IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 5, NO. 6, NOVEMBER 2018 1121

Disturbance Observer Based Control for Four WheelSteering Vehicles With Model ReferenceShuyou Yu, Member, IEEE, Jing Wang, Yan Wang, Hong Chen, Senior Member, IEEE

Abstract—This paper presents a disturbance observer basedcontrol strategy for four wheel steering systems in order to im-prove vehicle handling stability. By combination of feedforwardcontrol and feedback control, the front and rear wheel steeringangles are controlled simultaneously to follow both the desiredsideslip angle and the yaw rate of the reference vehicle model.A nonlinear three degree-of-freedom four wheel steering vehiclemodel containing lateral, yaw and roll motions is built up, whichalso takes the dynamic effects of crosswind into consideration.The disturbance observer based control method is provided tocope with ignored nonlinear dynamics and to handle exogenousdisturbances. Finally, a simulation experiment is carried out,which shows that the proposed four wheel steering vehiclecan guarantee handling stability and present strong robustnessagainst external disturbances.

Index Terms—Disturbance observer based control, four wheelsteering, handling stability, model reference.

I. INTRODUCTION

W ITH the rapid development of automobile industry,handling stability related to vehicle safety has attracted

more attention. The four wheel steering (4WS) vehicles havebeen studied in order to improve the maneuverability ofvehicle at low speed and enhance the stability at high speed.As rear wheels participate in steering process, the transientresponse of a vehicle can be improved [1], [2].

Traditional 4WS vehicles mainly refer to automobiles withactive rear wheel steering systems, and their control typesinclude both feedforward control and feedback control [3].Early 4WS vehicles intend to achieve zero sideslip angle toreduce the phase difference between lateral acceleration andyaw rate. This kind of 4WS vehicles usually use a feedforwardcontrol method. Sano proposes a ratio between rear wheels andfront wheels, which can guarantee a zero sideslip angle duringsteady steering [4]. This method is easy to implement, but it

Manuscript received June 17, 2016; accepted July 5, 2016. The work wassupported by the National Natural Science Foundation of China (61573165,61520106008, 61703178). Recommended by Associate Editor Yungang Liu.(Corresponding author: Shuyou Yu.)

Citation: S. Y. Yu, J. Wang, Y. Wang, H. Chen. “Disturbance observer basedcontrol for four wheel steering vehicles with model reference,” IEEE/CAA J.of Autom. Sinica, vol. 5, no. 6, pp. 1121−1127, Nov. 2018.

S. Y. Yu and H. Chen are with the State Key Laboratory of AutomotiveSimulation and Control, and with the Department of Control Science andEngineering, Jilin University, Changchun 130012, China (e-mail: {shuyou,chenh}@jlu.edu.cn).

J. Wang is with Bosch Automotive Products (Suzhou) Co., Ltd, Suzhou215021, China (e-mail: wangjing13@mails.jlu.edu.cn).

Y. Wang is with Huachen Auto Group, Shenyang 110141, China (e-mail:wyan14@mails.jlu.edu.cn)

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JAS.2016.7510220

increases the time it takes for yaw rate and lateral accelerationto reach steady states. Thus, it does not perform well duringtransient motion. For 4WS vehicles using the feedback controlmethod, two control strategies are presented. One also focuseson reaching a zero sideslip angle, the other focuses on trackingthe reference yaw rate. The feedback control method has thecharacteristic of fast response, which can effectively reducethe effect of external disturbances. Quite a few modern 4WScontrol methods are established based on it. In [5], a setof linear maneuvering equations representing 4WS vehicledynamics is used, which formulates the multi-objective vehiclecontrollers as a mixed H2/H∞ synthesis. A fuzzy controllerbased on the state feedback and the sliding-mode controlmethods is proposed for high-speed 4WS vehicles in [6].Considering the uncertain problems for 4WS control, robustcontrol schemes have been adopted in [7]−[10].

The development of the steer-by-wire (SBW) technique hasmade it possible for front and rear wheels to actively steer,simultaneously [11]. In this paper, motivated by this idea, thestrategy of model reference control is proposed in order toachieve both the desired sideslip angle and the yaw rate. Withthe combination of feedforward control and feedback control,front wheel active steering and rear wheel active steering areused to achieve the desired performance of a reference vehiclemodel. In [12], a similar model reference control strategy hasalso been used, where a H∞ output feedback controller isdesigned. Its simulation results show that the proposed 4WSvehicle can improve vehicle handling and stability, but it onlyfocuses on a linear 4WS model. In this paper, a nonlinearthree degree-of-freedom (3DOF) 4WS vehicle model whichcontains the dynamic effects of crosswind is built up. Themodel will be used to verify the effectiveness of the proposedcontrollers. In order to implement it easily, a relatively simplevehicle model, a linear two degree-of-freedom (2DOF) 4WSvehicle model, is used in the controller design stage. Whilethe feedforward and feedback controllers we designed do notconsider ignored nonlinear dynamics and external disturbancesexisting in 4WS system, a disturbance observer based control(DOBC) method is provided to attenuate disturbances.

The rest of this paper is organized as follows. 4WS vehiclemodels are built up in Section II. Section III describes the mainresults for 4WS control strategy. Disturbance observer basedcontrol scheme is illustrated in Section IV. The simulationresults are carried out in Section V. Section VI concludes thepaper with a short summary.

II. 4WS VEHICLE MODELS

In this section, 4WS vehicle models will be of interest. First

1122 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 5, NO. 6, NOVEMBER 2018

a 3DOF 4WS vehicle model with a nonlinear tyre model is setup, which also takes the crosswind disturbance into consider-ation. Such a model captures the most relevant characteristicsassociated to lateral and longitudinal stabilization of 4WSvehicle. For simplicity and easy implementation, a linear 4WSvehicle model is used for the design of controllers.

A. Nonlinear 4WS Vehicle Model

A nonlinear 3DOF 4WS vehicle model is shown in Fig. 1.The body frame (x, y, z) is fixed at the vehicle’s center ofgravity (CG). In Fig. 1, β and r are the sideslip angle and theyaw rate of vehicle at CG. m is vehicle mass, ms is sprungmass. hs is the height of sprung mass CG above roll axis, hg isthe height of sprung mass CG above ground. ϕ and ϕ are rollangle and roll velocity, Ix is the roll moment of inertia of thesprung mass about x axis, Iz is the yaw inertia moment aboutz axes, and Ixz is the product of inertia of the sprung massabout x and z axes. Cϕ is roll damping of vehicle suspension,and kϕ is roll stiffness of vehicle suspension. v is the vehiclelongitudinal velocity, a and b are longitudinal distances fromthe front and rear axle to the CG, and L = a + b is thewheelbase of vehicle. Ff and Fr denote the lateral forces offront and rear wheel respectively.

Fig. 1. The diagram of 3DOF vehicle model.

Considering lateral, yaw and roll motions, the 3DOF vehiclemodel of the 4WS system is given by [5], [13]

mv(β + r) + mshsϕ = ΣFy (1)

Iz r − Ixzϕ = ΣMz (2)

Ixϕ− Ixz r = ΣLx (3)

where Fy , Mz and Lx are the sums of the external forces andmoments acting on the vehicle measured at CG along the caraxes.

As crosswind has some influences on dynamics of the 4WSsystem, the following equations are

ΣFy = Ff + Fr + Fw (4)

ΣMz = aFf − bFr + Fwlw (5)

ΣLx = mshsgsinϕ + mshsv(β + r)cosϕ−Cϕϕ− kϕϕ− Fwhw

(6)

where Fw is the crosswind force, hw is the height of crosswindforce acting point above roll axis, and lw is the horizontaldistance from crosswind force acting point to vehicle’s CG.

The magic formula tyre model [14] is adopted here todetermine the lateral force of front and rear wheels

Fi = Dsin{Carctan[B(1− E)αi + Earctan(Bαi)]} (7)

where Fi and αi represent the lateral force of a wheel and itscorresponding slip angle. The parameters B, C, D and E aredetermined by the tyre structure and the tyre vertical load.

Assume that β is small and v varies slowly, the front slipangle αf and the rear slip angle αr can be derived by

αf =−β − ar

v+ δf

αr = −β +br

v+ δr

(8)

where δf and δr are respectively the steering angles of thefront and rear wheels.

The vertical load of each tyre includes both the static loadand the lateral load transfer caused by the vehicle rollingmotion

Fzfl =mgb

2L+4Fzf

Fzfr =mgb

2L−4Fzf

Fzrl =mga

2L+4Fzr

Fzrr =mga

2L−4Fzr.

(9)

The lateral load transfer is given by [5]

4Fzf =ay

Ef(mslrshf

L+musfhuf )+

1Ef

(−kϕfϕ−Cϕf ϕ)

4Fzr =ay

Er(mslfshf

L+musrhur)+

1Er

(−kϕrϕ−Cϕrϕ)(10)

where musf and musr are the front and rear axle loads ofunsprung mass, huf and hur are the heights of unsprungmass CG above the front and rear axles, lfs and lrs are thelongitudinal distances from the front and the rear axles to thesprung mass CG. Ef and Er are the front and rear wheeltracks, Cϕf and Cϕr are the roll damping of the front andrear suspensions, kϕf and kϕr are the roll stiffness of thefront and rear suspensions, respectively.

B. Linear 4WS Vehicle Model

In order to implement easily, it is reasonable to use a rela-tively simple vehicle model in the stage of controller design.Two degree-of-freedom vehicle model contains sideslip andyaw rate, which captures the most relevant dynamics of a 4WSvehicle. As is shown in Fig. 2, the coordinate frame is fixed

YU et al.: DISTURBANCE OBSERVER BASED CONTROL FOR FOUR WHEEL STEERING VEHICLES WITH MODEL REFERENCE 1123

at the vehicle’s CG. According to Newton’s second law, thevehicle dynamic equations can be written as

mv(β + r) = Ff + Fr

Iz r = aFf − bFr.(11)

While the tyre slip angle is small, the front and rear lateraltyre forces vary linearly with their slip angles

Ff = kfαf

Fr = krαr

(12)

Fig. 2. The diagram of 2DOF vehicle model.

where kf and kr are the cornering stiffness of the front andrear wheels. The front slip angle αf and rear slip angle αr

are obtained from (8). Then, (11) can be rewritten as

mv(β + r) = −(kf + kr)β − akf − bkr

vr + kfδf + krδr

Iz r = −(akf − bkr)β − a2kf + b2kr

vr + akfδf − bkrδr.

(13)From (13), the state space representation of the linear 4WS

vehicle model can be expressed as

x = Ax + Bu (14)

where x = [β r]T is the state vector, u = [δf δr]T the inputvector. The coefficient matrices are given as follows

A =

−

kf +kr

mvbkr−akf

mv2 − 1

bkr−akf

Iz−a2kf +b2kr

Izv

B =

[kf

mvkr

mvakf

Iz − bkr

Iz

].

Taking the external disturbances caused by the crosswindinto consideration, (11) can be rewritten as

mv(β + r) = Ff + Fr + Fw

Iz r = aFf − bFr + Fwlw.(15)

In accordance with (14), the state space representation of(15) can be written as

x = Ax + Bfwf + Bu (16)

where wf = Fw/mv is the normalized disturbance input, and

Bf =[

1mvlw

Iz

]. The key parameters of 4WS vehicle model

used in this paper are collected in Table I [13].

TABLE IPARAMETERS OF 4WS VEHICLE MODEL

Model parameters Valuesvehicle mass (m) 1500 kgsprung mass (ms) 1300 kg

roll inertia moment (Ix) 450 kg· m2

yaw inertia moment (Iz) 6000 kg· m2

the product of inertia (Ixz) 1000 kg· m2

longitudinal distance from front axle to CG (a) 1.1 mlongitudinal distance from rear axle to CG (b) 1.4 m

roll stiffness of vehicle suspension (kϕ) 47250 N·/radroll damping of vehicle suspension (Cϕ) 2587.5 N·/(rad/s)

height of sprung mass CG above roll axis (hs) 0.37 mcornering stiffness of front wheel 64000 N/radcornering stiffness of rear wheel 52000 N/rad

III. CONTROL OF 4WSThe performance requirement of a 4WS vehicle is to

improve the vehicle maneuverability at low speed and toenhance vehicle handling stability at high speed. The strategyof 4WS control is proposed in Fig. 3, where the 4WS controlproblem is simplified as a yaw rate and a sideslip angletracking problem. δ∗f is the front wheel angle input givenby the driver. In order to improve steering characteristics,both the front wheel angel δf and the rear wheel angle δr

are chosen as controlled quantities. The desired yaw rate andsideslip angle are provided by the reference model, while thefeedback controller is designed to minimize tracking errors.The feedforward controller proposed here guarantees thattracking errors are not too large, which is derived based onthe reference model as well. Hence the total control inputu is composed of the feedforward control input uf and thefeedback control input ue, that is

u =[δf

δr

]= uf + ue. (17)

Note that crosswind disturbances here are not considered.In the next section, a disturbance observer will be introducedso as to attenuate the effects of crosswind disturbances.

Fig. 3. Model reference control of 4WS vehicle.

A. Reference Model

In order not to change the driver’s perception of vehiclehandling, the steady state value of desired reference yaw rateshould be the same as that of the front wheel steering (FWS)vehicle with the same parameters. Based on vehicle theory,the desired yaw rate can be described as follows [15]:

r∗ =v

L(1 + kv2)1

1 + τrsδ∗f

k =m

L2

(a

kr− b

kf

) (18)

1124 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 5, NO. 6, NOVEMBER 2018

where r∗ represents the desired reference yaw rate, δ∗f thefront wheel angle, k the vehicle stability factor, and τr thetime constant of yaw rate. The desired reference sideslip isalso approximated as a first order inertia transfer function ofthe front wheel angle [15],

β∗ =kβ

1 + τβsδ∗f (19)

where β∗ represents the desired reference sideslip angle, andτβ is the time constant of the sideslip angle. The term kβ is aproportional gain, which is usually a small constant and canbe set to zero in the ideal situation. Thus, kβ = 0 is chosenin this paper.

Define the state vector xd = [β∗ r∗]T and the control inputvector ud = δ∗f . The state space equation of the referencevehicle model can be expressed as

xd = Adxd + Bdud (20)

where

Ad =

[− 1

τβ0

0 − 1τβ

], Bd =

[kβ

τβ

kr

τr

], kβ =0, kr =

11 + kv2

v

L.

B. Feedforward Controller

Applying Laplace transform on the linear 4WS vehiclemodel (14), the transfer function matrix can be obtained

G = (sI −A)−1B. (21)

The role of feedforward controller Cf (s) is to keep thedynamic response of the linear 4WS vehicle model and thereference model the same. If a given front wheel anglegoes through the feedforward controller and the 4WS vehiclemodel, cf. Fig. 3, the desired reference sideslip angle and yawrate could be obtained as

GCf · δ∗f =[β∗

r∗

]. (22)

Substituting (18), (19) and (21) into (22), the feedforwardcontroller can be obtained

Cf =

Izvs+akf L+bmv2

vkf L · vL(1+kv2) · 1

1+τrs

−Izvs−bkrL+amv2

vkrL · vL(1+kv2) · 1

1+τrs

. (23)

Then the feedforward control input can be expressed as

uf = Cf · δ∗f . (24)

C. Feedback Controller

A state feedback controller will be proposed in this part.Define the following error vector as

xe = x− xd =[β − β∗

r − r∗

]. (25)

According to (16), (17) and (20),we get

xe =Axe+Bue+Bfwf +(Axd+Buf )−(Adxd+Bdud)(26)

Substituting (24) into (22), we have

G · uf = xd. (27)

Substituting (21) into (27) and then applying Laplace in-verse transform, we get

xd = Axd + Buf . (28)

Substituting (28) and (20) into (26), the last two terms of(26) are eliminated. Thus a simple state space description ofthe error system is achieved

xe = Axe + Bfwf + Bue. (29)

Here we design a linear quadratic regular (LQR) controllerwithout considering disturbances that will be dealt with in thenext section. The performance index J is shown as follow

J =∫ ∞

0

(xTe Qxe + uT

e Rue)dt (30)

where Q ≥ 0 and R ≥ 0 are the weighting matrices

Q =[400 00 180

], R =

[1 00 1

].

The obtained LQR feedback control gain is

Ke =[14.06 7.9612.22 −10.67

].

IV. DISTURBANCE OBSERVER BASED CONTROL

In this section, a disturbance observer is designed to dealwith the crosswind. Furthermore, this disturbance observeralso takes the ignored nonlinear dynamics into account. Asthe linear vehicle model inevitably neglects some high ordernonlinear terms compared with nonlinear vehicle model, anonlinear function O(xe, ue, wf ) is added to (29)

xe = Axe + Bfwf + Bue + O(xe, ue, wf ). (31)

The DOBC approach [16] is used to deal with both distur-bances and uncertainties. For simplicity, the effect caused bynonlinearities is merged into the disturbance term, i.e.,

wd = Bfwf + O(xe, ue, wf ). (32)

Substituting (32) into (31), the full dynamic model of thenonlinear 4WS vehicle is described as

xe = Axe + Bue + Bdwd (33)

where Bd = I is a 2× 2 identity matrix. For system (33), thefollowing disturbance observer is employed to estimate thedisturbances{

p = −LBd(p + Lxe)− L(Axe + Bue)wd = p + Lxe

(34)

where wd is a disturbance estimate vector, p is an auxiliaryvector and L is the observer gain matrix to be designed. Theestimates of the disturbance observer (34) can asymptoticallytrack disturbances if the observer gain matrix L is chosen suchthat −LBd is a Hurwitz matrix. According to the disturbanceestimate, we attempt to design a disturbance compensationgain vector Kd to attenuate the disturbances in the outputchannel. Suppose that A + BKe and C(A + BKe)−1B are

YU et al.: DISTURBANCE OBSERVER BASED CONTROL FOR FOUR WHEEL STEERING VEHICLES WITH MODEL REFERENCE 1125

invertible, the disturbance compensation gain vector Kd canbe selected as [17]

Kd = −[C(A + BKe)−1B]−1 × C(A + BKe)−1Bd. (35)

The disturbance observer control law of system (33) isdesigned as [16]

ue = Kexe + Kdwd (36)

where Ke is the LQR feedback control law. Here we selectthe observer gain matrix as

L =[0.1 00 0.1

]. (37)

The control structure of the nonlinear 4WS vehicle with theproposed control strategy is shown in Fig. 4. The disturbancecompensation gain vector can be obtained based on (35),where

Kd =[−0.93 −0.10−0.67 0.10

]. (38)

V. SIMULATION

In this section, some simulation results are given toshow the effectiveness of the proposed control strategiesfor 4WS systems. Choose the vehicle forward velocity asv = 20 m/s (72 km/h) and the step front wheel input as δ∗f =0.087(5◦), cf. Fig. 5. The crosswind disturbance is presentedin Fig. 6. In the same condition, the experimental results forthe 4WS vehicle with or without the disturbance observer (thecorresponding control strategies are shown in Fig. 3 and Fig. 4)are discussed, while a conventional front wheel steering (FWS)vehicle and a 4WS vehicle with proportional controller are also

implemented for comparison. The classic proportional controlmethod uses a speed dependent ration between rear and frontwheels in an open-loop controller in order to achieve a zerosideslip angle during directional maneuvers. The ratio betweenthe front and the rear wheel is described as [1]:

K =δf

δr=−b + mav2

krL

a + mbv2

kf L

. (39)

As shown in Fig.7, the steady-state gain of sideslip angle of4WS vehicles with or without disturbance observer is muchsmaller than the one of the proportional control, but thecontrol strategy with disturbance observer is slightly better.In cases where crosswind disturbances occur, both proposedstrategies show great advantages. However, only the yaw rateof the 4WS vehicle with disturbance observer can achieve zerotracking error in the steady-state without overshoot Comparedwith the proportional control for 4WS vehicle, the proposed4WS control strategies can provide drivers a familiar drivingsensation. In addition, the proposed controller both work verywell against the crosswind disturbance that occurred from 2 sto 4 s, while the 4WS vehicle based on DOBC is clearlysuperior.

On the basis of analyses, it is shown that the proposeddisturbance observer based control strategy not only copeswith the ignored nonlinear dynamics, but also presents strongrobustness against external disturbances. Since there are ig-nored nonlinear dynamics, it can not achieve desired track-ing performance. Although the feedback structure effectivelyreduces the effect of crosswind disturbances, the additionalDOBC can have a better control effect.

Fig. 4. 4WS vehicle with the proposed control strategy.

Fig. 5. Front wheel angle input.Fig. 6. Crosswind disturbance input.

1126 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 5, NO. 6, NOVEMBER 2018

Fig. 7. The responses of step steering angle input: 4WS vehicle with disturbance control (cf. Fig. 4), 4WS vehicle without disturbancecontrol (cf. Fig. 3), classic proportional control 4WS vehicle, FWS vehicle.

VI. CONCLUSIONS

In this paper, a 4WS control strategy based on DOBCapproach was presented. First, a nonlinear 3DOF 4WS vehiclemodel involving dynamic effects of crosswind was intro-duced. In the stage of controller design, a relatively simplevehicle model, the linear 2DOF 4WS vehicle model, wasused. Taking the ignored nonlinear dynamics and crosswinddisturbances into consideration, a disturbance observer wasdesigned. Finally, step response experiments were carried out,which showed that the proposed 4WS vehicle with distur-bance observer is superior to the 4WS vehicle without thedisturbance observer, FWS vehicle, and 4WS vehicle withthe classical proportional control. The proposed scheme notonly copes with ignored nonlinear dynamics, but also presentsstrong robustness against external disturbances.

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Shuyou Yu (M’12) received the B.S. and M.S.degrees in control science and engineering at JilinUniversity, China, in 1997 and 2005, respectively,and the Ph.D. degree in engineering cybernetics atthe University of Stuttgart, Germany, in 2011. From2010 to 2011, he was a Research and TeachingAssistant at the Institute for Systems Theory andAutomatic Control at the University of Stuttgart. In2012, he joined the Faculty of the Department ofControl Science & Engineering at Jilin University,China, where he is currently a Full Professor. His

current research interests include model predictive control, robust control,and applications in mechatronic systems.

Jing Wang received the B.S. degree in automationfrom Jilin University, Changchun, China, in 2013,and the M.S. degree in system engineering fromJilin University, Changchun, China, in 2016. From2016 to 2018, she worked at Hirain TechnologiesCo., Ltd, Beijing, China. She is currently workingat Bosch Automotive Products (Suzhou) Co., Ltd,Suzhou, China. Her research interests include autoelectronics, H∞ control and 4WS control.

Yan Wang received the B.S. degree in electronicinformation engineering from Changchun Universityof Science and Technology, Changchun, China, in2013, and the M.S. Degree in System Engineeringfrom Jilin Unviversity, Changchun, China, in 2017.She is currently working at Huachen Auto Group,Shenyang, China. Her research interests include H∞control and model predictive control.

Hong Chen (M’02–SM’12) received the B.S. andM.S. degrees in process control from Zhejiang Uni-versity, Zhejiang, China, in 1983 and 1986, re-spectively, and the Ph.D. degree from the Univer-sity of Stuttgart, Stuttgart, Germany, in 1997. In1986, she joined Jilin University of Technology,Changchun, China. From 1993 to 1997, she wasa wissenschaftlicher Mitarbeiter at the Institut fuerSystemdynamik und Regelungstechnik, Universityof Stuttgart. Since 1999, she has been a Professorwith Jilin University, where she is presently a Tang

Aoqing Professor. Her current research interests include model predictivecontrol, optimal and robust control, nonlinear control, and applications inprocess engineering and mechatronic systems.