A study on µ synthesis control for four wheel steering system to enhance vehicle lateral stability

China e-mail: lywu@seu.edu.cn to Enhance Vehicle Lateral Stability This paper presents the design of ␮-synthesis control for four-wheel steering 4WS vehicle and an experimental study usi

e-mail: ygdl@seu.edu.cn

Nan Chen

e-mail: nchen@seu.edu.cn

Jin-Xiang Wang

e-mail: wangjx@seu.edu.cn School of Mechanical Engineering,

Southeast University, Nanjing 210096, P.R China

Ling-Yao Wu

School of Automation, Southeast University, Nanjing 210096, P.R China e-mail: lywu@seu.edu.cn

to Enhance Vehicle Lateral Stability

This paper presents the design of ␮-synthesis control for four-wheel steering (4WS)

vehicle and an experimental study using a hardware-in-the-loop (Hil) setup First, the robust controller is designed and the selection of weighting functions is discussed in the framework of ␮-synthesis control scheme, considering the varying parameters induced by

running vehicle condition Second, in order to investigate the feasibility of the four-wheel steering control system, the 4WS vehicle control system is built using dSPACE DS1005 platform The experimental tests are performed using the Hil setup which has been constructed using the devised rear steering actuating system The dynamics performance

is evaluated by experiment using the Hil setup under the condition of parameter varia-tions Finally, experimental results show that the ␮-synthesis controller can enhance

good vehicle lateral maneuverability. 关DOI: 10.1115/1.4002707兴

Keywords: four-wheel steering, robust control, µ-synthesis, hardware-in-the-loop setup

1 Introduction

Four-wheel steering共4WS兲 technique is one of the most

effec-tive methods of vehicle aceffec-tive control systems, which aims to

enhance handling and comfort characteristics ensuring stability in

critical manoeuvring situations Several control strategies have

been applied in 4WS vehicles, such as the LQG control, the fuzzy

control, etc 关1–3兴 In recent years, a great deal of attention has

been paid to the H⬁control because it not only provides a unified

and general control framework for all control structures, but also

yields a controller with guaranteed margins 关4,5兴 However, the

H⬁control models all uncertainties as a single complex full block,

which results in a rather conservative design 关6,7兴 Under such

circumstances, the␮-synthesis technique, which involves the use

of H⬁optimization for synthesis and structured singular value共␮兲

for analysis, has been developed关8兴 Literature survey shows that

most results related to␮-synthesis are simulations, and there is no

sufficient experimental evidence for four-wheel steering using

␮-synthesis

In this paper, an active four-wheel steering controller is

de-signed with the framework of ␮-synthesis control scheme,

con-sidering that the handling dynamic responses of 4WS vehicle can

be affected by parameter variations resulting from cornering

stiff-ness on different road conditions The control performance is

evaluated by the optimal control and␮-synthesis control

simula-tions, based on the linear vehicle model, considering the

param-eter variations Finally, using the hardware-the-loop setup

in-cluding the prototype control system, the performance of 4WS

vehicle system is investigated by experiment test, which is carried

out based on the dSPACE DS1005 PPC digital system

2 Mathematical Vehicle Model

2.1 Linear Vehicle Model In developing the active

control-ler, it is not desirable to use the complex vehicle model because of

sampling time and implementation of the control system In this paper, the linear vehicle model is used for the design of a control-ler Figure 1 shows a two-degree-of-freedom model including the yaw and lateral motion dynamics of 4WS vehicle, related to driver steering maneuvers, traveling on a road surface at a constant speedv In this model, the coordinate frame is fixed on the vehicle

body in the center of gravity which is denoted as CG

In a yaw plane representation, the sideslip angle␤, at the center gravity of the vehicle, is assumed to be small,兩␤兩Ⰶ1 The slip angles of front and rear tires共␣fand␣r兲 are, respectively, written

as关4兴

␣f=␦f−␤ −L f

v r and ␣r=␦r−␤ +L r

where L f and L rare the distances from the CG to the front and

rear axles, respectively, and L = L f + L r is the wheel base r is yaw

angular velocity.␦fand␦rare the steering angle of front and rear wheels, respectively

In general, lateral tire force is a nonlinear function of slip angle

In this paper, under the assumption that the lateral tire forces F f and F rare linear functions with slip angles ␣f and ␣r, respec-tively, the following equations are used:

F f= −␮K f␣f and F r= −␮K r␣r 共2兲

where K f = K cf K fn 共mgL r /L兲, K r = K cr K rn 共mgL f /L兲, m is the total

mass of vehicle,␮ is the adhesion coefficient between road sur-face and the tire ranging from 0.8共dry road兲 to 0.25 共icy road兲, and the cornering stiffness of the front共rear兲 tire is denoted by

K f 共K r 兲, K fn and K rn are normalized cornering stiffnesses, and K cf and K crare cornering stiffness coefficients

Considering the previous forces, equations of motion including lateral and yaw motions are written as关8兴

mv ␤˙ = − 共K f + K r兲␮␤ −再mv + ␮L f K f−␮L r K r

v 冎r + ␮K f␦f

+␮K r␦r

1

Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the

J OURNAL OF D YNAMIC S YSTEMS , M EASUREMENT , AND C ONTROL Manuscript received

May 13, 2008; final manuscript received June 13, 2010; published online November

23, 2010 Assoc Editor: Hemant M Sardar.

I z r˙ = − 共L f K f − L r K r兲␮␤ −L2f K f + L r2K r

v ␮r + ␮L f K f␦f−␮L r K r␦r

共3兲

where I zdenotes the yaw moment of inertia about its mass center

z-axis.

In addition, the lateral acceleration a yat the CG is obtained by

the yaw rate r and the sideslip angle␤ with the following relation:

␣y=v 共r + ␤˙兲 共4兲

2.2 State Space Representation From Eqs.共3兲 and 共4兲, the

state space representation can be expressed as

x˙ = Ax + Bu

where the state vector x =关␤ r兴T, the control input vector u

=关␦f ␦r兴T, and the output vector y =关␤ r ay兴T

A =冤 −␮K f + K r

mv ␮− L f K f + L r K r

mv2 − 1

␮− L f K f + L r K r

I z −␮L f

2K f + L r2K r

I z v 冥

B =冤 K f

mv␮ K r

mv␮

L f K f

I z ␮ −L r K r

I z ␮冥

−␮K f + K r

m −␮L f K f − L r K r

0 0

␮K f

m ␮K r

m冥

The key parameters of vehicle and tires used in this paper are

summarized in Table 1

3 Design and Evaluation of Robust Controller

3.1 Synthesis of the ␮-Controller To design the active

four-wheel steering controller K 共s兲, an output feedback ␮-synthesis

control scheme is applied In order to obtain robust stability and

performance, the additive modeling error resulting from parameter

variations is considered

In this system, the desired yaw rate gain G f−ris selected as关9兴

G f−r 共s兲 = s/300 + 3.75

s/10 + 1 共6兲

where G f−r is corresponding to a yaw rate of vehicle response which is of agility

See Fig 2 Consider the steer disturbance␦f, controlled output

z1, z2and control input u which denotes the rear wheel angle␦r,

and y is the measured output containing only the yaw rate r n is

the measurement noise The transfer function䊐, which represents the uncertainties between the nominal model and the actual plant,

is assumed to be stable and unknown, except for the norm condi-tion,储⌬储⬁⬍1 关7,8兴 In the diagram, e is the input of the perturba-tion, d is its output The main performance objective is that the

transfer function from␦f to z should be small, in the储•储⬁sense, for all possible uncertainty transfer functions 䊐 The weighting

functions W p and W rreflect the relative importance of the differ-ent frequency domains in terms of the tracking error The

weight-ing function W nrepresents the impact of the different frequency domains in terms of the sensor noise

Necessary and sufficient conditions for robust stability and ro-bust performance can be formulated in terms of the structured singular value denoted as␮ 关6,7兴 Now, the design setup in Fig 2 should be formalized as a standard design problem In order to analyze the performance and robustness requirements, the closed-loop system, which is illustrated in Fig 3, is expressed by using

the feedback effect u = Ky.

Note that the system P consists of recognizing three pairs of

input/output variables The complete vehicle model for the control system is described by

冤e

z

y冥=冤P11 P12 P13

P21 P22 P23

P31 P32 P33冥

P 共s兲

冤d

w

u冥

共7兲

Fig 1 A half vehicle dynamic model

Table 1 Parameters of the vehicle and the tires

m 1740 kg

I z 3048 kg m 2

Fig 2 The 4WS closed-loop interconnection structure

Fig 3 The P − K structure with uncertainty

M 共s兲 = F L 共P共s兲,K共s兲兲 =冋P11 P12

P21 P22册+ P13K 共I − P33K兲−1关P31 P32兴

共9兲

which is actually obtained by substituting u = Ky into Eq.共7兲

The LFT paradigm can be used to describe and analyze the

uncertain vehicle system, where M corresponds to what is

as-sumed as the constant in the control system and⌬ is the block

diagonal matrix Then, the matrix M is partitioned as

冋e

z册=冋M11 M12

M21 M22册

M 共s兲

冋d

w册

共10兲

Moreover, the upper LFT connects w and z, which is obtained

by combining Eq.共8兲 with Eq 共10兲 being expressed as

z = F U 共M,⌬兲w = 关M22+ M21⌬共I − M11⌬兲−1M12兴w 共11兲

where F U 共M ,⌬兲 is the upper LFT The robust performance of the

closed-loop system about nominal plant perturbation is equivalent

to储F U 共M ,⌬兲储⬁⬍1

The goal of the␮-synthesis is to minimize over all stabilizing

controllers K the peak value ␮⌬共·兲 of the closed-loop transfer

function F L 共P,K兲 The formula is as follows:

min

K

stabilizing

sup

␻苸R␮⌬关F L 共P,K兲共j␻兲兴 共12兲

Obviously, this is the standard ␮-control problem, and the design

can be based on theMATLAB␮-toolbox, in which the D−K

itera-tion is adopted to perform the synthesis procedure D − K iteraitera-tion

is a two-step minimization process: the first step is a minimization

of the H⬁norm over all stabilizing controllers K, while the scaling

matrix D is held fixed, and the second step is a minimization over

a set of scaling D, while the controller K is held fixed关7,10兴

weightings are included in the controller synthesis instead of in

the control system implementation to yield robust performance

and stability In general, in order to find a controller, they should

be properly selected in advance

The weighting functions W p and W rrepresent the performance

outputs, which are related to the components of z Now, the

per-formance weighting function is used to define design

specifica-tion The inverse of the performance weight indicates how much

the external disturbances should be rejected at the output, or how

much steady state tracking due to the external input is allowed

W p 共j␻兲 for the sideslip angle and W r 共j␻兲 for the yaw rate are the

weights specifying system performance The upper bounds on

兩1/W p 共j␻兲兩 and 兩1/W r 共j␻兲兩 are the weights for the tolerable

maxi-mum angle␤ and the maximum tracking yaw rate; the weights are

assumed to be constant over all frequencies and are set to

W p=0.3s + 0.5

s + 0.01

W r= s + 0.5

The weighting function W nrepresents the impact of the

differ-ent frequency domains in terms of sensor noise n In order to

account for the fact that system outputs can never be sensed

with-W n=

500s + 1 共14兲 where the upper bound of兩1/W n 共j␻兲兩 represents the maximal

ex-pected noise gain

qua-dratic optimization control is to seek an optimization control

sig-nal u 共t兲 which minimizes the following performance index J with

reference to the system described by Eq.共5兲

J =冕0

⬁

共x T Qx + u T Ru 兲dt 共15兲

Here, Q and R are the weighting matrices, where Q ⱖ0, Rⱖ0,and

共A,B兲 is assumed to be controllable and 共A,C兲 is assumed to be

observable The purpose of control is to minimize the sideslip

angle; thus, Q and R take the following values:

Q =冋103 0

0 1册, R =冋1 0

0 1册

The control input u, which minimizes Eq 共15兲, is u=−K op x,

where K opis called an optimal feedback coefficient matrix given

by K op = −R−1B T P Here, P which is a positive definite matrix is

the solution of the following Riccati matrix equation:

− P 共t兲A − ATP 共t兲 + P共t兲BR−1BTP 共t兲 − Q = 0 共16兲

3.4 Simulation With Full Vehicle In this section, the

dy-namic performance of both versions of the controller will be com-pared in order to validate the approximation put forward In what follows, the 4WS robust controllers are evaluated in time domain using the ␮-toolbox 关11兴 As shown in the ␮-design procedure

with the D − K iteration, the robust controller is synthesized and

designed for 4WS vehicle at a velocity with 30 m/s

To achieve the desired performance and cover the uncertainty for the considered vehicle, a set of frequency-dependent weight-ings have to be included, so the order of the generalized 4WS control system is increased, resulting in a high order controller It

is difficult to implement a high order controller because the con-troller normally is ill-conditional A lower order model can lead to

a lower order compensator By adopting the balanced model re-duction via the truncated method关12兴, which can preserve stabil-ity and gives an explicit bound on frequency response error, the 14-order controller obtained by the above iteration could be re-duced to a three-order controller

The transient responses to the steering wheel angle input which changes from 0 deg to 35 deg共gear ratio=15兲 So the given front wheel steering angle ␦f is 0.04 rad共step signal兲, approximately equivalent to 2.29 deg The simulation results are obtained as illustrated in Fig 4

Results obtained from the computer simulation indicate that the vehicle with the robust controller has a superior performance compared with that with the optimal controller Figure 4共a兲

illus-trates that the steady state values of the yaw rate of the two con-trollers are almost equal to which of the desired yaw rate How-ever, the yaw rate response of the robust controller is more rapid than which of the optimal controller and the peak value of the robust controller is lower than that of the optimal controller This means that the lower sensitivity of the steering system with the robust controller is achieved at high speed Furthermore, Fig 4共b兲

indicates that the reduction in the vehicle sideslip angle is an important safety criterion, which could certainly be achieved more

reduction in the robust controlled vehicle.

So the comparison of the robust and optimal controls for

im-proving vehicle performance shows that the robust controller can

certainly improve the vehicle handling when its performance is

compared with the optimal controller Next experiment work is to

further validate the superiority of the robust controller

4 Experimental Analyses

Faced with the need to reduce the development time and cost, the hardware-in-the-loop simulation 关13,14兴 increasingly proves

to be an efficient tool in the automotive industry Hardware-in-the-loop simulation is characterized by the operation of real compo-nents in connection with real-time simulated compocompo-nents Usu-ally, the control-system hardware and software are the real system,

as used for series production The controlled process, consisting of actuators, physical processes, and sensors, can then be either fully

or partially simulated Hence, hardware-in-the-loop simulators may also contain partially simulated共emulated兲 control functions The complexity of practical vehicles, large on-line real-time synthesis is required for developing the control system HILS of-fers the possibility to investigate new chassis control systems with fewer expensive chassis dynamometer experiments and test drives

delivered by dSPACE GmbH and consists of a DSP-processor board and I/O board connected by a fast PHS-bus The main rea-son to use this system is that the main parts of the system already exist and earlier experiences are good The DSP-processor offers a reasonable calculation power 共50MFlops兲, although much more powerful solutions nowadays exist A digital waveform output board is used to simulate the incremental encoder This special board is capable to pulse widths from 250 ns to 26 s with 25 ns accuracy and so clearly satisfies the requirements of 4WS control realization

simulation tool is used in the modeling of the system.SIMULINK

together with the MathWorks’ real-time workshop and dSPACE’s real-time interface makes it possible to generate the whole Hil-model from theSIMULINKmodel The digital control system con-sists ofSIMULINKmodeling software and a dSPACE DS2210 con-troller in a pentium computer The dynamics of the real system is first modeled as a SIMULINK block diagram Required I/O:s are determined by copying corresponding blocks from the real-time interface block library to the simulation model After that, a

c-code is automatically generated, compiled, linked, and

down-loaded into the DSP board The measurements are made using the dSPACE’s trace tool

4.3 Experiment Using Hil Setup In order to evaluate the

performance of the four-wheel steering control system, a series of experiments are performed using Hil setup The Hil simulation

(a)

(b)

Fig 4 „a… Yaw rate response for robust and optimal control

laws.„b… Sideslip angle response for robust and optimal

con-trol laws.

Fig 5 4WS vehicle HILS platform

technique is an efficient way to realistically test the dynamic

ve-hicle behavior in a laboratory A schematic diagram of the

experi-ment is shown in Fig 5 and a Hil simulation platform based on

MATLAB/SIMULINK/dSPACE is developed in Figs 6 and 7

The dynamic behavior in a vehicle induced by a driver steering

maneuver is simulated in the computer The real rear wheel

steer-ing angle is calculated ussteer-ing the feedback signal of the yaw

an-gular velocity The control signal corresponding to the desired

yaw rate is transmitted to the servo motor through an interfacing

board with a D/A converter of dSPACE The real rear wheel

steer-ing angle is measured by an absolute encoder and delivered to the

computer through an interfacing board with an A/D converter The

actual yaw rate and real steering angle are exerted on the vehicle

handling dynamic model

The most interesting results are yaw rate and sideslip angle

response through the HILS platform system during handling

op-eration The results in Figs 8–10 are obtained at a constant

for-ward speed of 30 m/s, with driver steering input as a lane change

maneuver in Fig 8 The four-wheel steering vehicle sideslip angle

and yaw rate are shown in Figs 9 and 10 for comparison with the

HILS and robust control simulation, to illustrate the scale of the

change, which is brought about by the HILS platform using the

␮-synthesis robust controller The results of the Hil setup have

some deviations from the simulation results, and Fig 9 is shown

for comparison with trajectories in yaw rate, to represent vehicle

lateral stability The resulting maximum value of sideslip angle is

0.04 rad共2.29 deg兲 in Fig 10, which explains that the four-wheel

steering vehicle has better steering and active safety performance

Furthermore, the change current of which shows that it follows

the desired yaw rate better

5 Conclusions

In this paper, the robust␮-method has been applied in

design-ing the four-wheel steerdesign-ing system, and proper selection of the

weightings is discussed In order to investigate the robustness of

the synthesized controller for active control, the case of parameter

variation by cornering stiffness to the 4WS is studied Meanwhile,

an optimal controller is also implemented for comparison From the Hil experimental tests, the following conclusions can be drawn that the␮-synthesis method is proved to be effective to cope with the possible vehicle system perturbation and disturbance A four-wheel steering prototype vehicle comprising electric motor, rear steering mechanism and sensors, etc., is constructed Finally, by

Fig 6 Four-wheel steering ECU development platform

Fig 7 dSPACE and control platform

Fig 8 The front wheel steering angle under lane change maneuver

Fig 9 Yaw rate response

Fig 10 Sideslip angle response

experimental work using the Hil setup, a useful method to

inves-tigate the characteristics of a prototype hardware system, it is

shown to produce that the␮-synthesis robust controller can

en-hance the vehicle lateral stability

Acknowledgment

This research is sponsored by the NSFC Fund共Contract Nos

50975047, 60904026, and 50575041兲 and the Southeast

Univer-sity Technology Foundation共Contract No KJ2009346兲

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