Feedforward and Feedback Control for Vehicle Handling Improvement by Active Steering

Copyright @ IFAC Advances in Automotive Control, Karlsruhe, Germany, 2001

FEEDFORWARD AND FEEDBACK CONTROL FOR VEHICLE HANDLING IMPROVEMENT BY ACTIVE STEERING Said Mammar

>,1

• INRETS / LIVIC, 94114, Arcueil, Cedex, France lax .. 33 1 40 43 29 08 e-mail .. [email protected]

Abstract: This paper presents coprime factors and linear fractional transformations (LFT) based feedforward and feedback Hoc control for vehicle handling improvement. Control synthesis uses a linear vehicle model which includes the yaw motion and disturbance input with speed and road adhesion variations. The synthesis procedure allows the separate processing of the reference signal and robust stabilization problem or disturbance rejection. The control action is applied as an additional steering angle, by combination of the driver input and feedback of the yaw rate. The synthesized controller is tested for different speeds and road conditions on a nonlinear model in both disturbance rejection and driver imposed yaw reference tracking maneuvers. Copyright ($12001 1FAC

Keywords: Vehicle handling, Feedforward control, Feedback control, Linear Matrix Inequalities, Hoc control

dling while ensuring similar steady state behavior for both the controlled and the conventional car.

1. INTRODUCTION

As pointed out in (Ackermann and Biinte, 1999), vehicle active steering has attractive benefits with regard to vehicle handling improvement. In fact additional generated lateral forces can be used in order to reject yaw and roll torque disturbances even on decreased road adhesion conditions. Active steering of 2 wheel steering vehicles has been principally studied by Ackerman (see (Ackermann and Biinte, 1999) and reference therein). In (Ackermann and Biinte, 1996) an analytical method which allows robust unilateral decoupling of the yaw rate from the lateral dynamics has been presented. The controller output consists in an additional steering angle obtained by integration of the difference between the reference yaw rate value as commanded by the driver and the actual achieved vehicle yaw rate. Several refinement have been introduced in (Ackermann and Biinte, 1998) in order to improve vehicle han-

1

In this paper a vehicle handling scheme is developed using combination of feedforward and feedback controllers. This configuration allows robust model matching against parameters variations and rejection of lateral forces and torque disturbances which may rise from wind forces. The controller configuration also allows a simple and a direct implementation on actual electrically steered vehicles. Control synthesis is conducted on an LFT model (Mammar, 1998) in order to represent vehicle speed and road adhesion variations while simulation tests are conducted on a nonlinear model. The paper is organized as follow: section 2 introduces the car models used for active steering controller synthesis and analysis. Active steering objectives and synthesis methodology are presented in section 3 which ends with controller implementation. Simulation results are given in section 4.

Also supported by CEMIF, universite d'Evry

125

~

·, ....

7

"'1 .~ ~ . /

89,d!rem! /

lya

.. ..... "..... ' ;........ ::..+..... ....... ~

.............

..····f

, .... .

~~.~ . . . . .f · .. ;· ..:·· .. ·. .•.. ······l ·::.:~.i .... , .-,: .......... . ... ~. ~. .' .. ~ ... + ........... . . ,'-.

' ... ."f

'

··· ... r· :::::· · .. !..... "..... .......-

..... '.~... ",

'"

-,.:~,( .•.. ;

"'

.

. .... ~.

. .. l,

.. r;."

V,hic " rUin.

""

f.:;a

Fig. 2. Damping factor function of speed and road adhesion 2.2 Nominal linear model

Fig. 1. Vehicle and external forces 2. VEHICLE LATERAL MODEL 2.1 Nonlinear model

The model used for control synthesis is derived from the previous model in which the longitudinal velocity is assumed to be constant (xa = v, xa = 0) and all the angles are assumed to be small (Ackermann and Steinhauser, 1993) . One

As we are concerned with lateral control, a simple nonlinear model of a vehicle is obtained by neglecting the roll and pitch motions. This model includes the two translational motion and the yaw motion. According to figure 1, the vehicle wheels are numbered from 1 to 4. The interaction between the tire i and road surface is decomposed into longitudinal forces fz . (A;) and lateral forces f y. (a;). These forces will be detailed below. The nonlinear model is obtained by writing the translational and rotational equations. The following model is finally obtained in the vehicle fixed frame

if;-)

can then use the side slip angle (,8 = as a state variable. It is also assumed equal cornering stiffnesses for the two front wheel (cl Cl C2) and the rear ones (c r C3 C4) . When the wheel base is neglected, the tire side slip angles are reduced to (a1 = a2 = a/) and (a3 = a4 = a r ) . These angles are also considered small and thus the total front and rear lateral forces !t (a I) and fr(a r ) are respectively given by

4

m (ia - Yar)

=L

fz . - kzxa

Ixal

;=1

4

m (fia

+ xar)

=

L fy. -

kyYa Iyal

+ fw

= =

= =

(1)

(8

!t(a/) = fYl

+ f Y2 =

fr(a r ) = lU3

+ fY4 = -Cr (,8 -

Cl

1

Jr

= l, (fYl + fy~) -lr (fY3 + f Y4) + lwfw

The distances , the longitudinal and lateral forces are appearing on figure 1. The nominal values and range of the parameters are given in the appendix. They correspond to an understeering medium class european vehicle.

I: r)

-,8 -

;=1

0

r)

(2) (3)

The nominal forward speed for controller synthesis is chosen as Vo = 15m.s- 1. However v is considered as a varying parameters in the range [1,40] m.s- 1 . The derived model is of second order with state vector (x [,8, rjT) , disturbance input (w fw) and control input, the steering angle (u = 8, ), Simulation of this model shows that the speed and road adhesion variations affect both the transient and the steady state behavior of the vehicle in response to driver steering angle input or lateral wind force input. Increasing speed and road adhesion reduction have the same effect and lead to damping reduction (figure 2).

=

Table 1 Nonlinear formulas of tire side slip angles Tire side slip angles

=

2.3 Vehicle LFT modelling

The longitudinal forces depend directly on the tire slip coefficient (A;) while the lateral forces depend on the tire side slip angles (a;) . The side forces are related to the wheel side slip angles by the cornering stiffness. In the following we will assume linear dependance between the forces and the variables (Jy. (a;) = ¥a;). In reality the forces saturates and the linearity assumption is no more valid. Table 1 summarizes the expression of the tire side slip angles. The longitudinal component are not considered here.

We assume the following parameter variations

+ 01 82) , 1181 11 S 1 Cr = c ro(l + or8d, 1182 11 S 1 v = vo(l + ov 83), 1183 11 S 1

Cl = c/o(l

(4) (5) (6)

The positive scaling factors 01, Or and Ov are used to reflect the magnitude of the deviation from the nominal values c/o, Cro and VD . In order to transform the parameter varying system (equations 1,

126

~

2 and 3) in the form of a minimal order LFT model, 6 fictitious inputs and outputs are needed. One fictitious input and output for each cornering stiffnesses variation, and four fictitious inputs and outputs for the variation of the longitudinal speed. The fictitious input and output vectors are related by (p = 6.q) where the diagonal perturbation matrix is 6. = diag{81 ,82,83,83 , 83,83,}. After some algebra manipulations, the LFT model takes the following form, where the system output is chosen to be the yaw rate T.

~

K.

.T o

Fig. 3. Feedforward control with reference model

(7)

with Fig. 4. Steering mechanism with DC motor steering actuator C fO

Uf

Bp=

[

-Uv 0

mvO

mvo cfO:flf

CrOurlr ---- 0 J

o

Uv 00 0

o

-a22

Dqp

=

0

0

0

0

0

0

0

0

0

0

0] T 1

The above two points summarize the needed vehicle handling improvement (Mammar, 2000) .

'f -Uv vo

'r

Considering vehicle active steering, the tire steering angle u is set in part by the driver through the vehicle classical steering mechanism while an additional steering angle 8c is set by the controller as shown on figure 4 using a DC motor.

--uv vo 0

0 0 0 -Uv 0 cfouf CrOUr -(Tv -O'v 0

--

mvO

• enhancement of vehicle damping responses at all operational speed while ensuring practically identical transient behavior for all speeds and road adhesion. The vehicle has also to present similar steady state behavior as the conventional car without active steering.

00 0]

0

mvo

o o

0

0

o

o

o o

-Uv 0

0

Dqv.= [100 cfO 0 o]T , Dqw

(8)

= [000

m~

_1_ Oo]T

where Ud = ~ , 8d is the steering wheel angle set by the driver and R is the steering gear ratio (R = 21) .

m~

where crO+CfO

all=----

mvO

a21=

Ir crO-lfcfO J

aI2=-I+ a22=-

IrcrO-lfCfO

2

CfO

l~crO+l}c fO JvO

The controller combines a feedforward and a feedback part, its output is thus splitted into two parts Kland K2 and takes the following form

bl=-

mvo

mvO

cfOlf

b2=-J

1 hl=-

(9)

mvo

Controller K 1 acts as a prefilter of the reference signal by adding the feedforward action (up = K1Ud) while the feedback controller K2 ensures robust stability and damping between the reference signal Ud and the output r (figure 3). The feedforward part Kl of the controller has to keep the error signal z small for an entire family of perturbed plants described by the previous LFT model. This fact constitutes robust model matching (Limebeer and Perkins, 1993) .

All other element of 7 are zero. Finally the corresponding transfer matrix from [P'/w , u]T to [q, r] is denoted P (figure 3). 3. ACTIVE STEERlNG DESIGN METHODOLOGY The control objectives are primarily twofold: • steady state rejection of step input disturbances such as side wind force or yaw torque disturbance

In the following a two stages approach is adopted. At the first stage the feedback part K2 of the con-

127

troller is computed using the loop shaping coprime synthesis of (McFarlane and Glover, 1992). 3.1 Feedback part synthesis z

Let G(s) be the nominal transfer function from the steering angle to the yaw rate. In order to reject a constant step input perturbation on the yaw rate, a weighting compensator W of the form of a high DC gain lag filter is introduced according to the loop shaping design methodology (McFarlane and Glover, 1992) (Mammar, 2000). O.ls + 1 W(s) = I000100s + 1

Fig. 5. Feedforward control Assuming that a state-space realization of G / / is given by

(14)

(10)

Let now G. be the shaped plant (G. = WG), using normalized left cop rime description, one can write G. = M;l N•. The closed loop part K2 of the controller is directly computed using the procedure in (McFarlane and Glover, 1992) such that 11

[i.]

a necessary and sufficient condition for the existence of such controller is given by the 3 LMIs (Gabinet and Chilali, 1995) (Giusto and Paganini, 1999)

(I - G.K.)-l [I G.] 1100 ::; 'Y (11)

r

with 'Y = 1.1 { 1 [N. M.] II~ / is a relaxed value of the inverse of the stability margin. The final feedback controller is thus K2 = W Ks.

-11

1 2

zci

ZAT AZ BX ZCfj X;T -x XLi [

3.2 Feedforward part synthesis

G(vo) 0.2s + 1

L2X

0

-I

The controller is finally implemented as shown on figure 6. As the weighting filter W is included in the feedback controller K 2 , disturbances are thus filtered . However, in order to obtain a similar steady state behavior for the conventional and the controlled car, a speed scheduled gain a(v) has to be added to the prefilter part K 1 . The controller implementation needs three type of measurement: measurement of the yaw rate by a gyro, the speed by an odometer and the steering wheel angle by an encoder.

(12)

4. SIMULATION RESULTS

The control up = K 1 Ud will be designed such as the Hoo robust performance level 'Y/ is ensured sup IITzud IIL2--+L2 < 'Y/

0

C2Z

3.3 Controller implementation

This model will ensure the same steady state value for the controlled and the conventional car. The settling time is about 0.8(sec). The reference model is chosen as a first order in order to avoid any overshot on vehicle responses. System of figure 3 is thus updated to the one of figure 5 where G / / includes the reference model and is the transfer matrix from [P, Ud, upf to [q, zf·

~EB~

-x

where .NtR = diag{RR, I} and RR is a matrix whose columns constitute a basis of the kernel of [MT NT Ni], Rand Z are symmetric definite matrices and X is a diagonal matrix which the same structure as 1::... The controller is then reconstructed using classical Hoo optimization (Giusto and Paganini, 1999). The optimization procedure achieves to a 7 order controller.

The reference model To is chosen as first order transfer function with the same steady state gain as the conventional car of the form

=

LIX

R-Z~O

Since the wind force is rejected by the closed loop part of the controller, it will not be further considered for the feed forward part synthesis. The loop is closed by controller K2 and system input reduces to [p, Ud, upf . Let now To be the desired transfer function between Ud and r. The open loop part K 1 of the controller has to keep the error signal z small for an entire family of the LFT modeled perturbed plants. The error signal z is computed from (z = r - Toud) .

To

xLf < 0

CIZ

All the simulations are conducted on the nonlinear model. In all figures, solid lines correspond to the controlled car responses and dotted ones to the conventional car responses.

(13)

128

(a) : yaw rate deg.s- 1

Fig. 6. Active steering controller implementation (a) : yaw rate

(b) : side slip angle (deg)

I:~II---~0"-'-'-'-'"------lj

deg.s- 1

-~~--,.---..--~.---.--~~

(c) : steering angle (deg)

---"10

(b) : side slip angle (deg)

j~!""----------'

rl------'----" \

~: :lf-------'

\'

- : : 0 - - - - - - - - - 1_

_ D · -O~---,----",-.l.... '~':"'-'_-_Q>O"O------''--~

Fig. 8. Wind force rejection for v = 30m.s- 1 and halfroad adhesion (solid: controlled car, dotted : conventional)

(c) : steering angle (deg)

for v = 30m.s- 1 and Ji. = 1. Due to the speed scheduling of the gain parameter a(v), we ensure that the controlled vehicle and the conventional one present the same steady state behavior.

Fig. 7. Wind force rejection for nominal system (solid: controlled car, dotted: conventional)

Firstly a step disturbance wind force of 5000N is applied to the vehicle at nominal speed and full road adhesion. The wind force appears at time tl = 1 sec and disappears at t2 = 2 sec. It is assumed that the driver doesn't react to this disturbance (6 w = 0) . One can note from figure 7 that the yaw rate goes to zero in steady state, i.e. the yaw angle remains constant in steady state. In comparison, the conventional car presents a non zero steady state yaw rate. This means that the yaw angle presents a ramp increasing without driver reaction. In addition, the maximum value of yaw rate during the transient phase is smaller than the one of the conventional car and the disturbance is rejected within driver reaction time.

On the other hand, the controlled yaw rate response of the controlled car closely follows the response of the reference model To. Responses are degraded for high speed and low road adhesion values, however, they practically not present overshot (figures 9-a to l1-a) . The robust model matching of the previous section makes the controlled vehicle to robustly follow the specified first order reference model To. Figures show also a reduction of side slip angle during the transient phase (figure 9-b and 10-b). One can notice that the control effort is limited (figures 9-c to l1-c) . The controller subtracts or adds a steering angle to the driver command. When the road adhesion is at its nominal value even when the speed varies, the control effort vanishes within driver reaction time. When the road adhesion is decreased, there is a remaining steering angle.

Responses for v = 30m.s- 1 and Ji. = 0.5 are given in figure 8. The controller exhibits good stability and performance robustness, in fact responses are still well damped.

5. CONCLUSION

4.1 Disturbance rejection

In this paper, combination of feed forward and feedback Hex; controllers applied to active steering for vehicle handling improvement has been presented. The synthesis methodology simply allows direct specification of time domain objectives. The obtained controller is tested on two typical maneuvers. The controller exhibits good performances and robustness properties face to parameters variations.

4.2 Lane change maneuver The handling improvement is now investigated in case of driver steering angle which corresponds to lane change maneuver (figure 9-c, dotted line). The dash-dot line corresponds to the response of the reference model. Figure 9 shows results obtained at nominal speed with common road adhesion equal to 1. Figure 10 shows results obtained

129

6. REFERENCES

(a) : yaw rate deg.,-l (solid: cont., dotted : conv.)

If\: [ \J --.-c_>

1

Ackermann, J., A. Bartlett D. Kaesbauer W . Sienel and R. Steinhauser (1993) . Robust control: Systems with uncertain physical parameters. Springer. London. Ackermann, J. and T . Biinte (1996) . Handling improvement for robustly decoupled car steering dynamics. 4th IEEE Mediterranean Symposium on New Directions in Control fj Automation. Ackermann, J. and T . Biinte (1998) . Handling improvement of robust car steering. International Conference on Advances in Vehicle Control and Safety. Ackermann, J., D. Odenthal and T. Biinte (1999) . Advantages of active steering for vehicle dynamics control. 92nd International Symposium on A utomotive Technology and A utomation pp. 263-270. Gahinet, P., A. Nemirovski A. J . Laub and M. Chilali (1995). Lmi control toolbox. The MathWorks. Giusto, A. and F. Paganini (1999). Robust synthesis of feedforward compensators. IEEE Transactions on Automatic Control 44, 1578-1582. Limebeer, D. J .N., E. M. Kasenally and D. Perkins (1993). On the design of robust two degree of freedom controllers. Automatica 29.1 , 157-168. Mammar, S. (1998) . Robust automatic steering using fuzzy modelling and lmi control. IFAC Conference on System Structure and Control. Mammar, S. (2000). Two-degree-of-freedom formulation of vehicle handling improvement by active steering. ACC'2000, American Control Conference. M cFarlane , D. and K. Glover (1992) . A loop shaping design procedure using hoc synthesis. IEEE Transactions on Automatic Control 37, 759- 769.

..

(b) : side slip angle (deg) (solid: cont., dotted: conv.)

.0._

.. .

't ._

J~~ ... -. .

~..

-...-c_>

r 9

(c) : steering angle (
llb: --.;

..

·u...--.()"

..

•

Fig. 9. Lane change maneuver, nominal system (a) : yaw rate
1~b?

t .g

(b) : side slip angle (
lE: s::tJ +1:: L .g

(c) : steering angle (
Fig. 10. Lane change maneuver, for nominal road adhesion and speed at 3Om.s- 1 (a) : yaw rate deg.,-l (solid: cont., dotted: conv.)

Nomenclature

lE . L .9 :E::S:d

(b) : side slip angle (deg) (solid: cont., dotted: conv.)

..I.:

\

~

",

:

~/

. ..

./

~

(c) : steering angle (
..

~ / h

I.:

°i

'

lw kz, kll

I . . . . ..

:~

(xGdia)

f3

Fig. 11. Lane change maneuver for v = 3Om.s- 1 and I' 0.5

=

130

vehicle center of gravity (CG) mass and inertia (991Kg, 1574Kg.m2 ) distance from CG to front axle (l.oOm) distance from CG to rear axle (l.46m) wheel-base front cornering stiff. (49.4K N.rad- 1 ) rear cornering stiff. (54.4K N .rad- 1 ) road adhesion (scaling factor [0, 1]) tire-road length contact (l.3cm) long. and lateral forces of the ith tire wind force side slip angle of the ith tire distance of wind force action (O.4m) long. and lateral air drag coefficients CG speed written in vehicle frame vehicle side slip angle yaw rate steering angle