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www.etasr.com Duc et al.: Imperfect Roll Arrangement Compensation Control based on Neural Network for Web … 

 

Imperfect Roll Arrangement Compensation Control 

based on Neural Network for Web Handling Systems 

Dinh Nguyen Duc 

School of Electrical Engineering 
Hanoi University of Science and 

Technology 

Hanoi, Vietnam 
dinh.nguyenduc201297@gmail.com  

Ly Tong Thi 

Hanoi University of Science and 
Technology and 

Hanoi University of Industry 

Hanoi, Vietnam 
ttly.haui@gmail.com 

Tung Lam Nguyen 

School of Electrical Engineering 
Hanoi University of Science and 

Technology 

Hanoi, Vietnam 
lam.nguyentung@hust.edu.vn 

 

 

Abstract—The speed and tension control problem of a web 

handling system is investigated in this paper. From the system 
equations of motion, we developed a backstepping-sliding mode 

control for web speed and tension regulation tasks. It is obvious 

that the designed control depends heavily on roll inertia 

information. Dissimilar to other researches that were based on 

the assumptions of rolls with perfect cylindrical form with the 

rotating shafts of the rolls considered properly aligned, the 

novelty of this paper is the presentation of a neural network to 
compensate the effects of imperfect roll arrangement. The neural 

network design is based on the Radial Basis Function (RBF) 

network estimating the uncertainty of roll inertia. The 

information on estimated inertia is fed into a backstepping-

sliding mode controller that ensures tension and velocity 

tracking. The control design is presented in a systematical 
approach. Closed loop system stability is proven mathematically. 

The tracking performance is shown through several simulation 
scenarios. 

Keywords-roll-to-roll; web handling system; backstepping; 

sliding mode control; radial basis function (RBF); adaptive control 

I. INTRODUCTION  

Numerous practical applications associated with web 
handling systems such as flexible displays, color-shifting, 
lighting, solar cells, etc. utilizing Roll-to-Roll (R2R) systems 
are becoming prevalent in industry [1-5]. Several papers have 
been published to enhance speed and tension control with 
various control algorithms based on the system dynamic 
modeling illustrated in [2, 3]. Sliding Mode Control (SMC) [6] 
is employed in [7], in which tension observers are introduced to 
eliminate the requirement of loadcells. Backstepping control is 
applied in [8–11] to counter the nonlinearity effects in R2R 
systems. An indirect tension control is proposed in [12], where 
the web tension is manipulated indirectly via distance 
information between two consecutive rolls. The experimental 
results show a moderate level of tension while tracking 
performance and high position control accuracy. Authors in 
[13] designed a tension coupling compensator based on active 
disturbance rejection control. The main advantage of this 
method was its simple approach. The elasticity of the web 
material is the most challenging problem in tension control. In 

[14], the phenomenon is tackled by a H-infinity robust control 
with varying gains. The control shows robustness against 
varying roll radius and inertia. Some effects of the transmission 
system such as backlash and flexible motor-roll coupling are 
considered in [15, 16]. 

It is evident that roll inertia information is essential for 
control system design [17–19]. Conventionally, the roll inertia 
depends on roll shape and web thickness. In practice, roll 
eccentricity and deformation exist and lead to false estimations. 
The robustness of the closed loop system against roll 
eccentricity is presented in [20]. However, when the parameters 
of the system are accompanied by system uncertainties, the 
proposed controllers can hardly guarantee efficiency and 
quality control in production. Hence, it is strictly essential to 
develop an adaptive controller that deals with the problem of 
varying roll inertia for the control of the R2R systems. A neural 
network is a powerful tool which can address the complex 
requirements of the system, requirements that other techniques 
could hardly solve. In this paper, by taking the advantages of 
Backstepping and Sliding Mode Control (BSMC) [21], a new 
control approach which uses BSMC integrated with an RBF 
neural network [22] (RBFN-BSMC) is proposed for the 
building of an R2R adaptive control system considering the 
presence of imperfection roll arrangements. The contribution of 
this paper can be summarized as: i) the proposal of a nonlinear 
backstepping-sliding mode control for web handling systems, 
ii) the realization of an RBF based adaptive mechanism for 
compensating roll inertia uncertainty. 

II. BSMC FOR ROLL TO ROLL WEB SYSTEMS 

A. R2R Web System Modeling 

The single-span R2R web control system shown in Figure 1 
contains an unwinder, a rewinder and a loadcell subsystem 
with idle rollers. Input torqueses are applied on the unwinder 
and rewinder. The following assumptions are made: 

• Web slippage and deformation are totally ignored. 

• Loadcell dynamics and friction are not taken into account. 

• The web obeys Hook’s law.  

Corresponding author: Tung Lam Nguyen



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• Actuator dynamics are ignored. 

 

 
Fig. 1.  Single-span web control system 

The non-linear dynamic equations of the single-span R2R 
web control system are [3]: 

1 2 3u u uc c T cω ω τ= + +�     (1) 

4 5 6u r rT c c T cω ω ω= + +
�     (2) 

7 8 9r r rc T c cω ω τ= + +�    (3) 

where 1 ,
u

u

B
c

J
=  2c ,

u

u

R

J
=  3

1
,

u

c
J

= −  4 ,uc KR= −  

5 ,
Rr

c
L

= −  6 ,rc KR=  7 ,
r

r

R
c

J
=  8 ,

r

r

B
c

J
= −  9

1
.

r

c
J

=  

In the above equations of motion, (1) and (2) present the 
dynamics of unwinding and rewinding section, and (3) 
describes tension dynamics between the two rolls. The relations 
between the total moment of inertia, operating radius, web 
thicknes, and roll speed are described by:   

0( )
2

u
u u

h
R t R

θ
π

= −  

( )4 40
0

2

u u

u u

R R
J J πρω

−
= +

 

0( )
2

r
r r

h
R t R

θ
π

= +  

( )4 40
0

2

r r

r r

R R
J J πρω

−
= +  

where T is web tension, Ju and Jr are the moments of inertia of 
the unwind and rewind rolls, Ru and Rr are radii of the unwind 
and rewind rolls, Bu and Br are the coefficients of vicious 
friction of the unwind and rewind rolls, K is the spring constant 
of the web, L is the total web length, h is web thickness, ρ is 
web density, ωu, ωr are the unwinding and rewinding roll 
speeds, and θu, θr, are unwinding and rewinding roll positions. 

B. Backstepping Sliding Mode Control 

The controller’s purpose is keeping web tension and web 
speed at the desired values. Web speed is controlled through 
tracking the angular velocity. This section uses the sliding 
algorithm based on backstepping technique to design the 
controller. The design steps are: 

Step 1: The tracking error variables are defined as bellow: 

dT T T= −     (4) 

u u udω ω ω= −     (5) 

r r rdω ω ω= −     (6) 

where Td and ωrd are the desired web tension and the desired 
rewind web angular velocity respectively, ωud is an unknown 
function serving as a virtual control that will be designed in 
such a way that T tends to Td. 

Step 2: The control signal ωud is determined such that the 
web tension tracks the desired value. Equation (2) can be 
rewritten as: 

4 5 6u r rT c c T cω ω ω= + +
�

    (7) 

To regulate the tension error 0T → , virtual control ωud is 
chosen as:  

( )5 6 1
4

1
ud r rc T c k T

c
ω ω ω= − + +     (8) 

where k1
 
is a positive gain. The proposed Lyapunov candidate 

function is: 

21

2
TV T=     (9) 

Using (7) and taking the time derivative of (9) we obtain:  

( )4 5 6T u r rV T c c T cω ω ω= + +�     (10) 

Replacing ωu by ωud from (8) results in: 

2
1 0TV k T= − ≤

�  

Step 3: Based on backstepping technique, the control signal 
τu, τr will be determined in order to drive ωu to track ωud, and 
ωr to track ωrd asymptoticaly. From (1), (3), (5), and (6) we 
obtain the error equations: 

1 2 3u u u udc c T cω ω τ ω= + + −
� �     (11) 

8 7 9r r r rdc c T cω ω τ ω= + + −
� �     (12) 

The sliding surfaces are defined as: 

u uS ω=     (13) 

r rS ω=     (14) 

The proposed Lyapunov candidate function is: 

21

2
u uV S=     (15) 

Differentiating (15) gives: 

u u uV S S=
��     (16) 

The control signal is calculated as: 



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u ueq uswτ τ τ= +     (17) 

where ueqτ is the control component making 0uS =
� and uswτ  is 

the control component ensuring 0.uS ≤
�  From (11), it is 

straightforward to get ueqτ as: 

( )1 2
3

1
ueq u urc c T

c
τ ω ω= − + − �     (18) 

In order to guarantee 0uS ≤
� , the signal 

usw
τ  is proposed 

to be: 

( )2
3

satusw u
k

S
c

τ = −     (19) 

where k2 is a positive gain. From (11), (13), (16)-(19) we get: 

( )2 sat 0u u uV k S S= − ≤�  

Similarly, the control signal rτ  is generated by taking the 
total reqτ and rswτ , where: 

( )8 7
9

1
req r rrc c T

c
τ ω ω= − + − �     (20) 

( )3
9

satrsw r
k

S
c

τ = −     (21) 

where k3 is a positive gain. It should be highlighted that the 
designed controls depend on the calculated roll inertia which is 
based on the assumption of a perfectly prismatic roll shape and 
ideal rotating shaft alignment. This assumption does not always 
hold in practical applications. 

III. ADAPTIVE BSMC FOR THE UNCERTAINTIES OF THE ROLL 

TO ROLL WEB SYSTEM 

In practice, roll eccentricity and deformation as 
demonstrated in Figure 2 might occur [3]. 

 

  

Fig. 2.  Roll eccentricity and deformation 

This phenomenon leads to miscalculated moments of inertia 
of the unwind and rewind rolls, hence some system parameters 

are unknown: 1 2 3 7 8, , , , ,c c c c c and 9.c The designed input torques 

given in (22)-(25) indicate a strong dependence on the accuracy 
of acquired inertia. Inaccurate information on roll inertia can 
deteriorate control performance. It is necessary to develop a 
strategy to retrieve unwinding and rewinding sections. In order 
to tackle this problem, a mechanism based on the RBF network 

is proposed to estimate these parameters. The inputs of the 

neural network are ωu, ωr and the outputs are 
ˆ ˆ,u rJ J which are 

the estimated inertia values. The neural network is trained 
online in order to force the error between the actual and the 
estimated values to approach zero. W is defined as the ideal 

weight and Ŵ as its estimation The error of ideal and estimated 
weights is: 

ˆW W W= −�     (22) 

Then, J and Ĵ  can be writen as:  

ˆ ˆ,   

ˆ

T T

T

J W h J W h

J J J W h

= =

= − =� �
    (23) 

where h  is the vector output of the hidden layer with its 

transfer function defined as: 

2 2

1 2

2

2 2

1 2

2
1

exp

exp

u i r i

i

i
n

u i r i

jj

c c

b
h

c c

b

ω ω

ω ω

=

 − + −
 −
 
 =
 − + −
 −
 
 

∑
 

where n is the number of neurons of the neural network. Under 
the condition of uncertain roll inertia, control signals that are 
derterminded in the above section can be rewritten as: 

( )( )1 2 2
3

1
ˆ ˆ ˆ sat

ˆ
u u ud uc c T k S

c
τ ω ω= − + − +�     (24) 

( )( )8 7 3
9

1
ˆ ˆ ˆ sat

ˆ
r r rd rc c T k S

c
τ ω ω= − + − +�     (25) 

where 1 2 3 7 8ˆ ˆ ˆ ˆ ˆ, , , , ,c c c c c and 9̂c are the estimated values of 

1 2 3 7 8, , , , ,c c c c c and 9,c respectively, since these constants 

depend on roll inertia. The objective of this control design is to 
identify the laws for unwinding and rewinding rolls. Replacing 

,uτ rτ  with ˆ ,uτ r̂τ  in (11) and (12) we obtain: 

( )2
ˆ

satu uu ud u
u u

J J
S k S

J J
ω= −
�

� �     (26) 

( )3
ˆ

satr rr ur r
r r

J J
S k S

J J
ω= −
�

� �     (27) 

The Lyapunov candidate function for 
u

S  becomes: 

( )2 11 1 Tr
2 2

T
u u u u u uV J S W F W

−= + � �     (28) 

where Fu
 
is a fit positive matrix. Taking the time derivative of 

(28) we get: 

( )2 11 ˆTr
2

T
u u u u u u u u uV J S J S S W F W

−= + +
�

�� � �     (29) 

d

i
O rO O



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Substituting (26) into (29) yields: 

( ) ( )2 12
ˆ1 ˆsat Tr

2

Tu u
u u u u u ud u u u u

u u

J J
V J S J S k S W F W

J J
ω −

 
= + − +  

 

�
�

� � ��

After some fundamental operations, it can be shown that: 

( )

( )

2
2

1

1 ˆ sat
2

ˆTr

u u u u u u

T
u u u ud u

V J S J k S S

W F W h Sω−

= −

 + + 
 

� �

�
� �

    (30) 

Equation (30) suggests that the updated law can be selected 
as: 

ˆ
u u u ud uW F h Sω= −
�

�     (31) 

With the selected updated law uV
�  can be rewritten as: 

( )2 2
1 ˆ sat
2

u u u u u uV J S J k S S= −
� �  

It is noted that 0uJ ≤
� , thus 0.uV ≤

�  The proposed 

candidate Lyapunov function for rS  is: 

( )1
2

Tr1

2 2

T
r r r

r r
r

W F W
V S

J

−

= +
� �

    (32) 

where Fr is a fit positive matrix. Taking the time derivative of 
(36) yields: 

( ) ( )( )11
2

2

TrTr

2

T
T

r r r rd r
r r r r

r r r

r r r

W F W h SW F W J
V J

J J J

ω
ω

−
− +

= − − +

�
�� �� � �

� � (33) 

Similarly, the updated law for neural network weight is 
chosen as: 

ˆ
r r r rd rW F h Sω= −
�

�     (34) 

The update law renders rV
� becomes: 

( )1
2

3 2

Trˆ

2

T
r r r

r
r r

r r

J W F WJ
V k S

J J

−

= − −
� � �

�  

In addition, 0rJ ≥
� , thus rendering 0.rV ≤

�  

IV. SIMULATION RESULTS 

In this section, the performance of the proposed control and 
estimation mechanism is demonstrated through a numerical 
simulation. To adequately evaluate the effectiveness of the 
control system, there will be two simulation cases: with 
constant inertia and with added disturbance to the inertia. The 

parameters of the R2R system are utilized as: 
0

0.04m,
u

R =

0.00002533kgms/rad, 1m,
u r

B B w= = =
0 0

1kg/m/s,
u r

J J= =  

0
0.3m, 0.015m, 0.00002m,

r
L R h= = = and K = 200kg/m. 
The parameters of the controller are chosen as: k1=5, k2=10, 

k3=10. In the simulation, the performance of the BSMC 
controller was compared with the adaptive RBFN-BSM 
controller’s. The reference web speed was 0.2m/s and the 
desired web tension Tref = 2N. 

 
BSMC receives full information 

of the web transport system while RBFN-BSMC must estimate 
the information of the model.  

 

(a) 

 

(b) 

 

Fig. 3.  System response with constant roll inertia. (a) Web tension, (b) 

web speed 

 

(a) 

 

(b) 

 

Fig. 4.  Input torques with constant roll inertial. (a) Rewinding roll, (b) 

unwinding roll 

When the roll inertia is pre-identified, both BMSC and 
RBFN-BMSC produce accurate and fast tension and speed 
responses as shown in Figures 4 and 5.  

 



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Fig. 5.  Roll inertia disturbance 

(a) 

 

(b) 

 

Fig. 6.  System response with varying roll inertia. (a) Web tension, (b) web 

speed 

(a) 

 

(b) 

 

Fig. 7.  Input torques with varying roll inertia. (a) Rewinding roll, (b) 

unwinding roll 

However, the required input torque for RBFN-BMSC is 
lower than BMSC’s. It is interesting to note that in this 
situation, RBFN-BMSC provides better tension tracking 
performance. The effectiveness of the proposed RBFN-BMSC 

is validated when the system is subject to unknown varying 
inertia. It is assumed that the roll inertia disturbance is the one 
given in Figure 5. As can be seen in Figure 6, the tracking 
result of RBFN-BMSC is still guaranteed. Without precise 
information of roll inertia, the BMCS shows its weakness in 
following the reference. Figure 7 indicates that under varying 
roll inertia, BMSC input torque fluctuates with high amplitude. 
Keeping in mind that full information of the roll to roll model 
is difficult to be obtained in real conditions, thus RBFN-BSMC 
law is more suitable for industrial applications than BSMC.  

V. CONCLUSION 

In this paper, a controller which combines the backstepping 
method with sliding mode control was proposed for web 
transport systems. In the proposed controller the estimator 
based on the RBF neural network was utilized for 
approximating system uncertainties. Simulation results show 
that the control system achieves both tracking and adaptation. 
Furthermore, when using RBF neural network, the controller 
does not require the precise description of the plant, thus the 
proposed controller is highly capable to be used in industrial 
applications. Future research focus on system generalization to 
multi-section web tension and a tension sensorless control will 
be proposed along with the experimental work.  

ACKNOWLEDGEMENTS 

This research is supported by the Hanoi University of 

Science and Technology under the research grant T2018-PC-

053. 

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