A multivariable centralized controller design methodology for a Steer-by-Wire system

A Multivariable Centralized Controller Design Methodology for a Steer-by-Wire System F. Claveau, and Ph. Chevrel

Abstract—This paper presents a multivariable centralized control law for a Steer-by-Wire system. The controller is design by solving an H2 problem, the H2 criterion being made thanks to a model-matching approach. The use of a reference model of a steering column is motivated to define a priori the performances to be reached by the Steer-by-Wire system. A driver and a vehicle models are also included in the H2 criterion. This strategy ensures that the designed control law leads to the best performance of the closed-loop Steer-by-Wire system in the usual working conditions.

T

I. INTRODUCTION

HE Steer-by-Wire (SbW) technology substitutes the usual mechanical steering column, linking the steering wheel and the steering system, by a "by-wire" interconnection. Two motors are involved (cf. Fig. 1); on one side a motor generates a force-feedback on the Steering Wheel (SW), and on the other side a motor actuates the steering system composed basically of the rack-and-pinion, the tie-rods and finally the Road Wheels (RW). Each subsystems SW and RW are equipped of several sensors to measure at least the angular positions of the road wheels and the steering wheel, and possibly several torques such as the aligning torque applied to the RW and the driver torque applied to the SW. SbW technology is an active field of research since 2000 in several academic and industrial labs [1]-[9]. Several advantages motivate the SbW solution; at first, such technology can make easier the design and the assembly of the vehicle. Secondly, the loss of the steering column gives security in case of a head-on collision. Furthermore, full electric steering is more in accordance with the new environmental and ecological constraints than a hydraulic system. Most of all, the SbW technology permits the integration of different kind of vehicle stability controls and driving assistances. One can consider as examples [5]-[9]. In particular, SbW is very useful in the implementation of lateral driving assistances. A lateral driving assistance is used to improve lane keeping and yaw dynamic under

F. Claveau and Ph. Chevrel are with the Ecole des Mines de Nantes, Department of Automatic and Industrial Engineering, La Chantrerie, 4 Rue Alfred Kastler, BP 20722, 44307 Nantes Cedex 3, France (corresponding author, phone: +33-2-51-85-83-26; fax: +33-2-51-85-83-26; e-mail: [email protected]). F. Claveau and Ph. Chevrel are with the Institute of Research in Communications and Cybernetic of Nantes – IRCCyN, UMR CNRS 6597, 1 Rue de la Noë, BP 92101, 44321 Nantes Cedex 3, France.

external disturbances (such as lateral wind) [10]-[11]. The lateral controller acts on the vehicle lateral dynamics by using individual wheel braking, but also by active steering. Concretely, the controller output consists in an additional steering torque added to the driver one. The usefulness of a SbW system is therefore obvious.

Steering Wheel motor

θsw , Γd

usw

Controller θrw , Γal

urw

Road Wheel motor + reducer

Fig. 1. The Steer-by-Wire system

The main objective of this paper is to design an efficient control law ensuring a behavior as close as possible to the one of a (ideal) mechanical and hydraulic steering column. To reach this objective, the model-matching strategy is used to design in one step a centralized controller driving both the SW and RW subsystems. This control strategy based on a centralized controller for the whole SbW system distinguishes from most of the literature where local controllers are designed separately for the SW and RW subsystems, and then the global control strategy is build "piece-by-piece" by interconnecting in different ways the local controllers (e.g. [1]-[3]). To the authors' knowledge, design of centralized controller for SbW systems is only considered in papers taking inspiration from teleoperation results [12]-[13]. Several studies in lateral driving assistance have underlined the necessity to consider a model of the driver in the design process of the controller [10], [11]. In the same way that a model of the vehicle is often considered to design a SbW system (e.g. [3]-[5]), the design methodology proposed here considers the whole driver-steering system-vehicle-road interactions chain to design the SbW controller. Several driver models are available in the literature [14]-[16], but a simple one is used, only to validate the design strategy. A

simplified model is also used to represent the vehicle. All this leads finally to the design of a H2 centralized controller. The H2 design strategy is motivated by the inherent observer – state feedback form of the controller solution of the problem. Such form will be of interest in future works; estimation of the driver or vehicle states will be very useful for a potential lateral driving assistance. The paper is organized as follows; section II presents the models of the different parts of the SbW system, as well as the ones of the driver, the vehicle, and the steering column of reference. Section III described the control objectives, as well as the H2 model-matching design methodology. In section IV the results obtained in simulation are shown. Conclusions and perspectives are presented in section V. II. DEFINITION OF THE MODELS A. General architecture of the SbW system As describe in the introduction, a SbW system substitutes the usual mechanical link between the steering wheel and the steering system, both subsystems being actuated by an electric motor (cf. Fig. 1); the SW motor is used for forcefeedback, the RW motor to actuate the mechanical steering system. In this paper, the steering wheel angular position θsw is supposed to be measured, as well as the driver torque Γd . For the RW subsystem, the angular position of the wheels θrw and the aligning torque Γal measured on the motor axis are also supposed available. Models of both SW and RW subsystems are defined in the following subsections. Both come from the works of Coudon et al. [3][4]. The parameters used in these models are summed up in the Tab. I. The whole SbW system is the interconnection of both SW and RW subsystems and the centralized controller to be designed as shown on Fig. 1. Γd

SW

θsw

usw

controller

Γd Γal

RW

θrw

Fig. 1. The complete Steer-by-Wire system

B. Steering Wheel Subsystem Model The steering wheel (linked to the force-feedback motor) is represented by the second order model in eq. (1). Γd is the driver torque, and usw the torque provided by the SW motor which gives a force-feedback of the aligning torque.

(1)

⎧ 0 0 ⎡ ⎤ ⎤ 1 ⎤ ⎡θ ⎤ ⎡ ⎪⎪⎪⎡θrw ⎤ ⎢⎡ 0 ⎢ ⎥ ⎥ ⎥ ⎢ rw ⎥ ⎢ ⎢ ⎥ ⎢ ⎪ ⎢ ⎥ ⎢ ⎥ ⎥ = + Γ + 1 R − 2 ⎥ urw ⎥ al ⎢ ⎪⎪⎪⎢⎢θ ⎥⎥ ⎢ 0 − Brw ⎥ ⎢⎢θ ⎥⎥ ⎢ rw (2) ⎢ ⎥ ⎢ ⎥ rw ⎥ ⎢ ⎥ ⎢ ⎪ ⎣ ⎦ R d M R d M M ⎦⎥ ⎣⎢ pc biel ⎦⎥ Grw (s ) ⎪⎨⎣ ⎦ ⎣⎢ ⎣⎢ pc biel ⎦⎥ ⎪⎪ ⎡θ ⎤ ⎡1 0⎤ ⎡θrw ⎤ ⎡ 0⎤ ⎪⎪⎪⎢ rw ⎥ = ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎪⎪⎢Γ ⎥ ⎢ 0 0⎥ ⎢θ ⎥ + ⎢1⎥ Γal ⎥⎦ ⎢⎣ rw ⎥⎦ ⎢⎣ ⎥⎦ ⎪⎪⎩⎢⎣ al ⎥⎦ ⎢⎣ 2

2

⎛ R ⎞⎟ ⎛ R ⎞⎟ with M = M c + J 2 ⎜⎜⎜ 2 ⎟⎟ , Brw = Bc + ⎜⎜⎜ 2 ⎟⎟ B2 . ⎜⎝ Rpc ⎠⎟ ⎜⎝ Rpc ⎠⎟

θrw

TABLE I. PARAMETERS OF THE SW AND RW SUBSYSTEMS Wheels RW motor urw angular torque position

θsw

Steering wheel angular position

usw

SW motor torque

ref θsw

Reference SW angular position

Γal

Aligning torque

Bsw =

M c = 66kg

Bc = 500kg.s −1

urw

1 ⎤ ⎡θ ⎤ ⎡ 0 ⎤ ⎡ 0 ⎤ ⎥ ⎢ sw ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ + 1 Γ + 1 ⎥u B − sw ⎥⎥ ⎢⎢θsw ⎥⎥ ⎢⎢ ⎥⎥ d ⎢⎢ ⎥⎥ sw J sw ⎥⎦ ⎣ ⎦ ⎢⎣J sw ⎥⎦ ⎢⎣J sw ⎥⎦ ⎡θsw ⎤ 0⎤⎥ ⎢⎢ ⎥⎥ ⎦ ⎢θsw ⎥ ⎣ ⎦

C. Road Wheel Subsystem Model The steering system is based on a classical architecture; the RW motor is linked to the rack-and-pinion system through a reducer. The front wheels are actuated by the rack through the tie-rods. The model is defined in eq. (2). Inputs are urw the torque generated by the SbW controller, and Γal , the aligning torque due to the lateral reaction of the road on the tires. Here it is supposed measured in the reducer.

0.2N .m.s SbW

Γal

⎧ ⎡0 ⎪⎪⎡θsw ⎤ ⎢ ⎪⎪⎪⎢⎢ ⎥⎥ = ⎢ ⎪⎪⎢⎢θsw ⎥⎥ ⎢⎢0 Gsw (s )⎪⎨⎪⎣ ⎦ ⎢⎣ ⎪⎪ ⎪⎪ ⎪⎪θsw = ⎡⎢1 ⎣ ⎪⎪ ⎪⎩

J2 = 10−4 kg.m 2 R2 = 16.5

SW viscous friction coefficient Rack mass

Rack viscous coefficient RW motor axle inertia

J sw = 0.02kg.m

2

Rpc = −3

8.5.10 m dbiel = 0.142m

B2 = 10−4 N .m.s

Steering wheel+mot or inertia Radius of the pinion Tie rod length RW motor viscous coefficient

RW motor reducer ratio

D. Driver Model Many driver models can be found in the literature, more or less detailed, depicting different human capacities; e.g. the neuromuscular dynamics, the visual or vestibular systems (see e.g. [14]-[16]).

In this paper, a simple PI is used as model (cf. eq. (3)). Its control objective is to pilot the angular position θsw of the steering wheel to follow the non-measurable signal reference ref θsw . The output is Γd the driver torque applied on the steering wheel. ⎧ ⎡ ⎤ ⎪ ⎪⎪x = [ 0 ] x + ⎢ 1 ⎥ (θ ref − θ ) d d sw ⎪ ⎢T ⎥ sw (3) Gdriv (s ) ⎨ ⎣ i⎦ ⎪ ⎪ ref ⎪ Γ = [K ] x d + [K ](θsw − θsw ) ⎪ ⎪ d ⎩ Parameters of the driver (see Tab. II) have been tuned thanks to simulations on the column reference model, interconnected to the vehicle model (see subsections E and F).

1 ⎤ ⎡θcol ⎤ ⎡ 0 ⎤ ⎡0⎤ ⎪⎪⎧⎡ θcol ⎤ ⎡ 0 ⎢ ⎥ ⎢ sw ⎥ ⎢ sw ⎥ ⎢ ⎥ ⎪⎢ ⎪⎪⎢ ⎥⎥ = ⎢ k ⎥ kv ⎥ ⎢ col ⎥ + ⎢⎢ 1 ⎥⎥ Γal + ⎢⎢ 1 ⎥⎥ Γd p col ⎢  ⎪ − ⎥ ⎢⎢θsw ⎥⎥ ⎢− ⎥ ⎪⎢⎢θsw ⎥⎦⎥ ⎢− ⎢⎣ J ⎥⎦ ⎪⎣ J ⎦⎣ ⎦ ⎣ J⎦ ⎣ J (5) Gcol (s ) ⎪ ⎨ ⎪ col ⎤ ⎡ ⎤ 1 0 col ⎤ ⎡ ⎪⎡ θ θ ⎪ ⎥ ⎢ sw ⎥ ⎪⎪⎢ sw ⎥ = ⎢⎢ ⎥⎢ ⎥ ⎢ col ⎥ ⎢ 1 col ⎪ 0⎥⎥ ⎢⎢θsw ⎥⎥ ⎪ ⎢θrw ⎥ ⎢ ⎪ ⎣ ⎦ ⎣ ⎦ d ⎪ ⎣ ⎦ ⎩ TABLE III. PARAMETERS OF THE REFERENCE STEERING COLUMN

kv = 0.8N .m.s

J = 0.03kg.m 2

Viscous coefficient

Inertia

k p = 3.9N .m

Steering ratio

d = 16.5

θsw θrw

TABLE II. PARAMETERS OF THE DRIVER MODEL

K p = 30

Gain

Ti = 0.1s

Integral term

E. Vehicle Model As the paper focuses on the lateral behavior of a vehicle, a single track (bicycle) model is used [10]-[11]. Its states are the vehicle side slide angle β and the yaw rate r . The state-space model is proposed in eq. (4). c f = c f 0v and cr = cr 0v are the front and rear wheels cornering stiffness. fw is the disturbance crosswind force (units Newton). Only the aligning torque Γal is considered as output. The definition of the parameters and their values can be found in [10]-[11]. Notice that for all this paper the longitudinal velocity of the vehicle is supposed to be constant equal to v = 20 m.s −1 . ⎧⎪ ⎡ 2 (c + c ) 2 (lr cr − l f c f )⎤⎥ ⎪⎪ r f ⎢ −1 + ⎪⎪⎡β ⎤ ⎢ − ⎥ ⎡β ⎤ mv mv 2 ⎥⎢ ⎥ ⎪⎪⎢ ⎥ = ⎢ ⎢ ⎥ ⎥ ⎪⎪⎢ r ⎥ ⎢⎢ 2 (l c − l c ) 2 (lr2cr + l f2c f ) ⎥ ⎢⎢⎣ r ⎥⎥⎦ f f ⎪⎪⎣ ⎦ ⎢ r r − ⎥ ⎪⎪ ⎢⎣ ⎥⎦ J veh J veh v ⎪⎪ ⎪⎪ ⎡ 2c f ⎤ ⎡ 1 ⎤ Gveh (s ) ⎪⎨ (4) ⎢ ⎥ ⎢ ⎥ ⎪⎪ ⎢ mv ⎥ ⎢ mv ⎥ ⎥θ ⎪⎪ +⎢ ⎥f +⎢ ⎢ lw ⎥ w ⎢ 2c f l f ⎥ sw ⎪⎪ ⎢ ⎥ ⎢ ⎥ ⎪⎪ ⎢⎣ J veh ⎥⎦ ⎣⎢ J veh ⎦⎥ ⎪⎪ ⎪⎪ ⎪⎪Γ = ⎡⎢ 2c f ηt 2c f l f ηt ⎥⎤ ⎡⎢ β ⎤⎥ + ⎢⎡ 2c f ηt ⎥⎤ θ sw al ⎢⎣ d ⎪⎪⎪ dv ⎥⎦ ⎢⎢⎣ r ⎥⎥⎦ ⎢⎣ d ⎥⎦ ⎩⎪

F. Reference Column Model The reference column model is also based on works of Coudon et al. [3], [4]. It is a second order model, based on three physical parameters, the inertia J , the viscous friction coefficient kv and the stiffness k p , chosen to define an "ideal" steering column (cf. eq. (5)). As for the vehicle model, they have been tuned under the assumption of a vehicle velocity of v = 20 m.s −1 . The parameters values are shown in Table III.

Stiffness coefficient

Finally, the interconnection scheme of all the model of the SW and RW subsystems, driver, vehicle, and the SbW controller to be designed is sum up in Fig. 1 and Fig. 2. The same complete interconnection scheme with the reference column is easily obtained by exchanging the "SbW system" bloc. ref θsw

Driver

Γd

θsw SbW system

θrw

Vehicle

fw

Γal

Fig. 2. Driver - SbW - Vehicle interconnection

III. MODEL-MATCHING CONTROLLER DESIGN A. Control requirements The main concern is obviously to make matching the SW and RW angular positions. But the force-feedback to the driver is also critical, a loss of information about the aligning torque and all the potential disturbances such as ruts on the road being too dangerous. Quantization of the performance to be reached by a steering system is difficult to find in the literature. One can consider [5], [17]. The most significant values are; - a bandwidth for the transfer function θrw / Γd at least of 5H z , - the driver torque should not exceed 10 N .m ( 15 N .m in an emergency maneuver), - an accuracy of 0.5° or 2% between the RW angle θrw and its reference signal (i.e. the SW angle θsw divided by the reducer ratio).

B. H2 Model-Matching problem 1) Definition of the H2 model-matching problem Briefly stated, a model-matching control problem consists in designing a controller K that makes the system behave as a given model. This control problem has been frequently studied in the literature, with several approaches [18]-[20]; design of a static state feedback or a dynamic controller, polynomial approaches, resolution of the Nevanlinna-Pick interpolation problem. In this paper, the model-matching problem is formulated through a H2 criterion defined as follows; let's consider Gref (s ) as the (stable) model to achieve and G (s ) the system to be controlled. The problem to be solved is

(

min Gref (s ) − Fl (G (s ), K (s )) K (s )

2

)

(6)

where K (s ) is the controller to be design. Fl (G (s ) , K (s )) denotes the lower Linear Fractional Transformation (LFT) on G (s ) and K (s ) , −1

Fl (G, K ) = G11 + G12K (I − G22K ) G21 ,

(7)

T w = ⎡⎢wx wy ⎤⎥ , wx = ⎣ ⎦ ⎡ ze = ⎢eθsw eθrw e Γd ⎣

with wy ∈ \ p

T

⎡θref ⎢⎣ sw

dis ⎤ Γal ⎥⎦ ,

T e Γal ⎤⎥ , ⎦ the output noises on each measured output

y , z u ∈ \m the weighted controlled outputs constraining col − θsw , the three other the control inputs u , and eθsw = θsw

ref controlled outputs being defined in the same way. θsw represents the reference input of the driver model (i.e. the SW angular position reference), whereas Γdis al represents a disturbance input applied directly on the output Γal of the vehicle model (4).

P (s ) wy

Ru

Qy

z

w

wx

Qx

Mp

Gref (s )

w

z

P (s ) y

u

K (s ) Fig. 3. H2 standard problem

The design problem can be redefined as a H2 standard problem. Consider Fig. 3, with P (s ) the standard model define as in (8), and K (s ) the researched dynamical controller. ⎡x ⎤ ⎡⎢A B1 B2 ⎤⎥ ⎡ x ⎤ ⎢ ⎥ ⎡P11 (s ) P12 (s )⎤ ⎥ ⎢⎢ ⎥⎥ ⎥ ⇔ ⎢z ⎥ = ⎢⎢C D P (s ) = ⎢⎢ 1 11 D12 ⎥ ⎢w ⎥ , (8) ⎢ ⎥ ⎥ ⎥⎢ ⎥ ⎢ ⎥ ⎢⎢ ⎢⎣P21 (s ) P22 (s )⎥⎦ ⎢⎣y ⎥⎦ ⎢C 2 D2 D22 ⎥⎥ ⎢⎣ u ⎥⎦ ⎣ ⎦ where u ∈ \m , w ∈ \mw , y ∈ \ p , and z ∈ \ pz are respectively the control inputs, the reference or disturbance inputs, the measured outputs and the controlled outputs. As demonstrated on Fig. 4, these signals are composed of; T T u = ⎡⎢usw urw ⎤⎥ , y = ⎢⎡θsw θrw Γd Γal ⎤⎥ , ⎣ ⎦ ⎣ ⎦

+

Rz −

⎡G11 (s ) G12 (s )⎤ ⎥. with G (s ) = ⎢⎢ (s ) G (s )⎥ G 21 22 ⎣⎢ ⎦⎥ In this paper, Gref (s ) is defined by the interconnection

of the driver, reference column and vehicle models, respectively Gdriv (s ) , Gcol (s ) , and Gveh (s ) in the same way than in Fig. 2 (by substituting the SbW system by Gcol (s ) ). In the same manner, G (s ) is defined by the interconnection of Gdriv (s ) , the SW and RW models Gsw (s ) and Grw (s ) , and the Gveh (s ) . 2) Reformulation through a H2 standard problem

zu

ze

+

G (s ) u

+

y K (s )

Fig. 4. H2 Standard model

Remark: To define the standard model, Γdis al is considered instead of fw (see eq. (4)) as exogenous disturbance input. This is done to avoid the controller solution of the design problem to reject the disturbances such as crosswinds or ruts instead of the driver. In fact, such interference would prevent the driver to be informed of these disturbances, which is dangerous. M p (s ) in Fig. 4 contains some predictor models associated to the disturbance and reference inputs ( Γdis al and ref θsw ). For instance, the more classical model associated to a

constant signal v (.) whose best prediction is vˆ(t + τ ) = v (t ) is given by v (t ) = 0 . Qy , Qx , Rz , Ru are static weighting matrices pertinent to the control objectives. Finally, the problem to be solved is; Problem 1: H2 model-matching problem Under the classical assumptions [21], find the controller K (s ) which; i. internally stabilizes Fl (G (s ), K (s )) , ii. minimize J (K ) = Tzw

2

,

withTzw = Fl (P (s ), K (s )) (cf. Fig. 4). Remark: As the predictor models contained in M p (s ) are

Bode Diagram

-50 -60 -70 -80

Magnitude (dB)

instable, the complete standard model P (s ) is nonstabilizable. Despite this, there is a solution to the problem under the relaxed constraint that the closed-loop system Fl (G (s ), K (s )) (and not Fl (P (s ), K (s )) ) is internally stable [22]-[23].

-90 -100 -110 -120 45

IV. RESULTS

-45

Phase (deg)

A. Controller tuning The tuning of the SbW controller K (s ) is done as follows. The weighting matrixes are; Qy = 0.01.I 4 ,

-90

-135

-180 -1 10

V. CONCLUSION This paper has proposed a multivariable centralized control law for a Steer-by-Wire system. This controller task is to drive both the SW and RW subsystems. The design methodology is based on a H2 criterion made thanks to the definition of a model-matching problem. A reference model

0

10

1

10

2

10

3

Fig. 5. Bode Diagram of θrw / Γd for reference column and SbW system Road Wheel Angle - Trw 0.6 SbW System Reference Column

of two constant predictor models. The parameter values of the reference column (cf. Tab. III) leads to a bandwidth of 8.5H z for the transfer function θrw / Γd .

0.4

Angle (deg)

0.2

0

-0.2

-0.4

0

1

2

3

4

5 time (s)

6

7

8

9

10

SbW system

10

SW Angle

Angle (deg)

5

0

-5

-10 0

1

2

3

4

5

6

7

8

10

9

10

RW Angle x R2

5 Angle (deg)

amplitude is 10° . Simulation results are demonstrated on Fig. 6. The first diagram demonstrated the good matching col between the RW angular positions θrw and θrw respectively of the reference column and the SbW system. The result is the same for any other measured signals. By consequence, the accuracy of the SbW system is perfect, as shown on the second diagram where the angular position θrw fits exactly θsw (considering the reducer ratio R2 ). Next, a straight line trajectory is considered (i.e. ref θsw = 0° ), and a crosswind disturbance of fw = 500N is applied to the vehicle at t = 3s during 3s (see Fig. 7). In this case some small differences between the SbW system and the reference column behaviors can be observed, mainly for the RW angular position θrw . However it leads to an accuracy between θrw and θsw which is still acceptable. A more accurate tuning of the controller (through the several weighting matrices) should improve the results.

10

Frequency (rad/s)

Qx = I 2 , Ru = 0.01.I 2 , and Rz = I 4 . M p (s ) is composed

B. Time performance Firstly, the model-matching performance is analyzed. To demonstrate its efficiency, the bode diagrams of the transfer function θrw / Γd of the reference column and the SbW system, both interconnected to the vehicle models are demonstrated on Fig. 5. As expected, the two transfer functions match, especially in the useful bandwidth. This result is verified by simulation. Two scenarios are ref considered. At first, a sinusoidal reference signal θsw is considered, without any crosswinds fw or other disturbance. The frequency of the sinusoidal signal is 0.3H z , its

SbW System Reference Column

0

0

-5

-10 0

1

2

3

4

5 time (s)

6

7

8

9

10

Fig. 6. Simulation results for a sinusoidal trajectory

of an ideal mechanical steering column is defined, which enables to tune all the expected characteristics of the final SbW system. Furthermore, the design of the controller is made while considering the steering system in interaction with both the driver and the whole vehicle. The multivariable controller and its design methodology have been shown to be powerful in that they lead to a SbW system behaving exactly as the ideal steering column used as reference, i.e. with both a good synchronization between SW and RW angles and a good force-feedback of the aligning torque. The control law does not interfere on the force-feedback to the driver when a crosswind occurs. There are several perspectives of this work. At first, a design methodology such as the H2 Standard State Control Methodology could be used to construct the final H2 criterion, leading especially to a very systematic manner to tune the weighting matrices through only two metaparameters [22]-[23]. Secondly, more complete models of the different subsystems could be used, for instance the test bed model proposed in [24] for the RW system. Another driver model based on a more realist control strategy, e.g.

Steering Wheel Angle - Tsw 0.8

0.4

Angle (deg)

0.2

-0.4

-0.6

-0.8 0

[5] [6] [7] [8] [9] [10] [11] [12] [13]

[14] [15] [16]

3

4

5 time (s)

6

7

8

9

10

RW Reference Angle SbW System Reference Column

0.04

0.03

0.02

0.01

0

-0.01

-0.02

-0.03

-0.04

-0.05 0

1

2

3

4

5 time (s)

6

7

8

9

10

6

7

8

9

10

6

7

8

9

10

6

7

8

9

10

Driver Torque - Ucond

1 torque (N)

[4]

2

Road Wheel Angle - Trw

SbW System Reference Column

0.5 0 -0.5 -1 0

1

2

3

4

5 SW motor Torque - Usw

1 torque (N)

[3]

1

0.05

0.5 0 -0.5 0

1

2

3

4

5 RW motor Torque - Urw

0.05 torque (N)

[2]

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REFERENCES [1]

Tsw Reference SbW System Reference Column

0.6

Angle (deg)

based on models of the visual and vestibular systems will be also useful [14]-[16]. Furthermore, if in this paper the longitudinal velocity of the vehicle was supposed to be constant, at the end the SbW controller will have to be robust to the variation of this parameter. One solution is to design a gain-scheduling control law based on interpolation. The estimated state feedback structure of the proposed controller will be useful here, leading to consistent realizations obtained at different operating velocities [23], [25]. The final step will be to adapt a lateral driving assistance to this SbW system. One strategy could be to modify the H2 criterion to consider also the driving assistance objectives. Moreover, the inherent estimated state feedback form of the H2 controller will be one more time of interest to develop an assistance strategy based on the estimation of the driver and vehicle states.

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1

2

3

4

5 time (s)

Fig. 7. Simulation results for crosswinds disturbance [17] R. Sebastian, T. Kaufmann, F. Bolourchi, and H.S. Tan, "Design of an automated highway systems steering actuator control system", in Proc. IEEE Intel. Transport. Sys. Conf., pp. 254-259, 1998 [18] J.C Doyle, B.A. Francis, and A.R. Tannebaum, Feedback Control Theory. MacMillan, New-York, USA, 1992 [19] S. Morse, "Structure and design of linear model following systems", IEEE Trans. Auto. Contr., vol. 18, n°4, 1973 [20] B.C. Moore, L.M. Silverman, "Model matching by state feedback and dynamic compensation", IEEE Trans. Auto. Contr., vol. 14, n°4, pp. 491-497 [21] J.C. Doyle, K. Glover, P.P. Khargonekar, and B.A. Francis, "Statespace solutions to standard H2 and H∞ control problems", IEEE Trans. Auto. Contr., vol. 34, n°8, pp. 831-846, 1989 [22] Ph. Chevrel, "Commande des systèmes Linéaires", In Méthodologie de la commande par l’approche d’état, Ph. de Larminat, Ed. Paris: Hermès, 2002, pp. 151-192 (in french) [23] F. Claveau, Ph. Chevrel, and D. Knittel, "A 2 DOF gain-scheduled controller design methodology for a multi-motor web transport system", IFAC Contr. Eng. Pract., vol. 16, pp. 609-622, 2008 [24] S. Bolognani, D. Ciscato, M. Tomasini, M. Zigliotto, "Virtual mechanical load setup for Steer-by-Wire – a case study", in Proc. IEEE Intern. Symp. Indus. Elec., Dubrovnik, Croatia, 2005 [25] D.J. Stillwell, and W.J. Rugh, "Interpolation of observer state feedback controllers for gain-scheduling" IEEE Transactions on Automatic Control, vol. 44, N°6, pp. 1225–1229, 1999