Fault-Tolerant Control of Four-Wheel Independently Actuated Electric Vehicles with Active Steering Systems★

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Fault-Tolerant Control of Four-Wheel Fault-Tolerant Control of Four-Wheel Fault-Tolerant Control of Fault-Tolerant Control Electric of Four-Wheel Four-Wheel Independently Actuated Vehicles Independently Actuated Electric Vehicles Independently Actuated Electric Vehicles ⋆⋆ Independently Actuated Electric Vehicles with Active Steering Systems ⋆ with Active Steering Systems with with Active Active Steering Steering Systems Systems ⋆ Hui Hui Hui Hui

Jing ∗∗ Rongrong Wang† ∗∗ Mohammed Chadli ∗∗ ∗∗ ∗ Rongrong ∗ Mohammed ∗∗ Jing Wang† ∗∗∗ ∗∗∗∗ Jing Wang† Chadli ∗ Rongrong ∗ Mohammed ∗∗ Chuan Hu ∗∗∗ Fengjun Yan Cong LiChadli ∗∗∗ ∗∗∗ ∗∗∗∗ Jing Rongrong Wang† Mohammed Chadli ∗∗∗ Cong Li ∗∗∗∗ Chuan Hu Hu ∗∗∗ Fengjun Yan Chuan Fengjun Yan Cong Li ∗∗∗ ∗∗∗ ∗∗∗∗ Chuan Hu Fengjun Yan Cong Li ∗ School of Mechanical Engineering, Southeast University, Nanjing ∗ ∗ School of Mechanical Engineering, Southeast University, Nanjing of Engineering, University, Nanjing ∗ School211189, P.R. China (e-mail: Southeast [email protected] School of Mechanical Mechanical Engineering, Southeast University,). Nanjing 211189, P.R. China (e-mail: [email protected] ). ∗∗ 211189, P.R. China (e-mail: [email protected] ). University of Picardie Jules Verne, MIS (E.A. 4290). 80039, ∗∗ 211189, P.R. China (e-mail: [email protected] ). ∗∗ University of Picardie Jules Verne, MIS (E.A. 4290). 80039, of Picardie ∗∗ University Amiens, University of France([email protected]). Picardie Jules Jules Verne, Verne, MIS MIS (E.A. (E.A. 4290). 4290). 80039, 80039, Amiens, France([email protected]). ∗∗∗ Amiens, France([email protected]). Department of Mechanical Engineering, McMaster University, ∗∗∗ Amiens, ofFrance([email protected]). ∗∗∗ Department Mechanical Engineering, McMaster University, Mechanical Engineering, McMaster ∗∗∗ Department Hamilton, of ON, L8S 4L7 Canada ([email protected]). Department of Mechanical Engineering, McMaster University, University, ON, L8S 4L7 Canada ([email protected]). ∗∗∗∗ Hamilton, Hamilton, ON, L8S 4L7 Canada ([email protected]). Department of Mechanical Engineering, Guilin University of ∗∗∗∗ Hamilton, ON, L8S 4L7 Canada ([email protected]). ∗∗∗∗ Department of Mechanical Engineering, Guilin University of Department of Guilin ∗∗∗∗ Aerospace Technology, Guilin,Engineering, China, ([email protected]). Department of Mechanical Mechanical Guilin University University of of Aerospace Technology, Guilin,Engineering, China, ([email protected]). ([email protected]). Aerospace Technology, Guilin, China, Aerospace Technology, Guilin, China, ([email protected]). Abstract: This paper presents a fault-tolerant control (FTC) strategy to improve lateral Abstract: paper presents fault-tolerant control (FTC) strategy to improve lateral Abstract: This paper a fault-tolerant control strategy to lateral stability andThis maneuverability of aa independently actuated (FWIA) electric ground Abstract: This paper presents presents afour-wheel fault-tolerant control (FTC) (FTC) strategy to improve improve lateral stability and maneuverability of a four-wheel independently actuated (FWIA) electric ground stability and maneuverability of a four-wheel independently actuated (FWIA) electric ground vehicle with front steering Frontindependently wheel steeringactuated angle and external yaw moment stability and active maneuverability of a(AFS). four-wheel (FWIA) electric ground vehicle with active front steering (AFS). Front wheel steering angle and external yaw moment vehicle with front steering (AFS). Front steering and external moment generated byactive the tire difference the two sidesangle of the areyaw adopted to vehicle with active frontforce steering (AFS).between Front wheel wheel steering angle andmotors external yaw moment generated by the tire force difference between the two sides of the motors are adopted to generated by the tire force difference between the two sides of the motors are adopted to simultaneously regulate the vehicle yaw rate and slip angle. Considering the high cost of generated by the tire force difference between the slip two angle. sides of the motorsthe arehigh adopted to simultaneously regulate the vehicle yaw rate and Considering cost of simultaneously regulate the vehicle yaw rate and slip angle. Considering the high cost currently available sensors for vehicle slip angle measurement, a robust H dynamic output∞ simultaneously regulate theforvehicle yaw rate and slip angle. aConsidering the high outputcost of of currently available sensors vehicle slip angle measurement, robust H dynamic ∞ currently available sensors for vehicle slip angle measurement, a robust H dynamic outputfeedback designed control theangle vehicle, which can be without outputthe slip currently controller available is sensors for to vehicle slip measurement, a implemented robust H∞ ∞ dynamic feedback controller is designed to control the vehicle, which can be implemented without the slip feedback controller is to vehicle, which be without slip angle or lateral speed measurement. Boththe vehicle parameter uncertainties and actuators faults feedback controller is designed designed to control control the vehicle, which can can be implemented implemented without the the slip angle or lateral speed measurement. Both vehicle parameter uncertainties and actuators faults angle or lateral speed measurement. Both vehicle parameter uncertainties and actuators faults are considered, making the proposed control method robust to the uncertainties and actuator angle or lateral speed measurement. Both vehicle parameter uncertainties and actuators faults are considered, making the proposed control method method robust to validate the uncertainties uncertainties and actuator actuator are considered, making proposed control to the and faults. Simulation resultsthe based on full-vehicle model inrobust Carsim the effectiveness of the are considered, making the proposed control method robust to validate the uncertainties and actuator faults. Simulation results based on full-vehicle model in Carsim the effectiveness of the faults. Simulation results based on full-vehicle model in Carsim validate the effectiveness of proposed control approach. faults. Simulation results based on full-vehicle model in Carsim validate the effectiveness of the the proposed control approach. proposed control approach. proposed control approach. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Robust control, fault tolerant control, output feedback, electric vehicle, in-wheel Keywords: Robust control, fault tolerant tolerant control, output output feedback, electric electric vehicle, in-wheel in-wheel Keywords: Robust motor, active front control, steering.fault Keywords: Robust control, fault tolerant control, control, output feedback, feedback, electric vehicle, vehicle, in-wheel motor, active front steering. motor, motor, active active front front steering. steering. 1. INTRODUCTION 1. INTRODUCTION INTRODUCTION 1. 1. INTRODUCTION Electric vehicles have several advantages in terms of high Electric conversion vehicles have have several advantages advantages in terms terms of high high Electric vehicles several in of energy efficiency, environmental friendliness, Electric vehicles have several advantages in terms of high energy conversion efficiency, environmental friendliness, energy conversion efficiency, environmental friendliness, and performance [Wang, R. et al (2011), Wu, energy conversion benefits efficiency, environmental friendliness, and performance performance benefits [Wang, R. et et al al (2011), (2011), Wu, and benefits [Wang, R. F.K. et al (2013)]. The four-wheel independently-actuated and performance benefits [Wang,independently-actuated R. et al (2011), Wu, Wu, F.K. et al (2013)]. The four-wheel F.K. et al (2013)]. The four-wheel independently-actuated (FWIA) in which independently-actuated each wheel is indepenF.K. et alelectric (2013)].vehicle, The four-wheel (FWIA) electric vehicle, in which each wheel is indepen(FWIA) electric vehicle, which wheel is dently actuated an in in-wheel (or hub) is an (FWIA) electric with vehicle, in which each each wheelmotor, is indepenindependently actuated actuated with anSince in-wheel (or hub) hub) motor, is an an dently with an in-wheel (or motor, is emerging configuration. the torque of each wheel in dently actuated with anSince in-wheel (or hub) motor, is an emerging configuration. the torque of each wheel in configuration. Since the torque of each wheel in aemerging FWIA electric vehicle is independently controlled, an emerging configuration. Since the torque of controlled, each wheel an in a FWIA electric vehicle is independently a FWIA electric vehicle is independently controlled, an external moment canisbeindependently generated as acontrolled, result of the a FWIA yaw electric vehicle an external yaw moment can be generated as a result of the external yaw can a the result of torque difference between thegenerated two sidesas in-wheel external yaw moment moment can be be asof result of the the torque difference difference between thegenerated two sides ofahandling the in-wheel torque between the two sides of the in-wheel motors, and thus can improve the vehicle and torque difference between the two sides of the in-wheel motors, and and thusM.can can improve the (2003), vehicle Wang, handling and motors, thus the vehicle handling stability [Shino, andimprove Nagai, M. R. and motors, and thusM.can improve the (2003), vehicle Wang, handling and stability [Shino, and Nagai, M. R. and stability [Shino, M. and Nagai, M. (2003), Wang, R. Wang, J.[Shino, (2012),M. Geng,C. et al (2009)]. Active steering is stability and Nagai, M. (2003), Wang, R. and and Wang, J. (2012), Geng,C. et al (2009)]. Active steering is Wang, J. (2012), Geng,C. et al (2009)]. Active steering is another effective way of improving drivers comfort and Wang, J. (2012), Geng,C. et al (2009)]. Active steering is another effective way of improving drivers comfort and another effective improving and handling. If both way the of external yaw drivers momentcomfort and active another effective way of improving drivers comfort and handling.angle If both both the external external yaw moment moment and lateral active handling. If the yaw active steering are adopted, the vehicle handling and handling. If both the external yaw moment and active steering angle are adopted, the vehicle handling and lateral steering angle are adopted, the vehicle handling and lateral stability can be the samehandling time. and lateral steering angle arecontrolled adopted, at the vehicle stability can be controlled at the same time. stability stability can can be be controlled controlled at at the the same same time. time.

Many researches have been devoted to either the FWIA Many researches been devoted to either the FWIA Many researches have been devoted to the FWIA electric vehicle orhave active steering system control. fuzzy Many researches have been devoted to either either theA FWIA electric vehicle or active steering system control. A fuzzy electric vehicle or active steering system control. A control for Electric Power Steering system (EPS) with electric vehicle or active steering system control. A fuzzy fuzzy control for Electric Power Steering system (EPS) with control for Electric Power Steering system (EPS) assist motor current input constraints is discussed in control for Electric Power Steering system (EPS) with with assist motor current input constraints is discussed assist motor current input constraints is discussed in [Saifia, motor D., etcurrent al (2015)]. Inconstraints [Geng,C. etis al (2009)], in assist input discussed ina [Saifia, D., et al (2015)]. In [Geng,C. et al (2009)], a [Saifia, D., et al (2015)]. In [Geng,C. et al (2009)], a direct yaw moment control of an in-wheel motored EV is [Saifia, D., et al (2015)]. In [Geng,C. et al (2009)], a direct yaw moment control of an in-wheel motored EV is direct yaw moment control of an in-wheel motored EV is presented. In [Doumiati, M. et al (2013)], an integrated direct yaw moment control of an in-wheel motored EV is presented. In [Doumiati, et al is (2013)], an integrated presented. In M. et (2013)], an vehicle dynamic control M. method designed based on presented. In [Doumiati, [Doumiati, M. et al al is (2013)], an integrated integrated vehicle dynamic control method designed on vehicle dynamic control method is designed based on active front-wheel steering method and rear isbraking. In [based Du, H.P. vehicle dynamic control designed based on active front-wheel steering and rear braking. In [ Du, H.P. active front-wheel steering and rear braking. In [ Du, H.P. et al (2010)], a robust yaw control method with control active front-wheel steering and rear braking. In [ Du, H.P. et al (2010)], a robust yaw control method with control et al (2010)], yaw method with control input consideration is presented, tire force et al saturation (2010)], aa robust robust yaw control control method the with control input saturation consideration is presented, the tire force input saturation consideration is presented, the tire force modeling error is also considered. The tracking control input saturation consideration is presented, the tire force modeling error is also considered. The tracking control modeling error is also considered. The tracking control problem iserror studied in [ Zhang, H. et al (2013a), Zhang, H. modeling is also considered. The tracking control problem is studied in [ Zhang, H. et al (2013a), Zhang, H. problem is studied in [[ Zhang, H. al (2013a), H. et al (2013b) and Aouaouda, S., et (2014b)]. Zhang, Note that problem is studied in Zhang, H. et al (2013a), Zhang, H. et al (2013b) and Aouaouda, S., et al (2014b)]. Note that et al (2013b) and Aouaouda, S., et al (2014b)]. Note that performances of an FWIA electric vehicle mainly depend et al (2013b) and Aouaouda, S., et al (2014b)]. Note that performances ofoperations an FWIA FWIA electric electric vehicle mainly mainly depend performances an vehicle on the properof the in-wheel motors.depend When performances ofoperations an FWIA of electric vehicle mainly depend on the proper of the in-wheel motors. When on the proper operations of the in-wheel motors. When an in-wheel motor fault occurs, the faulty motors. wheel will not on the proper operations of the in-wheel When an in-wheel motor fault occurs, the faulty wheel will not an in-wheel motor fault occurs, the faulty wheel will not provide the expected torque and will result in unpredicted an in-wheel motor fault occurs, the faulty wheel will not provide the expected torque and will result in unpredicted provide the expected torque and will result in unpredicted consequences [Wang, R. et al (2014)]. Thus, FTC is provide the expected torque and will result in unpredicted consequences [Wang, R. et al (2014)]. Thus, FTC is consequences [Wang, R. et al (2014)]. Thus, FTC is of critical importance to FWIA electric vehicles. Some consequences [Wang, R. et al (2014)]. Thus, FTC is of critical importance to FWIA electric vehicles. Some of critical importance to FWIA electric vehicles. Some FTC methods for ground vehicleselectric have been previously of critical importance to FWIA vehicles. Some FTC methods for ground vehicles have been previously FTC methods for ground have been proposed in the however, of previously them are FTC methods for literature, ground vehicles vehicles havefew been previously proposed in the literature, however, few of them are proposed in the literature, however, few of them specifically designed for for FWIA electric vehicles or proposed indesigned the literature, however, few ofvehicles them are are specifically for for FWIA electric or specifically designed for for FWIA electric vehicles or very limited types of actuator faults are considered, see specifically designed for for FWIA electric vehicles see or very limited types of actuator faults are considered, very limited types of actuator faults are considered, see [Zhang, Y. and Jiang, J. (2008),Wang, R. and Wang, very limited typesJiang, of actuator faults are considered, see [Zhang, Y. and J. (2008),Wang, R. and Wang, [Zhang, Y. and J. R. Wang, J. (2012), B. and Anwar,S. (2008), S. [Zhang, Y. Zheng, and Jiang, Jiang, J. (2008),Wang, (2008),Wang, R. and andYim, Wang, J. (2012), Zheng, B. and Anwar,S. (2008), Yim, S. J. (2012), Zheng, B. and Anwar,S. (2008), Yim, J. (2012), Zheng, B. and Anwar,S. (2008), Yim, S. S.

⋆ This work is partly supported by National Natural Science ⋆ This partly by Natural Science ⋆ This work workof is isChina(Grant partly supported supported by National National Natural Science Foundation nos. 51205058, 51375086, 61403252), ⋆ This workof isChina(Grant partly supported by National Natural Science Foundation nos. 51205058, 51375086, 61403252), Foundation of China(Grant nos. 51205058, 51375086, 61403252), and Jiangsu Province Science Foundation for Youths, China(Grant Foundation of China(Grant nos. 51205058, 51375086, 61403252), and Jiangsu Province Science Foundation for Youths, China(Grant and BK20140634),the Jiangsu Province Science Foundation for Youths, no. Foundation of Education OfficeChina(Grant of Guangxi and Jiangsu Province Science Foundation for Youths, China(Grant no. BK20140634),the BK20140634),the Foundation of Education Education Office of Guangxi Guangxi no. Foundation of of Province of China (Grant no. KY2015YB101), theOffice Fundamental Reno. BK20140634),the Foundation of Education Office of Guangxi Province of (Grant no. KY2015YB101), the ReProvince of China China (Grant no. Universities KY2015YB101), the Fundamental Fundamental Research Funds for the Central and Jiangsu Postgraduate Province of China (Grant no. Universities KY2015YB101), the Fundamental Research Funds Funds for the the Central and Jiangsu Jiangsu Postgraduate search for Central and Postgraduate Innovation Programm (GrantUniversities no. KYLX-0102). search Funds for the Central Universities and Jiangsu Postgraduate Innovation Programm (Grant no. KYLX-0102). Innovation Programm (Grant no. KYLX-0102). Innovation Programm (Grant no. KYLX-0102). Copyright 2015 IFAC 1165Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2015, IFAC (International Federation of Automatic Control) Copyright 2015 IFAC 1165 Copyright ©under 2015 responsibility IFAC 1165Control. Peer review© of International Federation of Automatic Copyright © 2015 IFAC 1165 10.1016/j.ifacol.2015.09.684

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steering angle is small, the vehicles handling dynamics in the yaw plane can be expressed as:  1  v˙ y = (Fyf + Fyr ) − vx r + d1 (t) m , (1) 1 1  r˙ = (lf Fyf − lr Fyr ) + ∆Mz + d2 (t) Iz Iz

Fig. 1. Schematic diagram of a vehicle planar motion model. (2014) and Aouaouda, S., et al (2014a)]. In [Wang, R. et al (2014)], an linear parameter-varying system based control method considering actuator fault is designed for a FWIA electric vehicle with active steering system. However, vehicle slip angle measurement is required. Since the sensors for vehicle slip angle measurement are typically considered expensive for commercial vehicle applications, the proposed control method may hard to be implemented in practical applications. In this paper, a robust output feedback control based FTC strategy is proposed for a FWIA electric vehicle with AFS to improve the handling stability despite of active steering system and/or in-wheel motor faults. The main contributions of this paper lie in the following aspects. First, both AFS and external yaw moment are adopted to simultaneously regulate the vehicle yaw rate and track the vehicle slip angle references. Second, considering that the measurement of vehicle slip angle or lateral speed requires expressive sensors, a robust H∞ dynamic outputfeedback controller is designed to regulate the lateral motion without using the vehicle slip angle or lateral speed information. Third, actuator faults, external disturbances and vehicle parametric uncertainties are simultaneously considered in the controller design, making the proposed output-feedback controller robust to the different types of faults, uncertainties and external disturbances. The rests of the paper are organized as follows. Vehicle model considering parametric uncertainties and actuators faults is presented in Section 2. The proposed robust FTC dynamic output-feedback controller is designed in Section 3. Simulation results based on a full-vehicle model are provided in Section 4 followed by conclusive remarks.

where vx and vy are the vehicle longitudinal and lateral speeds, respectively. r is the yaw rate, m and Iz are the vehicle mass and yaw inertia, respectively. d1 (t) and d2 (t) are external disturbances which can be used to describe the unmodeled terms such as the cross wind, tire rolling resistance, and so on. Fyf and Fyr are the front and rear tire lateral forces, respectively. ∆Mz is the external yaw moment, which is generated with the longitudinal tire force difference between the left and right side wheels. With the small front-wheel steering angle assumption, ∆Mz can be written as,

∆Mz =

(−1)i Fxi ls ,

(2)

1

where Fxi is the longitudinal force of the ith tire, ls is half of the track width. Supposing that the right tire slip angle is identical to the left one, only front tire slip angle αf and rear tire slip angle αr are used in the paper. The front and rear tire lateral forces Fyf , Fyr are functions of tire slip angles and can be modelled as, Fyf = Cf αf , Fyr = Cr αr ,

(3)

where Cf,r are the front and rear tire cornering stiffness values, αf,r are the tire slip angles which can be calculated as lf r vy lr r vy αf = δ − − , αr = − , (4) vx vx vx vx with δ being the front wheel steering angle. Denote x = [vy r]T and u = [δ ∆Mz ]T , based on (3) and (4), the vehicle model (1) can be rewritten in the statespace form as x(t) ˙ = Ax(t) + Bu(t) + d(t),

(5)

with 

2. SYSTEM MODELLING AND PROBLEM FORMULATION 2.1 Vehicle Model A schematic diagram of a full-vehicle model is shown in Figure 1. The most special features of a FWIA electric vehicle is that an in-wheel (or hub) motor is equipped in each wheel. An external yaw moment can be easily generated to regulate the vehicle yaw and lateral motions since each wheel torque can be independently controlled. The paper mainly focus on vehicle lateral dynamics, therefore vx is assumed to be constant. Assuming the front wheel

4 �

 Cf + Cr Cr lr − Cf lf −vx +  − mvx  mvx  A=  Cr lr − Cf lf Cf lf2 + Cr lr2  , − Iz vx Iz vx   Cf � � 0  d1  . B =  Cml 1  , d = f f d2 Iz Iz

Remark 1 : Small front wheel steering angle assumption is used in the vehicle modelling. This is because the front wheel steering angle is usually small at high speed. Note that the vehicle motion control is more necessary when the vehicle speed is high. The small front wheel steering angle assumption facilitates the controller design and thus is beneficial for the real-time implementation purpose. A robust controller will be designed to attenuate the effects of unmodeled dynamics and disturbance.

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2.2 Vehicle Modelling with Parameter Uncertainties And Actuator Faults

Thus the vehicle model (5) can be rewritten as

Due to the change of road conditions and vehicle states, the tire cornering stiffness is always varying and unknown. Thus, the cornering stiffness can be presented as (6) Cf = C0f + λf C˜f , Cr = C0r + λr C˜r ,

Based on the definitions of ∆A and ∆B, we have

where λf,r are time-varying parameters and satisfy |λi | < 1, (i = f, r), C0f and C0r are the nominal values of Cf and Cr , respectively. The vehicle mass m may change due to the payload change. Assuming m is bounded by its minimum value mmin and its maximum value mmax , then 1/m can be represented by 1 1 1 = (7) + λm , m m0 m ˜ where m0 is the nominal value of m, λm is a unknown parameter satisfying |λm | ≤ 1, and 1/m ˜ is the maximal value of |1/m − 1/m0 |. Similarly, assuming Iz is bounded by its minimum value Iz min and its maximum value Iz max , 1/Iz can be represented by 1 1 1 = + λIz , (8) Iz Iz0 I˜z where Iz0 is the nominal value of Iz , |λIz | ≤ 1, and 1/I˜z is the maximal value of |1/Iz − 1/Iz0 |. Hence, we can get 2mmax mmin 2Iz max Iz min m ˜ = , I˜z = . mmax − mmin Iz max − Iz min Since |λm,f | ≤ 1, based on (6) and (7) the varying Cf /m can be further written as, C0f λf C˜f λm C0f λm λf C˜f Cf = + + + m m0 m ˜ m0 m ˜ C0f + λ1 ϕ1 , = m0 where λ1 is an unknown parameter satisfying |λ1 | ≤ ˜ + C˜f /m0 + C˜f /m. ˜ Similarly, we have 1, ϕ1 = C0f /m Cr /m = C0r /m0 + λ2 ϕ2 , Cf /Iz = C0f /Iz0 + λ3 ϕ3 , and Cr /Iz = C0r /Iz0 + λ4 ϕ4 , where λi , (i = 2, 3, 4) are unknown parameters and satisfy |λi | ≤ 1, ϕ2 = C0r /m ˜ + ˜ ϕ3 = C0f /I˜z + C˜f /Iz0 + C˜f /I˜z , and C˜r /m0 + C˜r /m, ϕ4 = C0r /I˜z + C˜r /Iz0 + C˜r /I˜z . So we have A = A0 + ∆A, B = B0 + ∆B. (9) with   C0f + C0r C0r lr − C0f lf −vx +  − m0 vx  m0 vx , A0 =  2 2  C0r lr − C0f lf  C0f lf + C0r lr − Iz0 vx Iz0 vx   C0f 0   B0 =  Cm0l 1 , 0f f

 Iz0 Iz0 λ1 ϕ1 + λ2 ϕ2 λ2 ϕ2 lr − λ1 ϕ1 lf  − vx vx ∆A =   λ4 ϕ4 lr − λ3 ϕ3 lf λ3 ϕ3 lf2 + λ4 ϕ4 lr2 − vx vx   λ1 ϕ1 0 λIz  . ∆B =  λ3 ϕ3 lf I˜z



 , 

x(t) ˙ = (A0 + ∆A)x(t) + (B0 + ∆B)u(t) + d(t). [∆A ∆B] = HΛ1 [E1 E2 ],

(10) (11)

where, H=

�

� ϕ1 ϕ2 0 0 0 , 0 0 ϕ3 ϕ4 1

Λ1 = diag {λ1 , λ2 , λ3 , λ4 , λIz } , � �T −1/vx −1/vx −lf /vx lr /vx 0 , E1 = −lf /vx lr /vx −lf2 /vx −lr2 /vx 0 �T � 1 0 lf 0 0 . E2 = 0 0 0 0 1/I˜z The performance of an FWIA vehicle heavily depends on the proper operations on the in-wheel motors, the unwanted steering effects caused by the actuator fault in the active steering system can also jeopardize the vehicle motion. Thus the actuator faults should also be considered in the controller design. The actuator faults can be modeled as u(t) = ηud (t),

(12)

where ud is the desired control signal generated by the controller, and η = diag{η1 , η2 } with ηi being unknown parameters. Assuming ηi is bounded by its minimum value ηi min and its maximum value ηi max , η can be represented by η = ηm + Λ2 η¯,

(13)

where ηmax − ηmin ηmax + ηmin , η¯ = , 2 2 = diag{η1 max , η2 max } and ηmin = diag{η1 min ,

ηm = with ηmax η2 min }.

Remark 2 : To avoid computational burden and possible design conservation, the number of the uncertainty parameters in matric in (11) should be small. As the road conditions are usually uniform to the front and rear wheels, it is reasonable to assume that the road coherent coefficients for both tires are identical, which means that λf = λr can be assumed. With the above assumption, we have λ1 = λ2 and λ3 = λ4 after some calculation. In addition, the vehicle yaw inertia is approximately proportional to the vehicle mass, i.e. λm = λIz can be assumed, which indicates we can further have λ1 = λ3 . Since λ1 = λ2 and λ3 = λ4 can be obtained with the assumption that the road coherent coefficients are identical for both front and rear tires, we have λ1 = λ2 = λ3 = λ4 . Therefore, ∆A and ∆B can be rewritten as   −λ1 (ϕ1 + ϕ2 ) λ1 (ϕ2 lr − ϕ1 lf )   vx vx  ∆A =  2 2  λ1 (ϕ4 lr − ϕ3 lf ) −λ1 (ϕ3 lf + ϕ4 lr )  , vx  λ1 ϕ1 0 λIz  , ∆B =  λ1 ϕ3 lf I˜z

vx



which means that the matrices in (11) can be rewritten as, 1167

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where C2 = diag{1, 1}. Therefore, the vehicle model can be rewritten as,  x(t) ˙ =(A0 + ∆A)x(t) + (B0 + ∆B)u(t) + d(t)    y(t) =C x(t) 1 . (16) z(t) =C  2 x(t)   u(t) =ηur (t)

� � � λ1 0 0 1 0 0 H= , Λ1 = 0 λ2 0 , 0 1 1 0 0 λ3 � � −(ϕ1 + ϕ2 )/vx (ϕ4 lr − ϕ3 lf )/vx 0 E1 = , (ϕ2 lr − ϕ1 lf )/vx −(ϕ3 lf2 + ϕ4 lr2 )/vx 0 � � ϕ1 ϕ3 lf 0 . E2 = 0 0 1/I˜z �

Remark 3 : By applying the conclusion in Remark 2, the number of uncertainty parameters in ∆A has been reduced from 4 to 1, thus design conservation can be avoided. Note that λf and λm may actually approximately equal to λr and λIz , respectively. This means that modelling error will be introduced if the conclusions in Remark 2 are used in the controller design. Note that the two assumptions in Remark 2 facilitate the controller design and reduce control conservativeness. Robust H∞ controller will be designed to attenuate the effects of disturbance and unmodeled dynamics.

The control objective is to design a robust FTC dynamic output-feedback controller to generate a control signal u(t), such that the system in equation (16) is asymptotically stable and has the following H∞ disturbance attenuation performance in the presence of parameter uncertainties, actuator faults and an external disturbance � t � t z T (t)z(t)dt ≤ γ 2 dT (t)d(t)dt, (17) 0

0

where γ is the prescribed attenuation level.

3. FAULT-TOLERANT CONTROLLER DESIGN

Remark 4 : Generally speaking, the parameter ηi in the fault model (12) is supposed to vary in the range of [0, 1]. However, if it is so, the solution of the controller design will be infeasible. This is because ηi = 0 means that none of the actuator will generate any effective control efforts or the single actuator in the system is in complete failure, the system will be totally uncontrollable in those cases. Note that for over-actuated system ηi > 0 will always hold even if some of actuators completely fails. For instance, the external yaw moment in a FWIA electric vehicle is generated by torque difference between the left and right sides in-wheel motors. If one of the in-wheel motors completely fails or the motor torque stuck at a fixed level, vehicle motion can still controlled with the remaining healthy in-wheel motors. For the in-wheel motor faults, if only one in-wheel motor is in fault, we can assume that η2 varies in the range of [0.75, 1] based on the external yaw moment definition (2), if two in-wheel motors are simultaneously in fault, we can assume η2 varies in the range of [0.5, 1]. As hardware redundancy design, for example the dual-motor steering systems, is usually adopted to improve the robustness of the active steering systems [Zheng,B. et al (2005), Zheng, B. and Anwar,S. (2008)], we can assume η1 varies in the range of [0.5, 1].

In order to achieve the desired control performance with only the yaw rate measurement, we propose the following dynamic output feedback controller � xˆ˙ (t) = Ac x ˆ(t) + Bc y(t) , (18) ˆ(t) + Dc y(t) ud (t) = Cc x where x ˆ(t) is the states of the controller, Ac , Bc , Cc , and Dc are the control gains to be designed. Substituting (14) and (18) into the actuator fault model (12), one obtains � ˆ(t) + Bc C1 x(t) xˆ˙ (t) = Ac x , (19) u(t) = ηCc xˆ(t) + ηDc C1 x(t) Then, the closed-loop system can be written as, � ¯ 1 E)¯ ¯ x(t) + Bd(t) ¯ x ¯˙ (t) = (A¯ + HΛ , (20) z = C¯ x ¯ where � � � � � � x ¯= I , H ¯ = H , x¯ = , B xˆ 0 0 � � A0 + B0 ηDc C1 B0 ηCc A¯ = , (21) Bc C1 Ac ¯ = [E1 + E2 ηDc C1 E2 ηCc ] , E C¯ = [C2 0] .

2.3 Problem Statement

In order to deal with the actuator faults, parameter uncertainties and external disturbances, we introduce the following lemma.

The vehicle yaw rate can be measured with vehicle onboard sensors such as gyroscope, or it can be synthesized from accelerometers. However, the vehicle slip angle or lateral speed cannot be directly measured with cheap sensors currently. Thus, only the vehicle yaw rate is taken as the measured output in this study, and the measured output y(t) is defined as y(t) = C1 x(t), (14) where C1 = [0 1]. To the improve handling performance of the FWIA electric ground vehicle, the vehicle yaw rate and slip angle should be simultaneously controlled. Therefore, the controlled output can be defined as z(t) = C2 x(t), (15)

Lemma 1 [Zhang, H. et al (2010)]: Let Y = Y T , Γ and Ψ be ˜ satisfies the real matrices with proper dimensions, and N T Λ Λ < I, then the following conditions: Y + ΓΛΨ + ΨT ΛT ΓT < 0, (22) holds if and only if there exists a positive scalar ε > 0 such that Y + ǫΓΓT + ǫ−1 ΨT Ψ < 0. (23) Theorem 1. Given positive constants γ, the closed-loop system in (20) satisfy the H∞ performance index (17), if there exists a symmetric positive definite matrix P and positive scalars ǫ1 and ǫ2 , such that   ΨT1 Y1 Γ1 ¯ 1 =  ∗ −ǫ−1 I 0  < 0, Π (24) 2 ∗ ∗ −ǫ2 I

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where

Y1

Γ1 Ψ1 A¯0 J

  ¯ 1 E} ¯ PB ¯ C¯ T sys{A¯T P + P HΛ  ∗ −γI 0  ∗ ∗ −γI     ¯ ¯ C¯ T PH sys{A¯T P } P B ¯ 0 0] = ∗ −γI 0  +  0  Λ1 [E 0 ∗ ∗ −γI   E¯ T � T � ¯ P 0 0 < 0. +  0  Λ1 H 0

 ¯ C¯ T P H ¯ E ¯0T  sys{A¯T0 P } P B  ∗ −γI 0 0 0    ∗ ∗ −γI 0 0 , =  ∗ ∗ ∗ −ǫ−1 0  1 I ∗ ∗ ∗ ∗ −ǫ1 I � T � T = J P 0 0 0 E2T , = [ η¯L 0 0 0 0 ] , � � A0 + B0 ηm Dc C1 B0 ηm Cc , = Bc C1 Ac � � B0 = , 0

L = [ Dc C1 Cc ] , ¯ E0 = [ E1 + E2 ηm Dc C1

E2 ηm Cc ].

Proof. Define a Lyapunov function for the system in (20) as V =x ¯T (t)P x¯(t),

(25)

where P is a symmetric positive matrix. Evaluating the time derivative of the above Lyapunov function, we obtain V˙ =x ¯˙ T (t)P x¯(t) + x ¯T (t)P x¯˙ (t) � �T ¯ 1 E)¯ ¯ x(t) + Bd(t) ¯ = (A¯ + HΛ Px ¯(t) � � T ¯ ¯ ¯ ¯ x(t) + Bd(t) +x ¯ (t)P (A + HΛ1 E)¯ � � ¯ 1E ¯+E ¯ T Λ1 H ¯ T x¯(t) =¯ xT (t) A¯T P + P A¯ + HΛ ¯T P x ¯ ¯(t) + x¯T (t)P Bd(t). + dT (t)B

(26)

(27)

Π=

 1 ¯T ¯ ¯ C C PB , γ T ¯ −γI B P

Θ+

(32)

It follows from Lemma 1 that the condition is equivalent to,   Y1 ǫ2 Γ1 ΨT1  ∗ −ǫ2 I 0  < 0. (33) ∗ ∗ −ǫ2 I

The above inequality indicates that (24) is to equivalent to Π < 0, which means that the condition (24) ensures that the closed-loop system (20) is asymptotically stable and the H∞ performance (17) can be satisfied. The proof is finished.

Note that there are nonlinear terms involved in (24), and these nonlinear terms cannot be removed by the change of variable which is usually used in the state-feedback controller designs. Since the matrix P is nonsingular, we can partition P and its inverse as � � � � R N S M −1 P = , P = . NT W MT V Without loss of generality, we can assume that both M and N are full rank � Let � � � matrices. S I I R , F . = F1 = 2 MT 0 0 NT

where 

As ΛT1 Λ1 < I, it follows from Lemma 1 that the condition Π < 0 is equivalent to  ¯ C¯ T ǫ1 P H ¯ E ¯T  sys{A¯T P } P B  ∗ −γI 0 0 0    (30) ∗ ∗ −γI 0 0  < 0,   ∗ ∗ ∗ −ǫ1 I 0  ∗ ∗ ∗ ∗ −ǫ1 I where ǫ1 are a positive constant. Note that there are ¯ based on the matrices uncertainties exist in A¯ and E, ¯ ¯ can be rewrite as definitions (13) and (21), A and E ¯=E ¯0 + E2 Λ2 η¯L, A¯ = A¯0 + JΛ2 η¯L, E (31) Therefore, the inequality (30) is equivalent to Y1 + Γ1 Λ2 Ψ1 + ΨT1 ΛT2 ΓT1 < 0,

¯ 1E ¯ +E ¯ T Λ1 H ¯ T P and by Denoting Θ = A¯T P + P A¯ + P HΛ 1 T T adding γ z (t)z(t) − γd (t)d(t) with γ > 0 to both sides of (26), we have 1 V˙ + z T (t)z(t) − γdT (t)d(t) γ 1 =V˙ + z T (t)z(t) − γdT (t)d(t) γ � 1 ¯ = x¯T (t)(Θ + C¯ T C)x(t) γ � ¯ x ¯ ¯(t) + x¯T (t)P Bd(t) − γdT (t)d(t) +dT (t)BP � � �T � x ¯(t) x ¯(t) Π , = d(t) d(t)

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(28)

It can be observered that if Π < 0, then the controlled system (20) is asymptotically stable and the H∞ performance defined in (17) can be satisfied. Using Schur complement, Π < 0 is equivalent to   ¯ 1 E} ¯ PB ¯ C¯ T sys{A¯T P + P HΛ  (29) ∗ −γI 0  < 0. ∗ ∗ −γI

Then, we can get the following Theorem. Theorem 2. Given positive constants γ, the closed-loop system (20) satisfy the H∞ performance index (17), if ˆ B, ˆ there exists positive scalars ǫ1 , ǫ2 , general matrices A, ˆ ˆ C, D, symmetric positive matrix R and S such that   Π1 Ω1 ΘT1 ¯ 1 =  ∗ −¯ ˜2  < 0, Ξ (34) ǫs E ∗ ∗ −¯ ǫs where

Based on the definition of Θ, the above inequality can be further written as

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�

� − ǫ−1 I 0 s ǫ¯s = , s = 1, 2, ∗ −ǫs I

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  sys(L) Aˆ + K I SC2T  ∗ ˆ 1 ) R C2T  sys(RA0 + BC , Π1 =   ∗ ∗ −γI 0  ∗ ∗ ∗ −γI   T T T ˆ H SE1 + C ηm E2 RH E T + C T D ˆ T ηm E T   2 , 1 1 Ω1 =   0 0 0 0 � � T B0 B0T RT 0 0 Θ1 = ˆ 1 0 0 , η¯Cˆ η¯DC � � ˜2 = 0 0 , E E2 0 ˆ L =A0 S + B0 ηm C,

and

F1T P A¯0 F1 = F2T A¯0 F1 � � ˆ 1 A0 S + B0 ηm Cˆ A0 + B0 ηm DC . = ˆ 1 Aˆ RA0 + BC

Therefore the condition (24) is equivalent to (34). This finishes the proof.

ˆ 1. K =A0 + B0 ηm DC with Aˆ = R (A0 + B0 ηm Dc C1 ) S + N Bc C1 S      + RB0 ηm Cc M T + N Ac M T  ˆ = RB0 ηm Dc + N Bc B   ˆ C = Dc C1 S + Cc M T   ˆ D = Dc

.

(36)

Proof. Performing a congruence transformation with ¯ 1 , we Υ1 = diag{F1 , I, I, I, I, I, I} to the matrix Π have   ˜1 ˜T Ψ Y˜1 Γ 1 ˜ 1 = ΥT Π ¯  ∗ −ǫ−1 I 0  , Π (37) 1 1 Υ1 = 2 ∗ ∗ −ǫ2 I where  ¯ F T C¯ T F T P H ¯ FTE ¯T  Φ1 F1T P B 1 1 1 0  ∗ −γI 0 0 0    ∗ −γI 0 0 , Y˜1 =  ∗ ∗ −1 ∗ ∗ −ǫ1 I 0  ∗ ∗ ∗ ∗ −ǫ1 I  T  F1 P J  0   ˜1 =  Γ  0 ,  0  E2 ˜ 1 = [ η¯LF1 0 0 0 0 ] , Ψ Φ1 = F1T A¯T0 P F1 + F1T P A¯0 F1T . Based on the definition of F1 and F2 , we have, P F1 = F2 , � � � � S I R I = , F1T P F1 = I S RS + N M T R � �� � � � ¯= I 0 I = I , F1T P B R N 0 R � �� T� � T� S M C2 SC2 , F1T C¯ T = = I 0 0 C2T � �� � � � ¯ = I 0 H = H , F1T P H R N 0 RH � � �� T S M E1 + (E2 ηm Dc C1 )T T ¯T F1 E0 = I 0 (E2 ηm Cc )T � � SE1T + Cˆ T ηm E2T = ˆ T ηm E T , E1T + C1T D 2

� � � S I � LF1 = Dc C1 Cc T M 0 � � ˆ ˆ = C DC1 � � � � �� B0 I 0 B0 T = F1 P J = 0 RB0 R N

It can be observed from (36) that matrices M and N are required to calculate Ac , Bc , Cc , and Dc . Since P P −1 = I, we can get M N T = I − SR. Making singular value decomposition for I − SR, we can get M and N , then the matrices Ac , Bc , Cc , and Dc can be calculated as,  ˆ Dc = D     −T  Cc = (Cˆ − Dc C1 S)(M ) −1 ˆ . (40) Bc = N (B − RB0 ηm Dc )   −1 ˆ −T  A = N [ A − R(A + B η D C )S]M 0 0 m c 1  c   − Bc C1 SM −T − N −1 RB0 ηm Cc Note that the matrices Ac , Bc , Cc , and Dc are calculated off-line. Thus the proposed controller has low on-line computational complexity and can be easily implemented. 4. SIMULATION STUDIES In this section, simulation results with Carsim and Simulink are presented to illustrate the advantage of the proposed control method. The vehicle parameters are listed in Table 1. In the lateral stability control, the vehicle slip angle is required to be as small as possible, the desired lateral speed is given as zero. The desired vehicle yaw rate can be generated from the drivers steering angle and vehicle longitudinal speed as [Du, H. et al (2011)] vx rr = δ(t), l(1 + kus vx2 ) where l = lf + ls is the distance between the front and rear axles, kus = 0.002 is the stability factor. The vehicle speed was chosen as 30m/s. For the actuator faults, we assume η1 ∈ [0.5, 1], η2 ∈ [0.75, 1]. With the selected parameters, Table 1. Vehicle parameters in the simulation Parameters m Iz ls lf lr Cf Cr

Nominal value 1500 kg 2000 kg · m2 0.8 m 1.4 m 1.4 m 45000 N/rad 42000 N/rad

Uncertainty ± 20% ± 20% None None None ± 30% ± 30%

the matrices Ac , Bc , Cc , and Dc in the dynamic output feedback controller (18) can be calculated and are given as � � � � 0.016 1.090 −4037.2 698.7 −4 , Cc = 10 × Ac = −0.019 −0.326 116.5 −19.7 Bc = 108 × [5.4695 − 0.1582]T ,

Dc = [−0.1053 − 0.0719]T . 1170

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20

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10 Controlled (no fault) Reference Controlled (with fault) Uncontrolled (no fault)

8

15

6

10

4 Yaw rate (deg/s)

Steering

sw

(deg)

5 0 −5

2 0 4

−2

3.5

−4

−10 −15 −20 0

1

2

3 Time (s)

4

5

−6

3

−8

2.5 1.3

−10 0

6

Fig. 2. Hand-wheel steering angle in the single-lane change simulation.

1.5

1.6 2

3 Time (s)

4

5

6

Fig. 5. Vehicle yaw rates in the single-lane change simulation. 1.5

0.8 No fault With fault (desired value) With fault (actual value)

0.6

1.4

1

−0.1 −0.2

1

−0.3

0.4

0

0.6

−0.2 −0.4

−0.4 1.4

Beta (deg)

Steeringwheel (deg)

0.5 0.2

1.6

0

−0.5 0.4

Controlled (no fault) Reference Controlled (with fault) Uncontrolled (no fault)

−1 −0.6 −0.8 0

0.2 1.4

1.6 1

2

3 Time (s)

4

5

−1.5 0

6

Fig. 3. Front-wheel steering angles in the single-lane change simulation.

1

2

3 Time (s)

4

5

6

Fig. 6. Vehicle slip angles in the single-lane change simulation. 8

50

Controlled (no fault) Reference Controlled (with fault) Uncontrolled (no fault)

7

No fault With fault (desired value) With fault (actual value)

6 5

∆ Mz (N ⋅ m)

Y (m)

0

4 3

0.8 0.7

2 −50

0.6

1 0.5 60

65

70

0 −1 0 −100 0

1

2

3 Time (s)

4

5

6

Fig. 4. External yaw moments in the single-lane change simulation. In the simulation, the vehicle is controlled to make a singlelane change at a high speed. The longitudinal speed is 30m/s in the simulation. The hand-wheel steering angle is shown in Figure. 2. Both the steering faults and in-wheel motor faults were applied as follows: (1) At 1.5 s, the loss-of-effectiveness fault is introduced to the active steering system, this fault makes the frontwheel steering angle become δ = 0.6δd , where δd is the desired steering angle. (2) At 2.0s, the loss-of-effectiveness fault is introduced to the front-right in-wheel motor, the fault makes the in-wheel motor torque decrease to half of its desired value. (3) At 3.0s, the fault which makes the rear-left in-wheel motor torque stuck at a fixed value is introduced. This fault makes the motor torque stuck at 40Nm after 3.0s.

20

40

60

80 X (m)

100

120

140

160

Fig. 7. Vehicle trajectories in the single-lane change simulation. Vehicle front-wheel steering angle and external yaw moments are plotted in Figure.3 and Figure.4, respectively. From Figure.3, it can be seen that as the faults occurs at 1.5s, the front-wheel steering angle can only become 60% of the desired steering angle, then the desired steering angle (generated by the controller) automatically rises to adjust the faults occurred. The similar trend can be seen in Figure.4. When the fault occurs on the in-wheel motor at 2s and 3s, the desired external yaw moments automatically adjusts the faults to regulate the vehicle dynamics, respectively. Vehicle yaw rate control results are plotted in Figure.5. In order to better show the performance of the proposed FTC controller, the states of an uncontrolled vehicle with the same hand-wheel steering input are also given in the figure. It can be observed that the vehicles controlled by the proposed FTC controller could track the desired yaw rate well, while the yaw rate of the uncontrolled vehicle

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deviates from the desired value. We can also see from this figure that the yaw rate of the controlled faulty vehicle was very close to that of the controlled healthy one, which proves that the vehicle under the proposed FTC controller is robust enough to the actuator faults. Figure. 6 shows the vehicle slip angles. It can be observed that although the slip angle of the controlled vehicle does not converge to zero, it has been tremendous reduced, compared to the slip angle of the uncontrolled vehicle. The results plotted in Figure.5 and Figure. 6 indicate that the proposed control method simultaneously improve the vehicle handling and stability. The vehicle global trajectories are shown in Figure.7, where one can observe again that the proposed FTC controller ensures the vehicle tracking performance no matter which kind of faults are applied. 5. CONCLUSION The FTC control problem of a FWIA electric ground vehicle with AFS is investigated in this paper. Both the in-wheel motor faults and steering motor faults are considered. Since the costs of currently available sensors for vehicle slip angle measurement are usually high, a robust H∞ dynamic output-feedback controller is designed to control the vehicle without using the vehicle slip angle information. Parameter uncertainties and actuator faults are simultaneously considered in the controller design, making the controlled vehicle under the proposed FTC controller robust to the tire force modelling error and actuator faults. Simulation results based on a high-fidelity, CarSim, full-vehicle model show the effectiveness of the control approach. REFERENCES Aouaouda, S., Chadli, M. and Karimi, H. R.(2014a). Robust static output-feedback controller design against sensor failure for vehicle dynamics, IET Control Theory and Applications, Volume 8, Issue 9, pp. 728-737. Aouaouda, S., Chadli, M., Boukhnifer, M. and Karimi, H. R.(2014b). Robust fault tolerant tracking controller design for vehicle dynamics: A descriptor approach, Mechatronics, Doi:10.1016/j.mechatronics.2014.09.011. Doumiati, M., Sename,O., Dugard, L., Martinez-Molina, J. Gaspar, P. and Szabo,Z(2013). Integrated vehicle dynamics control via coordination of active front steering and rear braking, European Journal of Control, Vol. 19, Issue 2, pp.121-143. Du, H.P., Zhang, N. and Dong, G. M.(2010). Stabilizing vehicle lateral dynamics with considerations of parameter uncertainties and control saturation through robust yaw control, IEEE Transactions on Vehicular Technology, Vol. 59, No. 5, pp.2593-2597. Du, H., Zhang, N. and Naghdy, F.(2011). Velocitydependent robust control for improving vehicle lateral dynamics, Transportation Research Part C, , Vol. 19, pp. 454-468. Geng, C., Mostefai, L., Dena¨ı,M., and Hori, Y.(2009). Direct yaw moment control of an in-wheel motored electric vehicle based on body slip angle fuzzy observer, IEEE Transactions on Industral Electronics, Vol. 56, Issue 5, pp.1411-1417.

Saifia, D., Chadli, M., Karimi, H. R. and Labiod, S.(2015). Fuzzy control for electric power steering system with assist motor current input constraints, Journal of the Franklin Institute, Vol. 352, Issue 2, pp.562-576. Shino, M. and Nagai, M.(2003). Independent wheel torque control of small-scale electric vehicle for handling and stability improvement, JSAE Review, Vol. 24, Issue 4, pp. 449-456. Wu, F.K., Yeh,T.J. and Huang, C. (2013). Motor control and torque coordination of an electric vehicle actuated by two in-wheel motors, Mechatronics, Vol. 23, Issue 1, pp.46-60. Wang, R., Chen, Y., Feng, D. Huang, X. and Wang, J. (2011). Development and performance characterization of an electric ground vehicle with independently actuated in-wheel motors, Journal of Power Sources, Vol. 196, Issue 8, 3962-3971. Wang, R. and Wang, J.(2012). Fault-tolerant control for electric ground vehicles with independently-actuated inwheel motors, ASME Transactions Journal of Dynamic Systems, Measurement, and Control, Vol. 134, Issue 2, pp.021014. Wang, R., Zhang, H. and Wang, J.(2014). Linear parameter-varying controller design for four wheel independently-actuated electric ground vehicles with active steering systems, IEEE Transactions on Control Systems Technology, Vol. 22 , Issue 4, pp. 1281-1296. Yim, S.(2014). Fault-tolerant yaw moment control with steer- and brake-by-wire device, International Journal of Automotive Technology, Vol. 15, No. 3, pp. 463-468. Zheng, B., Altemare,C. and Anwar, S.(2005). Fault tolerant steer-by-wire road wheel control system, Proceedings of American Control Conference, pp. 1619-1624. Zheng, B. and Anwar, S.(2008). Fault-tolerant control of the road wheel subsystem in a steer-by-wire system, International Journal of Vehicular Technology, vol. 2008, pp. 1-8. Zhang, H., Shi, Y. and Mehr, Y.S.(2010). Robust energyto-peak filtering for networked systems with timevarying delays and randomly missing data ,” IET Control Theory Application , Vol. 4, No. 12, pp. 29212936. Zhang,H., Shi,Y. and Mehr,A.S. (2011). Robust static output feedback control and remote PID design for networked motor systems, IEEE Transactions on Industrial Electronics, Vol. 58,No.12, pp. 5396-5405. Zhang,H., Shi,Y. and Wang,J. (2013a). Observer-based tracking controller design for networked predictive control systems with uncertain Markov delays, International Journal of Control, Vol.86, No.10, pp. 1824-1836. Zhang, H., Shi,Y. and Mu, B. (2013b). Optimal H∞ based linear-quadratic regulator tracking control for discrete-time Takagi-Sugeno fuzzy systems with preview actions, Journal of Dynamic Systems, Measurement, and Control, Vol. 135, No. 4, pp. 044501-1-044501-5. Zhang, Y. and Jiang, J. (2008). Bibliographical review on reconfigurable fault-tolerant control systems, Annual Reviews in Control, Vol. 32, no. 2, pp. 229-252.

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