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AUTOMATIC CAR STEERING CONTROL

BRIDGES OVER THE DRIVER REACTION TIME

Jurgen Ackermann and Tilman B unteDLR, German Aerospace Research Establishment

Institute for Robotics and System Dynamics, Oberpfaffenhofen, D-82230 Wessling, Germanyemail: [email protected]

May 24, 1995

Keywords Automobiles, automatic steering, robust con-trol, similarity mechanics

Abstract: A car driver needs at least ve hundred milliseconds beforehe can react to unexpected yaw motions. During this timethe uncontrolled car may produce a dangerous yaw rateand sideslip angle. Automatic steering control for distur-bance rejection is designed such that it bridges over thedriver reaction time, but returns the full steering authorityto the driver thereafter. The solution is robust with respect

to uncertainties in the road-tire contact and in the mass and velocity of the vehicle. The model representation is scaled by methods of similarity mechanics.

1 Introduction

Critical car driving situations arise, when a disturbance tor-que acts on the vehicle. Examples are crosswind, at tire,braking on a slippery road and gas release in a curve. Ittakes the driverat least ve hundred milliseconds to react tothe resulting yaw motion of his car. During thistime the tiresideforce may already reach its physical limits. A delayed

overreaction of the driver may also cause driver-inducedoscillations. Such dangerous situations can be avoided byrobust decoupling of car steering [1, 2]. On the other hand,the driver should not be cheated by the feedback controlsystem. In particular he should feel a similar response tohis steering commands as in the uncontrolled car. This ap-plies both to the immediate reaction after a step input at thesteering wheel and to the required steering-wheel angle forsteady-state cornering. These requirements call for a modi-cation of the decoupling control law.In section 2 of this paper the car steering model is rst sca-led by application of similarity mechanics [3] in order tosimplify the further analysis. In section 3 the properties of

the robust decoupling control law are analyzed. Section 4introduces the modied control law. In section 5 we discusssome of the benets that come from the application of simi-

larity mechanics and the meaning of the various similaritynumbers. Some simulationresults are shown in section 6 tocompare the conventional car with both controlled cars.

2 Scaling of the car steering dynamics

Consider thevehicle of Fig. 1. Vehicledata are usually given

Fig 1: Vehicle with arbitrary mass distribution

in the formof `r

; `

f

; vehicle mass m , and moment of inertiaJ w.r.t. a vertical axis through the center of gravity (CG).

These quantities are related with the quantitiesm

1; m

r

; `

1of Fig. 1 by

m = m 1 + m r (1)J = m 1 `

21 + m r `

2r

(2)m 1 ` 1 = m r ` r (3)

(2) may be expressed using (3) as

J = m

r

`

r

` 1 + m 1 ` r ` 1

and with (1)

J = m `

r

` 1

The resulting length is

` 1 = J

m `

r

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There is a one-to-one relationship between the parametersets f m ; J ; `

r

; `

f

g and f m ; ` 1 ; ` r ; ` f g . We will use ` 1 in thederivation of the model. is the yaw angle between thevehicle center line x and an inertially xed direction x 0 .Input to the system is the front-wheel steering angle

f

,output is the yaw rate r = measured by a gyro. For smallsideslip angle and small steering steering angle

f

thestandard single-track model of car steering [4] is

m v (

+ r )

J r=

f

y

m

z

+

0m

d

(4)

Only a disturbance torquem

d

is assumedand no disturbancelateral force, because the latter is easily compensated by thedriver in the course of his normal steering.The steering force f

y

and torque m z

are generated by thelateral tire forces f

f

(

f

) and f r

(

r

) via

f

y

m

z

=

1 1`

f

? `

r

f

f

(

f

)

f

r

(

r

)

(5)

The tire forces depend on the tire sideslip angles f

and r

as illustrated in Fig. 2 for the front wheel.

Fig 2: Variables of the tire model

The sideslip angle at the front mass 1 will be introdu-ced as a state variable. Thesideslipangles at theCG ( ) andat the axles (

r

) ; (

f

) are related with 1 by the kinematicrelations for small angles

= 1 ? ` 1

v

r

f

= 1 + `

f

? ` 1

v

r

r

= 1 ? ` 1 + ` r

v

r (6)

The local velocity vector ~ vf

forms the (chassis) sideslipangle

f

with the car body and the tire sideslip angle f

with the tire direction, thus

f

=

f

?

f

=

f

? 1 ? `

f

? ` 1

v

r

r

=

r

?

r

=

r

? 1 + ` 1 + ` r

v

r (7)

In this paper we do not use rear wheel steering, i.e.r

0.The tire force characteristics are linearized as

f

f

(

f

) = c

f

f

f

r

(

r

) = c

r

r

(8)

where the cornering stiffnesses cf

and cr

are uncertain pa-rameters that vary with the road tire contact. We assumec

f

= c

f 0 ; c r = c r 0 where c f 0 and c r 0 arenominal valuesfor the dry road and 2 ? ; 1 ] ; ? > 0 is an uncertainparameter. Other uncertain parameters are the vehicle massm 2 m

? ; m + ] and velocity v 2 v ? ; v + ] . Themodel (4),(5) with the above equations substituted becomes

2

6

4

m v (

1 ?

` 1v

r + r )

m `

r

` 1 r

3

7

5

=

2

4

1 1

`

f

? `

r

3

5

2

6

4

c

f 0 ( f ? 1 ? `

f

? ` 1v

r )

c

r 0 ( ? 1 + `

1+ `

r

v

r )

3

7

5

+

2

4

0

m

d

3

5

and, solving for 1 and r ,

2

4

1

r

3

5

=

2

6

6

6

4

`

f

+ `

r

m `

r

v

0

`

f

m ` 1 ` r?

1m ` 1

3

7

7

7

5

2

6

4

c

f 0 ( f ? 1 ? `

f

? ` 1v

r )

c

r 0 ( ? 1 + ` 1 + ` r

v

r )

3

7

5

?

?

2

4

1

0

3

5

r +

2

6

6

6

4

1m `

r

v

1m ` 1 ` r

3

7

7

7

5

m

d

(9)

This form shows that 1 does not depend directly on theforces at the rear tires. An indirect coupling occurs throughthe r -terms in the equation for 1 . These r -terms will be

cancelled by thedecouplingcontrol law. Thestate equationsof the system are

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2

4

1

r

3

5

=

2

4

c 1 c 2

c 3 c 4

3

5

2

4

1

r

3

5

+

+

2

6

6

6

4

c

f 0?

`

f

+ `

r

m `

r

v

c

f 0 ` f

m ` 1 ` r

3

7

7

7

5

f

+

2

6

6

6

4

1m `

r

v

1m ` 1 ` r

3

7

7

7

5

m

d

(10)

c 1 = ? c

f 0?

`

f

+ `

r

m `

r

v

c 2 = ? 1 ? c

f 0?

`

f

+ `

r

?

`

f

? ` 1

m `

r

v

2

c 3 =

?

c

r 0 ` r ? c f 0 ` f

m ` 1 ` r

c 4 = ?

` 1?

c

f 0 ` f ? c r 0 ` r ? c f 0 `2f

? c

r 0 `2r

m ` 1 ` r v

This equation contains seven independent parameters( ` 1 ; ` f ; ` r ; m ; v ; c f 0 ; c r 0) involving three basic units (m,kg, s). Thus we know by Buckinghams theorem [3] that7 ? 3 = 4 dimensionless similarity numbers sufce to

characterize the system, provided that also the variablesand the time are scaled appropriately. There are many waysto choose such similarity numbers. We have chosen a set of numbers with some physical meaning and the property thatthe uncertain operating point m ; v ; enters only into onenumber. Let

P

J

=

` 1

`

f

mass distribution number

P

`

=

`

r

`

f

CG location number

P

c

=

c

r 0

c

f 0 cornering stiffness ratio

P

v

= v

r

m

c

f 0 ` foperating point number

(11)

Also introduce the dimensionless state and input variablesrespectively

r = `

f

v

r

1 = 1

f

=

f

m d

=

1c

f 0 ` fm

d

(12)

and scale the time by

t = t

r

c

f 0

m `

f

(13)

In the Laplace transformed equation (10) with s 1 ( s ) ands r ( s ) on the left hand side, the scaled Laplace variablebecomes

s = s

s

m `

f

c

f 0(14)

such that s

t = s t

. With the above substitutions(11),(12),(14), the Laplace transformed stateequations (10)become

2

4

s 1 ( s )

s r ( s )

3

5

=

2

4

c 5 c 6

c 7 c 8

3

5

2

4

1 ( s )

r ( s )

3

5

+

+

2

6

6

6

4

1P

v

1 + 1

P

l

1P

J

P

l

P

v

3

7

7

7

5

f

( s ) +

2

6

6

6

4

1P

l

P

v

1P

J

P

l

P

v

3

7

7

7

5

m d

( s ) (15)

c 5 = ? 1

P

v

1 + 1

P

l

c 6 = P

J

? 1 + P l

?

P

J

? 1 ? P 2v

P

l

P

v

c 7 = 1

P

v

P

J

P

c

?

1P

l

c 8 = P

J

? 1 ? P c

P

l

( P l

+ P

J

)P

J

P

l

P

v

3 Robust decoupling control law

Robust decoupling [1] is achieved by the feedback controllaw

f

( s ) =

1s

F ( v ) i

L L

( s ) ? r ( s ) +

`

f

? ` 1

v

s r ( s )

(16)

L is the steering wheel input, i L denotes the transmissionratio of the steering gear. F ( v ) is the steady-state yaw rateresponse to a unit step input for a nominal model ( = 0

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and m = m 0). It can be computed by (10) with 1 = r = m

d

= 0 andf

= 1.

F ( v ) =

c

f 0 c r 0 ( ` f + ` r ) v

c

f 0 c r 0 ( ` f + ` r )2

+ ( c

r 0 ` r ? c f 0 ` f )m 0

0v

2(17)

Fora general investigation, again the scaled (dimensionless)representation is employed. Therefore we introduce the di-mensionless variable

L

= i

L L

(18)

The introduction of a nominal virtual mass m 0 = 0 makes itnecessary to dene a new similarity number.

P

m

=

m 0

0

m

virtual mass number (19)

The dimensionless form of (17) is

F =

`

f

v

F ( v )

=

P

c

( 1 + P l

)

P

c

( 1 + P l

)

2+ ( P

c

P

l

? 1 ) P m

P

2v

(20)

With the above substitutions (11),(12),(14),(18),(20) the

control equation (16) becomes

f

( s ) = 1s

((1 ? P J

) s ? P v

) r ( s ) + P v

F

L

( s ) (21)

Consider now the lateral acceleration a 1 at the front massm 1 , see Fig. 1. It is related with the model quantities by

a 1 = a C G + ` 1 r = f

f

(

f

) + f

r

(

r

)

m

+ ` 1 r (22)

We substitute (7) and (8) to obtain the output equation fora 1 (Laplace-transformed)

a

1( s ) = ?

( c

f 0 + c r 0 )

m

1( s ) +

c

f 0

m

f

( s ) +

+

` 1 s + c

r 0 ( ` r + ` 1 ) ? c f 0 ( ` f ? ` 1 )

m v

r ( s ) (23)

The dimensionless lateral acceleration at the front mass isdened as

a 1 = `

f

v

2a 1 (24)

Equation (23) now can be rewritten in its dimensionlessrepresentation

P

2v

a 1 ( s ) = ? ( 1 + P c ) 1 ( s ) + f ( s ) +

+ ( P

J

? 1 + P c

( P

J

+ P

l

) + P

J

P

v

s ) r ( s ) (25)

The transfer function from the yaw disturbance torque tothe yaw rate (

L

( s ) 0) is obtained by substituting (21)into (15) and solving for r ( s ) .

rd e c

( s ) = R

d e c

( s )D

d e c

( s )m

d

( s ) (26)

R

d e c

( s ) = s R ( s ) = s ( P v

s + 1 + P c

)

D

d e c

( s ) = ( P l

P

v

s + 1 + P l

)?

P

J

P

v

s 2 + ( P J

+ P

l

) P c

s + P c

P

v

The properties of the decoupling control law are

The coupling from the yaw mode into the lateral ac-celeration a 1 has been removed. Thereby the steeringtransfer function from

L

( s ) to a 1 ( s ) is of rst order( m

d

( s ) 0).

a 1 ( s ) = ( 1 + P

l

)

F

1 + P l

+ P

l

P

v

s

L

( s ) (27)

The inuence of the uncertain parameters m , v and

has been drastically reduced by decoupling. Theore-tically the steering dynamics can be made arbitrarilyfast by additional feedback of a 1 , measured by an ac-celerometer, to

L

. The task of the driver is only tokeep the point mass m 1 on top of his planned path bycommanding a 1 . He does not have to care about thestable yaw motion [1].

For a step input at the yaw disturbance m d

(e.g. fromcrosswind, -split braking) the conventional car hasa nonzero steady state value of the yaw rate, whereasfor the decoupled car the yaw rate quickly returns tozero [2]. The steady-stateeffect can be seen from(26),

where lim s ! 0R

d e c

( s )

D

d e c

( s )

= 0. Considerable safety ad-

vantages have been shown in [2].

On the other hand, the decoupling control law (16) also hassome disadvantages that motivate the modications in thenext section.

By the integration in the control law there is no imme-diatereaction of a 1 aftera step command at L .Thedri-ver feels that the car is not reacting as promptly as theconventional car. Therefore a direct throughput from

L

tof

will be provided, i.e.f

= i

L L

+

c

, wherec

is produced by the controller (16). This change does

not affect the feedback path from r to f and thereforethe disturbance attenuation properties of the controllaw (16) are preserved.

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The yaw damping is decreased at high velocity. In [2]a good yaw damping was achieved by rear-wheel stee-ring, which is not available under the assumptions of this paper. The driver is not so much concerned aboutthe rst overshoot in r (which is similar to the conven-tional car) but about the following undershoot, whichoccurs about 0 : 8 seconds after a steering wheel stepinput. One remedy is a velocity scheduled feedback of the lateral rear axle acceleration a

r

[5]. Another oneis to provide the decoupling action only for the rst0 : 5 seconds after a step input and thereafter to returnsmoothly to the properties of the conventional car, i.e.

c should follow the integral action only initially andthen return to zero. Regarding a disturbance step m d

this concept helps thedriverduring therst half secondby providing an additional steering angle but thereaf-ter it returns the full authority gradually back to thedriver. Reasonable yaw damping can be achieved bytuning the parameters of the lter which is employedto implement the fading decoupling action.

The feedback from r provides an additionalterm tof

.In steady-state cornering there is no need for this term,the controlled car should behave like the conventionalcar in this situation, i.e. the driver should apply thesame

L

. The above modication takes care of thisrequirement.

4 The modied control law

The modied control law is

f

( s ) = i

L L

( s ) +

c

( s ) (28)

c

( s ) =

s

s

2+ 2 D ! 0 s + ! 20

x 1 ( s )

x 1 ( s ) = F ( v ) i L L ( s ) ? r ( s ) + `

f

? `

1v

s r ( s )

Now there is a direct throughputof the input by the steeringwheel

L

to the front wheel steering angle. The pure inte-grator has been replaced by a dynamical lter. The initialbehaviour ( s ! 1 ) of this lter to any input is the sameas the response of an integrator. But the steady-state outputof the lter ( s ! 0) is zero. In the meantime the lter isunloaded so that after some time the responsibility is softlyreturned to the driver. It also leads to the same steady-statecornering as of the conventional car.Hence by the modied control law both the immediate andthe long term reaction to any driver input is the same for

the controlled car as for the uncontrolled car. But the tran-sient behaviour is different as well as the disturbance rejection property of both systems. Since we add additional

dynamics to the system by applying the modied controllaw, new parameters are introduced to the system. They areindependent of each other and of the already introducedparameters. This increases the number of independent pa-rameters to ten ( ` 1 ; ` f ; ` r ; m ; v ; c f 0 ; c r 0 ; m 0 = 0 ; D ; ! 0).According to Buckinghams theorem the number of simila-rity numbers which sufce to characterize the system nowis 10 ? 3 = 7. In addition to (11) and (19) we dene twomore similarity numbers

P

D

= D lter damping number

P

!

= ! 0

s

m 0 ` f

0 c f 0lter bandwidth number

(29)

With the substitutions (11),(12),(14),(18),(29) the controlequation (28) becomes

f

( s ) = L

( s ) + s

s 2 + 2 P

D

P

!

p

P

m

s + P

2!

P

m

x 1 ( s ) (30)

x 1 ( s ) = ((1 ? P J ) s ? P v ) r ( s ) + P v F L ( s )

5 Meaning of the similarity numbers

Using a dimensionless representation of a systems equati-ons and introducing dimensionless similarity numbers hasseveral benets. Most interesting with respect to robust con-trol theory is the fact, that the number of uncertain pa-rameters of a system possibly can be reduced. Thus theanalysis and design of controlled systems are simplied.Without employing similarity mechanics, for the analysisof the model (10) a two-dimensional operating domain hasto be checked. The three uncertain parameters ( ; m ; v )have to be varied in order to cover the whole operating

domain. Since

andm

only occur in the combination of m = , the variation of uncertain parameters of this plant isa two-dimensional problem.After making use of Buckinghams theorem, we succee-ded to put all uncertain parameters of the conventional carinto one similarity number P

v

. Hence all possible operatingpoints of the system in itsdimensionless form can be repre-sented by one single number. The other similarity numbers( P

J

; P

l

; P

c

) are invariant for a specic car due to construc-tion, tire properties etc.If for one value of P

v

an analysis is performed (e.g. a simu-lation or an eigenvalue computation), the results stand foran innite number of uncertain parameter values, i.e. one

can obtain the same P v for either large values of v or m ora small value of . The dimensionless time-domain resultsof a simulation can be transformed to original time-domain

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0 5 100

0.1

0.2

0.3

0.4

0.5

0.6Steering Wheel Step Input

Dimensionless Time

Y a w

R a

t e

0 5 100.05

0

0.05

0.1


Dimensionless Time

A d d i t i o n a

l S t e e r i n g

A n g

l e

0 5 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4Yaw Disturbance Torque Step Input

Dimensionless Time

Y a w

R a

t e

0 5 100.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0Yaw Disturbance Torque Step Input

Dimensionless Time

A d d i t i o n a

l S t e e r i n g

A n g

l e

3 2 1 0

2

1

0

1

2

3Eigenvalues

I m

Re

conventional car

decoupled car

filterequipped car

Pc = 2.1, Pl = 0.874, PJ = 1

Pv = 1, Pm = 1

Pomega = 0.2, PD = 0

Time Scale: 0.2

Fig 3: Response to steering wheel step and yaw disturbance torque step for P v

= 1

by rescaling the time and variables.With the robust decoupling control law a new similaritynumber was needed, because the use of a nominal modelintroduced a new scale to the system. Every new indepen-dent parameter which is added to the system brings with ita new similarity number, unless it brings a new physical ba-sic unit with it also (Buckingham). By the modied controllaw two more lter parameters and similarity numbers areneeded to sufciently characterize the system.In the following simulation study we used the data of aBMW 735i passenger car to determine the values of thecer-

tain similarity numbers ( P l = 0 : 874 ; P c = 2 : 1 ; P J = 1 : 0)as wellas theoperatingdomain of the car (1 : 0 P

v

8 : 0).A lower limit for P

v

is necessary since thecontroller is swit-ched on only after a certain velocity is reached. The car isnot controllable for v = 0, i.e. P

v

= 0. The virtual massnumber P

m

is a new uncertain similarity number althoughit can be inuenced by the choice of the nominal values( 0 ; m 0). Concerning the modied control law, the valuesof P

!

and P D

can be set in the controller design for a speci-c car. We choose a lter bandwith ! 0 = 1 = s e c to providethe above described properties (the decoupling action issupposed to fade out after ca. 0 : 5 seconds). With the dataof the car we have P

!

= 0 : 2. The damping of the lter istuned by the value of D . It turned out, that a scaling withvelocity of this parameter is useful to achieve a reasonable

damping at all velocities.

D = f ( v ) = f

v

q

m 0 = ( 0 c f 0 ` f )

P

D

= f

P

v

p

P

m

(31)

6 Simulation results

In the following simulation plots of the scaled yaw rate rand additionalsteering angle

c

the three controllerversionsare compared.

The conventional car. Equation (15) is employed tocompute the yaw rate response to yaw disturbancetorque step inputs or steering wheel step inputs res-pectively. The "control law" is just

f

( s ) = L

( s ) .The additional steering angle is zero. Dotted line styleis used for the dimensionless time response plots andcircles denote the eigenvalue position in the dimen-sionless s -plane.

The decoupled car with direct throughput. To obtaincomparable results with the two other versions of the

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0 10 200

0.02

0.04

0.06

0.08

0.1

0.12


Dimensionless Time

Y a w

R a

t e

0 10 200.6

0.5

0.4

0.3

0.2

0.1

0

0.1


Dimensionless Time

A d d i t i o n a

l S t e e r i n g

A n g

l e

0 10 200.05

0

0.05

0.1


Dimensionless Time

Y a w

R a

t e

0 10 201

0.8

0.6

0.4

0.2


Dimensionless Time

A d d i t i o n a

l S t e e r i n g

A n g

l e

0. 8 0. 6 0. 4 0. 2 0 0 .2

0

0.5

1

1.5

Eigenvalues

I m

Re

conventional car

decoupled car

filterequipped car

Pc = 2.1, Pl = 0.874, PJ = 1

Pv = 8, Pm = 1

Pomega = 0.2, PD = 1.5

Time Scale: 0.2

Fig 4: Response to steering wheel step and yaw disturbance torque step forP

v

=

8

0 20 400

0.05

0.1

0.15

0.2

0.25

0.3


Dimensionless Time

Y a w

R a

t e

0 20 400

0.1

0.2

0.3

0.4

0.5


Dimensionless Time

A d d i t i o n a

l S t e e r i n g

A n g

l e

0 20 400.05

0

0.05

0.1

0.15


Dimensionless Time

Y a w

R a

t e

0 20 401

0.8

0.6

0.4

0.2


Dimensionless Time

A d d i t i o n a

l S t e e r i n g

A n g

l e

0. 8 0. 6 0. 4 0. 2 0 0 .2

0

0.5

1

1.5

Eigenvalues

I m

Re

conventional car

decoupled car

filterequipped car

Pc = 2.1, Pl = 0.874, PJ = 1

Pv = 4.5, Pm = 0.5

Pomega = 0.2, PD = 1

Time Scale: 0.2828

Fig 5: Response to steering wheel step and yaw disturbance torque step for P m

= 0 : 5

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car, the direct throughput is combined with the robustdecoupling control law (21). Thus the steering angleequation reads

f

( s ) = L

( s ) + 1s

((1 ? P J

) s ? P v

) r ( s ) +

+ P

v

F

L

( s )

Dashed line style for the resonses and star markers forthe eigenvalues are employed.

The lter-equipped car. It is described by equation

(15) extended by the steering angle equation of themodied control law (30). Plots use solid line style forthe responses and x-marks for the eigenvalues.

The time scale t = t = P !

= ( ! 0p

P

m

) (see (13)) is printedbelow the eigenvalue plot. Original time scale is obtainedby multiplying the dimensionless time of the plots with thespecied time scale and original eigenvalue scale is obtai-ned by dividing the dimensionless s by the same time scalesuch that s t = s t .In the rst simulationof Fig. 3 a steering wheel step inputisapplied to the three cars at a small value of P

v

= 1 (low carmass at low velocity on dry road). For the choosen value of P

D

there is not much difference between the three versions.Both the decoupled and the lter-equipped car perform alittle overshoot in the yaw rate. With the simulation in Fig.3 for a disturbance torque step input the concept of fadingout the decoupling action is well recognizeable. Within therst 0 : 5 seconds both the decoupled and the lter-equippedcar provide the same disturbance rejection activity. Howe-ver after 0 : 5 seconds for the lter-equipped car the steeringauthority is softly returned to the driver and the additionalsteering angle

c

( s ) = f

( s ) ? L

( s ) goes to zero. Onlynow the driver is supposed to react to the yaw disturbance.The right upper plot of Fig. 3 shows the eigenvalues forP

v

= 1. They are well damped for all three controllers.

ConcerningFigs.4 and 5, forevery complex eigenvaluepairof the lter-equipped car three more curves (dash-dotted)are added to the eigenvalue plot to simplify comparison.One is a natural frequency circle around the origin throughthe eigenvalues. The other curves are damping lines whichconnect the origin with the eigenvalues.Fig. 4 shows the responses for P

v

= 8 which correspondse.g. to higher speed on wet road. The yaw damping forall controllers is worse than at lower values of P

v

. Theworst damping occurs for the decoupled car. The oscilla-tions of the lter-equipped car after a steering wheel stepinput decrease as quickly as those of the conventional caralthough the damping of the corresponding eigenvalues is

less but their bandwidth is higher. The yaw disturbance re-sponses show again how the initial decoupling action isfading out. Fig. 5 shows the result of deviations ( P

m

6= 1)

of the real car from thenominal model w.r.t. the virtual massat a medium velocity ( P

v

= 4 : 5). Here P m

equals 0 : 5. Thesteady-state responsesof the car with lter-feedback are t hesame as for the conventional car, whereas the decoupled carhas a nonzero steady-state value of the additional steeringangle.

7 Conclusions

The lter-equipped car can be considered as a compromisebetween the conventional car which has bad yaw distur-bance attenuation properties and the decoupled car whichexhibits poor yaw damping at higher velocities. Within theminimum driver reaction time of 0 : 5 seconds, the lterequipped car behaves like the decoupled car and then itreturns smoothly to the steady-state behaviour of the con-ventional car. By using themodied control law of thelter-equipped car instead of the pure decoupling control law theyaw damping properties are considerably improved.

References

[1] J. Ackermann, Robust decoupling of car steering dy-namics with arbitrary mass distribution, in Proc. Ame-rican Control Conference , vol. 2, (Baltimore, USA),pp. 19641968, 1994.

[2] J. Ackermann and W. Sienel, Robust yaw damping of cars with front and rear wheel steering, IEEE Trans. onControl Systems Technology , vol. 1, no. 1, pp. 1520,1993.

[3] G. Rumpel and H. Sondershausen,Ahnlichkeitsmechanik, in Dubbel, Taschenbuch f ur den Maschinenbau, 15. Auage (W. Beitz and K.-H.Kuttner, eds.), pp. 175178, Berlin: Springer Verlag,1986.

[4] M. Mitschke, Dynamik der Kraftfahrzeuge , vol. C. Ber-lin: Springer, 1990.

[5] W. Sienel, Design of robust gain scheduling control-lers, in Proc. Robust and Adaptive Control TutorialWorkshop , (Dublin), 1994.

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