WHEEL SLIP CONTROL IN ABS BRAKES USING GAIN
SCHEDULED CONSTRAINED LQR
Idar PetersenW , Tor A. JohansenW*WW , Jens KalkkuhlWWW and Jens LüdemannWWW
SINTEF Electronics and Cybernetics, N-7465 Trondheim, Norway. e-mail: Idar.Petersen@ecy.sintef.no
Department of Engineering Cybernetics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway.
DaimlerChrysler, Research and Technology, Alt-Moabit 96a, D-10559 Berlin, Germany.
Keywords: Automotive control, gain scheduling, nonlinear
systems, robustness, stability
Abstract
A wheel slip controller for ABS brakes is formulated using an
explicit constrained LQR design. The controller gain matrices
are designed and scheduled on the vehicle speed based on local
linearizations. A Lyapunov function for the nonlinear control
system is dervied using the Riccati equation solution in order
to prove stability and robustness with respect to uncertainty
in the road/tyre friction characteristic. Experimental results
from a test vehicle with electromechanical brake actuators and
brake-by-wire show that high performance and robustness are
achieved.
1 Introduction
An anti-lock brake system (ABS) controls the slip of each
wheel to prevent it from locking such that a high friction
is achieved and steerability is maintained. ABS controllers
are characterized by robust adaptive behaviour with respect to
highly uncertain tyre characteristics and fast changing road surface properties [1].
The introduction of advanced functionality such as ESP (electronic stability program), drive-by-wire and more sophisticated
actuators offers both new opportunities and requirements for
higher performance in ABS brakes. The contribution of the
present work is a study of a model-based design of ABS controllers, see also [2], [3]. We consider electromechanical actuators [4] rather than hydraulic actuators, which allow continuous adjustment of the clamping force. Here, we design
the wheel slip controller based on a recently developed explicit
LQR design method that takes into account input and state constraints [5], see also [6]. The wheel slip dynamics are highly
nonlinear. Despite this, our control design relies on local linearization and gain-scheduling. In order to analyse the effects
of this simpli¿cation, we develop a Lyapunov based stability
and robustness analysis. Results from experiments using a test
vehicle are also included. Other studies of model-based wheel
slip control in ABS can be found in [7], [8], [9], [10].
2 Modelling
In this section, we review a mathematical model of the wheel
slip dynamics, see also [1], [8] and [7]. The problem of wheel
slip control is best explained by looking at a quarter car model
as shown in Figure 1. The model consists of a single wheel
attached to a mass p. As the wheel rotates, driven by inertia
of the mass p in the direction of the velocity y> a tyre reaction
force I{ is generated by the friction between the tyre surface
and the road surface. The tyre reaction force will generate a
torque that initiates a rolling motion of the wheel causing an
angular velocity $. A brake torque applied to the wheel will
act against the spinning of the wheel causing a negative angular
Figure 1: Quarter car forces and torques.
acceleration. The equations of motion of the quarter car are
pyb @ I{
M $b @ u I{ We vljq+$,
where
y
$
I}
I{
We
u
M
(1)
(2)
horizontal speed at which the car travels
angular speed of the wheel
vertical force
tyre friction force
brake torque
wheel radius
wheel inertia
The tyre friction force I{ is given by
I{ @ I} +> K > ,
(3)
where the friction coef¿cient is a nonlinear function of
K
tyre slip
friction coef¿cient between tyre and road
slip angle of the wheel
and the longitudial slip de¿ned by
@
y $u
y
(4)
describes the normalised difference between horizontal
speed y and the speed of the wheel perimeter $u. The slip value
of @ 3 characterises the free motion of the wheel where no
friction force I{ is exerted. If the slip attains the value @ 4>
then the wheel is locked which means that it has come to a
standstill.
The friction coef¿cient can span over a very wide range,
but is differentiable with the property +3> K > , @ 3 and
+> K > , A 3 for A 3. Its qualitative dependence on
slip is shown in Figure 2. The friction coef¿cient increases
with slip up to a value where it attains its maximum K .
1
α =0
Proof. Note that +w, is a continuous trajectory. Hence, the
possibile escape points are @ 3 and @ 4. Consider ¿rst
u W 3
@ 3. Since +3, @ 3, it follows from (5) that b @ yM
e
due to We 3. Hence, +3, 3 implies +w, 3 for all w 3.
Consider next @ 4. Then, $ @ 3 and from (2) it follows that
$b 3 due to the discontinuity vljq+$, in (2). From (4), we
conclude that b 3 and +3, 4, which implies +w, 4 for
all w 3. Finally, note that yb 3 from (1) because I{ 3 for
5 ^3> 4`.
o
0.8
α = 10
o
µ
0.6
0.4
0.2
0
0
0.2
0.4
0.6
λ
0.8
1
3 Control problem
1
The control problem is essentially to control the value of the
longitudinal slip to a given setpoint that is either constant or commanded from a higher-level control system such as
ESP (electronic stability program). The controller must be robust with respect to uncertainties in the tyre characteristic and
variations in the road surface conditions.
0.9
µ H =0.95
0.8
µ H =0.8
0.7
0.5
x
µ (λ)
0.6
µ H =0.5
0.4
The control input is the clamping force Ie that is related to
the brake torque as We @ ne Ie . A maximum or a minimum
force can be applied to the brake pads by the actuator during
braking. The (small) minimum is to ensure that the pads are
positioned close to the brake disk (no air-gap). Maximum is
what the actuator is capable of. There is also a rate limit at how
fast the torque can be changed by the actuator.
µ H =0.3
0.3
0.2
µ H =0.1
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
λ x (longitudinal slip)
0.8
0.9
1
Figure 2: Tyre slip/friction curves +> K > ,
For higher slip values, the friction coef¿cient will decrease to
a smaller value where the wheel is locked and only the sliding friction will act on the wheel. The dependence of friction
on the road condition is shown in the bottom part of Figure 2.
For wet or icy roads, the maximum friction K is small and the
right part of the curve is Àatter. The tyre friction curve will also
depend on the brand of the tyre. In particular, for winter tyres,
the curve will cease to have a pronounced maximum.
If the motion of the wheel is extended to two dimensions, then
the lateral slip of the tyre must also be considered. The slip angle is where the wheel moves with velocity y{ in the longitudinal direction and with a velocity y| in the lateral direction.
In this case, the longitudinal slip { @ y{ y $u and the lateral
slip | @ vlq are distinguished as well as the corresponding friction coef¿cients { and | . The top part of Figure 2
shows the dependence of the friction coef¿cient on the side
slip angel = The side force I| depends greatly on the side slip
angle . For large side slip angles, the force gets smaller. In
the sequel, for simpli¿cation purposes unless otherwise stated,
the side slip angle will be considered to be zero with { @ and y{ @ y=
Using (1)-(4) we get for y A 3 and $ 3
5
b @ 4 4 +4 , . u I} +> K > , . 4 u We
y p
M
yM
(5)
4
(6)
yb @ I} +> K > ,
p
Note that when y $ 3, the dynamics of the open loop system
(from We to ) becomes in¿nitely fast with in¿nite gain. This
leads to a loss of controllability and the slip controller must be
switched off for small y.
Proposition 1 Consider the system (5)-(6) with We +w, 3 for
all w 3. If y+3, A 3 and +3, 5 ^3> 4`> then +w, 5 ^3> 4` and
y+w,
b 3 for all w 3 where y+w, A 3=
Integral action or adaptation must be incorporated to remove errors due to model inaccuracies, in particular the road/tyre friction coef¿cient K . It is essential that the controller maintains
a high performance and is robust w.r.t. to any road/tyre friction
curve that can be encountered cf. Figure 2.
4 Gain-scheduled LQRC controller design and
analysis
The dynamics of the wheel and car body are given by (5) and
(6), respectively. Due to large differences in inertia, the wheel
dynamics and car dynamics will evolve on signi¿cantly different time scales. The speed y will change much more slowly
than the slip and is therefore a natural candidate for gain
scheduling. Thus, for the control design, we consider only
(5) and regard y as a slow time-varying parameter. A constrained LQR controller [5] is applied. This requires a nominal
linearized model for design.
4.1 Linearized dynamics
Let ( > Wae ) be an equilibrium point for (5) de¿ned by the constants > I} and K , where is the desired slip
M
+4 , . u I} + > K > ,
Wae @
pu
The linearized slip dynamics are given by
4
4
+ , . +We Wae , . h.o.t.
b @
y
y
where 4 and 4 are linearization constants given by
u5 C 4
+4 , .
+ > K > ,
4 @ I}
p
M C
4
. I} + > K > ,
p
u
4 @ A 3
M
(7)
(8)
(9)
namely
The wheel slip dynamics (5) can be written in the form
{b 5 @
where
!+{5 , @ .
!+{5 , 4
. +We We ,
y
y
5
4
u
+4 {5 , .
p
M
(10)
I} +{5 . > K > ,
u W
M e
(11)
and {5 @ and
M
We @
+4 , . u I} + > K > ,
pu
(12)
It can be seen that this system has an equilibrium point given
by {5 @ 3, We @ We and !+3, @ 3, and the linearized slip
model (7) with a perturbation term written as follows
{b 5 @
4
4
+{5 ,
{5 . +We We , .
y
y
y
(13)
4.2 Wheel slip control with integral action
Let the system dynamics be augmented with a slip error integrator {b 4 @ @ {5
{b 4
{4
@
D+y,
{b 5
{5 . E+y, +x We , . Z +y, +{5 ,
(14)
D+y, @
3
3
4
4
y
> E+y, @
3
4
y
> Z +y, @
3
4
y
{b @ D+y, +{ { , . E+y, +x x , . Z +y, +{5 ,
(15)
Next, de¿ne the quadratic cost function for the purpose of local
LQ design based on the nominal part of (15):
] 4
W
M+{+w,> x^w> 4,, @
++{+ , { , T+y, +{+ , { ,
w
. +x+ , x ,W U +x+, x ,,g
(16)
and the controller gains are
4@5
N4 +y,@ T4>4 +y,U4
4 . 54 . 45 U4 T5>5 +y, .
N5 +y,@
4
5+
T4>4 +y,U,4@5 y
4
4@5
(19)
for some Wepd{ A Weplq 3. Furthermore, we de¿ne the error
(note that both x and x
a depend on y and {)
v +{> y, @ 4 +x xa,
(20)
With the de¿nition +{> y, @ v +{> y, . +{5 , and assuming
{4 @ We @N4 +y,, the closed loop dynamics can be written as
(21)
with { @ { { .
Proposition 2 Consider the system (5) with controller (19).
Assume U A 3 and the smooth matrix-valued function T satis¿es T4>5 +y, @ T5>4 +y, @ 3> T4>4 +y, A 3> T5>5 +y, A 3>
gT4>4 +y, 3> gT5>5 +y, 3 for all y A 3. Moreover, suppose
gy
gy
S5>4 +y,F
+4 ,
(22)
T4>4 +y,A
# y
$
{5 ,
5 +
{5 ,S5>5 +y,
{5 S5>4 +y,5 +
T5>5 +y,
.
{55A
+4 ,
y
yF
(23)
Assuming constant y, the optimal control law is given by
3
(17)
where the gain matrix N+y, @ U4 E W +y,S +y, and we
choose to neglect the unknown { and x which will be accounted for due to the integral action. The symmetric matrix
S +y, A 3 is de¿ned by the solution to the algebraic Riccati
equation for the design:
S +y,D+y, . DW +y,S +y,
S +y,E+y,U4 E W +y,S +y, @ T+y,
4@5
+T4>4
4
y
4@5
S4>5 +y, @
+T4>4 +y,U,
4
4
S5>5 +y, @ 5 4 y
4 U
4@5 4@5
45 . 45 U4 T5>5 +y, . 5+T4>4+y4,U, y
.
y
45 U4
{
b @ +D+y, . E+y,N+y,, {
 . Z +y,+
{5 > y,
Since We is assumed unknown, we have de¿ned x @ We , and
the equilibrium point to be { @ +{4 > 3,W and x @ We where
{4 is still unspeci¿ed and we get
xa @ N+y,{
T4>4 +y,U,4@5 y
4
4@5
+y,U,
5+
We introduce the saturated control
+ pd{
We > xa A Wepd{
x@
Weplq > xa ? Weplq
x
a>
otherwise
where +{5 , @ !+{5 , 4 {5 .
where
S4>4 +y, @
54 . 45 U4 T5>5 +y, .
(18)
3
3
3
G+y,@S4>4 +y,S5>5 +y, S4>5 +y,S5>4 +y, A 3
(24)
are satis¿ed for all y A 3, {
5 5 ^ > 4 ` and some F A 3
pd{
A We A Weplq , the equilibrium {
@3
and 5 +3> 4,. If We
is uniformly exponentially stable.
Proof. Consider a neighbourhood [ 3 of the origin in U5 such
that v +
{> y, @ 3 for all { 5 [ 3 and y A 3. Let a Lyapunov
function candidate be
Y +
{, @ {W S +y,
{
Its time-derivative along trajectories of (21)
CS +y,
W
yb {
.{
b S { . {
W S {
b
Yb @ {
W
Cy
3
x 10
7
Left hand side
2.5
(25)
2
1.5
1
is found by substituting for (15), (17) and (18) in (25):
CS +y,
y
b{ {
W T+y,
{
Yb @ {
W
Cy
{5 ,+Z W +y,S +y,
{.{
W S +y,Z +y,,
. +
0.5
0
Right hand sides
-0.5
-1
(26)
From Proposition
1, it
is clear that yb 3. Thus, the negaCS
+y ,
CS +y,
W
3
tivity of {

Cy yb { requires S +y, @ Cy A 3 for all
0
3
x 10
0.1
0.2
0.3
0.4
0.5
λ
0.6
x
0.7
0.8
0.9
1
0.9
1
(longitudinal slip)
7
Left hand side
2.5
2
1.5
3
y A 3= For S +y, to be positive de¿nite, it is suf¿cient that
3
S4>4 +y, A 3 and G+y, @A 3 which follows due to (24). Note
that since T4>4 +y,> T5>5 +y,> 4 A 3, it follows immediately
3
that S4>4 +y, A 3. Then, (26) becomes:
{5 ,+Z W S +y,
{ . {W S +y,Z +y,, {W T+y,
{
Yb +
5
5
@ T4>4 +y,
{4 T5>5 +y,
{5
5
. +
{5 ,+S5>5 +y,
{5 . S5>4 +y,
{4 ,
(27)
y
1
0.5
Right hand sides
0
-0.5
-1
0
0.1
0.2
0.3
0.4
0.5
λ
0.6
x
0.7
0.8
(longitudinal slip)
Figure 3: Illustration of the robust stability requirement, i.e.
the left and right hand sides of eq. (23). The upper part is for
y @ 4 m/s and the bottom part is for y @ 65 m/s.
To obtain all {
4 -terms in (27) in a quadratic form, we apply
Young’s inequality 5de d5 @F . Fe5 . Hence,
S5>4 +y, 5
{
4 F T5>5 +y,
Yb T4>4 +y,
{54 .
{55
y
{5 ,
5
S5>4 +y, 5 +
. +
{5 ,S5>5 +y,
{5 .
y
y
F
taking S +y, from the solutions of the Riccati equation are possibly conservative.
(28)
Due to (22) and (23) it follows that
{54 T5>5 +y,
{55
Yb T4>4 +y,
and we conclude that the equilibrium is uniformly exponentially stable using Corollary 4.2, [11].
Inequality (22) states that the error weight T4>4 +y, must be suf¿ciently large, leading to a suf¿eciently large controller gain.
Note that S5>4 +y, depends
s on T4>4 +y,> but it is evident that
S5>4 +y, increases with T4>4 +y, such that (22) will indeed
hold for a suf¿ciently large T4>4 +y,, except when y $ 3=
Inequality (23) states that the error weight T5>5 +y, must also be
suf¿ciently large, leading to a suf¿ciently high s
gain to stabilize
the system. Note that S5>5 +y, increases with T5>5 +y, such
that this is also possible, except for y $ 3.
{5 m is essentially chosen in (23) to dominate
Note that T5>5 +y, m
the perturbation +
{5 ,. Inequality (23) should be checked with
respect to the perturbations that are generated by all possible
friction curves +, to ensure robust stability.
Inequality (24) can be seen to be non-restrictive since it will
always be satis¿ed for 4 close to zero [3]. This corresponds
to generating the nominal model by linearizing near the peak of
the friction curve. Experience shows that high performance is
indeed achieved this way. Note that for 4 @ 3, no information
on the friction curves is actually utilized in the control design.
The constant F A 3 should be chosen to minimize conservativeness. However, the choice T4>5 +y, @ T5>4 +y, @ 3 and
The controller gain N+y, depends on the speed (gain scheduling). From a practical point of view, a useful gain schedule is
gT >4 +y, A 3 and gT5>5 +y, A 3, as this reachieved by letting 4gy
gy
duces the gain as y $ 3. As mentioned earlier, this is necessary
to avoid instability due to the unmodelled (actuator) dynamics
since these tend to dominate as y $ 3.
An important aspect is the initialization of the integrator state
{4 . Note that {4 is a controller state that can be initialized arbitrarily, while {4 is unknown since it depends on the road/tyre
friction coef¿cient K . Hence, an intelligent initialization of
{4 +3, based on any a priori information on K and possibly on
the known {5 +3, may signi¿cantly improve the transient performance [2].
4.3 An idealized design example
We consider a design example with the following parameters
p @ 783 nj, I} @ 7747 Q, u @ 3=65 p, M @ 4=3 nj p5
and the friction model in the right part of Figure 2. Assuming
@ 3=47 and the nominal design K @ 3=; and @ 3,
we get 4 @ 43=5 and 4 @ 3=65. We choose U @ 4 and
 4>4 @ 9 43< and T
 5>5 @ 73 439 . Note
 6@5 with T
T+y, @ Ty
the scaling due to the different magnitudes of We and . The
choice for T+y, leads to a gain schedule with reduced gain as
y $ 3, that is useful to avoid instability due to unmodelled
acatuator dynamics as y $ 3.
Figure 3 shows that the robust stability requirement (23) is satis¿ed for all {
5 5 ^ > 4 ` for all the friction curves in
the right part of Figure 2. Although curves are shown only for
y @ 4 m/s and y @ 65 m/s, we have veri¿ed that (24) is ful¿lled
for intermediate values of y. The control design is also satis¿ed the stability requirements (22) and (24), and we conclude
robust uniform exponential stability of the equilibrium.
5 Implementation
slip (lambda)
1
0.9
The experimental vehicle is a Mercedes E220 equipped with
electromechanical disc brake actuators and a brake-by-wire
system. A previous version of this system is described in [4].
The system consists of four independent servo controllers, one
for each brake, and a central electronic control unit (ECU)
where the wheel slip controllers run. With reference to the
theoretical study presented in previous sections, we note the
following implementation details:
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
2.5
3
2
2.5
3
2
2.5
3
t, [s]
The actual control implementation is based on the discretization of the models and a discrete-time control design. The discrete-time model also takes into account
communication delays on the brake-by-wire electronics.
The sampling rate for the wheel slip control loop is 7 ms.
x 10
2
clamping force (Fb, N) from driver (dashed) and ABS (solid)
4
1.8
1.6
1.4
1.2
1
0.8
The electromechanical brake actuator is assumed to be linear,
0.6
0.4
0.2
Ib e @ Ie . Ie
0
(29)
0
0.5
1
1.5
t, [s]
speed (v, m/s) and wheel speed (w r, m/s)
25
where is the time-constant and Ie is the commanded
clamping force. This is reasonable since it contains a local feedback loop. The model used for control design is
augmented with a discretized version of (29) and the measured clamping force Ie is used for feedback.
The controller actually computes the change of the clamping force, rather than the clamping force itself. This is implemented in order to enjoy the bene¿ts of velocity-based
gain scheduling [12], as well as it easily account for the
actuator constraint (see [5])
Ibepd{ Ibe Ib epd{
(30)
Anti-windup is implemented on the integrator state {4 .
20
15
10
5
0
0
0.5
1
1.5
t, [s]
Figure 4: Experimental results with braking on dry asphalt.
Gain scheduling is implemented by computing gain matrices for a ¿nite number of operating points and then
switching gain matrices. To achieve bumpless transfer the
integrator is reset at the switching instants. The gain matrices are scheduled on both speed and slip. The scheduling on slip only has an effect for very small slip values, and will improve the transient performance when the
wheel slip controller is activated.
clamping force does not oscillate, we conclude that this is actually sensor noise that is known to increase at the speed goes
to zero. We also note that the initial transient is not satisfactory
as the claming force does not increase fast enough so that the
slip is too low and the resulting friction force is too low in the
interval w 5 +3=5> 3=:, leading to increased braking distance.
This is due to the signi¿cant model inaccuracy in the low-slip
region, and redesign of the slip controller is necessary for this
region.
The slip and speed y are estimated online using an extended Kalman ¿lter based on wheel speed ($) and acceleration measurements.
The second test, Figure 5, is braking on snow, starting at
y+3, @ 55=3 m/s and without any steering manoeuvers. The
slip setpoint is @ 3=3: and we note that the regulation is
satisfactory. There are some small oscillations that we believe
are due to unmodeled actuator phenomenon (non-linearities)
that are expected to be more pronounced when operating at low
clamping force levels due to low friction on snow.
The ABS system monitors the commands given by the
driver using the brake pedal. Essentially, we set Wepd{ @
ne Ie where Ie is the clamping force commanded by the
driver using the brake pedal. The slip controller is deactivated when the speed is below 1 m/s.
The tuning of the implemented controller is similar to the
tuning in the idealized set-up in section 4.3. More speci¿cally, the dominant pole of the nominal linear closed loop
is almost the same in both cases (about 12.0 for a typical
y).
6 Experimental results
The ¿rst test, Figure 4, is braking on dry asphalt, starting at
y+3, @ 54=8 m/s and without any steering manoeuvers. The
slip setpoint is @ 3=3< and we note that the regulation is
highly accurate and satisfactory. When the speed approaches
zero, signi¿cant variability in the slip emerges. Since the
The third test, Figure 6, is braking on a wet inhomogeneous
surface (asphalt that is partially covered by a plastic coating
and water) without any steering manoeuvers. The intial speed
is y+3, @ 55=3 m/s and the slip setpoint is @ 3=3<. We
note that in this case there are signi¿cant transients in the slip
and claming force. We still conclude that the regulation performance is highly satisfactory, since the surface is characterized
by very large variations in the friction coef¿cient that cause
large disturbances on the system.
7 Discussion and Conclusions
Using Lyapunov analysis and experimental veri¿cation we
have shown good performance and robustness of a modelbased nonlinear wheel slip controller for ABS. In order to
slip (lambda)
slip (lambda)
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
1
2
3
4
5
6
7
8
0
9
0
1
2
3
4
t, [s]
x 10
2
5
6
7
8
9
5
6
7
8
9
5
6
7
8
9
t, [s]
clamping force (Fb, N) from driver (dashed) and ABS (solid)
4
x 10
2
1.8
1.8
1.6
1.6
1.4
1.4
clamping force (Fb, N) from driver (dashed) and ABS (solid)
4
1.2
1.2
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
1
2
3
4
5
6
7
8
0
9
0
1
2
3
4
t, [s]
t, [s]
speed (v, m/s) and wheel speed (w r, m/s)
speed (v, m/s) and wheel speed (w r, m/s)
25
25
20
20
15
15
10
10
5
0
5
0
1
2
3
4
5
6
7
8
9
0
0
t, [s]
Figure 5: Experimental results with braking on snow.
achieve the robustness, the approach does not rely on knowledge of the tyre/road friction curve. Static uncertainty (due
to unknown K ) is eliminated using integral action, while dynamic uncertainty (due to unknown shape of +,) is handled
by a robust design with suf¿cient stability margin.
The results are somewhat prelimenary, and can be possibly be
extended and improved. In particular, the analysis might take
into consideration some of the details that are present in the
implementation, but omitted from the current analysis. Also,
the use of the Riccati-solution to de¿ne the Lyapunov function
is not always the best choice. Finally, means for improving the
transient performance of the controller (especially for low slip)
are under investigation.
Although a detailed comparison with existing off-the-shelf
ABS has not been conducted, the present results are encouraging, in particular when taking into account the modest time
taken to design, tune and comission this model-based approach.
Acknowledgements
2
3
4
t, [s]
Figure 6: Experimental results with braking on a wet inhomogeneous suface.
[4]
[5]
[6]
[7]
[8]
[9]
The work was sponsored by the European Commission under
the ESPRIT LTR-project 28104 H5 C.
[10]
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