This document discusses a flatness-based cascade control system for an automatic clutch. The system uses an actuator to position the clutch bearing instead of a clutch pedal. The system is modeled with two subsystems: 1) a low-level actuator model describing the motor dynamics and 2) a high-level vehicle model relating clutch position to vehicle speed. This model is differentially flat with vehicle speed as the flat output. A cascade control approach is proposed using this structure. A reference trajectory for vehicle speed is generated based on driver inputs. This is used to compute reference trajectories for clutch position and actuator current via flatness. A feedback controller tracks these references while maintaining the cascade structure for robustness.

1. Flatness Based Control
of an Automatic Clutch∗
Jean L´EVINE‡
Bernard R´EMOND§
‡
Centre Automatique et Syst`emes, ´Ecole des Mines de Paris,
35 rue Saint-Honor´e, 77305 Fontainebleau Cedex, France.
E-mail: levine@cas.ensmp.fr
§
VALEO Clutch, 15 rue des Rosiers
93585 Saint-Ouen, France.
E-mail: bernard.remond@valeo.com
Keywords. Automatic clutch system, nonlinear sys-
tems, flatness, trajectory planning, cascade control de-
sign.
Abstract
In this paper, we study the control of an automatic
clutch where the pedal has been replaced by an actuator
positionning the clutch disk (more precisely the clutch
bearing), without modifying the gear box and gear stick.
We present a cascade regulation of the clutch system
based on the flatness property of its model and based
on measurements of the vehicle speed, the thermal en-
gine speed, the gear number, the force applied to the
gear stick (push-pull sensor), and the throttle. The aim
is to control the vehicle speed to prevent from abrupt
acceleration changes and from over- and under-speeds
of the thermal engine, while complying to the driver’s
requests on the gear stick and the acceleration pedal (or
throttle). We show how to design reference trajectories
for the vehicle speed and a two-level feedback to fulfill
these demands.
1 Introduction
A traditional clutch is needed to progressively transmit
motor torques to the wheels at start and while changing
gear (see figure 1). It may be partially or fully automa-
tized according to various technologies [3]. The present
paper concerns a technologically simple approach, devel-
oped by VALEO, where the traditional clutch and gear
stick are kept untouched, but where the clutch pedal
∗Work supported by VALEO Electronics and VALEO Clutch.
The authors wish to express their warm thanks to Antonin DE-
MAZY who significantly contributed to this project at the oc-
casion of his graduation memoir of Electrical Engineer from the
Catholic University of Louvain-la-Neuve, Belgium, under the su-
pervision of the authors.
xb
Cc
Thermal
motor
Clutch system
Gear boGear box
Wheels
torque
Cc
Cw
Cm
Motor
torque
Clutch
torque
xb
Figure 1: The clutch function: full or partial transmis-
sion to the wheels of the torque delivered by the thermal
motor.
is removed. It is replaced by an actuator controlling
the clutch position (more precisely the clutch bearing
position–see figure 2), and therefore the torque trans-
mission. Consequently, the feedback regulation of the
vehicle speed, traditionnally manually realized by the
driver, may be replaced by a computer controller that
assigns the clutch bearing position in view of several
measurements: vehicle speed, thermal engine’s speed,
gear number, force applied to the gear stick (push-pull
sensor), actuator position, throttle.
The actuator’s controller must be designed to prevent
from abrupt acceleration changes, to improve the equip-
ment reliability, increase its duty cycle, and satisfacto-
rily manage the thermal engine over- and under-speeds
as well as various failure modes. Moreover, it must re-
alize a tradeoff between power of the actuator and size
and energy consumption, which must be kept as low as
possible. Another tradeoff concerns lowering the con-
trol algorithm complexity, to decrease the cost of the
computer, versus keeping high control performances.
In driving operations, various sequences consisting in
engaging and releasing the clutch are required, in par-
ticular at starts, stops, or gear shifts.
Disengaging the clutch (that corresponds to engaging

2. CompensatingCompensating
sprspringing
Actuator's positionActuator's position
Forork
BearBearing positioning position
Motor
Powerer
electronicselectronics
HydrHydraulicaulic
transmissionansmission ScreScrew
ClutchClutch
PlatePlate
BearBearinging
NutNut
Figure 2: The actuator, the hydraulic transmission and
the clutch system.
the clutch pedal in traditional clutch systems) means
that, according to the driver’s action on the gear stick or
if the engine speed is too low, the actuator fastly moves
the clutch bearing out of its engaged position (10mm in
less than 0.1s) to decouple the wheels from the thermal
engine.
Engaging the clutch (that corresponds to releasing the
clutch pedal in traditional clutch systems) means that,
at start or when leaving the neutral, the actuator fastly
moves the clutch bearing to a position called “touching
point” where the plate surfaces arrive in contact, then
progressively synchronizes the engine and wheel speeds,
to finally lock the clutch. This sequence must be done
smoothly, i.e. with gentle speed and acceleration varia-
tions, but more or less quickly, depending on the throt-
tle.
These two phases may be combined in several se-
quences composed of an initialisation step (due to a
driver’s action on the gear stick or on the throttle), an
end (locked clutch), and/or a change (a new driver’s re-
quest on the gear stick or on the throttle before the end
has occured).
A model, composed by two subsystems, is presented.
The first part (low-level) expresses the force balance be-
tween the actuator and the screw, or equivalently the
clutch throwout bearing. The second part (high-level)
describes the thermal engine torque transmission to the
wheels. The current in the actuator is the system input.
This model is shown to be flat, with the vehicle speed
as a flat output [5, 6]. Moreover, its cascade structure
allows us to control the vehicle speed by the gear bearing
position and the latter (at a faster rate) by the actuator
drive current.
Depending on the driver’s action on the gear stick and
the throttle, a reference trajectory of the vehicle speed
is computed on-line as a polynomial of time, starting
from the present measured vehicle speed, and then ref-
erence trajectories of the clutch bearing and of the actu-
ator drive current are deduced from Flatness. Moreover,
a feedback is designed to stabilize the system around
the computed reference trajectories while preserving the
cascade structure. The reference trajectories are recom-
puted on a receding horizon each time a change is re-
quired by the driver, and their tracking is ensured by
the same fixed feedback, thus combining flatness and
predictive control.
Simulations on a validated numerical model are pre-
sented. We show that the performances of this controller
are not significantly deteriorated under sampling with a
large sampling period, more than ten times larger than
the one obtained by a linear approach and ensuring a
comparable tracking behavior.
2 Modelling
Let us denote by αm the angle of the actuator (the elec-
trical motor driving the clutch bearing, not to be con-
fused with the thermal engine) and let ωm be its angular
velocity, ωm = ˙αm. We denote by Im the motor current
and by Cr the resisting torque, a non linear function
of the actuator’s position αm and of the sign of its ve-
locity sign (ωm). It corresponds to the torque on the
actuator’s screw resulting from the difference between
the force exerted on the clutch bearing, transmitted by
the hydraulic system, and the force exerted by the com-
pensating spring (see figure 2). The function Cr is ob-
tained offline from real data. Its detailed expression is
not needed here and is omitted.
The actuator dynamics are thus described by
˙αm = ωm
Jm ˙ωm = −fvωm + KmIm − Cr(αm, sign (ωm))
(1)
where
• Jm is the motor inertia,
• Km the torque constant,
• fv the coefficient of viscous friction, assumed linear
in the velocity.
Here, the current Im may be considered as an input
to this subsystem.
We also need a model describing the effects of the
clutch on the car’s speed. By Newton’s second princi-
ple, the vehicle acceleration, multiplied by the vehicle’s
mass, is equal to the torque exerted on the wheels di-
vided by the radius of the wheel, the frictions being
neglected.
M ˙V =
1
r
Cw(xb, j, τ) , xb = µ αm (2)
where
• M is the vehicle mass, r the wheel radius and µ a
constant,
• V is the vehicle speed (measured),
• Cw is the torque exerted on the wheels,
• xb is the clutch bearing position,

3. • j is the gear number (measured),
• τ is the throttle valve opening percentage (mea-
sured).
Cw is a nonlinear function of the bearing position xb,
and thus of αm, of the gear number j and of the throt-
tle τ. Practically speaking, it is obtained once for all,
offline, by direct measurements on the vehicle. Indeed,
the function Cw depends on the vehicle. We do not give
here its detailed expression, unnecessary for the control
design. We will only use the fact that Cw is an increas-
ing function of xb for each value of the pair (j, τ), which
implies in particular that the clutch bearing position xb
can be deduced from the torque Cw knowing j and τ:
xb = C−1
w (rM ˙V , j, τ).
Note that this model is deliberately simplified: it
does not take into account the road profile, frictions on
the tyres and the transmission, or additional loads such
as trailers, which are not supposed to be measured in
practice. These quantities may be considered as distur-
bances.
In this subsystem (2), xb can be considered as the
input of this subsystem.
3 Flatness
A system is said to be differentially flat [5, 6] if there
exists a set of independent variables referred to as flat
output such that every other system variable (including
the input variables) is a function of the flat output and
a finite number of its successive time derivatives. More
precisely, the system
˙x = f(x, u),
with x ∈ Rn
and u ∈ Rm
is differentially flat if one can
find a set of variables (flat output)
y = h(x, u, ˙u, ¨u, . . . , u(p)
), y ∈ Rm
with p finite integer, such that
x = α(y, ˙y, ¨y, . . . , y(q)
)
u = β(y, ˙y, ¨y, . . . , y(q+1)
)
with q a finite integer, and such that the system equa-
tions
dα
dt
(y, ˙y, ¨y, . . . , y(q+1)
) =
f(α(y, ˙y, ¨y, . . . , y(q)
), β(y, ˙y, ¨y, . . . , y(q+1)
))
are identically satisfied.
The reader is invited to refer to [5, 6] for further infor-
mation concerning the various implications of this prop-
erty on motion planning and feedback design.
In our case, if we assume that the gear number j and
the throttle τ are measured, and that τ is a piecewise
twice continuously differentiable function of time, the
system (1), (2) is flat with the vehicle’s velocity V as a
flat output. We now briefly show. Assuming that the
vehicle velocity V is known, as well as j and τ, from (2)
and the invertibility of the function Cw, we have
xb = C−1
w (rM ˙V , j, τ) , αm =
1
µ
C−1
w (rM ˙V , j, τ). (3)
Moreover, by (1),
Im =
1
Km
(Jm ¨αm + fv ˙αm + Cr(αm, sign ( ˙αm))) . (4)
We have thus proven that xb, αm and Im are functions
of V, ˙V , ¨V and V (3)
= d3
V
dt3 and that V is a flat output.
4 Cascade control
The system (1)-(2) has a natural cascaded structure,
which is moreover compatible with the system flatness:
the subsystem (1), whose input is the current Im, de-
scribing the behavior of the actuator, is much faster than
the subsystem (2) describing the vehicle behavior that
may be considered as controlled by the clutch bearing
position xb, or equivalently by the actuator’s position
αm.
More precisely, if we are given a reference trajectory
of the car’s speed V , it can be realized by suitably po-
sitionning the clutch bearing, using xb as a fictitious
input thanks to (3), and the clutch bearing position, in
turn, may be realized by (4). For this reason, we call
the subsystem (2) the high-level subsystem and (1) the
low-level subsystem.
This structure is not only useful for trajectory plan-
ning but also for the tracking of the corresponding tra-
jectories by feedback.
4.1 trajectory planning
According to (3), it suffices to define a reference tra-
jectory for the vehicle speed V to obtain all the other
system variables. Moreover, V can be chosen arbitrarily
in the set of piecewise 3 times continuously differentiable
functions of time. Let us consider a time interval [ti, tf ]
(which may depend on the required objective) and let
us denote by V ∗
the searched reference trajectory on
this interval. We are given the measurement of the ini-
tial speed Vi at the initial time ti, with its (estimated)
derivative ˙Vi at ti (initial acceleration). At the final time
tf , we may be given a final speed Vf and acceleration
˙Vf . These data highly depend on the driver’s objec-
tives: start or stop, gear upward or downward shifts,
which involve engaging or disengaging the clutch more
or less fastly, depending on the thermal engine’s speed
and throttle. In any case, the final acceleration ˙Vf can

4. be chosen equal to 0. For Vf , the following formula gives
satisfactory results in practice:
Vf = Vi + k1
τ(ti)
j(ti)
+ k2
1
1 + τ(ti)
(5)
where τ(ti) and j(ti) are the measured throttle value and
gear ratio at time (ti) (momentarily assumed to remain
constant on [ti, tf ] – this assumption will be removed in
section 5) and where k1 and k2 are tunable parameters,
whose tuning may be different for gear up or down. Note
that the term 1
1+τ prevents Vf from being to close to
Vi which would prevent the clutch from locking when
the throttle is closed (τ = 0), and that the ratio τ
j is
introduced to produce a larger acceleration for low gears.
By polynomial interpolation, V ∗
(t) is given by the
following 3rd degree polynomial:
V ∗
(t) = Vi + T ˙Vis(t)
+s2
(t) (∆V − T ˙Vi) + (2∆V − T ˙Vi)(1 − s(t))
(6)
with
∆V = Vf − Vi , T = tf − ti and s(t) =
t − ti
T
. (7)
Clearly, the clutch bearing reference x∗
b results from (3)
by replacing V by V ∗
given by (6). Accordingly, we may
deduce the reference trajectories of the actuator’s posi-
tion αm and of the current Im by (3-4) but the cascade
approach described in the next section, which recom-
putes on-line, at a faster rate, the motor and current
references has been preferred for robustness reasons.
Moreover, the duration T is chosen according to the
following formula:
T = Tmin +
Tmax − Tmin
100
τ (8)
where Tmax (resp. Tmin) is the duration necessary to
realize the whole clutch engagement phase with τ = 0%
(resp. τ = 100%).
4.2 Cascaded feedback design
Let us consider the above defined reference trajectory
V ∗
and the corresponding reference x∗
b by (3). The high-
level tracking error is thus given by
M( ˙V − ˙V ∗
) =
1
r
(Cw(xb, j, τ) − Cw(x∗
b , j, τ)) .
Therefore, it suffices to adress the bearing position as
xb = C−1
w (Cw(x∗
b , j, τ) − rMkV (V − V ∗
), j, τ) (9)
for which the tracking behavior is given by
˙V − ˙V ∗
= −kV (V − V ∗
) (10)
which is indeed exponentially stable for every kV > 0.
At the low-level, we define
α∗
m =
1
µ
xb (11)
xb being defined by (9), and I∗
m by (4) with α∗
m and ˙α∗
m
in place of αm and ˙αm. Then, setting
Im = I∗
m − k
Jm
Km
(αm − α∗
m)
+
1
Km
(Cv(αm, sign ( ˙αm)) − Cv(α∗
m, sign ( ˙α∗
m)))
(12)
it results that the low-level closed-loop subsystem reads
¨αm − ¨α∗
m = −k(αm − α∗
m) −
fv
Jm
( ˙αm − ˙α∗
m) (13)
which is exponentially stable for every k > 0. Note
that the low-level subsystem (1) is supposed to be much
faster than (2) which means in particular that fv
Jm
is
large. Therefore, if we set k = fv
2Jm
2
, the low-level
closed-loop subsystem (13), whose transfer function be-
comes (s+ fv
2Jm
)2
, is exponentially stable and much faster
than (10) if kV is chosen small enough. The overall
stability of the system (10)-(13) follows from standard
singular perturbation theory [4, 8, 9, 10, 12, 14].
The resulting two-level controller architecture is
shown in figure 3.
VEHICLEActuator
Low-level
loop
High-level
loop
Driver
Im xb Vαm
*V*
j, τ
Figure 3: The controller architecture.
Remark In practice, compared to the actuator’s rate,
the clutch bearing position xb is a slowly evolving signal,
that can be even considered piecewise constant. There-
fore, in place of recomputing it at each time according
to (9), it suffices to consider its discretized version at
a sampling period Tl with Tl T and to approximate
the reference (11) by a 3rd degree polynomial. These
implementation details are omitted here.
5 The driver in the loop
As was seen in formula (5), the throttle τ and gear
number j were assumed constant over the time inter-
val [ti, tf ]. These measured inputs however are not con-
trolled by the system and can be modified at any time
by the driver. Therefore, the system must be able to
include the new driver’s requirements as fast as possi-
ble without risking to destabilize the clutch positionning

5. system. This is done in the following way: assume that
at any moment ti the driver modifies τ or j.Then, since
the car’s speed is measured, one can use Vi and ˙Vi at
that time in (5)-6-8) and deduce the new reference tra-
jectory on the interval [ti, ti +T], which guarantees that
the junction between the previous reference trajectory
and the new one is smooth (see figure 4).
t t+T
.
Vi
Vf
Vi
.
Vf
reference
defined
before
time t
new reference
smooth
change of
reference
trajectory
Figure 4: Smooth modification of the reference trajec-
tory according to a new driver’s request on the gear stick
or the throttle.
According to the above analysis, the controller is such
that the initial error on the car’s speed will exponentially
decrease at a rate larger than or equal to kV and the
error on the clutch bearing position at the rate 2Jm
fv
(whose magnitude in this application is approximately
0.1s). Thus, if we refresh the values of τ and j at a
lower rate, the low-level loop cannot be destabilized and
therefore, the car speed error will not grow, while the
vehicle will be able to react to the driver’s request.
Remark Note that the fact that the speed reference
trajectory is recomputed on a receding horizon each time
a change is required by the driver, and that their track-
ing is ensured by the same fixed feedback, may be viewed
as an attempt to combine flatness and predictive control
[1, 13]. Other examples of this nature may be found in
[11]. See also [7].
6 Simulations
The following simulation has been realized on a vali-
dated simulator, namely a simulation program which
satisfactorily reproduces the measurements done on a
given car. It describes the evolution of the thermal en-
gine speed, the car speed and acceleration, the clutch
bearing position and the motor current when starting
from 0 speed to the third gear ratio (j = 3) at approx.
80 km/h, with the throttle evolving in a large range
(from 0 to 100%).
One can see that, even if the driver pushes the ac-
celerator pedal at its maximal position while changing
speed, the clutch locks fastly enough to avoid an engine
overspeed, and without too abrupt variation of acceler-
ation.
0 2 4 6 8 10 12
0
50
100
TIME (s)
0 2 4 6 8 10 12
0
500
1000
TIME (s)
0 2 4 6 8 10 12
0
TIME (s)
0 2 4 6 8 10 12
0
TIME (s)
0 2 4 6 8 10 12
-5
0
5
10
TIME (s)
Throttle (%)
Gear ratio
Thermal motor speed
Gearbox speed
Actuator position
Motor current
Car acceleration
Figure 5: Simulation from start (0 km/h) to gear num-
ber 3 (approx. 80 km/h) in 12s.
7 Concluding remarks
We have presented here a method combining flatness
and two-level cascaded feedback design to control the
vehicle speed with an automatic clutch system. The
nonlinearities of the system are mainly due to frictions
which are known with mediocre precision. However, the
robustness of the controller is sufficient to produce sat-
isfactory responses even when the clutch is old or when
the various coefficients describing the motor and the fric-
tions are corrupted by errors. Moreover, the sample-
time version of this controller can be used directly with
a large sampling time (up to 20ms) without significant
deterioration of the performances [2]. Note also that
the complexity of the real-time calculations is reason-
able and the controller can be easily implemented on a
cheap 8-bit microprocessor [2].
References
[1] D.W. Clarke, C. Mohtadi and P.S. Tuffs. General-
ized predictive control - Parts I and II. Automat-
ica, 23(2), p. 137–148 and 149–160, 1987.

6. [2] A. Demazy. Mise au point de la commande de
l’embrayage automatique compact EAC. M´emoire
en vue de l’obtention du grade d’Ing´enieur Civil
Electricien, Universit´e Catholique de Louvain-la-
Neuve, 1997.
[3] B. Derreumaux. Transmissions. Editions Tech-
niques pour l’Automobile et l’Industrie, 1991.
[4] N. Fenichel. Geometric singular perturbation the-
ory for ordinary differential equations. J. Diff.
Equations, 31:53–98, 1979.
[5] M. Fliess, J. L´evine, P. Martin, and P. Rouchon.
Flatness and defect of non-linear systems: intro-
ductory theory and examples. Internat. J. Con-
trol, 61, pp. 1327–1361, 1995.
[6] M. Fliess, J. L´evine, P. Martin, and P. Rouchon.
A Lie-B¨acklund approach to equivalence and flat-
ness of nonlinear systems. IEEE Trans. Automatic
Control, 44, N. 5, pp. 922–937, 1999.
[7] M. Fliess and R. Marquez. Continuous-time linear
predictive control and flatness: a module-theoretic
setting with examples. Int. J. Control. To appear.
[8] H. K. Khalil. Nonlinear Systems. Prentice Hall,
Englewood Cliffs, NJ, 1996.
[9] P.V. Kokotovi´c. Application of singular pertur-
bation techniques to control problems. SIAM Re-
view, 26(4):501–550, 1984.
[10] P. Kokotovi´c, H.K Khalil and J. O’Reilly. Singu-
lar perturbation methods in control: analysis and
design. Academic Press, London, 1986.
[11] J. L´evine. Are there new industrial perspectives
in the control of mechanical systems? in “Ad-
vances in Control, Highlights of ECC’99”, P.M.
Frank (ed.) Springer-Verlag, London, p. 197–226,
1999.
[12] J. L´evine and P. Rouchon. An invariant manifold
approach for robust control design and applica-
tions. in “Systems and Networks: Mathematical
Theory and Applications”, Vol. II, U. Helmke, R.
Mennicken, J. Saurer (eds.), Akademie Verlag, p.
309–312, 1994.
[13] J. Richalet, A. Rault, J.L. Testud and J. Papon.
Model predictive heuristic control: application to
industrial processes. Automatica, 14, p. 413–428,
1978.
[14] A.N. Tikhonov, A.B. Vasil’eva, and A.G. Svesh-
nikov. Differential equations. Springer-Verlag,
1985.