1. Robust control strategies for an electric motor driven accumulator
with elastic webs
David Kuhm a,b,c
, Dominique Knittel a,b,n
, Marie-Ange Bueno c
a
Web handling research Center, UFR Physique et Inge´nierie, Campus Meinau, University of Strasbourg, 17, rue du Mare´chal Lefebvre, 67100 Strasbourg, France
b
Laboratoire de Ge´nie de la Conception INSA Strasbourg, 24, boulevard de la Victoire, 67084 Strasbourg, France
c
Laboratoire de Physique et Me´canique Textiles ENSISA, 11, rue Alfred Werner, 68093 Mulhouse, France
a r t i c l e i n f o
Article history:
Received 12 July 2011
Received in revised form
26 January 2012
Accepted 19 May 2012
Available online 15 June 2012
Keywords:
Industrial accumulator
Roll-to-roll
Dynamic modelling
H1 controllers
Multi-model control
PI control
Robustness
a b s t r a c t
This paper concerns the modelling of an accumulator used in industrial elastic web processing plant to
allow changing material roll while the rest of the line remains at a constant web velocity. A nonlinear
model of a motor actuated accumulator is first summarized. This model is derived from the physical
relationships describing web tension and velocity dynamics in each web span of this accumulator. A
linear model is deduced from the nonlinear one around a working point for frequency domain analysis.
Thus the effect of some mechanical accumulator parameter variations are analyzed. In a second part,
multi-model industrial PI controllers, adjusted with evolutionary algorithm on our realistic nonlinear
model are compared with multi-model H1 controllers. Both controllers allow good robustness against
mechanical parameter variations.
& 2012 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction
Continuous process lines increase the productivity of the plant.
Accumulators (Fig. 1) are used to permit rewind or unwind core
changes while the process continues at a constant velocity. Accu-
mulators are composed of many free rollers and of a carriage moved
to store or release web during the production phases. Most of the
time, the accumulator carriage is maintained at its nominal position
and the web just follow the geometrical path made by the free
rollers. When the unwinder wound roll is almost empty, the carriage
is moved up in order to increase the web stored in the accumulator.
During the unwinder roll change, the unwinder is stopped and the
accumulator carriage is adequately moved down to restore the web.
The objective is to maintain constant web line velocity. Finally, after
the unwinder roll change, the carriage is moved back to its nominal
position. The unwinder roll changing time has to be as short as
possible being limited by the accumulated web length. Web tensions
inside an accumulator are easy to control during the regular
production phase (when the carriage is at its nominal position).
However web tension variations often occur in the transition phases
and therefore can generate web folds or breaks due to the inertia and
friction of the free rollers. Different works in web tension control of
roll-to-roll systems have been published in the past years. For
example Zalhan et al. [1], Brandenburg [2,3], Benlatreche et al. [4],
Knittel et al. [5,6], Gassmann et al. [7] , Giannoccaro et al. [8], Chen
et al. [9] present the modelling and control of a continuous proces-
sing line composed only with tractors and free rollers, whereas
Wolfermann [10], Olsen [11], Knittel et al. [12] describe the unwind-
ing and the winding process. Only few results present the modelling
and the control of industrial accumulators. Important results can be
found in Koc- et al. [13,14], Pagilla et al. [15,16], Kuhm et al. [17]. The
last reference concerns the modelling and robust control of an
industrial accumulator including an ideal carriage (the carriage
actuator dynamics are neglected) and [18]compare the accumulator
presented in this paper to one including a pneumatic actuated
carriage. Knittel et al. [19] concerns the accumulator controllers
optimization using genetic algorithm. This paper presents for the
first time robust control strategies for a complete modelled industrial
accumulator including the dynamics of the motor actuated carriage.
2. Accumulator modelling
2.1. Nonlinear model of the accumulator web dynamics
Equations describing web tension behavior between two
consecutive rollers and the velocity of each roll enable to build
the nonlinear model of a web transport system. Our accumulator
is modelled by a succession of web span tensions and velocities
between each roll and by the accumulator carriage motion.
Contents lists available at SciVerse ScienceDirect
journal homepage: www.elsevier.com/locate/isatrans
ISA Transactions
0019-0578/$ - see front matter & 2012 ISA. Published by Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.isatra.2012.05.004
n
Corresponding author at: Web handling research Center, UFR Physique et
Inge´nierie, Campus Meinau, University of Strasbourg, 17, rue du Mare´chal
Lefebvre, 67100 Strasbourg, France. Tel.: þ33 6 71 20 31 30;
fax: þ33 3 68 85 49 72.
E-mail address: knittel@unistra.fr (D. Knittel).
ISA Transactions 51 (2012) 732–742

2. 2.1.1. Web tension calculation
Assuming no sliding between web and rollers, the equation of
continuity applied to a web span between two consecutive rollers
(Fig. 2), is given by [20,21]
d
dt
Lk
1þet
k
!
¼ À
Vk þ 1
1þet
k
þ
Vk
1þet
kÀ1
ð1Þ
where Lk is the web length between the kth and the (kþ1)th
rollers. This web length can vary due to the accumulator carriage
motion. Vk and Vk þ 1 are respectively the velocity of the kth and
the (kþ1)th rollers and et
k the strain of the kth web span. Web
tension Tk
t
is then deduced from general web tension relation (1)
by applying Hooke’s law.
The equivalent strain et
k in the web (see relation (2)) is composed
of the strain ew
k induced by the web weight (relation (4)) and by the
strain ee
k induced by the web elongation without the web weight:
et
k ¼ ee
k þew
k ð2Þ
The weight WLk
of a web span is given by the following relation [22]:
WLk
¼ rgSLk ¼ ESew
k ð3Þ
with r the web mass density, g the gravity and Lk the span length.
The strain ew
k in the web span resulting from the gravity is then
obtained from (3):
ew
k ¼
rgLk
E
ð4Þ
In our accumulator (web with a Young modulus of 3000 MPa, a
mass density of 750 kg mÀ3
, a section of 9.35 Â 10À5
m2
and a
nominal tension of 400 N, corresponding to a paper web) we have
ee
k bew
k (ee
k ffi387ew
k ).
2.1.2. Web velocity calculation
The velocity dynamic of the kth roller is calculated using the
torque balance. Assuming there is no slippage between the web
and the roll, the web velocity Vk is equal to the linear velocity of
the roll. This velocity dynamic is given by
d
dt
ðJkOkÞ ¼ CmkÀCrkÀCfk ð5Þ
where Ok ¼ Vk=Rk is the angular velocity of the kth roll, Jk the roll
inertia and Rk the roll radius. Cmk is the motor torque for a driven
roller. Crk ¼ RkðTkÀTkÀ1Þ stands for the torque introduced by the
web and Cfk is the friction torque between the roll and its shaft
(composed of a term proportional to the roll velocity and a
constant term). Eqs. (2) and (5) are applied to each web span
and roller of the studied accumulator and enable the construction
of a nonlinear longitudinal web dynamic simulator.
2.2. Nonlinear model of the accumulator carriage motion
This part presents the nonlinear model of a motorized carriage,
including static frictions representation.
2.2.1. Displacement of the accumulator carriage
In the phase of wound roll change, the output speed Vout accu is
maintained constant at nominal speed Vref and the input speed
Vin accu decrease to zero. In order to maintain the output web
tension and velocity constant during this change, it is necessary to
move adequately the accumulator carriage. In our study, the
accumulator is composed of nþ1 rollers and therefore has n
web spans. The accumulator carriage displacement speed refer-
ence during the wound roll change is given by the following
Motor
Unwinder Welding table
1 n+1
Accumulator
ChainCarriage
Vin accu Vout accu
Tout
L
Fig. 1. Example of a motor actuated accumulator.
Vk+1
Vk
Lk
k-1 Volume of control
Roll k+1
Roll k
k = k+ k
e wt
t
Fig. 2. Indices of web span between two consecutive rollers.
D. Kuhm et al. / ISA Transactions 51 (2012) 732–742 733

3. relation:
Vaccu ¼
1
n
ðVin accuÀVout accuÞ ð6Þ
This speed reference enables to find the reference carriage
position.
2.2.2. Unwinding speed during the accumulation phase
During the web accumulation phase, the unwinder speed has
to be increased to keep a constant web tension. This unwinding
speed reference can be deduced from Eq. (6):
Vin accu ¼ nVaccu þVout accu ð7Þ
2.2.3. Motor actuated carriage
In order to move the accumulator carriage, in some industrial
accumulator, a motor can be used. A representation of the motor
actuator is shown in Fig. 3. The carriage is maintained at its
nominal position thanks to a brake during the regular production
phase. During the carriage motion phase, the brake is deactivated
and the motor moves adequately the accumulator carriage. Due to
the inclusion of the brake, there is no device to attenuate web
tension variations into the motor actuated accumulator. The
equation of the carriage motion is deduced by applying a torque
balance on the carriage motor (8). The carriage position xc is related
to wm (angular velocity) by the radius of the motor/chain gear Rm.
In order to provide the carriage motion dynamic, a special
attention has to be paid to static friction force Ffs.
Static friction sign depends obviously on the direction of the
carriage velocity. But the key point that has to be handled is when
carriage velocity is zero. One has to make sure that the carriage
remains immobile until the forces applied on the carriage are
sufficient to exceed the static friction force. These conditions are
summarized in (9):
Jm _wm ¼ udmkdmÀRmðFweb þFfvð _xc Þ
þFfs þmtgÞÀFfvmwm if B1 or B2 satisfied
Jm
_wm ¼ udmkdmÀRmðFweb þFfvð _xc Þ
ÀFfs þmtgÞÀFfvmwm if B3 or B4 satisfied
Jm
_wm ¼ 0 else
8
>>>>>><
>>>>>>:
ð8Þ
B1 : udmkdmÀRmðFweb þmtgÞ4FfsRm
B2 : _wm 4e _wm
B3 : udmkdmÀRmðFweb þmtgÞoÀFfsRm
B4 : _wm oÀe _wm
8
>>>><
>>>>:
ð9Þ
with Jm is the motor inertia, udm is the reference voltage, kdm is the
ratio from reference voltage to torque, Fweb is the force applied
to the carriage by the web span and Ffvm is the motor dynamic
friction force. The accumulator carriage motor is position con-
trolled in order to move the carriage to the desired position. The
control scheme is given in Fig. 3.
2.3. Linear model of the accumulator
2.3.1. Linearization
The linear model is deduced from the nonlinear one and can be
described by a state space representation. The web span dynamic
linearization is detailed in Koc- [20] and Kuhm et al. [17]. The
linear model is obtained by linearizing an accumulator, around a
setting point, including an ‘‘ideal’’ carriage actuator (neglecting
nonlinearities such as static friction, dead band, pressure depen-
dant bulk modulus and saturations), according to the scheme
given in Fig. 4. The linear model is useful for Bode diagrams
calculation and for the frequency analysis of the transfer function.
In steady state, ee
k bew
k in our application and therefore the web
weight is neglected. However in other case, such as accumulators
designed for metallic webs, the web weight is significative and
has to be taken into account by keeping the web weight terms
during the linearization.
By derivating relation (1) describing the tension and by
keeping only the first order terms, the relation after simplification
mc
web
Chain
Motor
Carriage
Rm
xc
fMtg
Cm
-
+
Xref
Controller
Fig. 3. Generic motor representation including position control.
Unwinder Welding table
1 n+1
Accumulator
Carriage Actuator
Vin accu Vout accu
Tout
L
Fig. 4. Accumulator scheme representation used to build a linear model.
D. Kuhm et al. / ISA Transactions 51 (2012) 732–742734

4. can be written as follows:
Lk
dek
dt
¼ Vk þ1ð1þekÞÀVkð1þ2ekÀekÀ1Þþ
dLk
dt
ð1þekÞ ð10Þ
With the notation ek ¼ ee
k. An example of the developed relation,
approximated before derivation (less accurate), including web
weight, can be found in [22]. Hooke’s law gives the relation
between the web tension and its elongation:
Tk ¼ ESek ð11Þ
Including the web tension (11) into relation (10) leads finally to
the tension dynamic description:
Lk
dTk
dt
¼ Vk þ 1ðESþTkÞÀVkðESþ2TkÀTkÀ1Þþ
dLk
dt
ðESþTkÞ ð12Þ
The relation (12) is then linearized around an operating point,
while posing Tk ¼ T0 þtk, TðkÀ1Þ ¼ T0 þtðkÀ1Þ, Vk ¼ V0 þvk and
Vðk þ1Þ ¼ V0 þvðk þ 1Þ. By preserving only the first order terms
Eq. (12) gives
dtk
dt
¼
E0
Lk
vkþ 1À
E0
Lk
vkÀ
V0
Lk
tk þ
V0
Lk
tkÀ1 þ
E0
Lk
dLk
dt
ð13Þ
whereas the web speed can be calculated from relation (5):
dvk
dt
¼ À
R2
k
Jk
tkÀ1À
Fk
Jk
vk þ
R2
k
Jk
tk ð14Þ
where respectively Rk is the roll radius of the kth roll, Fk the
friction coefficient between the kth roll and its shaft and Jk the
inertia of the kth roll.
2.3.2. State space representation of the accumulator
The linear model of the accumulator can be described by a
state space representation given in (15). The states are the tension
and velocity in each web span. The state space model has tree
inputs uþv and one output Tnþ1, the accumulator output tension.
The input u is the carriage displacement velocity, Vd is the
unwinder speed, L is the web span length in the accumulator
between two consecutive rollers.
_X ¼ Aþ
A1
L
X þ
B
L
uþB1v
Y ¼ CX þDu
8
:
ð15Þ
where
X ¼
Td
Va
Ta
Va1
Ta1
^
Vnþ 1
Tn þ1
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
, u ¼
dL
dt
, v ¼
Vd
Voutaccu
!
ð16Þ
C ¼ ð0 0 0 0 0 0 0 . . . 0 0 1Þ ð17Þ
D ¼ 0 ð18Þ
Simulation in Matlab/Simuling software environments shows that
the linear model fit on the nonlinear model, even if we neglect
some nonlinearities (Fig. 5). The maximum relative error between
the two model, during fast tension variations is 9.9%, therefore the
linear model is used on the one hand to study the frequency
response of the system and on the other hand to synthesize H1
controllers.
3. Accumulator performances analysis
3.1. Influence of the mechanical parameters
Depending on the type of accumulator, mainly two control
strategies are applied: one using Vin accu as control signal (con-
troller output), the other using L. The synthesis of both control
strategies need to calculate two transfer functions: a first one
35 40 45 50 55 60 65 70
0
100
200
300
400
Time (s)
N
Web tension
Reference tension
Tension using non linear model
Tension using linear model
35 40 45 50 55 60 65 70
0
5
10
Time (s)
Error(%)
Tension error between linear and non linear model
Tension error
Fig. 5. Linear and nonlinear model comparison during web span length variation.
D. Kuhm et al. / ISA Transactions 51 (2012) 732–742 735

5. between the accumulator output tension Tout and the accumula-
tor input velocity Vin accu named W1 and a second one between
Tout and the web span length L named W2. In the following parts,
the influences of some physical parameter variations around their
nominal values are analyzed on simulated Bode diagrams.
3.1.1. Influence of the elasticity modulus and web span length
The web elasticity influences the web dynamics (tension and
velocity) in the transient phases. One can observe in Fig. 6 that the
static gain and resonances (gains and frequencies) are depending
on the Young modulus. The same observation have been made for
the transfer functions W1 and W2. Very often in the industry,
Young modulus changes occur during the manufacturing process
and therefore the control performances decrease if the controllers
are not robust enough for web elasticity variations. Consequently,
the controllers have to be adjusted/calculated for each range of
web elasticity, or robust for a given web elasticity range.
Like the Young modulus, the web span length in the accumu-
lator (between two consecutive rollers) has a significant influence
on the web dynamic (Fig. 7). A long web span will have a weaker
resonance frequency and a lower gain. Such observations have
been made for both transfer functions W1 and W2. Therefore the
dynamic sensitivity to the web length variation has to be taken
into account during controller synthesis because the web span
length is varying each time when a wound roll change occurs.
Moreover the nominal carriage position has to be fixed ade-
quately in order to increase the system bandwidth without
resonances. The study of the influence of other parameters like
roller inertias can be found in [17,18].
3.2. Control strategy
As mentioned previously, the accumulator can be controlled in
two different ways: the controller can operate either on the web
span length (by using the accumulator carriage as actuator) or the
input velocity (by using the unwinder as actuator). More pre-
cisely, the accumulator output web tension controller is synthe-
sized using the input velocity of the accumulator as control signal,
whereas the second control strategy uses the web length (by
moving the accumulator carriage) as control signal. In industrial
applications, both control schemes are implemented. Neverthe-
less this paper studies a combination of the two strategies.
During the regular production phase (when the accumulator
carriage is located at its nominal position), we use the unwinder
(accumulator input velocity) as actuator, whereas, during the
wound roll change, the carriage is chosen as actuator. The on/off
switching strategy of the two controllers is performed by weight-
ing the controller output with a coefficient continuously varying
from 1 to 0. The control scheme is represented in Fig. 8, and the
weighting coefficient evolution is given in Fig. 9.
3.2.1. Linear time invariant (LTI) optimized PI controller
In this part, industrial controllers (in our case a PI controller)
are employed to control the output web tension. This controllers
have the following form:
CiðsÞ ¼
Kið1þtisÞ
s
, i ¼ 1 or 2 ðin Fig: 8Þ ð19Þ
The PI parameters Ki and ti are determined with an optimization
approach for our realistic nonlinear model using ITAE cost func-
tion (20) and constraint (21) given as follows. The optimization
was performed in modeFRONTIER software environment with
MOGA-II genetic algorithm (population of 120 individuals, 100
generations, probability of directional cross-over equal to 0.5 and
Frequency (rad/s)
Gain(dB)
E
2E
3E
4E
5E Young
m
odulus
Fig. 6. Bode diagram of W2 for different Young modulii.
Gain(dB)
Frequency (rad/s)
W
eb
span
length
(m
)
Fig. 7. Bode diagram of W2 for different spans of web length.
D. Kuhm et al. / ISA Transactions 51 (2012) 732–742736

6. probability of mutation equal to 0.4).
J ¼
Z tmax
0
t9XpÀXpref 9 dt ð20Þ
Tmin oTi oTmax, 1rirn ð21Þ
with Ti the web tension of the ith web span.
Other cost function for PI optimization are presented in [23].
3.2.2. Multi-model PI controller
As shown previously, the web length has a significant influ-
ence on the tension; static gain and bandwidth depend on the
web span length. Therefore this LTI controller should not be
optimal for full web length range and the controllers have to take
into account the web length variation. To improve the accumu-
lator output web tension controller, an another approach consists
of synthesis a multi-model controller. A first controller is opti-
mized for a working point corresponding to the minimum web
span length in the accumulator. Then, a second controller is
optimized for the maximum web span length. The output of each
controller is weighted by a coefficient a (22) depending on the
web length L stored in the accumulator. To avoid problem of
integration during the controllers interpolation, the controllers
integral terms are frozen when the controllers are not used. The
evolution of a during web span length variation can be found in
Fig. 10 The control scheme is given in Fig. 11 As observed in
Fig. 11, controlling an accumulators using a multi-model struc-
ture is more complex; in fact, we have to synthesis four con-
trollers by optimization, and so the computational time will be
longer. This control structure (Fig. 11) is similar to the LPV control
structure, but the synthesis methods of the controllers is differ-
ent. For LPV controllers [24,25], controllers are synthesized
simultaneous to guaranty quadratic stability. LPV controllers a
priori may be robust. With multi-model controllers, each con-
troller is synthesized separately, so that the quadratic stability
cannot be guaranty [26]. We have to take into account the
controllers robustness during the optimization process (by pena-
lizing unstable controller parameters set) and check the robust-
ness range of our controllers, by simulation, after synthesis:
a ¼
1
L
À
1
Lmin
1
Lmax
À
1
Lmin
ð22Þ
Accumulator
C1
Controller
switching
strategy
C2
Vref
Lref
Vin accu
Vout accu
Tout accu
L
Tref +
+
+
+
-
-
Fig. 8. Control scheme for linear time invariant PI controllers.
0 10 20 30 40 50 60 70 80 90 100
0
2
4
6
8
10
12
Time (s)
Velocity(m/s)
Web velocity
0 10 20 30 40 50 60 70 80 90 100
0
0.2
0.4
0.6
0.8
1
Actuator switching strategy
Time (s)
Fig. 9. Evolution of the controller on/off weighting coefficient.
D. Kuhm et al. / ISA Transactions 51 (2012) 732–742 737

7. 3.2.3. Linear time invariant H1 controller
To improve the control performances and the robustness to
the mechanical parameter variations shown in the previous part,
a H1 controller has been synthesized. The H1 controller is
calculated using the linear state space described in (15), according
to the framework S/KS/T proposed in Fig. 12. The weighting
functions Wp, Wu and Wt used for loop-shaping appear in the
closed-loop transfer function between the exogenous input r, and
the performances outputs z in the following manner:
Tr-z ¼
WpS
WuKS
WtT
2
6
4
3
7
5 ð23Þ
where S is the sensitivity function S ¼ ð1þKGu-yÞÀ1
and T ¼ IÀS.
Exogenous input r is the web tension reference. The control
scheme is the same as for PI controllers (see Fig. 8). The H1
problem consists to find a stabilizing controller K which mini-
mizes the H1-norm of the transfer function Tr-z between
external input r and performance outputs z
JTr-zJ1 ¼ sup
o
ðsðTr-zÞÞrg ð24Þ
0 10 20 30 40 50 60 70 80 90 100
1
1.5
2
2.5
3
3.5
Web length
Time (s)
Length(m)
0 10 20 30 40 50 60 70 80 90 100
0
0.2
0.4
0.6
0.8
1
Alpha
Time (s)
Fig. 10. Evolution of the multi-model PI controllers weighting coefficient a.
Accumulator
C1
Controller
switching
strategyC2
Vref
Lref
Tref +
+
--
+
+
L
Vin accu
Vout accu
Tout accu
1-
C3
C4
1-
Lref
Lref
+
+
+
+
Fig. 11. Control scheme for multi-model PI controllers.
K G
Wp
Wu
Wt
z1
z2
z3
y
u
e
r +
-
Fig. 12. H1 framework.
D. Kuhm et al. / ISA Transactions 51 (2012) 732–742738

8. The weighting function Wp is usually taken with a high gain at
low frequency in order to reject low frequency disturbances. The
form of Wp is as follow [20]:
WpðsÞ ¼
s
M
þo0
sþo0e0
ð25Þ
where M is the maximum peak magnitude of the sensitivity S, o0
is the required frequency bandwidth, e0 is the steady state error
allowed. The weighting function Wu is used to avoid large control
signals. Wt is selected to attenuate high frequencies. As for PI
controllers, the web length has a significant influence on the web
tension, therefore a synthesis of a multi-model H1 controller will
be presented in the next part.
3.2.4. Multi-model H1 controller
The synthesis of a multi-model H1 controller is based on the
same approach as for a LTI H1 controller. For each actuator one
H1 controller is first synthesized for the minimum web length
0 10 20 30 40 50 60 70 80 90 100
0
100
200
300
400
500
600
700
Time (s)
Tension(N)
Web tension (N)
Toutref
ToutPILTI
ToutPImulti_model
0 10 20 30 40 50 60 70 80 90 100
0
2
4
6
8
10
12
Time (s)
Velocity(m/s)
Web velocity (m/s)
Vout
VdPILTI
VdPImulti_model
0 10 20 30 40 50 60 70 80 90 100
1
1.5
2
2.5
3
3.5
Web length (m)
Time (s)
Length(m)
Lref
LPILTI
LPImulti_model
a b c d e f g
Fig. 13. Accumulator simulation results for industrial PI controllers.
D. Kuhm et al. / ISA Transactions 51 (2012) 732–742 739

9. and second controllers for the maximum web length. The same
control strategy as for a multi-model PI controller (Fig. 11)
is used.
3.3. Nonlinear realistic simulation results
3.3.1. Industrial PI controllers simulation results
The simulations represented in this part result from the realistic
nonlinear model programmed in Matlab/Simulink software envir-
onment. Our principal accumulator performance criteria concerns
the maximum amplitude of tension variations in relation to the
reference tension, during the wound roll change phase (when the
unwinder is stopped).
Accumulator simulation sequence:
(a) from t¼25 to t¼35 s: the accumulator is at its nominal web
length, velocity and tension.
(b) from t¼35 to t¼41 s: the accumulator carriage is moved up
to reach the maximum web length, the velocity is increased to
maintain nominal web tension.
(c) from t¼41 to t¼60 s: the accumulator is charged and main-
tained at maximum web length.
(d) from t¼60 to t¼64 s: the input velocity decreases and the
accumulator carriage is moved down to restore web and
maintain the nominal web tension.
(e) from t¼64 to t¼66 s: the input velocity is equal to zero, the
accumulator carriage is moved down to restore web and
maintain the nominal web tension.
(f) from t¼66 to t¼70 s: the input velocity increase to reach the
nominal velocity, the accumulator carriage is moved down to
restore web and maintain the nominal web tension.
(g) from t¼70 to t¼100 s: the accumulator is at its nominal web
velocity and tension, the accumulator is maintained at a
constant position to maintain a constant web length in the
accumulator.
One can observe in Fig. 13 that optimized PI controller allows to
ensure moderate tension peaks and variation, even during the
wound roll change phase (d, e and f). As expected, multi-model PI
controllers lead to better performances during the unwinder roll
change phase (d, e and f). This performances improvement can be
observed in Fig. 14 which represent the relative absolute tension
error for both control strategies (LTI and multi-model) during the
wound roll change phase. One can observe that for multi-model PI
controllers, the maximum tension error is significantly reduced
(up to 57%).
3.3.2. H1 controllers simulation results
Fig. 15 presents LTI H1 controllers simulation results. The
simulation sequence is the same as indicated previously. Using
H1 controllers leads to ensure moderate tension variation, even
during the wound roll change phase. As expected, H1 controllers
lead to better performances than PI controllers: LTI H1 controller
reduces significantly the maximum web tension peaks in relation
to the reference tension, during all the simulation sequence,
especially during the wound roll change phase (see phases d, e
60 62 64 66 68 70 72
0
20
40
60
80
100
Time (s)
Error(%)
Relative absolute tension error for PI control strategies
PI LTI
Fig. 14. Relative absolute tension error for LTI and multi-model PI control strategies during wound roll change phase.
0 10 20 30 40 50 60 70 80 90 100
0
100
200
300
400
500
600
700
Time (s)
Tension(N)
Web tension (N)
Toutref
ToutPILTI
Tout a b c d e f g
Fig. 15. Accumulator simulation results for linear time invariant H1 controllers.
D. Kuhm et al. / ISA Transactions 51 (2012) 732–742740

10. and f in Fig. 15). The little static error at the beginning of the
simulation results from the terms neglected during the model
linearization. Indeed, the H1 controller is calculated for the linear
model and applied on the nonlinear one. Fig. 16 presents multi-
model H1 controllers simulation results. Unlike PI controllers,
multi-model H1 do not increases significant performances, indeed,
linear time invariant H1 performances are already improved
compared to PI controllers. A minor performances improvement
(tension peaks reduction) can be observed in Fig. 17 which
represents the relative absolute tension error for both control
strategies during the wound roll change phase.
4. Conclusion
This paper presents the physical equations and the modelling
of an industrial accumulator. The detailed nonlinear model,
programmed in Matlab/Simulink software environment, enables
0 10 20 30 40 50 60 70 80 90 100
0
100
200
300
400
500
600
700
Time (s)
Tension(N)
Web tension (N)
Toutref
Tout
Tout multi_model a b c d e f g
0 10 20 30 40 50 60 70 80 90 100
0
2
4
6
8
10
12
Time (s)
Velocity(m/s)
Web velocity (m/s)
Vout
Vd H
Vd H
0 10 20 30 40 50 60 70 80 90 100
1
1.5
2
2.5
3
3.5
Time (s)
ref
H
H
a b c d e f g
a b c d e f g
Fig. 16. Accumulator simulation results for multi-model H1 controllers.
D. Kuhm et al. / ISA Transactions 51 (2012) 732–742 741

11. a model based optimization of PI controllers. The linear model is
used to study the effect of some mechanical parameters on the
accumulator, with the help of Bode diagrams, and enables to
synthesize H1 controllers. For both controllers, linear time invar-
iant and multi-model control strategies are compared. Simulation
results comparing both control strategy have also been presented.
As expected, multi-model control strategy improves significantly
the performances.
Acknowledgment
The authors wish to thank Monomatic France Company for
their helpful discussions. Also thanks to the Re´gion Alsace for
having partially founded this work.
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D. Kuhm et al. / ISA Transactions 51 (2012) 732–742742