 (𝑁𝑚𝑠/𝑟𝑎𝑑) 12

𝐽𝑠𝑚 (𝑘𝑔.𝑚2) 0.02129

𝐵𝑠𝑚 (𝑁𝑚𝑠/𝑟𝑎𝑑) 0.038

𝐽ℎ (𝑘𝑔.𝑚2) 0.0791

𝐵ℎ (𝑁𝑚𝑠/𝑟𝑎𝑑) 0.15

𝐶ℎ (𝑁𝑚/𝑟𝑎𝑑) 0.2 𝑁2𝑁1

3 𝑟 6

𝑁𝜃 12

𝑟𝑔 8.5

𝐹𝑠(𝑁𝑚) 2.68

TABLE III

PARAMETERS OF VEHICLE DYNAMICS AND MOTOR HARMONIC TORQUE FOR SIMULATION

Parameter Value

𝑙𝑐 , 𝑙𝑝 (𝑚) 0.016, 0.023

𝑙𝑓 , 𝑙𝑟 (𝑚) 1.2, 1.05

𝐼𝑧 (𝑘𝑔.𝑚2) 1300

𝑀 (𝑘𝑔) 2000

𝐶𝑓 ,𝐶𝑟 (𝑁/𝑟𝑎𝑑) for wet asphalt road

45000, 45000

𝐶𝑓 ,𝐶𝑟 (𝑁/𝑟𝑎𝑑) for snowy road 12000,12000

𝐶𝑓 ,𝐶𝑟 (𝑁/𝑟𝑎𝑑) for dry asphalt road

80000,80000

𝜏𝑠𝑚6, 𝜏𝑠𝑚12 (𝑁𝑚) for wet asphalt road

0.022, 0.005

𝜏𝑠𝑚6, 𝜏𝑠𝑚12 (𝑁𝑚) for snowy road

0.01,0.003

𝜏𝑠𝑚6, 𝜏𝑠𝑚12 (𝑁𝑚) for dry asphalt road

0.038, 0.007

��∆𝑖𝑎2� + ∆𝑖𝑎∆𝑖𝑏 + �∆𝑖𝑏2� ≤ 𝜉𝑐 = 0.2 𝐴 (63)

Furthermore, the yaw motion of the road vehicle based on the bicycle model in Fig. 2 can be used to calculate the vehicle yaw rate 𝛾, which is described by the following state-space equation [1], [12]:

�̇� = 𝑨𝒚 + 𝒃𝛿𝑓 (64)

where 𝒚 = [𝛽 𝛾]𝑇

𝑨 =

⎣⎢⎢⎢⎡−𝐶𝑓 − 𝐶𝑟𝑀𝑉𝐶𝐺

−1 +𝐶𝑟𝑙𝑟 − 𝐶𝑓𝑙𝑓𝑀𝑉𝐶𝐺2

𝐶𝑟𝑙𝑟 − 𝐶𝑓𝑙𝑓𝐼𝑧

−𝐶𝑓𝑙𝑓2 − 𝐶𝑟𝑙𝑟

2

𝐼𝑧𝑉𝐶𝐺 ⎦⎥⎥⎥⎤

𝒃 = �𝐶𝑓

𝑀𝑉𝐶𝐺 𝐶𝑓𝑙𝑓𝐼𝑧�𝑇

8

B. Control Law Due to the gearhead connected to the front wheel steering motor, the model equation of the SbW system derived in (5) is re-written as the following state-space equation:

��̇�𝑓�̈�𝑓� = �

0 10 −𝐵𝑒𝑞

𝐽𝑒𝑞� �𝛿𝑓�̇�𝑓� + �

0𝑘𝑟

𝜏𝑑𝑖𝑠𝐽𝑒𝑞

−𝐹𝑠𝑠𝑖𝑔𝑛��̇�𝑓�+𝜏𝑒

𝐽𝑒𝑞�

+ �0

𝑘𝑟1𝐽𝑒𝑞� 𝜏𝑠𝑚∗ (65)

where 𝑘𝑟 is defined as 𝑘𝑟 = 𝑟𝑁2𝑁1𝑟𝑔.

The corresponding SMC is of the form: 𝜏𝑠𝑚∗ = − 1

𝑘𝑟0𝑠𝑖𝑔𝑛(𝑠) �𝐽𝑒𝑞1 ��̅̈�ℎ𝑟 + 𝜆|𝜀�̇�|� + 𝐵𝑒𝑞1��̇�𝑓�

+𝑘𝑟1𝜏�̅�𝑖𝑠 + 𝐹𝑠1 + 𝜏�̅�] (66) where 𝑘𝑟0 and 𝑘𝑟1 are the lower and the upper bounds of 𝑘𝑟, and defined as 𝑘𝑟0 = 𝑁1

𝑟0𝑟𝑔0𝑁2, and 𝑘𝑟1 = 𝑟1𝑁2

𝑁1𝑟𝑔1 . �̅̈�ℎ𝑟 is

selected as �̅̈�ℎ𝑟 = 10 𝑟𝑎𝑑/𝑠2 and the sliding mode parameter is chosen as 𝜆 = 12.

In addition, the control gains of the PD regulator for controlling the feedback motor are chosen as 𝑘𝑝 = 3.6 and 𝑘𝑑 = 1.5, respectively.

C. Simulation Environment In order to demonstrate the effectiveness and robustness of

the proposed SMC, the simulation environment is set up as follows:

• Driver’s input torque is a periodic sinusoidal signal as 𝜏ℎ = 1.6 𝑠𝑖𝑛(1.9𝑡) 𝑁𝑚.

• Three different road conditions (wet asphalt, snowy, and dry asphalt roads) are set for 0-15s, 15-25s, and 25-35s, respectively.

• The vehicle velocity is set as 𝑉𝐶𝐺 = 35 𝑚/𝑠. The Euler method with the sampling interval 𝛥𝑇 = 0.001 𝑠 is adopted to solve the closed-loop differential equations in this simulation.

D. Simulation Results The steering performance of the SMC is shown in Fig. 4(a)

and (b), while the associated control input is depicted in Fig. 4(c). It can be seen that the front wheel steering angle is driven to closely follow the hand-wheel reference angle in the whole period. Although the road conditions are suddenly changed at 15s and 25s, respectively, the good steering performance can still be achieved. Such an excellent steering performance indicates that the SMC is capable of eliminating the effects of uncertain road conditions on the steering performance. Fig. 4(d) and Fig. 4(e) show the upper bounds of the disturbances that are needed for the SMC design no matter how they change. Particularly, Fig. 4(f) shows that the steering performance at time 25s is not affected much owing to the robustness of SMC. Additionally, due to the discontinuous control input when crossing the sliding mode surface, the chattering occurs unavoidably, which can be solved by adopting the BL-SMC in (52). In order to further show the control performance, the root mean square (RMS) for the tracking error as a performance

evaluation index is utilized for the sake of clear comparison [36]:

𝜎𝑟𝑚𝑠 = ��∑ 𝜀𝜃2𝑛

𝑗=1 (𝑗)�

𝑛 (67)

where n is the number of the iterations. The RMS for the proposed SMC during the simulation period (35s) is 1.47 ×10−2.

In order to eliminate the chattering in the closed-loop system, BL-SMC is also used in the simulation. Fig. 5(a)-Fig. 5(f) show the steering performance with the BL-SMC, where the constant 𝛿 is chosen as 𝛿 = 0.8 and the RMS during the simulation period (35s) is 2.36 × 10−2 . It is observed from Fig. 5(c) that with the proper choice of 𝛿 , not only the undesired chattering in the control signal is effectively removed, but also the amplitude of the control torque is greatly reduced. Moreover, as shown in Fig. 5(f), the variations of the road conditions do not affect the performance of BL-SMC due to its robustness.

For further comparison, Fig. 6(a)-Fig. 6(f) show the steering performance of the steering system using a PD controller with feed-forward torque control method [1], [14], [15]:

𝜏𝑠𝑚∗ = 𝑘𝑝𝑠𝜀𝜃 + 𝑘𝑑𝑠𝜀�̇� + 𝐵𝑒𝑞𝑘𝑟𝐽𝑒𝑞

�̇�ℎ𝑟 + 𝐵𝑒𝑞𝑘𝑟𝐽𝑒𝑞

𝐹𝑠𝑠𝑖𝑔𝑛��̇�ℎ𝑟� (68)

where 𝑘𝑝𝑠 and 𝑘𝑑𝑠 are the proportional and derivative control gains, respectively. The last two terms on the right-side of (68) are adopted for the purpose of reducing the effects of disturbances on the steering performance. Based on the system model and motor characteristics described in section II, two most suitable control gains in (68) are determined as follows:

𝑘𝑝𝑠 = −4.5, 𝑘𝑑𝑠 = −1.4 (69) It is clearly seen from Fig. 6(a) to Fig. 6(f) that the steering

performance with the PD control is not as good as the ones as shown in Fig. 4(a)-Fig. 5(f) with the proposed SMC schemes in this paper. The reason is that the PD controller with fixed gains is unable to deal with the time-varying road conditions. This point can be easily seen, from Fig. 6(f) that, at 25s, the front wheels sideslip seriously due to the varying road condition and after that the tracking performance has greatly deteriorated. In addition, the RMS for the steering performance with the PD control in (68) is 6.92 × 10−2 that is much larger than the ones in the SbW systems with the SMC and BL-SMC presented in Fig. 4(a)-Fig. 5(f). Fig. 7(a)-Fig. 7(f) show the performance of the SbW system with the following 𝐻∞ controller [37]:

𝜏𝑠𝑚∗ = 𝐽𝑒𝑞𝑘𝑟�𝐵𝑒𝑞𝐽𝑒𝑞

�̇�𝑓 + �̈�ℎ𝑟 − 𝛼1𝜀�̇� − 𝛼2𝜀𝜃 + 𝐾𝑧� (70)

where 𝛼1 and 𝛼2 are the gains of the nominal feedback control, the error state vector 𝑧 = [𝜀𝜃 𝜀�̇�]𝑇 , and 𝐾 =−( 1

𝜌2)𝐵𝑧𝑇𝑃 that is the optimal control gain of the 𝐻∞ control

for minimizing the effects of the following lumped uncertainty 𝑤 on the steering performance:

𝑤 = −𝑘𝑟𝜏𝑑𝑖𝑠 + 𝜏𝑒 + 𝐹𝑠𝑠𝑖𝑔𝑛��̇�𝑓� + 𝑤0 (71) where 𝑤0 represents the modelling uncertainties.

9

Fig. 4. Control performance of SMC. (a) Tracking performance. (b) Tracking error. (c) Control torque. (d) Self-aligning torque and upper bound. (e) Torque pulsation disturbances and upper bound. (f) Tracking in the 25th second.

Fig. 5. Control performance of BL-SMC. (a) Tracking performance. (b) Tracking error. (c) Control torque. (d) Self-aligning torque and upper bound. (e) Torque pulsation disturbances and upper bound. (f) Tracking in the 25th second.

10

Fig. 6. Control performance of PD controller. (a) Tracking performance. (b) Tracking error. (c) Control torque. (d) Self-aligning torque. (e) Torque pulsation disturbances. (f) Tracking in the 25th second.

Fig. 7. Control performance of 𝐻∞ controller. (a) Tracking performance. (b) Tracking error. (c) Control torque. (d) Self-aligning torque. (e) Torque pulsation disturbances. (f) Tracking in the 25th second.

11

Fig. 8. Control performance of BL-SMC. (a) Tracking performance. (b) Tracking error. (c) Control torque.

Fig. 9. Control performance of PD controller. (a) Tracking performance. (b) Tracking error. (c) Control torque.

Fig. 10. Control performance of 𝐻∞ controller. (a) Tracking performance. (b) Tracking error. (c) Control torque.

It is noted from (70) that the hand-wheel angular acceleration �̈�ℎ𝑟 is required in the 𝐻∞ controller design. In fact, it is difficult to measure �̈�ℎ𝑟 in practice. In this simulation as well in the following experiments, �̈�ℎ𝑟 is obtained by using the filtering method that has been widely used in engineering practice [1].

The performance index for the 𝐻∞ control is given as

∫ ‖𝑧(𝑡)‖𝑄2𝑡𝑓0 𝑑𝑡 < ‖𝑧(0)‖𝑃2 + 𝜌2 ∫ ‖𝑤(𝑡)‖2𝑡𝑓

0 𝑑𝑡 (72)

where Q and P are the weighting matrices, 𝜌 is a prescribed attenuation level as 0 < 𝜌 < 1. P can be found by solving the following Riccati matrix equality:

A𝑧𝑇𝑃 + 𝑃𝐴𝑧 −

1𝜌2𝑃𝐵𝑧𝐵𝑧𝑇𝑃 + 𝑄 + 𝜉𝐼2 = 0 (73)

where 𝐴𝑧 = � 0 1−𝛼2 −𝛼1

�, 𝐵𝑧 = [0 1]𝑇 , and 𝜉 is a designed

positive constant. The parameters 𝜌, 𝜉,𝛼1,𝛼2 are set as 0.1, 0.1, 1, and 150, respectively. For the 𝐻∞ controller design, the matrix Q is selected to be 3𝐼2. The matrix P is found as

𝑃 = �25.2224 0.01030.0103 0.1669� (74)

and the control gain is 𝐾 = [−1.0298 − 16.6935] (75) It has been seen from Fig. 7(a)-Fig. 7(f) that, the steering

performance of the SbW system with 𝐻∞ control has been improved compared with the one of using the PD control, but still not as good as the ones with the SMC and BL-SMC. Fig. 7(f) shows that the 𝐻∞ control cannot eliminate the effects of

12

the variations of the road conditions on the steering performance. Further, the RMS for the 𝐻∞ control based SbW system is 6.21 × 10−2 that is larger than the ones of the SMC and BL-SMC and lower than the one of the PD controller. Remark 4.1: It is seen from the above simulation results that the influence of the self-aligning torque on the steering performance is much greater than that of the torque pulsation disturbances. For instance, the maximum values of the self-aligning torque and torque pulsation disturbances are about 150 Nm and 6 Nm for wet asphalt road, 40 Nm and 4 Nm for snowy road, and 220 Nm and 9 Nm for dry asphalt road, respectively. The advantage of the proposed sliding mode control methodology can eliminate not only the effect of the torque pulsation disturbances, but also the one of the self-aligning torque on the steering performance. This point has been clearly seen from both the stability analysis and the simulation studies in Section III and Section IV, respectively. Remark 4.2: It is noted from the above simulation studies and [3] that the self-aligning torque under three different road conditions behaves like three hyperbolic tangent disturbance signals of the steering angle with different amplitudes. We may thus use the following hyperbolic tangent signal to model the self-aligning torque 𝜏𝑒:

𝜏𝑒 = �𝜌1𝑡𝑎𝑛ℎ�2𝛿𝑓� for 0 < 𝑡 < 15𝑠

𝜌2𝑡𝑎𝑛ℎ�𝛿𝑓� for 15𝑠 ≤ 𝑡 < 25𝑠𝜌3𝑡𝑎𝑛ℎ�2𝛿𝑓� for 25𝑠 ≤ 𝑡 ≤ 35𝑠

(76)

where 𝜌1 , 𝜌2, and 𝜌3 are chosen as 𝜌1 = 520, 𝜌2 = 150, and 𝜌3 = 950, to ensure that the amplitudes of 𝜏𝑒 in three different road conditions are about 150 Nm, 40 Nm and 220 Nm, respectively. The corresponding bounds of 𝜌1, 𝜌2, and 𝜌3 are selected as 550, 180, and 980, respectively.

Fig. 8-Fig. 10 show the steering performance of the SbW system with the BL-SMC, the PD controller with feed-forward torque method, and the 𝐻∞ controller, respectively. The corresponding RMS for these three controllers are 3.12 ×10−2 ,7.75 × 10−2 , and 7.28 × 10−2 , respectively. It can be seen that the simulation results are similar to the ones presented in Fig. 5(a)-Fig. 7(f).

V. EXPERIMENTAL STUDIES

In this section, we will verify the effectiveness and the advantages of the proposed SMC on an SbW experimental platform in Fig. 11 in Robotics and Mechatronics Lab at Swinburne University of Technology.

A. Experimental System Setup It is seen from Fig. 11 that Mitsubishi HF-SP102 (A) AC

motors are used as the steering motor and the feedback motor, respectively. The steering motor is connected with the gearhead, and the corresponding servo-driver is selected as MR-J3100A manufactured by Mitsubishi Inc. An angle sensor (59006-10 turn, MoTeC) is installed on the pinion to measure the front wheel steering angle indirectly. In the hand-wheel side, the feedback motor is mounted on the steering column to provide the feeling of the interactions between the steered wheels and road surface.

Fig. 11. The SbW Experimental Platform.

TABLE IV

VALUES OF CONTROL PARAMETERS

Parameter Value 𝜆 12 𝛿 1.35 𝑘𝑝𝑠 -3.6 𝑘𝑑𝑠 -1.2

The nominal parameters of the SbW platform and AC servo motor in experiments are the same as the ones in the simulation section. Both the servo drivers are operated in torque control mode, driven by a +/- 8V reference signal. The servo motor is provided with a current by the servo driver, which is linearly proportional to the reference input voltage. Then, the torque generated by the servo motor is proportional to the input current. Thus, the torque generated by the servo motor is linear with the analogue torque command (input voltage). The proposed control algorithm is implemented on a HP personal computer using Matlab/Simulink/Real-time Workshop. The Advantech PCI 1716 Multifunction Card is installed in the PC for real-time control applications. The sampling period is chosen as 𝛥𝑇 = 0.001𝑠 , and the Euler method is adopted for this real-time experiment. In order to reduce the cost of the SbW system in real applications, the velocity of the front wheel steering angle is computed by differentiating and low-pass filtering the position signal measured by the position sensor [38].

B. Experimental Results The bound information of all the SbW system parameters is same as the one used in the simulation. To avoid the chattering in the control signal, the BL-SMC is utilized in the following experiment. The values of BL-SMC parameters (𝜆 and 𝛿) and PD controller parameters (𝑘𝑝𝑠 and 𝑘𝑑𝑠 ) are determined in Table IV, while the control parameters of 𝐻∞ controller are kept the same as the ones in simulations. On the other hand, the control gains of the PD regulator for the hand-wheel feedback motor are also set as same as the ones in the simulation section.

After the system is set up, the current offsets of phase a and phase b are measured with the values of 0 .1 𝐴 and 0 .05 𝐴, respectively. It should be emphasized that, although the current offsets are varying with time and temperature, the design of the proposed controller is not affected because only

13

Fig. 12. Control performance of BL-SMC. (a) Tracking performance. (b) Tracking error. (c) Control torque.

Fig. 13. Control performance of PD controller. (a) Tracking performance. (b) Tracking error. (c) Control torque.

Fig. 14. Control performance of H∞ controller. (a) Tracking performance. (b) Tracking error. (c) Control torque.

the bound information of the current offsets is required, as shown in (63). For confirming the robustness of the proposed scheme, the following voltage signal that models the self-aligning torque 𝜏𝑒 in (76) is added to the system:

𝑉 𝜏𝑒 =

⎩⎪⎨

⎪⎧

𝜌1𝜒𝑘𝑟

𝑡𝑎𝑛ℎ�2𝛿𝑓� for 0 < 𝑡 < 15𝑠

𝜌2𝜒𝑘𝑟

𝑡𝑎𝑛ℎ�𝛿𝑓� for 15𝑠 ≤ 𝑡 < 25𝑠𝜌3𝜒𝑘𝑟

𝑡𝑎𝑛ℎ�2𝛿𝑓� for 25𝑠 < 𝑡 ≤ 35𝑠

(77)

where 𝑘𝑟(= 18 ∗ 8.5)is the combined ratio including the steering ratio and the gearhead ratio defined in (65), 𝜌1 , 𝜌2, and 𝜌3 are the same as those determined in (76), and χ(= 1.8) is the nominal ratio between the motor torque and the input voltage of the servo driver given in the manual. The experimental results with the BLC, the PD controller, and the 𝐻∞ controller are shown in Fig. 12-Fig. 14, and the corresponding RMS are 5.41 × 10−2 , 9.31 × 10−2 , and 8.47 × 10−2, respectively.

It is observed that the steering performances of the three controllers are nearly the same on the snowy road at time period of 15-25s. This is because the equivalent disturbance is very mall and the three controllers are working closely at the idea condition. However, in the periods of 0-15s and 25-35s, the steering performances of both the PD controller and the 𝐻∞ controller are deteriorated seriously due to the large variation of disturbances. Only the SbW system equipped with the proposed SMC performs very well and behaves with a strong robustness against the large changes of the external disturbance.

14

Remark 5.1: It has been noted that the tracking performances in the simulations are better than the ones in the experimental results. The reasons are as follows: (i) All the mechanical parts, such as rack and pinion gearbox, are assumed to match perfectly, that is, no backlash exists in the mathematical model; however, the backlash indeed exists in the rack and pinion gearbox in practice, which is actually the main factor of affecting the steering accuracy; (ii) In the experiments, we have observed that the small structural resonances of mechanical parts occur sometimes during the operations, which have also affected the tracking precisions; (iii) It is noticeable that the low sampling rate of the microcontroller and the low resolution of the angle sensors are the other factors of degenerating the steering performance of the SbW system. Thus, the use of the high quality rack and pinion gearbox and the proper adjustment of the structure of the SbW system to avoid the structural resonances can play an essential role of further improving the tracking performance. In addition, an advanced microcontroller with a fast sampling rate and the sensors with higher resolutions are required to achieve more accurate tracking precision in SbW systems.

VI. CONCLUSION

In this paper, the mathematical modelling of the SbW systems has been further explored and a robust sliding mode steering controller has been proposed. It has been seen that the proposed sliding mode controller can efficiently alleviate the effects of uncertain system parameters and the variations of the road conditions as well as torque pulsation disturbances. Both the simulation and experimental results have verified the excellent steering performance of the proposed scheme. The further work on designing a sliding mode-based adaptive controller and the sliding mode observer-based diagnosis system are under the authors’ investigation.

ACKNOWLEDGEMENT

The authors would like to thank the Editor-in-Chief, the Associate Editor and the three anonymous reviewers for their invaluable comments for improving this work.

REFERENCES [1] P. Yih and J. C. Gerdes, “Modification of vehicle handling

characteristics via Steer-by-Wire,” IEEE Trans. Control Syst. Technol., vol. 13, no. 6, pp. 965-976, 2005.

[2] M. Bertoluzzo, G. Buja, and R. Menis, “Control schemes for Steer-by-Wire systems,” IEEE Ind. Electron. Magazine, vol. 1, no. 1, pp. 20-27, 2007.

[3] A. Baviskar, J. R. Wagner, and D. M. Dawson, “An ajustable steer-by-wire haptic-interface tracking controller for ground vehicles,” IEEE Trans. Vehi. Technol., vol. 58, no. 2, pp. 546-554, 2009.

[4] S. W. OH, H. C. Chae, S. C. Yun, and C.S. Han, “The design of a controller for the steer-by-wire system,” JSME International Journal, vol. 47, no. 3, pp. 896-907, 2004.

[5] C. J. Kim, J. H. Jang, and S. K. Oh, “Development of a control algorithm for a rack-actuating steer-by-wire system using road information feedback,” Proc. IMechE, Part D: J. Automobile Engineering, vol. 222 pp. 1559-1571, 2008.

[6] T. J. Park, C. S. Han, and S. H. Lee, “Development of the electronic control unit for the rack-actuating steer-by-wire using the hardware-in-

the-loop simulation system,” Mechatronics, vol. 15, no. 8, pp. 899-918, 2005.

[7] P. Setlur, J. R. Wagner, D. M. Dawson, and D. Braganza, “A trajectory tracking steer-by-wire control system for ground vehicles,” IEEE Trans. Vehi. Technol., vol. 55, no. 1, pp. 76-85, 2006.

[8] Y. Marumo and M. Nagai, “Steering control of motorcycles using steer-by-wire system,” Veh. Syst. Dyn., vol. 45, no. 9, pp. 445-458, 2007.

[9] Y. Marumo and N. Katagiri, “Control effects of steer-by-wire system for motorcycles on lane-keeping performance,” Veh. Syst. Dyn. vol. 49, no. 8, pp. 1283-1298, 2011.

[10] C. D. Gadda, S. M. Laws, and J. C. Gerdes, “Generating diagnostic residuals for Steer-by-Wire vehicle,” IEEE Trans. Control Syst. Technol., vol. 15, no. 3, pp. 529-540, 2007.

[11] R. Kazemi and A. A. Janbakhsh, “Nonlinear adaptive sliding mode control for vehicle handling improvement via steer-by-wire,” International Journal of Automotive Technology, vol. 11, no. 3, pp. 345-354, 2010.

[12] Y. Yamaguchi and T. Murakami, “Adaptive control for virtual steering characteristics on electric vehicle using Steer-by-Wire system,” IEEE Trans. Ind. Electron., vol. 56, no. 5, pp. 1585-1594, 2009.

[13] Y. Fujimoto, “Robust servo-system based on two-degree-of-freedom control with sliding mode,” IEEE Trans. Ind. Electron., vol. 42, no. 3, pp. 272-280, 1995.

[14] A. E. Cetin, M. A. Adli, and D. E. Barkana, “Implementation and development of an adaptive steering-control system,” IEEE Trans. Vehi. Technol., vol. 59, no. 1, pp. 75-83, 2010.

[15] A. H. Brian, D. Pierre, and C. D. W. Carlos, “A survey of models, analysis tools and compensation methods for the control of machines with friction,” Automatica, vol. 30, no. 7, pp. 1083-1138, 1994.

[16] Z. Man and X. Yu, “Terminal sliding mode control of MIMO linear systems,” IEEE Trans. Circuit Syst., vol. 44, no. 11, pp. 1065-1070, 1997.

[17] Z. Man, H. R. Wu, and M. Palaniswami, “An adaptive tracking controller using neural networks for a class of nonlinear systems,” IEEE Trans. Neural Networks, vol. 9, no. 5, pp. 947-955, 1998.

[18] H. B. Pacejka, Tire and vehicle dynamics, SAE, Warrendale, PA, 2002. [19] Y. H. J. Hsu, S. Laws, and J. C. Gerdes, “Estimation of tire slip angle

and friction limits using steering torque,” IEEE Trans. Control Syst. Technol., vol. 18, no. 4, pp. 896-907, 2010.

[20] H. Wang, Z. Man, H. Kong, and W. Shen, “Terminal sliding mode control for steer-by-wire system in electric vehicles,” Proceedings of ICIEA 2012, pp. 919-924, 2012.

[21] M. Gopal, Control systems: principles and design, McGraw-Hill, New Delhi, 2002.

[22] V. I. Utkin and K. D. Young, “Methods for constructing discontinuity planes in multidimensional variable-structure systems,” Automation and Remote Control, vol. 39, pp.1466-1470, 1978.

[23] V. I. Utkin, “Sliding mode control design principles and applications to electric drives,” IEEE Trans. Ind. Electron., vol. 40, no. 1, pp. 23-36, 1993.

[24] A. Izadian, L. A. Hornak, and P. Famouri, “Structure rotation and pull-in voltage control of MEMS lateral comb resonators under fault conditions,” IEEE Trans. Control Syst. Technol., vol. 17, no. 1, pp. 51-59, 2009.

[25] E. Kayacan and O. Kaynak, “Sliding mode control-based algorithm for online leanring in type-2 fuzzy neural networks: application to velocity control of an electro hydraulic servo system,” Int. J. Adapt. Control and Signal Processing, vol. 26, no. 7, pp. 645-659, 2012.

[26] E. Kayacan, O. Cigdem, and O. Kaynak, “Sliding mode control approach for online learning as applied to type-2 fuzzy neural networks and its experimental evaluation,” IEEE Trans. Ind. Electron., vol. 59, no. 9, pp. 3510-3520, 2012.

[27] Y. Lin, Y. Shi, and R. Burton, “Modeling and robust discrete-time sliding-mode control design for a fluid power electrohydraulic actuator (EHA) system,” IEEE Trans. Mechatronics, vol.18, no. 1, pp. 1-10, 2013.

[28] O. Kaynak, K. Erbatur, and M. Ertugrul, “The fusion of computationa-lly intelligent methodologies and sliding-mode control: A survey,” IEEE Trans. Ind. Electron., vol.48, no. 1, pp. 4-17, 2001.

[29] X. Yu and O. Kaynak, “Sliding mode control with soft computing: A survey,” IEEE Trans. Ind. Electron., vol.56, no. 9, pp. 3275-3285, 2009.

[30] T. R. Oliveira, L. Hsu, and A. J. Peixoto, “Output-feedback global tracking for unknown control direction plants with application to extremum-seeking control,” Automatica, vol. 47, no. 9, pp. 2029-2038, 2011.

15

[31] V. Utkin, Sliding mode control in electro-mechanical systems, Taylor & Francis, 2009.

[32] C. Edwards and S. Spurgeon, Sliding mode control: theory and applications, Taylor & Francis, 1998.

[33] H. K. Khalil, Nonlinear systems (3rd Edition), Prentice- Hall, New York, 2002.

[34] R. Rajamani, Vehicle dyamics and control, Springer, 2006. [35] G. Baffet, A. Charara, and D. Lechner, “Estimation of vehicle sideslip,

tire force and wheel cornering stiffness,” Control Engineering Practice vol. 17, no. 11, pp. 1255-1264, 2009.

[36] W. F. Xie, “Sliding-mode-observer-based adaptive control for servo actuator with friction,” IEEE Trans. Ind. Electron., vol. 54, no. 3, pp. 1517-1527, 2007.

[37] N. C. Shieh, “Robust output tracking control of a linear brushless DC motor with time-varying disturbances,” Electric Power Applications, IEE Proceedings, vol. 149, no. 1, pp. 39-45, 2002.

[38] M. Smaoui, X. Brun, and D. Thomasset, “High-order sliding mode for an electropneumatic system: A robust differentiator-controller design,” Int. J. Robust Nonlinear Control, vol. 18, no. 4-5, pp. 481-501, 2008.

[39] K. K. Shyu, C. K. Lai, Y. W. Tsai, and D. I. Yang, “ A newly robust controller design for the position control of permanent-magnet synchronous motor,” IEEE Trans. Ind. Electron., vol.49, no. 3, pp. 558-565, 2002.

[40] Y. X. Su, C. H. Zheng, and B. Y. Duan, “Automatic disturbances rejection controller for precise control of permanent-magnet synchronous motors,” IEEE Trans. Ind. Electron., vol.52, no. 3, pp. 814-823, 2005.

[41] I. C. Baik, K. H. Kim, and M. J. Youn, “Robust nonlinear speed control of PM synchronous motor using boundary layer integral sliding mode control technique,” IEEE Trans. Control Syst. Technol., vol. 8, no. 1, pp. 47-54, 2000.

[42] W. C. Gan and L. Qiu, “Torque and velocity ripple elimination of AC permanent magnet motor control systems using the internal model principle,” IEEE Trans. Mechatronics, vol.9, no. 2, pp. 436-447, 2004.

[43] J. X. Xu, S. K. Panda, Y. J. Pan, T. H. Lee, and B. H. Lam, “A modular control scheme for PMSM speed control with pulsating torque minimization,” IEEE Trans. Ind. Electron., vol.51, no. 3, pp. 526-536, 2004.

[44] G. J. Wang, C. T. Fong, and K. J. Chang, “Neural-network-based self-tuning PI controller for precise motion control of PMAC motors,” IEEE Trans. Ind. Electron., vol.48, no. 2, pp. 408-415, 2001.

[45] G. Ferretti, G. Magnani, and P. Rocco, “ Modeling, identification, and compensation of pulsating torque in permanent magnet AC motors,” IEEE Trans. Ind. Electron., vol.45, no. 6, pp. 912-920, 1998.

[46] W. Qian, S. Panda, and J. Xu, “Torque ripple minimization in PM synchronous motors using iterative learning control,” IEEE Trans. Power Electron., vol.19, no. 2, pp. 272-279, 2004.

[47] H. Liu and S. Li, “Speed control for PMSM servo system using predictive functional control and extended state observer,” IEEE Trans. Ind. Electron., vol.59, no. 2, pp. 1171-1183, 2012.

Hai Wang (S’12) was born in Hefei, China in 1984. He received his B.E. degree in Electrical Engineering from Hebei Polytechnic University, China, in 2007, and the M.E. degree in electrical engineering from Guizhou University, China, in 2010. He is currently pursuing the Ph.D. degree in the Faculty of Engineering and Industrial Sciences, Swinburne University of Technology, Melbourne, Australia. His research interests are in sliding mode control, adaptive control, robotics, neural networks, nonlinear systems, and vehicle dynamics & control.

Huifang Kong received her B.S., M.S. and PhD degrees from Hefei University of Technology, Hefei, China, in 1986, 1989 and 2008, respectively, all in electrical engineering. Since 1989, she has been with School of Electrical Engineering, Hefei University of Technology, where she is currently the Professor and Director of the Research Centre of Automotive Electronics and Control Technology. Her research interests are in control engineering, industrial automation, and vehicle dynamics & control.

Zhihong Man (M’94) received his B.E. degree from Shanghai Jiaotong University, China, in 1982, the M.Sc. degree from Chinese Academy of Sciences in 1987, and the Ph.D. degree from the University of Melbourne, Australia, in 1994, respectively. From 1994 to 1996, he was the Lecturer in the School of Engineering at Edith Cowan University, Australia. From 1996 to 2001, he was the Lecturer and then the Senior Lecturer in the School of Engineering at the University of Tasmania, Australia. From 2002 to 2007, he was the Associate Professor of Computer

Engineering at Nanyang Technological University, Singapore. From 2007 to 2008, he was the Professor and Head of Electrical and Computer Systems Engineering at Monash University Sunway Campus, Malaysia. Since 2009, he has been with Swinburne University of Technology, Australia, as the Professor and Head of Robotics and Mechatronics. His research interests are in nonlinear control, signal processing, robotics, neural networks, and vehicle dynamics & control.

Do Manh Tuan (S’12) received the B.E. degree with the first-class honors in electrical and automatic control engineering from the Center for Gifted Education, Hanoi University of Science and Technology, Vietnam, in 2008. From 2008 to 2010, he was a lab systems engineer with Intel Products Corporation. Currently, he is working towards his Ph.D. degree in the Faculty of Engineering and Industrial Sciences, Swinburne University of Technology, Melbourne, Australia. His research

interests include sliding mode control and its applications, nonlinear systems, robotics, and robust control of road vehicles.

Zhenwei Cao received the B.S. and M.E. degrees from the Southeast University, Nanjing, China in 1985 and 1988, respectively, and the Ph.D. degree from the University of Newcastle, Newcastle, Australia, in 2001, all in electrical engineering. She is currently a Senior Lecturer in the Faculty of Engineering and Industrial Sciences, Swinburne University of Technology, Australia. Her current research interests include robotics, mechatronics, control and automation.

Weixiang Shen (S’00–M’02) received his Ph.D. degree from the University of Hong Kong, China, in 2002. From 2002 to 2003, He was the Lecturer in Ngee Ann Polytechnic, Singapore. From 2003 to 2008, he was the Lecturer and then the Senior Lecturer in School of Engineering, Monash University Sunway Campus, Malaysia. He then worked as the Research Fellow for one year in the School of Electrical and Electronics Engineering, Nanyang Technological University, Singapore. Since 2009, he has been the Senior Lecturer of Electrical

Engineering in the Faculty of Engineering and Industrial Sciences, Swinburne University of Technology, Australia. His research interests are in electrical vehicles, renewable energy, power system and power electronics.