LATERAL CONTROL OF A WEB USING
ESTIMATED VELOCITY FEEDBACK
By
SHAIBAL SAILAZA MANDAL
Bachelor of Engineering
Sardar Patel College of Engineering
Mumbai, India
1998
Submitted to the Faculty of the
Graduate College of the
Oklahoma State University
in partial fulfillment of
the requirements for
the Degree of
MASTER OF SCIENCE
July, 2000
LATERAL CONTROL OF A WEB USING
ESTIMATED VELOCITY FEEDBACK
Thesis Approved:
Thesis Advisor JU( hrQ'~
,
11
Acknowledgements
I would like to express my gratitude to Dr. Prabhakar Pagilla, my thesis advisor, for his
guidance and advice throughout the development ofthis investigation. Without his help
and supervision, I would not have completed this work. I am also grateful to my
committee members, Dr. Gary Young and Dr. John J. Shelton, for their help and
guidance. My sincere gratitude to Fife Corporation in Oklahoma City and Ken HOPCllS
for providing us with the equipments necessary for conducting the experiments.
I wish to thank Yongliang for setting up the realtime program. Finally, I wish to thank
my colleagues Biao and Sun Yu for helping me with the experiments.
1ll
Table of Contents
Chapter Page
1 Introduction 1
1.1 Literature Review " 2
1.2 Thesis Contribution .5
1.3 Thesis Outline 6
2 Ba,ckgl"ound 7
2.1 Static behavior of web 7
2.2 Dynamic behavior ofweb 11
2.2.1 ldeal moving web 11
2.2.2 Real moving web .15
2.3 Lateral control of a web using nip rollers 26
2.3.1 Concept used in the use of rubber covered rollers 26
2.3.2 Lindley's analysis 27
2.3.3 Foreman's analysis 28
2~3A Shelton's analysis 30
3 Automatic Guiding Mechanism and Control Systems 34
3.1 Basic types of automatic control systems 34
3.1.1 Hydraulic types 34
3.1.2 Mechanical types 35
3.2 Guiding mechanisms 36
3.2.1 End pivoted guide ' .36
3.2.2 Center pivoted guide 37
3.2.3 Remotely pivoted guide 37
3.2.4 Offset pivoted guide 38
4 Observer Design and Simulation Results , .42
4.1 Present controller .42
4.2 The importance of innerloop velocity feedback , .43
4.3 Observer design 44
4.3.1 Developing the state space fonn .45
4.4 State observers 48
4.5 Lateral dynamics: Simulated results 51
4.5.1 Openloop response 51
4.5.2 PI controL 53
IV
5 Experimental Setup 64
5.1 Hardware '" 65
5.1.1 Lateral control system 65
5.1.2 Computer system 69
5.2 Software structure· 70
5.2.1 Realtime software 71
6 Experimental Results 74
6.1 Emulation ofFife A9 by computer 74
6.2 Solutions without tachometer 78
6.3 Comparison of controller 83
7 Conclusion and Future Research 87
7.1 Conclusion 87
7.2 Future work 88
Bibliography 90
A Derivation of Velocity Observer 93
B Matlab Script File 100
v
Figure
List of Figures
Page
2.1 Schematic diagram ofweb on rollers 8
2.2 Freebodies and symbols for steadystate analysis 9
2.3 Web passing over a series ofnonparallel rollers II
2.4 Symbols for the derivation of response at a fixed roller 12
2.5 Symbols for the derivation of response at a steering guide 13
2.6 Symbols for the derivation of response at a point between two rollers 14
2.7 First order time r,esponse at a fixed roller due to input at the previous roller 15
2.8 First order time response at a steering guide 16
2.9 First order response at a point between two parallel rollers 17
2.10 Steering action of a web with induced curvature 1R
2.11 Boundary conditions for translation of the end of a web 19
2.12 Boundary condition for rotation of the end of a web " 21
2.13 Translation '" 22
2.14 Rotation 23
2.15 Translation: Deflection ofa web between two parallel rollers 24
2.16 Rotation: Deflection of a web between two parallel rollers 25
2.17 Second order time response of a web at fixed roller to input at previous roller 26
2.18 Second order time response for a steering guide 27
2.19 Second order time response at a point between two parallel rollers .28
VI
Figure Page
2.21 Concept for rubber covered rollers 29
2.22 Foreman's experimental setup , 30
2.23 A free catenary , 31
2.24 A supported catenary " " " 31
2.25 Equivalent free catenary of the supported catenary .32
2.26 Schematic diagram ofroller 33
3.1 Pneumohydraulic guiding control system 35
3.2 Electrohydraulic guiding control system 36
3.3 Pneumomechanical guiding control system 37
3.4 Electromechanical guiding control system 38
3.5 End pivoted guide 39
3.6 Center pivoted guide 39
3.7 Remotely pivoted guide 40
3.8 Offset pivoted guide .41
4.1 Block diagram ofcontroller using tachometer .42
4.2 Web position: computer control without velocity feedback .44
4.3 Motor velocity: computer control without velocity feedback .45
4.4 Block diagram of controller using tachometer .46
4.5 Block diagram of controller using velocity estimator .46
4.6 A web system 51
4.7 Response at KamberoUer guide to an impulse disturbance 53
4.8 Response at Kamberoller guide to unit step disturbance .54
4.9 Response at Kamberoller guide to sinusoidal disturbance _ 55
VB
Figure Page
4.10 Block diagram oflatera~ control system .56
4.11 Response at Kamberoller guide to an impulse disturbance 57
4.12 Response at Kamberoller guide to unit step disturbance 57
4.13 Response at Kamberoller guide to sinusoidal disturbance 58
4. m4 Lateral control system with estimated motor velocity feedback 58
4.15 Simulink block diagram using tachometer feedback 59
4.16 Simulink. block diagram using estimated motor velocity feedback 60
4.17 Response using estimated motor velocity feedback (impulse disturbance) 6]
4.1 8 Response using estimated motor velocity feedback (unit step disturbance) 61
4.19 Response using estimated motor velocity feedback (sinusoidal disturbance) 62
4.20 Impulse reference response using estimated velocity feedback 62
4.21 Unit step reference response using estimated motor velocity feedback 63
4.22 Sinusoidal reference response using estimated motor velocity feedback 63
5.1 Experimental web platform 66
5.2 Lateral control of the experimental web pJatfonn 67
5.3 Schematic of analog lateral control system 67
5.4 Fife analog lateral control system 68
5.5 Schematic oflateral computer control system with velocity innerloop 69
5.6 Lateral computer control system with tachometer velocity 69
5.7 Lateral computer control system using velocity estimator 70
5.8 Software for web handling system 71
6.1 Web Position: Fife (A9) controller and computer controller 76
6.2 Motor Velocity: Fife (A9) controHer and computer controller 77
VIn
Figure Page
6.3 Web position: computer control using finite difference velocity feedback 79
6.4 Motor velocity: computer control using finite difference velocity feedback .....80
6.5 Finite difference velocity: computer ,control using finite difference velocity
feedback 80
6.6 Web position: computer controller using velocity observer 81
6.7 Tachometer velocity: computer controller using velocity observer '" 8]
6.8 Estimated velocity: controller using velocity observer 82
6.9 Comparison of velocities from three controllers: no disturbance 84
6.10 Comparison of velocities from three controllers: step disturbance 85
6.1] Comparison of velocities from three controllers: pulse disturbance 86
A.I Typical control system with estimated state feedback 93
IX
a
A
A
6
D
E
F
F/W
G
I
K
L
M
Mo
}.h
n
N
Q
R
S
t
T
y
v
Nomenclature
half contact width of rubber covered roUs
crosssectional area of the web
durometer (chapter 2 only)
radial deflection of rubber covered rolls
roll diameter
modulus of elasticity (Young's modulus)
force or load
effective nip load
modulus of elasticity in shear
moment of Inertia
constant for a given operating condition (JT/ EI)
the length of a free span of a web
bending moment
bending moment at upstream roller
bending moment at downstream roller
a stress averaging factor
shear force normal to the elastic curve of the web
shear force parallel to the original web centerline
roller radi us
undeflected ru bber covered roll radius
shape factor
rubber cover thickness
web tension
lateral posi tion of web
velocity of the web
x
YL response at a Kamberol1er guide
Yo( s) the positional disturbance from upstream roller
w side load per unit length
W width of web
z position of downstream roller relative to the ground
X2 estimated velocity of motor
U control input
() angle between web and roller
Br angle of rotation of roller
00 angular disturbance at upstream roller
(h angular input at downstream roller
f3 angle of arc of lateral slippage
fL coefficient of friction
Xl
Chapter 1
Introduction
A web refers to any material that is very long compared to its width and very
wide compared to its thickness. Examples include paper rolls, thin aluminum foils,
photo films, plastic films and metaI strip. In a processing line, a web travels over a
series of rollers that provide support, transport and control.
Web handling involves unwinding the material, feeding it to a processing plant
and then winding it back on to a rolL Generally the web must follow a predetermined
path which is in the longitudinal direction with a constant velocity. Deviation from
this may result in inferior product quality. Too much lateral movement can cause
slackness of the web.. When this happens the slack web can go through a nip or be
wound on to a roll causing wrinkles or creases to form. For this reason a tight control
on the lateral displacement of the web is essential. Optical, pneumatic or ultrasonic
sensors are used as edge detectors to sense the edge of the web.
The basic elements of an automatic system to control the lateral position of a
web are (1) a sensor to detect web position, (2) a controller to receive input signals
from the sensor and produce a higher power level output, (3) a guiding mechanism
including some kind of actuator to receive the output of the controller and translate it
into physical positioning of the web, and (4) the web itself, through which the sensor
1
detects lateral signals. Therefore a guiding installation is a system that consists of
several components selected to function together to obtain the desired control. Proper
selection and integration of the components is essential to obtaining optimum results.
The guiding system should be located as close as possible and immediately preceding
the point in the process where accurate control is required. It should be designed
and installed so that it is capable of correcting the maximum lateral deviation expected.
It is important to remember that once the web leaves the guiding system,
it is free to wander again, depending upon the many factors that may influence it,
such as web properties and machine and process disturbances. All commonly used
guide rollers today depend on a fundamental law of web behavior, that a web moves
laterally on a roll it is approaching until its upstream span becomes perpendicular to
that roll. The steeringtype guide roller utilizes this principle by inclining itself to its
entry span. As the web moves through the machine, it moves to align itself perpendicularly
to the inclined guide roll and is laterally displaced in the entry span an amount
depending upon the length of the span and the guide roll angle. A steering guide is
useful where a relatively long, free web span is available. Lateral displacement of a
web is also possible using differentially loaded nip guides.
1.1 Literature Review
This section will first review the statics and dynamics of a moving weh and then
briefly discuss nip rollers in the lateral control of web.
Modeling lateral dynamics is an important first step in an attempt to improve web
process control and endroll product quality. The model must provide an accurate
description of the system and facilitate the implementation of online parameter and
state estimation. Mathematical modeling of lateral web dynamics was introduced
by Campbell (1958). Campbell's model was ba.sed on the assumption that the web
2
behaves like a string and makes sharp breaks at every roller. This basic assumption
led to a first order model which was not very accurate. Shelton (1} , in his thesis
(1968) developed an improved model based on the assumption that the web between
two rollers behaves like a beam. The Shelton model can describe accurately the
response of the web at a roller due to an input from a pure displacement guide but
is limited to single span systems.
Shelton has been broad in his work on the lateral control of a web due to lack
of previous literature on this subject. The fundamental static beam theory has been
covered in considerable depth. He found the web lateral behavior to depend upon
the dimensionless parameter K L. If K L is small (less than unity) and the shear
modulus low, the static behavior was found to be quite dependent upon the shear
nT
deflection parameter AG' Shelton has analysed the web for conditions considering
only shear deflection (simplified model of first order) and for both bending deflection
and shear deflection (accurate model of second order). The transfer functions of
several practical web handling components and configurations were derived for each
case. The accurate method of dynamic analysis confirmed the trends found in the
simplified analysis to give a fair approximation at values of wT1 of less than two. If
K L is less than unity and ~~ is less than 0.001, as is generally the case, the second
order theory was found to be more accurate.
Although web conveyance systems have been widely used in the industry since
the days of industrial revolution, the technical hterature on the subject of lateral
dynamics is very limited. The most significant work on lateral dynamics was done by
Dr. J.J.Shelton. Shelton and Reid [2, 3] presented a comparison of the first order and
the second order web dynamics to illustrate the inadequacies of the former for certain
frequencies and operating conditions. Young and Reid [5] have given a clear insight
into the fundamentals of lateral control of a web. Young, Shelton and Kardamila,q [6]
3
use state estimation to predict lateral web position on a downstream sensor WiUl use of
the feedforward sensor to achieve improved control. The experiments provide with a
solution when placement of an edge sensor near the web guide is prevented by physical
constraints in the processing lines. Young and Karrlamilas [7] and Kardamilas [8] have
introduced a stochastic model, which represents nonideal webs and disturbances at
the entering span roU,. based on experimental data. Little significant work has been
done on the lateral control of web using differentially loaded nip rollers. Guiding
by means of a higher loading on one side of a nip than on the other side causes a
variation in velocity across the width of a web \vhich is indicative of a variation in
strain and therefore stress across the width. When integrated this stress will result in
a moment which will be shown to produce the lateral deformation of the web. Guiding
the web using differentially loaded nip guides would be particularly advantageous in
applications where the entering and exiting spans are collinear. Young, Shelton and
Fang [9J discuss the effects of low friction between the web and the roller between
two web spans. The resulting slippage may cause the tension distribution in the
downstream span to affect the tension distribution in the upstream span, resulting
in a large lateral displacement. The primary purpose for the static analysis on the
interactive web systems is to establish the governing function for each span. In [10],
Young, Shelton and Fang develop the dynamics of interacting web spans based on
the fundamental theory for web lateral motion developed by Shelton in his thesis.
Lindley [11 J and Foreman [12] have been successful in controlling the lateral displacement
using differentially loaded rubber covered rollers. Lindley presented a
loaddeformation relationship. Foreman used rubber covered rollers and found that
the velocity of the web is proportional to the load on these rubber covered rollers.
He stated that the increased velocity of the strip is due to the increased length of
contact between the compressed rubber and the strip passing through. Shelton [13]
4
in his report to Fife Corporation on guiding with differential forces on rubbercovered
nip rollers analysed the free and supported catenary conditions of the web for lateral
movement. He found inadequacies in Foreman's analysis as it did not readHy fit into
a manageable theory. A few of the problems as specified by Shelton are (1) Extensive
testing was done with hardness of rubber covers of 46, 50 and 58 Shore A durometer,
well below the range of wringer rollers to have satifactory life expectancy. (2) The
extensive testing was done with one rubbercovered and one steel roller. The thickness
of the steel strip was not reported, but unless it was thick enough to remain
planar within the nip, the nip behavior would be somewhat different from that with
two identical rubbercovered rollers. (3) The web was dry, and the importance of friction
was not determined. Ahmad [14] in his thesis experimentally verified the lateral
deformations of a web due to nonuniform nip loading with rubber covered rollers.
Shelton [15] has discussed the problems associated with cambered webs and how to
eliminate or minimize the effects of camber. In his paper, Hopcus {4] discusses the
practical application of terminal guides (unwind and rewind), the control system loop,
sensor configurations, sensor locations and response of the system and equipment for
the lateral control of a web.
1.2 Thesis Contribution
This research focuses on the control of lateral movement of the web and is an extension
of the work done by Shelton for his PhD. thesis. The major achievements of this
research are: (1) Investigation of existing models; (2) Investigation of the importance
of the innerloop velocity feedback; (3) Design and development of the experimental
web platform; (4) Design and development of an open architecture realtime software
system for easy implementation of lateral control strategies; (5) Implementation of
a PID controller using the computer, bypassing the analog controller from Fife and
5
using it as an edge sensor signal amplifier.
Presently the industry uses different guiding mechanisms for lateral control of a
web. A brief description of each of these guiding mechanisms and control system is
explained in Chapter 2. The guiding mechanism used for the experiments in this
research is a Fife Kamberoller guide, which is a remotely pivoted guide. The control
system for this guiding mechanism uses the velocity feedback from the motor. At
steady state, the position of the motor is directly proportional to the position of the
web. The motor velocity can therefore he estima.ted using the velocity of the web,
which can be estimated by using a minimum order observer design. The objective of
this research is to eliminate the innerloop velocity feedback in the motor using an
observer based controller, thus reducing the cost of the motor. It is experimentally
verified that the digital control using the computer gives a better response than the
A9 analog controller used by Fife.
1.3 Thesis Outline
The organization of this report is as follows. Chapter 2 gives a brief overview of
the work done by Shelton, Lindley and Foreman. Some of Shelton's work has been
reproduced in this chapter since it forms the fundamental basis for web lateral behavior.
Different types of automatic control systems and guiding mechanisms have been
described in Chapter 3. The observer design to estimate the velocity of the web for
eliminating the innerloop velocity feedback in the motor has been done in Chapter
4. Chapter 5 describes the openarchitecture experimental platform that is developed
for lateral control experiments. Chapter 6 shows the experimental results obtained
after conducting lateral control experiments. Chapter 7 lists some remarks of this
work and suggestions for future research.
6
'Chapter 2
Background
Dr. Shelton's [1] work has been extensively reviewed and a summary of this is presented
in this chapter. It clearly explains the statics and dynamics of a moving web.
Re assumed the following conditions for the lateral behavior (static and dynamic) of
a web. (1) Standard beam theory assumptions: 0) The web is initially straight and
uniform; (ii) All deflections are small. (2) The area of contact between the web and
the roller is small compared to the length of the web span. (3) Friction between the
web and the roller is sufficient to prevent slippage and moment transfer, so that a
moment in one span is isolated from its adjacent span. (4) Stress distribution in the
normal section of a web is linear.
2.1 Static behavior of a web
Using the elementary beam theory Shelton analyzed the static behavior of a web to
derive a fourth order differential that is used for web analynis. Fig. 2.1 shows a
schematic diagram of a section of the web with tension forces on it. The web span is
assumed to be long so that deflections due to shear stress can be neglected.
The moment and the normal forces acting on the web is given by the following
7
equations:
T
t
: ·ljPSTREAM _ _. _.. y
:_.._ .RQ~~J~B..__ _ .
T
Figure 2.1: Schematic diagram of web on rollers
Ai = _ Ed (IdY)
dx dx
IV = _ Ed (IcPY)
dx dx2
W = Ed
2
(Id'2 y )
dx2 dx2
(2.1 )
(2.2)
(2.3)
Fig. 2.2 shows the free body diagrams of a free web span, a horizontal section
and a normal section. These are used to derive the fourth order differential equation
8
)(
L
N o .. y
(c)
2 "'T'1 M + dM 2dxYdx2+., D?"' i: ,
; N f
(a)
; UPSTREAM :
~ .I3_9.'=!_~B j
M o'.)
(b)
T
M;+dM+ /
i lai !i:fnT
~~
. M~a/
T Not,e : a is shear
para,l1el to y axis
Figure 2.2: Freebodies and symbols for steadystate analysis
used for web analysis. Since all deflections are assumed to be small the tension T in
both (b) and (c) of Fig. 2.2 are equal. A summation of all moments about point 0
gives us the following relation:
Q = dM +Tdy
dx dx
(2.4)
Taking the derivative of equation (2.1) and substitution into equation (2.4) gives
the following equation
(2.5 )
Q is a constant since there is no side load assumed in the free span, and EI is a
constant because it is assumed that the entire span is taut.
9
Differentiating equation (2.5) and dividing it by EI yields the fourth order differential
equation of the elastic curvature of a web.
The solution for the linear differential equation (2.6) is
y = C1 sinh Kx + C2 cosh Kx + C3x + c't
(2.6)
(2.7)
Four boundary conditions are applied to evaluate the constants in equation (2.7):
• The lateral displacement at the upstream roBer is assume to be zero which
means x = 0 and y = o.
• Friction is assumed to be sufficiently large to prevent circumferential slippage
d
which means at x = 0, d~ = O.
• The web approaches a roller perpendicularly to the roller axis. Shelton has
experimentally verified this condition which lead to the following equation
C1 cosh K L + C2 sinh K L + C3 = fh
• The moment on the guide roller is assumed to be zero at steady state, that is
NIt = O.
Considering the above boundary conditions the constants are found to be:
C
j
= _ fh cosh K L
K cosh KL  1
C
2
= ()L sinh K L
K coshKL1
C
_ () cosh KL
3  L cosh K L  1
10
(2.8)
(2.9)
(2.10)
C1 = _ fh sinh K L
K cosh KL  1
2.2 Dynamic behavior of a web
2.2.1 Ideal moving web
(2.11)
In this case the physical properties of the web are ignored. The web is assumed to
have no shear strength and hence can be assumed to be a straight line between two
rollers. A web approaching a roller will align itself perpendicular to the roller as
shown in Fig. 2.3
WEB ..
TRAVEL
Figure 2.3: vVeb passing over a series of nonparallel rollers
On account of lateral movement of the roller the velocity of the web edge relative
to the ground is the summation of the velocity of the web relative to roller and the
velocity of the roller relative to the ground.
If the roller is moving laterally, the total velocity of the web edge relative to the
ground is equal to the sum of the velocity of steering of the web and the velocity of
lateral transport of the web. The following equation expresses the web velocity at the
downstrean roller.
11
dYL dz
 = ve+
dt dt
(2.12)
The negative sign in equation (2.12) accounts for the fact that a positive angle results
in a negative velocity.
Shelton [1] has given an explained! derivation of the transfer function and the
frequency response for each of the following cases:
• Response at a fixed roller to input at the previous roller.
• Steering guide response.
• Response at a point between two parallel rollers.
A time response for each of the above cases has been done in this thesis with
T = 0.625.
x
B
A

 YL f
1 ._. ~_. ~  _.
viL
I y
~ ' R y
Figure 2.4: Symbols for the derivation of response at a fixed roller
Figures 2.4, 2.5 and 2.6 show all symbols used in the derivation of the transfer
function for each of the above cases respectively. Their transfer functions are given
as follows:
12
x
T
E::::::3 Y
T
Figure 2.5: Symbols for the derivation of response at a steering guide
1. Response at a fixed roller to input at the previous roller.
2. Steering Guide Response.
YL(S)
}O(s)
YL(S)
Z(S)
1
(2.13)
(2.14)
3. Response at a point between two parallel rollers.
13
x
t... Y
v
r+ Sample
Point
·r
X2 __J,+l_.1
Figure 2.6: Symbols for the derivation of response at a point between two rollers
1';(s)
Yo(s)
(2.15)
Figures 2.7, 2.8 and 2.9 show the time response at a fixed roller due to input at the
previous roller, steering guide response and response at a point between two parallel
rollers respectively. Fig. 2.8 shows that the time to reach steady state depends on
the distance of the instant center of the guide roller from the guide roller. As the
distance decreases the amplitude of response increases. Fig. 2.9 indicates that as the
distance of the sampling point from the upstream roller increases the time required
to reach steady state increases.
14
0.9
O.B.
0.7
L1V ~ 0.625
0.6
0.5
0.5 1.5
Tlme (sec.)
2 2.5 3 3.5
Figure 2.7: First order time response at a fixed roller due to input at the previous
roller
2.2.2 Real moving web
Equation (2.6), applied to a web is independent of its steady state condition, if the web
mass is neglected. But only the boundary conditions change. Shelton has rewritten
the equation in the partial derivative form since time and location are both variables
in the dynamic condition. Therefore equation (2.6) becomes:
(2.16)
He developed partial differential equations showing a relation between the downstream
end of a web to its dynamics of steering. Dynamic analysis of the downstream
end of the web is done with the statics as the fundamental basis. The static condibons
are broken into two parts for dynamic analysis which are then superimposed to
develop a differential equation of dynamic steering.
15
1.9
1.8
1.7
LI L, =2
l/V = 0.625
Q) 1.6
"0
.~
Ci
E
~
0.5 '.5 2 2.5 3 3.5
Time (sec.)
Figure 2.8: First order time response of a steering guide
Equations of dynamic steering
Fig. 2.10 shows a shiftable roller with curved web passing over it. All roller and web
angles are assumed to be positive. The lateral velocity of the web is the summation
of the velocity of the web relative to the roller and the velocity of the roller relative
to the ground. As seen in Fig. 2.10 point A lies on the line of entering contact, hence
the subscript is used to identify it as a point on the downstream roller. The above
reasoning can be expressed in the form:
dYL = v (0 _ oy I) dz
dt r at ;L + dt (2.17)
where
oy
~ = the slope of the web evaluated at L.
uXL
As point B on the web passes the line of entering contact, the lateral velocity of
the web relative to the roller is equal to the product of slope of the web relative to
the roller at point B and the longitudinal velocity v of the web. As points A and B
16
Q)
"0
.~
a.
E «
0.2 '''''''' o 0.5 1.5 2 2.5 3 3.5
Time (sec.)
Figure 2.9: First order response at a point between two parallel rollers
enter the line of contact, the lateral velocity of the web relative to the roller in that
period is given by:
dYL I
dt A
(2.18)
Dividing the left hand side of equation (2.18) by 6.t and the right hand side by 6.~,
v
Shelton developed the equation of lateral acceleration to be:
(2.19)
which is the summation of the acceleration due to steering and acceleration due to
lateral transport.
Analysis of web mechanics for bending
Fig. 2.11 shows the symbols used for the derivation of relations for pure translation of
the end of web. Ignoring deflection due to shear forces and considering the following
boundary conditions:
17
Line of Entering
Contact
v
/ Lateral Velocity at A
!:..,
Lateral Velocity at B
~I  e
{tx A
"~I. e {tx B
B
Figure 2.10: Steering action of a web with induced curvature
18
x T
Y ya'= a o=0 l....__I. y
T
Figure 2.11: Boundary conditions for translation of end of web
1. Yo IS zero
2. Yo IS zero
3. !VIo = llh
the coefficients of equation (2.7) are evaluated to be:
ydl + cosh K L)
K L(l + cosh K L)  2sinh KL
C
_ YL sinh KL
2 K L(l + cosh f{ L)  2 sinh K L
19
(2.20)
(2.21)
C
3
= yL K(l + cosh K L) .
K L(l + cosh KL)  2sinh K L
C
4
= _ YL sinhKL
KL(l + coshKL)  2sinhKL
(2.22)
(2.23)
The primary parameter of evaluation of equation (2.19) is ih and Alt. Because
ih is equal to K 2(C[sinhKL + C2 coshKL) and EIK2 is equal to T, simplification
leads to the results:
.. __K 2 ( sinh K L )
YL  YL KL(cosh K L + 1)  2 sinh KL
M T (
sinhKL )
L = YL I KL(coshKL+ 1)  2 sinh KL
Considering
f (K L) = K L
2
sin h K L
1 KL(coshKL+ 1)  2sinhKL
so that,
Y..J, =  YL2Lf(JrL) I \.
(2.24)
(2.25)
(2.26)
(2.27)
(2.28)
A similar derivation is done for pure rotation. Fig. 2.12 shows the boundary
conditions. The constants of equation (2.7) are found out to be:
C
1
= eL . cosh KL  1
K KLsinhKL  2(coshKL  1)
20
(2.29)
x
1 1 Y
T
Figure 2.12: Boundary conditions for rotation of the end of web
c _ OL KL  sinhKL
2  K KLsinhKL2(coshKLl)
c __{} cosh K L  1
1  L KL sinh KL  2(C08h1\L  1)
c __(it K L  sinh K L
4  K KLsinhKL  2(coshKL  1)
The equations of interest similar to those for end translation are:
21
(2.30)
(2.31)
(2.32)
(2.33)
where,
TLfh
Ah = KL2 h(KL)
(
KLcoshKL  sinhKL )
12(KL) = K L K L sinh K L  2(cosh K L  1)
x
   I~,
_________ .J _
,
. ... t~'
v
t
 , ++ Y
.r~
Figure 2.13: Translation
(2.34)
(2.35)
Shelton used the fundamenta] equations derived here and applied them to the
following cases and evaluated their second order transfer functions to be:
1. Response at a fixed roller to input at the previous roller.
1
22
(2.36)
x
L
Figure 2.14: Rotation
t+ Y
v
Fig. 2.5 shows the arrangement and nomenclature. Figures 2.13 and 2.lQ shmv
2. Steering guide response.
the scheme of superposition of translation and rotation of the end.
yt.,(s)
2(s)
(2.37)
3. Response at a point between two parallel fixed rollers.
Figures 2.16 and 2.15 show the two components of web shape used ill this
23
x
v t
r1+ 
X 2 _J'___~__y_o_=:~~~~~~~~~~~~~~2~~~=,j y
rI Lt+
L 2
Figure 2.15: Translation: Deflection of a web between two parallel rollers
derivation. The transfer function of the system is given by:
12(s)
Yo(s)
(2.38)
where
(2.39)
1 S1.11h I\'L(x2 )
,(cosh K L  1) L
KLsinhKL  2(coshKL  1) KL
X2  sinh K L  K L (cos'h I\.'L(X2) 1)
L KL L
1 X2
[KL(cosh K L + 1) _ 2sinh K L][(cosh K L + 1)(J(L(L)
sinhKL(~))+sinhKL(coshKL(~)1)] (2.40)
Fig. 2.17 shows the second order time response of a web at fixed roller to inpu t
at previous roller. It can be seen that as at: value of K L increases the time required
24
x
y

Figure 2.16: Rotation : Deflection of a web between two parallel rollers
to reach steady state value increases. As f{ L increases the response is sirnilar to the
first order response.
Fig. 2.18 shows the second order responsE' of a steering guide. The amplitude of
the step response depends on the distance of the instant center from the guide roller.
As the distance decreases the response increases. The second order system reache~
steady state value faster than the first order system.
Fig. 2.19 shows the second order time response at a point between two parallel
rollers. It is seen that the system reaches steady state value faster when the sampling
point is near the upstream roller. \Vhen compared to the first order system it is seen
that the second order system reaches steady state faster at all conditions.
25

1.4,,r.,r.,r,
L1V = 0.625
2 2.5 3 3.5 4
Time (sec.)
1.2
Figure 2.17: Second order time response of a web at fixed roller to input at previous
roller
2.3 Lateral control of a web using nip rollers
Lateral control of a web can also be achieved by the use of nip rollers. These roHers
are usually rubber covered. A nip is any two rollers in contact as shown in Fig. 2.20.
2.3.1 Concept used in the use of rubber covered rollers
Consider Fig. 2.21. Let A, Band C be three chambers filled with water and let water
from chamber A be allowed to flow to chamber C through chamber B. The velocity of
flow of water in chamber B increases since the crosssectional area of flow decreases,
that is if the mass flow rate of incompressible fluid is assumed to be constant then the
velocity of the fluid must increase as it passes through the constriction, which in this
case is chamber B. This same principle could be applied to the nip rollers to deflect
the web laterally.
26

2.4,.,,.__,.__,__,__,_,
LtV =0.625
2.2
2
I.Kc /1.1 =2
CD 1.6
"0
.~
1i.
E «
0.5 1.5 2
Time (sec.)
2.5 3 3.5 4
Figure 2.18: Second order time response for a steering guide
Due to the differential loading of the nip rollers there is a differential velocity
across the width of the rollers and the web that passes through these rollers also
has a differential velocity. This variation in the velocity of the web will also cause a
variation in the strain and stress across the width of the web. When integrated, the
stress will result in a moment which will be shown to produce the lateral deformation
of the web.
2.3.2 Lindley's analysis
Lindley used the relations derived for load compression of rubber blocks at low strain
to derive relationships between rubber covered rollers at large deformation. Considering
Young's modulus to be independent of the strain in the material, the effective
nip load is:

F (kD ) W = EV(tD) CXR + t{3R
27
(2.41)
l/ V =0.625
0.1 L._'_'__'_'_l.__"_J...._'
o U U U U
Time (sec.)
Figure 2.19: Second order time response at a point between two parallel rollers
where
(};R = ~ln .( 1 + JU) _ 16VU
3 1JU 3
fJ  l (1 + JU) lOjU 4y'U R  n  + 2
1  JU 3(1  u) 3(1  u)
6
u=
t
VDJ
5 = (1  6)
The factor, k, is determined empirically from equation (2.42)
2.3.3 Foreman's analysis
(2.42)
...
In 1964, Foreman performed experiments to show the relationship between rubber
compression and velocity of the strip passing through the pinch rolls. Foreman used
28


Force
Force
Figure 2.20: Illustration of a nip
. .m.m._..., /rm m.__~
~~ f~_·_··M,·~r
'I I.
I I I,
I. I, A :: B :: C '1 I,
: j !i
n n _00_1'm  m    00nn n~n_mm_nn_;
Figure 2.21: Concept for rubber covered rollers
two sets of rubber covered rollers, one of which was 11 inches in diameter with a ~4
inch rubber covering. The second roller was 30 inches in diameter with 1 inch rubber
covering. The hardness of the rubber ranged from 60 to 70 durometer. The strip was
passed between these two rollers and the upper roller was loaded. The lower roller
was rotated using a hand lever by one revolution. After this, the length of movement
of the strip was measured. The length of movement of the strip without any load was
calculated using the diameter of the roller. A difference was found between these two
values. It was found that with increase in load, the length of movement of the strip
increased. Increases upto 2 percent were found. Foreman also showed relationships
29
......

Load
and rubber thickness. Fig. 2.22 shows the setup for Foreman's experiments.
Pointer for Rotation
indicating amount
of rotation
Sheet
between compression of rubber and the movement of the strip for different durometer
Figure 2.22: Foreman's experimental setup
2.3.4 Shelton's analysis
In one of his reports to Fife Corporation, Shelton gave an outline on web steering
due to a differentially loaded nip across the width of the web, Shelton presented a
relationship for calculating the effective nip load due to radial deformation of the
rubber.
The following assumptions were made by him:
1. The covering material behavior is similar to that of natural rubber and when
in the form of a roll covering it is relatively incompressible.
 30
2. Rubber covered roUs are identical in size and durometer (durometer is the measure
of hardness of rubber).
3. Small deflections of the rubber covered roUs occur when they are loaded.
4. The entering span prior to the nip guide is very long. So shear deformation can
be neglected.
Shelton [13] points out that, for the proper application of the differentially loaded
nip guide the entering web span needs to be long. Hence he analyzed the longitudinal
stiffness of a free catenary (see Fig. 2.23) and a supported catenary (see Fig. 2.24.
Fig. 2.25 shows an equivalent free catenary of the supported catenary.) which are
necessary for the proper application of the differentially loaded nip guide.
I~
=.,;;:~.x
L
Figure 2.23: A free catenary
I,
I
L ~i
~+T
I
I
I
I.I.....f
I
T
Figure 2.24: A supported catenary
31
T
Le
Aor. T
Ir
Figure 2.25: Equivalent free catenary of the supported catenary
Fig. 2.26 shows a portion of a roller and the symbols used by Shelton to find out
the expression for effective nip load.
The effective nip load formulated by Shelton was :
F ( (2R )0.5 (0 )1.5 __1O_5_.4_e_O_.04_8_A, (2Rto
_ = t 41.3eO.048A )(1.50.002A) (Otf. ) (2.50.002A)) _0 ....l +
'W t t (2.5  O.002A)
(2.43)
Shelton's frequency response plots and the time response plots shown indicate
that the second order system gives a more appropriate results. Lateral control of a
web lising nip rollers has been studied and further research in that area will come in
the future.
32
Steel
Roller
Rubber Covering
Figure 2.26: Schematic diagram of roller
33

J.
'Chapter 3
Automatic Guiding Mechanisms
and Control Systems
This chapter describes the basic types of guiding mechanisms and automatic control
systems used in the web handling industry for the lateral control of web. Hopcus [4]
has briefly discussed different types of control systems used for the Lateral control of
a web.
3.1 Basic types of automatic control systems
There are four types of automatic guiding control systems: Pneumohydraulic, Electrohydraulic,
Pneumomechanical and Electromechanical systems. AU systems are
closedloop proportional control systems.
3.1.1 Hydraulic types
The two hydraulic types function in a similar manner. A sensor monitors the lateral
position of the web. The sensor signal is transmitted either directly to the power unit
servo valve (Pneumohydraulic systems) or to a signal processor which then sends a
signal to the power unit servo valve (Electrohydraulic systems). Hydraulic output
from the power unit through the servo valve, proportional to the lateral error of
34

the web, positions the guide structure, which moves the web to the correct lateral
position. These systems are attractive for extremely heavy loads and harsh enyironments.
Figures 3.1 and 3.2 show a block diagram of the pneumohydraulic and the
electrohydraulic types of automatic guiding systems., respectively.
Guide
Structure
Web
Sensor
Air Lines
HydraUlic Lines
Pneumohydraulic
Servo Valve
PneumohydraUlic
power unit
Figure 3.1: Pneumohydraulic guiding control system
3.1.2 Mechanical types
The two mechanical types of control systems also function in a similar manner. A
sensor, either electronic for electromechanical systems or pneumatic for pneumomechanical
systems, monitors the lateral position of the web. The sensor signal is either
transmi tted directly to the processor (electromechanical system) or is first converted
from an air pressure signal to an electrical signal with a transducer (pneumomechanical
system). The processor then sends a signal, proportional to the amount of error
detected by the sensor, to the DC drive motor on the electromechanical actuator.
The actuator positions the guide structure which moves the web to the correct lateral
35
 . ~
Guide
Structure
Web
1 Hydraulic Unes
Electrohydraulic
Servo Valve
Electrohydraulic
Power Un~
I
Figure 3.2: Electrohydraulic guiding control system
position in the sensor. These systems are especially attractive for applications demanding
a high frequency response and where hydraulics are not desirahle. Figures
3.3 and 3.4 show a block diagram of the pneumomechanical and electromechanical
types of guiding control systems, respectively.
3.2 Guiding mechanisms
The basic guiding applications can be listed as: End pivoted guide, Center pivoted
guide, Remotely pivoted guide and Offset pivot guide. A brief description of each
guide is given in the following.
3.2.1 End pivoted guide
Fig. 3.5 shows a schematic diagram of an end pivoted guide. 'When the guide is
stationary, it does not affect the web position. As the guide is steered about its
pivoted end the web is displaced to the desired position since the web always travels
perpendicular to the roller.
36

Sensor Air lines
Air Riter
and Regulator
Guide
Structure
, Web
r
Ail
Electric Cables
Figure 3.3: Pneumomechanical guiding control system
3.2.2 Center pivoted guide
Fig. 3.6 shows a schematic diagram of the center pivoted guide. This guide is very
similar to the end pivoted guide except that the pivoted point is at the center of the
roHer.
The transfer function for the end pivoted and center pivoted guides for figures: 3.5
and 3.6 respectively is given by equation (3.1)
3.2.3 Relnotely pivoted guide
Fig. 3.7 shows a schematic: diagram of the remotely pivoted guide. These are steering
type guides in that the roller is moved laterally and angularly to accomplish lateral
37
I
Sensor
Guide
Structure
1
Electric Cables
Electromechanical
Actuator
Figure 3.4: Electromechanical guiding control system
web correction. This action steers the web laterally in the entering span. The point
about which the guide assembly rotates in reaching the angular position required for
a given correction is called the 'center of rotation'.
For a remotely pivoted steering guide shown in Fig. 3.7, the transfer function is
(3.2)
3.2.4 Offset pivot guide
Fig. 3.8 shows a schematic diagram of the offset pivoted guide. These are displacementtype
guides which provide web position co 'vc"ion with minimum entry and exit span
38
I
Pivot Point
Figure 3.5: End pivoted guide
u
Web
Pivot Point
Figure 3.6: Center pivoted guide
requirements. When the guide is centered, it does not affect the web position. As the
guide moves to a position other than the center, the web is displaced to the desired
position as it moves across the guide span.
For a displacement guide shown in Fig. 3.8, the transfer function is
39
Fixed Entering
Roller
Actuator
l· '.
Center of Rotation Guide Roller
of Guide roll
: : ;"
j ~ ~"
'. ~ : : .
___ r_ ~~.c...::..... .~:.:.: 
~ : : : ,.
: : : :; : ~ : :.
Servo· Center
Transducer
Figure 3.7: Remotely pivoted guide
Yc,(s)
(3.3)
40
· .
f·····f·
'·r,·' :: . ~ _ . .'.   _. .,::::,::::::::':::::::::::::::::~ ::'~
I
0 , = Enlering Span
D2 = Exiting Span
rwL ::::,r
Exiting
Roller
Figure 3.8: Offset pivoted guide
41
....
'Chapter 4
Observer Design and Simulation
Results
This chapter justifies the need for an observer based controller and explains the procedure
followed for the design.
4.1 Present controller
The present controller on the experimental platform which is the A9 signal processor
(courtesy: Fife Corporation, Oklahoma City) uses position feedback signal from the
edge sensor and motor velocity feedback signal from the tachometer. A block diagram
of the present controller is shown in Fig 4.1.
Reference
Web
Lateral
Posllion
I
Figure 4.. 1: Block diagram of controller using tachometer
42
4.2 The importance of innerloop velocity feedback
To investigate the importance of the velocity innerloop, a digital lateral controller
without the velocity innerloop is implemented for the following three conditions, i.e.
no disturbance, step disturbance and pulse disturbance. B~r tuning the control gains
for computer control, similar performance as that of Fife A9 controller is obtained.
Figures 4.2 and 4.3 show the experimental results of computer control. The following
observation can be made from the experimental results.
1. When there is no disturbance then the results with and without innerloop
velocity feedback are similar. But when there is step and pulse disturbances, the
performance of the control system deteriorates. Therefore, innerloop velocity
feedback is essential in maintaining stability of the guide system in the presence
of any lateral disturbances.
2. As shown in Figures 4.2 and 4.3, when the step disturbance is not that large,
it is still possible to push the web edge to the reference position. However,
the overshoot becomes very large, and the oscillations last longer. During this
oscillation period the motor may saturate.
3. If the disturbance magnitude is large, then the system can go unstable. From
the third row plot in Figures 4.2 and 4.3 the oscillation does not subside.
Thus, innerloop velocity feedback is critical to stable closedloop system performance
in presence of disturbances.
The cost of a motor increases because of the presence of the innerloop velocity
feedback. Therefore a need arises to circumvent the use of tachometer signals for
innerloop feedback with a motor velocity estimator.
43 

Web Position
Computer Conlml (no velocity feedback.Kp=15)
0.5
Q)
<>
0:: .e~ 0 1i'j
'i5
0z
0.5
0
0.5
~
0::
€'"
~ en 0
'i5
0.
Q)
U5
0.5
0
3
Q)
<c> 2 ~'"
~ en 1
'i5
Q)
Ul 0 "S
0.
11
0
2
2
2
3
3
3
4
4
4
Time (s.ec)
5
5
5
6
6
6
7
7
7
B
8
B
Figure 4.2: Web position: computer control without velocity feedback
4.3 Observer design
The analog controller (A9 signal processor) used Ly Fife for the lateral control of
a web uses velocity feedback from the tachometer of the motor as stated earlier.
Research was undertaken to replace the tachometer feedback signal since it would
greatly reduce the cost of the motor and the lateral guide on the whole.
Considering the angular and lateral disturbance to be zero, equation (3.2) can be
rewritten as
(
2 12(KL) ,12(,K'...L) )
3 + S +2
YL(s) = 3
2 + h(~L) s + fJ;~) Z(s)
T 7 2
(4.1)
Equation (4.1) shows that at steady state the lateral position of web and the guide
which means the position of the motor are proportionally related. Since the system is
observable the velocity of the web can be .:Jc.lmated using a minimum order observer
44
Motor Tachometer Output
Computer Control (no velocity feedback, Kp=15.Kd=5)
2 3 4 5 6 7 8
5 L L L__.JL L L__.JL__~L._______J
o
5
Bc::
~
.2 0
til
'6
o z
2 3 4 5 6 7 8
~ 5
c::
'e"
~ 0
'6
c..
0)
US
5 L L__.JL__.JL__.JL__.J'__'' L__'
o
<D 5
<.> c
'e" :::> W 0
:.0
:Jl
~
c.. 5 ' L__.J'__.J'__''__.J'__.J ''__'
o 2 3 4
Time (sec)
5 6 7 8
Figure 4.3: Motor velocity: computer control without velocity feedback
design. The velocity of the motor can then be estimated using the relation between
the motor position and web position.
Fig. 4.4 shows the analog controller used by Fife. Fig. 4.5 shows the block
diagram of the system used to replace the innerloop velocity feedback. Note that the
analog controller llsed by Fife has been replaced with a digital controller. The web
dynamics is given by equation (4.1).
4.3.1 Developing the state space form
From Fig. 4.5 the openloop actuator dynamics can be written as:
(4.2)
45

+
________ 1
Velocity feedback
Figure 4.4: Block diagram of controller using tachometer
Web YL
Dynamics Ir+
V,elocily
,1 1 EstimatorI+{
Figure 4.5: Block diagram of controller using velocity estimator
If an inverse laplace transform is applied to equation (4.2) then the following differential
equation results.
For simplicity equation (4.1) is written in the form:
(4.3)
where
!(KL)
Dj=
7 2
YL(S)
Z{s)
S2 + D2S + f3
S2 + G'2 S + Dj
46
(4.4)
....
h(KL)
Q2 = 'T
(3 = h(KL)
T2XI
Taking an inverse laplace transform of equation (4.4) yields the following differential
equation.
(4.5)
Consider Xl and X2 to be the state variables of equation (4.3) and, X3 and X4 to
be the state variables of equation (4.5). Let us assume Xl = z and X2 = .2, which give
the following relations:
(4.6)
(4.7)
Let us assume X3 = YL and X4 = ilL, and equation (4.5) and broken into two state
equations and written as:
(4.8)
(4.0)
Since Z, the position of the motor is expected to be a constant z and .i are zero.
Therefore equation (4.9) can be rewritten to include the above conditioll.
(4.10)
Putting equations (4.6), (4.7), (4.8) and (4.10) in matrix form yields the following
state space model of the system.
Xl (t) 0 1 0 0 Xl (t) 0
X2 (t) 0 am 0 0 X2(t) k +i m (4.11) Urn
X3(t) 0 0 0 1 Xl(t) 0
X4(t) ,8 0 al a2 X4 (t) 0
47
....

Note that Xl, X2, X3 and X4 are state variables which denote the position of the motor,
velocity of motor, position of web and velocity of web respectively. Since X3 is the
only state variable that can be measured the output equation is given by:
y(t) = (0 0 1 0)
4.4 State Observers
Xl {t)
X2(t)
X3(t)
X4(t)
+ (0) +u(t) (4.12)
In a practical system, not all state variables can be measured. Hence, it is necessary
to estimate the state variables that are not directly measurable. Such estimation is
commonly called observation. Estimation of the unmeasurable state variables can be
done using the output and control variables. State observers can be designed if and
only if the observability condition is satisfied.
Full order state estimation means that we observe (estimate) all n state variables
regardless of whether some state variables are available for direct measurement. Here
we design a minimum order observer to estimate the velocity of the motor. From Fig.
4.5, the state observer will have YL and Urn as inputs and .'];2 as output. For the system
under consideration the state and output equations are given by equations (4.11) and
(4.12) respectively. Suppose the state vector x is a nvector and the Y vector is a
mvector. Hence we need to estimate only (n  m) variables. The red Ileedorder
observer becomes an (n  m)thorder observer.
The minimumorder observer can be designed by first partitioning the state vector
into two parts, as shown
(4.13)
where Xa (t) is that portion of the state vector that can be directly measured and
48


Xb(t) is the unmeasureable portion. The partitioned state and output equations of
the system can then be written as follows
(4.14)
(4.15 )
Rewriting the above equations, the measured portion of the state equation becomes
or,
(4.16)
Equation (4.16) has all measurable quantities on the left hand side and unmeasurable
qua.ntities on the right hand side. Equation (4.16) is also the output equation for
minimum order observer design. The unmeasured portion of the state, which is also
the state equation for minimum order observer design is as follows
(4.17)
Let us assume that the state Xb(t), to be approximated by the state Xb(t) is of the
dynamic model
(4.18)
where
GXb is the approximated output.
]{e is a weighing matrix.
49

Substituting equation (4.16) as the output equation in equation (4.18) and C =
Aab , we get the following equation
:h(t) = (Abb  KeAab)ib(t) + Key(t} + (Aba  KeAaa)y(t) + (Bb  KeBa)u(t)
(1.19)
Let us define the observer error dynamics
(4.20)
(4.21)
Expanding equation (4.21) using equations (4.17), (4.19) and (4.20) we get the
fonawing equation
(4.22)
The characteristic equation for the minimumorder observer is obtained from equation
(4.22)
(4.23)
From equation (4.23) we see that the dynamic behavior of the error signal is determined
by the eigenvalues of Abb  KeAab . If Abb  KeAab matrix is a stable matrix, the
error vector will converge to zero for any initial error e(O).That is,_ i(t) will converge
to x(t) regardless of the values of x(O) and i(O)
Ke can be evaluated from equation (4.23) and the unmeasured states can be
estimated using the value Ke and equation (4.19).
A complete derivation of the minimum order observer has been given in appendix
A. The velocity observer is given by the following equation:
"y2 (8) = (33
83 + (32
82 + (31 8 + (30 Y (. '
8
3 + 1282 + 11 8 + 10
50
A2 82 + A1 8 + Ao U(8)
8
3 + 1282 + 11 8 + 10
(4.24)
4.5 Lateral dynamics
4.5.1 Openloop response
Simulated results
In this section, openloop response of the web lateral dynamics is investigated. A Fife
Kamberoller guide (a remotely pivoted steering guide) is used for this investigation.
Fig. 4.6 shows a web span with a Kamberoller guide and an edge sensor immediately
downstream of the guide roller.
Instant Center WEB MOVING
L
v •
Original
Center
Figure 4.6: A web system
The lateral dynamics for the web span shown in Fig. 4.6 is given by equation
(4.25), which is
Equation (4.25) can be rewritten in the form:
51
where Gds), G2 (s), G3 (s) are as given in equation (4.25) and
Yds) : response at a Kamberoller guide;
}o(s) : the positional disturbance from upstream roller;
00 (s) : angular disturbance at upstream roller;
Z(s) : positional input at steering guide;
Ods) : angular input at steering guide.
The following simulation investigates the response at a Kamberoller guide, under the
assumption that a disturbance Yo(s) is introduced at upstream roller. Other terms
in equation (4.25), which are the angular disturbance, Oo(s), at upstream roller and
input at Kamberoller guide, Z(s), are taken to be zero. Under this r.ondition, the
web dynamic behavior is described by
The parameter and experimental conditions used in this simulation are those of the
Kamberoller guide in our experimental web platform, and are given in the following
table.
K L(in) v(in/sec) xl('in)
0.0292 46 80 88
Simulation on a web system described by Fig. 4.6 is accomplished. The responses
at a KamberoHer guide to impulse, step and sinusoidal disturbance are investigated.
The simulation results are shown in Figures 4.7, 4.8 and 4.9 respectively. The .sirnulation
results show that the studied web system can follow the disturbance after
a short time period. Please note that when the disturbance is introduced, the web
edge first moves toward the direction contrary to that of the disturbance. This can
be understood from the term G3 (s) in equation (4.25), which has the initial value
as _h(KL). Moreover, from Fig. 4.9, the openloop dynamics of the system apT
pears to be characterized by a lowpass filter.The goal of lateral control of a web is
52
Response at gtir:le roller 10 impulse dislurbance (open loop)
'~
o j
2~
oJ >
I
2
3
2 2.5 3 3.5 4.5 5
Time (sec)
05 15
4'_'_"'_'_..L._'_'_'_'_L_J
o
Figure 4.7: Response at Kamberoller guide to an impulse disturbance
to minimize the lateral position error by applying appropriate control strategies to
the intermediate steering guide. The lateral control system to be discussed is given
in Fig. 4.10, where
Yds)
YR(s)
Yo(s)
Bo(s)
Cc(s)
GM(s)
UM(s)
VM(s)
em
response at the remotely pivoted guide;
web reference position at the remotely pivoted guide;
positional disturbance from upstream roller;
angular disturbance at upstream roller:
transfer function of controller;
transfer function of motor dynamics;
control input to the motor;
motor angular position;
constant relating motor angle to the lateral displacement of the guide.
4.5.2 PI Control
Proportional and integral control strategy is applied for web lateral control system.
To test the performance of PI control at steering guide, computer simulation was
53
Response at guide '011&110 unil Slep distur1:JallcE> (open loopl
1       .,~~I
1.5 2 2.5 ;>
Time (sec)
I===
3.5 4
Disturt:>ance
Response
4.5
Figure 4.8: Response at Kamberoller guide to unit step disturbance
done with different proportional and integral gains. Because in the real web system,
roller axes are parallel to each other, the angular disturbance at upstream roUer eo in
the equation (4.25) is taken as zero. Simulink block diagram for simulations is given
in Fig. 4.15. The parameters used in the simulations are computed in the Matlab
script file given in Appendix, from the parameters given in the following table.
K L(in) v(in/ser;) :rdin)
0.0292 46 80 88
4.5.3 Control with estimated motor velocity
According to modern control theory, for an observable system, the state variabl(~s can
be estimated based on the system input and output information. Following this idea,
a minimumorder estimator is designed which estimates the velocity of the DC motor.
Fig. 4.14 shows the block diagram of a lateral control system with estimated motor
velocity feedback, where the steering guide takes the form of remotely pivoted guide
(Kamberoller guide) whose dynamics is described by equation (4.25). where
54
0.8
,I
(
I
I
0.6 (
I
I
0.4 I
I
I
0.2 I
? I
" :.§. 0
>~
0.2
0.4
0.6
0.8
Response al guide roller to sin"soidaJ disturbance (open 100j)
I ,',' DislurtJanCe
Response
'' (
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\ I
\
\
\
\
\
1 '''''"~~'''~'
o 2 3 4 5 6 7 8 9 10
Time (sec)
Figure 4.9: Response at KamberoUer guide to sinusoidal disturbance
Yg(S)
Yr(S)
Yo(S)
fJo(S)
Gc(s)
GM(S)
G1IU (S)
Gvy(S)
\/,'1'[( S)
VM(s)
em
response at the guide;
reference position at the guide;
position disturbance from upstream roller;
angular disturbance at upstream roller;
transfer function of controller;
transfer function of motor;
transfer function from input 'U to estimated velocity;
transfer function from measured web position y to estimated velocity;
actual motor velocity (from tachometer);
estimated motor velocity;
constant relating motor angle to the guide displacement.
Simulink block diagram of the lateral control system given in Fig. 4.14 usmg
tachometer feedback is shown in Fig. 4.15. Simulink block diagram of estimated
motor velocity feedback is shown in Fig. 4.16. The variables used in the Simulink
block diagram can be obtained from the Matlab script file in appendix B.
To test the effect of motor velocity observer, computer simulation is accomplished
on a web lateral system with estimated motor velocity feedback. Three types of
55
Figure 4.10: Block diagram of lateral control system
disturbances are adopted, which are impulse disturbance, unit step disturbance. and
sinusoidal disturbance, respectively. Figures 4.17 through 4.19 compare the response
of web system with tachometer and with motor veloci ty observer, to different kinds of
reference input. From the simulation results it can be seen that the performance of the
closedloop system using estimated velocity in the innerloop is similar to that with
tachometer feedback. The simulation results reveal that a motor velocity ohserver
can functionally replace a tachometer,
56

Response at guide roller 10 impulse cisturbance (dosedloop)
3.,.....,r.~..,
0 I
>~
1
2
3
\
\ .
\
\
'.
~. ~ ~ ~'~" ' '.. _~_ ..._'._._._. .__._.
Kp=10,Ki"Q 1
Kp=40,Ki"Q1
Kp..so,Ki"Q,1
_4L'_~ ____' _'_ _'__ ___J
o 05 1.5 2 25 3
llme (sec)
Figure 4.11: Response at Kamberoller guide to an impulse disturbance
Response al guide rolle, 10 unil slep dislurbance (ctosedIoop)
12.,.,.,·,...,...,,,
o.e
t.
0.6 \
0.4
0.2
o
\
\
,
"
Disturbance
Kp.10,KieO.l
Kpe40,Ki.0 1
Kp.eO,Ki:O.l
_O.2l'.''"''....l'_ ____'__'__J
o 5 10 15 20 25 30 35 oW 45 50
Time (sec)
Figure 4.12: Response at Kamberoller guide to a unit step disturbance
57
Response al guide, roller 10 Sinusoidal dislurt>ance (closedlOOp)
6 9 10
Dislulbance
Kp= lO,Ki~O. 1
Kp~O.Ki=Ol
Kp..80.Ki=0.1
458
Time (sec)
,
I.,
I
, I·
,\
" I' "
"\.
...
\\ I,
\
"I'
\ I·.
1\ , \.
I "
\ '..
I
I
\.
I
0.6 I
I
I
0.6 I
I
I
0.4 I
I
I
0.2 I
? I
§. 0
~ >
0.2
0.4
~6
0.8
1
0
Figure 4.13: Response at Kamberoller guide to a sinusoidal disturbance
e (5) o
Y,(s)
.....
~(s)
Figure 4.14: Lateral control system with estimated motor velocity feedback
58
Figure 4.15: Simulink block diagram using tachometer feedback
59
...:.:
u
~
.D
'"0
Q) ..s
.;.>.,,
.~ u
.3
Q) >
I;
0 ...,
0S
'"0
.0..,,)
c<:l
S .~ ...,
fJ)
0,)
be
i=:
'Ci.l 0
:;l '0
S
c<:l
I;
~
.~
'"0
~u0
:E
...:.:
i=: :=
:;l
.5
Cf)
'0
...<
.q<
0,)
bo
;::l
be
.~
~
Gvu
Gvy
beta3.s3+beta2.s2tbeta1.s+betaO
s3+gamma2.s2tgamma1.s+gammao
lambda2.s2tlambda1.s+lambdaO
s3+gamma2.s2.tgamma1.s+gammaO
+1. I
+
Sum1
I
Kp.s+Ki
I~ ~I Km s  I
s+am ~~ Sum2 Controller
Integrator Gain Web Dynamics
~um
DC Motor

Response al guide ,oller to impulse disturbance (using estimaled motOl velocily leedbackl
2,,.,,,,
:_":.. 
2 Kp=10,Ki=0.1
Kp=40.Ki=O.l
Kp=80,Ki=O.1
3
0.5 2 2.5 3
_4' l ' .L ' ' ....J
o
Figure 4.17: Response using estimated motor velocity feedback (impulse disturbance)
Response at guide rO+ler 10 unit step dislurbance (usioO estimated motor velocily feedback)
12,.....,..,,.....,.,,,......,.,
O.B
0.6
'2
<> .:§.
>~
0.4
0.2
,
0
0.2
0 2 3 5 6
r,me (sec)
Disturbance
Kp=10,Ki=0.1
Kpm 40,Ki=0.1
Kp=80.Ki=O.1
7 8 9 10
Figure 4.18: Response usmg estimated motor velocity feedback (unit step disturbance)
61

Response at guide mUer 10 sinusoidal disturbance (using esnmaHad motor velocily leedtlaCk)
0.6
0.4
.... .:. \
''\ ,
\
\
\
\
\
\
Dislurnance
Kp.IO,Ki=O 1
Kp=40,Ki=O 1
Kp=80.Ki;Q.\
,,
\
\
\
\
\
\
\
\
\
\
\
\
0.8
0.4
0.6
I
f
I
I
f
f
I
I
I
,I
0.2 I
0.2
0.8
... >
5 6 8 9 10
nme(sec)
2 3
1 '_'__L__'__''''__'__''_''_L__J
o
Figure 4.19: Response using estimated motor velocity feedback (sinusoidal disturbance)
Comparsion of impulse responses between using lachomate, and eS1imeted velocUy feedbeck
1.8,.,,,.,,,,
16 1 Observe,
Tachometer I
1.4
12
0,6
0.4
05 15 2.5 3 35
Figure 4.20: Impulse reference response using estimated motor velocity feedback
62

Oomparsion 01 unil Slep responses be1We<!n using lachometer and eslimated velocily leedtlack
1.4
1.2
0.6
>'
0.6 Relerence
Observer
TaChomeler
0.4
0.2
2.00 250 300 350 400
T.me (sec)
50 100 \50
O'__' .l..__~ ..J_____' _'_ .l.____J
o
Figure 4.21: Unit step reference response using estimated motor velocity feedback
Comparslon 01 sinusoidal responses belween uSIng lachometer and eslimaled velocily feedback
10
\
\
\
\
\
\,
\
\,
\
\
\
\
\
\
\,
\
\
8
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
5 6
Time (sec)
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
2
Reference
Observer
Tachometer
I ,
I
I
I
I
I
1 '__'__'__.1__.1'_/'__'__'__'__'_'
o
0.6
0.4
I
I
I
I
,I
I
I
,I
I
0.2 I
0.6
0.6
0.2
0.4
0.6
£"
.§. 0
' >
Figure 4.22: Sinusoidal reference response using estimated motor velocity feedback
63

Chapter 5
Experimental Setup
This chapter describes the openarchitecture experimental platform that is developed
for conducting lateral control experiments. The platform consists of a closedloop web
line as shown in Fig. 5.1. The term closedloop web line refers to web line without
unwind and rewind roUs. This type of a platform mimics most of the features of a
process section of a real processing line.
The experimental platform can be divided into two parts: hardware and software.
The hardware part consists of the closedloop web line, signal processors, drivers for
the actuators, and computer for implementing control algorithms in realtime. Software
part consists of an open architecture realtime program written in C++ programming
language. In the following sections we describe the hardware and software
elements of the experimental platform. The machine section consists of a number
of rollers, with one large master speed roller which is used to transport the web in
the line. The main control elements are a Fife remotely pivoted guide and an active
dancer mechanism as shown in Fig. 5.1. A functional sketch of the experimental web
platform is shown in Fig. 5.2.
The Fife guide mechanism consists of an actuator and an edge sensor immediately
downstream of the guide roller. Lateral control of the web in the line is accomplished
64
using the Fife guide.
5.1 Hardware
The closedloop web line shown in Fig. 5.1 consists of 15 rollers. A master speed
roller, shown as large diameter roUer in Fig. 5.1, is used to transport the web in
the webline. Since each roller width is 8 inches, the maximum web width that can
be used in the web line is 6 inches. The diameter of each roller is 5 inches, except
for the master speed roller, which has a diameter of 10 inches. A nip roller for the
master speed roller is used to reduce slip during startup. An analog controller for
the master speed roller is available to obtain the desired transport velocity of the web
in the line. The closedloop web line, as shown in Fig. 5.1, consists of three main
control elements: (i) Lateral control system, (ii) Active dancer mechanism, and (iii)
Passive dancer mechanism.
5.1.1 Lateral Control System
Lateral control of the web is accomplished by a remotely pivoted Fife guide as shown
in Fig. 5.1. The guide mechanism consists of a guide roller on a base which is
actuated by a DC motor. An edge sensor downstream of the guide roller gives the
web lateral position. From a control viewpoillt, the Fife analog control system is givf'n
by the sketch shown in Fig. 5.4. The physical elements of the Fife guide and their
interaction is shown in Fig. 5.3. The analog lateral control system includes: (i) Fiff'
analog signal processor (A9), (ii) Sensors (edge sensor, tachometer), (iii) DC motor.
The A9 signal processor serves as an amplifier and an onboard analog controller. It
implements a velocity innerloop and a position outerloop as shown in Fig. 5.4. The
velocity innerloop is formed by feedback of the velocity signal of DC motor from
the tachometer, which is used to regulate motor velocity by applying proportional
65
Active Dancer System
~~_._._ .......•••
I
~I
I
I
j,
I
I
I,
Load cell
I
I
I
I
I
I
I
I
,
Passive Dancer ' System
...\ .. _..
i..'.T...j
(J'q
~
"('t)
CJl
I.."
M
X
"0
('t)
"......
en S en ('t)
::l
<+
~
~
('t)
cr'
~
~
<+ 0'
"a
Computer Control System I
1  Data Acquisition Board I
! 1
A9 Fife Controller
I
I
Edge Sensor Kamberoller Web Velocity
Guide Sensor
I
Web Dynamics I
Figure 5.2: Lateral control of the experimental web platform
A9 as Analog Lateral Controller I:
Lateral Position Sensor DC Motor
I
II Lateral Dynamics of Web System
Figure 5.3: Schematic of analog lateral controi system
67

Web
Lateral
Position
Figure 5.4: Fife analog lateral control system
control. The position outerloop is formed by feedback of the web lateral position
signal from the edge sensor, which regulates the web lateral position by applying
proportional and integral control. The edge sensor is a Fife optical position sensor.
The DC motor drives the guide roller based on the control signal from A9. To obtain
an openarchitecture computer control system, we bypass the analog controller used
in the A9 processor and instead use the control algorithm generated in the computer.
In the computer control system, Fife A9 processor simply serves as an amplifier
only. The openarchitecture computer control system can be used to implement any
desired control algorithm. A schematic of physical clements and their interaction
in computer control system is shown in Fig. 5.5. The main component of a Fife
guide system is the DC motor. A velocity innerloop is typically used to stabilize
the DC motor. This requires measurement of motor velocity using a tachometer.
It is typical that the tachometer may cost up to 25 percent of the cost of the DC
motor setup. Considerable reduction in cost can be achieved if other means can be
employed for generating a stable innerloop without using tachometer to mea.sure
velocity. To investigate an estimated motor velocity innerloop, we consider two
different conditions for the lateral computer control system. In the first case, shown
in Fig. 5.6, a tachometer velocity based innerloop is used. In the second case,
a velocity estimator is designed to estimate the motor velocity based on the input
to the motor and the web lateral positi\ '.1, dnd the estimated velocity is used for
68
computer Control System
Data Acquisition Board
Amplifier ( A9 )
Lateral Position Sensor DC Motor
Lateral Dynamics of Web System
Figure 5.5: Schematic of lateral computer control system with velocity innerloop
innerloop as shown in Fig. 5.7.
Reference
Web
Lateral
Position
Figure 5.6: Lateral computer control system wIth tachometer velocity
5.1.2 Computer System
The computer system consists of a 450 MHz Pentium computer with a digital data
acquisition board. The data acquisition board is a Keithley DAS 1601, which consists
of eight AID and two 01A channels. The two DIA channels are used to send control
input to the amplifiers of the guide actuator and the active dancer motor. The eight
AID channneis are used to acquire the sensor signals. The distribution of the AjD
and DIA channels are given in the following.
69
Reference
Web
Lateral
Position
Figure 5.7: Lateral computer control system using velocity estimator
A/D Channel Configuration
• Channel 0 : None
• Channell: Lateral Control Motor Tachometer
• Channel 2 : None
• Channel 3 : Upstream Loadcell
• Channel 4 : Dancer Motor Tachometer
• Channel 5 : Downstream Loadcell ( After Amplifier)
• Channel 6 : Upstream Loadcell
• Channel 7 : Edge Sensor
D / A Channel Configuration
• Channel 0 : Dancer Motor
• Channell: Lateral Control Motor
5.2 Software Structure
The software for realtime control and data analysis is written in C++ programming
language, and can be divided into offline software and realtime software as shown
70
in Fig. 5.8. MATLAB software and C++ programming language are used for data
analysis and offline simulation. The realtime software, which is written in C++
based on Windows platform, implements the following functions in a modular way:
data acquisition, data storage, realtime data display and plotting, contro} algorithm,
state observer algorithm, and control signal output.
Offline software
Offline
Analysis Database
Data
Acquisition &
Processing
Realtime software
Realtime
Display &
Plot
Timer
Interrupt
Control
Algorithm
Control
Output
,           ,,
State
 ; Observer
: Algorithm
,      1,     ·,·,···
Figure 5.8: Software for web handling system
5.2.1 Realtime Software
The individual blocks of the realtime software shown in Fig. :J.8 are explained in the
following.
• Timer interrupt: Timer interrupt serves as the" clock" of the realtime control
system. As shown in Fig.. 5.8, timer interrupt determines both the sampling
period and the control period of the computercontrol system. For all the control
experiments, the sampling period and the control period are the same, and
is taken to be 20 milliseconds. It is well known that when a continuoustime systern
is discretized, if the sampling frequency is not fast enough, then discretized
71
control system can go unstable. From the well known Shannon sampling condition,
we know that the sampling frequency should be at least twice the highest
frequency content of the sensed signal. Considering the dynamic characteristics
of the web system, the chosen sampling frequency of 50 Hz is fast enough.
Moreover, since computer control is used, the sampling period can be set at any
value by just a change of the variable in the realtime control program.
• Data acquisition and processing: At each sampling time, current information
on web lateral position (for lateral control) and web tension (for tension
control) is read from AID channels on the digital data acquisition board. More
specifically, the information collected include: the lateral positional signal from
edge sensor; the tachometer signal from the DC motor of Kamberoller guide:
force information from upstream loadcell and down stream loadcell, tachometer
signal from the DC motor of the active dancer mechanism. The supporting
software for this function module is DAS1600/1400 series standard software
package, which is shipped with the data acquisition board. This software package
includes support functions for Microsoft Windows and function libraries for
writing application programs under Windows™ in Borland C++ Builder.
• Control output: During each sampling period, after the control algorithm is
evaluated, the control signals are output through D/ A channels on the digital
data acquisition board, and then sent out to DC motors after amplification to
drive the active dancer (for tension control) and Kamberoller guide (for lateral
control).
• Realtime display and plot: Based on the data acquired through data acquisition
board, realtime information on web tension and lateral position is
plotted, so that the users can have a direct sense on the performance of the
72
control designs. Other parameters such as controller gains are displayed on
computer screen, and can be modified in real time.
• Database: Data from the sensor signals acquired from AID channels is written
into a database for later offline analysis, which is mainly performed using
MATLAB software package.
• Control algorithm: This block implements the control algorithm via a control
function. The function can be suitably modified based on the design of the
controller.
• State Observer Algorithm: This block contains function for implementation
of a minimumorder observer to estimate the motor velocity. The inputs to this
function at each sampling period is the web lateral position and the control
input to the DC motor. The motor velocity is estimated in realtime via this
functional block.
73

Chapter 6
Experimental Results
The main focus in this set of experiments is to study the analog control system
of Fife signal processors and to use computer control to emulate and improve Fife
control designs. Further, a motor velocity estimator is designed and is implemented
to circumvent the use of tachometer signals for innerloop feedback. The following
experimental conditions are used during lateral control experiments.
• Web velodty: 424 feet/min
• Average web tension: 9.71bf
• Computer control sampling period: 20 milliseconds
• Web material: polyester film
6.1 Emulation of Fife A9 by computer
In this set of experiments the control algorithm used in Fife A9 controller is emulated
in the computer, i.e. we bypass the analog A9 PI controller and implement the
PI control algorithm using the computer. Three conditions are considered while
emulating Fife A9 controller: no disturbance, pulse disturbance and step disturbance.
The step disturbance in our case is actuaP'· ~. long pulse disturbance since appropriate
74
mechanisms to create an actual step disturbance was unavailable. For the following
experiments a long pulse disturbance is considered as a step disturbance. The lateral
position signal from edge sensor under the three experimental conditions are shown
in Fig. 6.1. Fig. 6.2 shows the motor velocity (tachometer signal) for both computer
control and k9 controller implementation. The lateral pulse disturbance in the web is
obtained using a small width nontransparent tape on the side of web containing the
edge sensor. Since the edge sensor is an optical sensor, the nontransparent tape on
the web (polyester film) is perceived as a pulse disturbance. Increasing the width of
the nontransp,arent tape on the web results in a step of finite duration. In Fig. 6.1,
the dotted lines indicate the disturbance profile on the web. The disturbance profile
is also shown in Fig. 6.2 to indicate the time instance of the disturbance. In all the
experiments, the lateral web position shown is in inches and the motor velocity is in
volts. The following observations can be made from the experimental results (Figures
6.1 and 6.2):
• By tuning the control gains for computer control, similar performance as that
of the Fife A9 controller is obtained. In all three cases, i.e. no disturbance,
step disturbance and pulse disturbance, the system shows the same response
pattern.
• Due to the connection tape used to form the web in the experimental platform,
a periodic small pulse disturbance exists in an three cases in addition to the
introduced disturbances.
• Notice that with rising edge of the step disturbance, the edge sensor signal
increases, which is immediately pushed back to the reference zero position by the
control action of the guide. Also, with the falling edge of the step disturbance,
the edge of the web moves to the negative side, and then is pushed back to the
75
reference position due to control action of the guide.
• Also, when pulse disturbance appears on the web, the control action of the
web guide is to push the web to keep the reference position, and when the pulse
disturbance disappears, overshoot is observed due to control action and the web
8
8
8
6
6
6
4
4
4
Time (sec)
Web Position
Computer Controller
2
2
2
returns to the zero reference position.
Web Position
Fife(A9) Controller
0.2 0.2
<I> Q)
<> 0.1 <> c c 0.1
<to III
of? e
::> 0 ::> u; u;
'5 '6
~ 0.1 ~ 0.1
0.2 0.2
0 2 4 6 8 0
0.5 0.5
<I> Ql
<> <>
c c
m III e e
::> ::::l u; 0 .. u; 0
'6 '6
0. 0.
.$ Q)
(f) U5
0.5 0.5
0 2 4 6 8 0
2 2
~ ~
c c
ctI 1 III e e 1
::> ::> u; .£i
'6 "0
Q) 0 3l 0 .;...
<n ::; ::;
0 0..
1 1
0 2 4 6 8 0
Time (sec)
Figure 6.1: Web Position: Fife(A9) controller and computer controller
76
Motor Tachometer Output Motor Tachometer Output
Fife{A9) Controller Computer Controller
~ 0.5 ~ c c 0.5
'" co .e .c
u=>; 0 u:s; 0
'6 '6
~ 0.5 ~ 0.5
1 1
0 2 4 6 6 0 2 4 6 8
~ 2 ~ 2
c c
'" co
.0 € .2 0
=>
0 '" u;
u U
a. a.
Q) Q)
U5 2 U5 2
0 2 4 6 8 0 2 4 6 B
5 5
Q) Q)
u <.> c c
.'e" 'e"
.2 =>
'" 0 u; 01.'
'6 '6
Q) Q)
<J) ::; :':";
a. a..
5 5
0 2 4 6 6 0 2 4 6 6
Time (sec) Time (sec)
Figure 6.2: Motor Velocity: Fife(A9) controller and computer controller
77
6.2 Solutions without tachometer
For the purpose of stabilizing the system in the absence of motor tachometer, two
approaches are considered. One is using finite difference of edge sensor signal to obtain
lateral velocity of the web, which is used as a derivative action in the outer feedback
loop. The second solution is to design a minimumorder velocity observer using edge
sensor signal and motor control signal to get an estimation of motor velocity. This
estimated velocity is used for innerloop feedback. Figures 6.3 and 6.5 show the
experimental results using finite difference approach. The observations from these
experimental results are summarized below.
• Compared to the results obtained without innerloop feedback, which is shown
in Fig. 4.2, the edge sensor signals in Fig. 6.3 are smoother and the overshoot
due to the disturbance is smaller. The finite difference velocity in feedback acts
as a Daction and thus can stabilize the system.
• Although using the finite difference velocity as Daction in the outerloop feedback
has a stabilizing effect, it performs poorly in the presence of disturbances,
see second and third row plots in Fig. 6.5. Moreover, the finite difference velocity
signal depends on the noise level in the position signal, which makes it
unusable in some instances when the lateral position signal noise is high.
78
Web Position
Computer Control (tinite difference velocity feedback.Kp=15.Kd=5)
B
8
B
7
7
7
6
6
6
5
5
5
4
4
4
Time (sec)
3
3
3
2
2
2
05
<1>
<..> c
'e"
.2
''C"
0z
0.5
0
05
Q)
u
c:
'e"=> 12 0
"0
a..
Q)
Ci5
0.5
0
3
Q) u
c:: 2 'e"=>
<;; 1
'C
Q)
:'"; 0
c..
1
0
Figure 6.3: Web position: computer control using finite difference velocity feedback
The experimental results using velocity observer are shown in Figures 6.6 and 6.8.
The following observations can be made from these results.
• Compared with Fig. 6.3, the edge sensor signals in Fig. 6.6 are more stable.
In all the three cases, the oscillations and overshoot due to disturbances are
smaller. The use of estimate velocity in controller can significantly improve the
performance of the system without a tachometer.
• The velocity observer is capahle of effectively picking up the motor velocity
changes in presence of step disturbance and large pulse disturbance.
• Estimated velocity feedback in the innerloop can provide similar closedloop
performance as that of the system with tachometer feedback (compare Figures
6.1 and 6.6).
79
Motor Tachometer Outpu.t
Computer Control (finite difference velocity feedback,Kp=15.Kd=5)
2 3 4 5 6 7 8
5 ' ' 'L 'L ' ' ' ' '
o
5
2 3 4 5 6 7 8
~ 5
c:
'e" :::>
1lj
'6
a.
Q)
Ci5
5 ' ' ' ' ' L ..I... ' '
o
::l 5
c:
'e" :::>
1lj 0
'6
Q)
"'5"
a. 5
''''''''' o 2 3 4
Time (sec)
5 6 7 8
Figure 6.4: Motor velocity: computer control using finite difference velocity feedback
Finite difference velocity
Computer Controfler(no velocity feedback.Kp=15.Kd=5)
5
Q)
u
c:
ctI e
.2 0
''6"
0 z
5
0 2 3 4 5 6 7 8
2 3 4 5 6 7 8
::l 5
c:
'e"
~ 0 fTAA"AA,r"'\{'.J\, \
'6
a.
Q)
ii5
5 L ' ' L ' ' .....L. .....L. ...J
o
~ 20 c:
'e"
~ 0 h~/'VV'~".....~~' Irv'
6
Q) "'R" 20
a 2 3 4
Time (sec)
5 6 7 8
Figure 6.5: Finite difference velocity: computer control using finite difference velocity
feedback
80
Web Position
Compu.ter Control (observer velocity teedback,Kp=15)
0.5
~
c
'e" u:::>; 0
u0
z
0.5·
0
0.5
~
c
e'" ::::> u; 0
'5
0
2
en
0.5
0 1
3
OJ uc
2 e'"
.2 en
'5
OJ en 0 "S
0
1
0
2
2
3
3
3
4
4
Time (sec)
5
5
6
6
6
7
7
7
8
8
8
Figure 6.6: Web position: computer controller using velocity observer
Motor Tachometer Output
Computer Control (observer ve.locity teedback, Kp=15)
2 3 4 5 6 7 8
5 L... ~ ' L __'___ ___'_ .L._ ____'__ __'
o
5
~
c
'e" .2 OhlIlJII'...tV\lIJl"~1\
''5"
o z
2 3 4 5 6 7 8
~ 5
c
o'r">
'5 u; 0
'5
0
2
en
5 ' I. I. 'L ' 'L I. L '
o
~ 5
c
<tI e:::> u; 0
'0
OJ
"'"S
a. 5 L __'___ ___'_ ' ....L. l. ..L ' '
o 2 3 4
Time (sec)
5 6 7 8
Figure 6.7: Tachometer velocity: computer controller using velocity observer
81
Velocity Observer Output
Computer Controller(veloclty observer.Kp=15}
5
'<".:> c:
e'"
u:::;J 0
i5
0 z
5
0 2 3 4 5 6 7 8
'<".:> c:
'E"
:::J
<;;
i5
C>.
<l>
(j)
5
0 2 3 4 5 6 7 8
~ 5
c:
e'" :::J u; 0
i5
'e"n
::;
0 5
0 2 3 4 5 6 7 8
Time (sec)
Figure 6.8: Estimated velocity: controller using velocity observer
82
6.3 Comparison of controllers
To investigate the effects of finite difference velocity and observer estimated velocity
on the web system, the transient response of three different controllers are compared:
(1) Fife A9 controller with tachometer velocity feedback, (2) computer controller
using finite difference velocity and (3) computer controller using observer estimated
velocity. Figures 6.9, 6.10 and 6.11 show the velocity from the tachometer, finite
difference velocity, and observer estimated velocity, respectively, for the three cases.
Figures are enlarged to investigate the time shift.
• Finite difference velocity and velocity observer introduce a time shift. This
is expected and is confirmed by the experiment results of all three cases, l.e.
Figures 6.9, 6.10 and 6.11.
• It is surprising that the finite difference velocity is time ahead (as opposed to
time delay) of tachometer signal. Two possibilities may contribute to this effect:
 a time delay circuit has been inserted in A9 controller considering the
system delay;
 tachometer dynamics may introduce the time delay;
• When magnitude of disturbance is not large, i.e. for cases of no external disturbance
(Fig. 6.9) and step disturbance (Fig. 6.10)' both finite difference velocity
and observer velocity can provide estimation for motor velocity. However, the
velocity observer provides a better velocity estimate.
• When the disturbance is large, consider the case of pulse disturbance (Fig.
6.11)' finite difference velocity completely failed to follow the change in motor
velocity. Furthermore, the finite difference velocity estimation becomes worse if
web position sensor signal is noisy.
83

• The velocity observer provides a very good estimate of motor velocity. But
introduces some time delay when compared with tachometer velocity. This
time delay can be reduced by using a higher signal sampling frequency than the
control sampling frequency.
• The most attractive feature of the velocity observer is that it can provide a good
and relatively smooth motor velocity estimate. Thus, the control gains can be
chosen higher to provide faster response.
Velocity: Comparison of controllers (no dlisturbance)
2
1.5
Tachometer
Observer
Finite Differeno
1.5
2.5 3 3.5
2 L ' L 1
2
Time (sec)
Figure 6.9: Comparison of velocities from three controllers: no disturbance
84
Velocity: comparison of controllers (step disturbance)
6
4
2
2
Tachometer
Observer
Finite Differenc
4
2.5 3 3.5
6''''
2
Time (sec)
Figure 6.10: Comparison of velocities from three controllers: step disturbance
85
Velocity: comparison of controllers (pulse disturbance)
3
2
lS c
'e"
.~ 0
u
Q) en
:5 a...
1
2
Tachomeler
Observer
Finite Difterenc
\
\
\
\
\
\
\
\
3 3.5 4
3 L ' ' '
2.5
Time (sec)
Figure 6.11: Comparison of velocities from three controllers: pulse disturbance
86
Chapter 7
Conclusion and Future Research
7.1 Conclusion
The objective of this research was to circumvent the use of the tachometer signals for
the innerloop feedback with a motor velocity estimator. Specifically, the equations
developed by Shelton for a remotely pivoted steering guide and for lateral deflection
of web were to be verified. Once verified these equations could be used for other
web line application. Chapter 2 lists the basic types of automatic control systems
and guiding mechanisms used in the industry. The control system on the traction
machine in OSU is the electromechanical type coupled with the remotely pivoted
guide. Shelton's model for the remotely pivoted steering guide is rederived, which
forms a basis for aU the experiments in this thesis. It has been experimentally verified
in Section 4.2 that the velocity innerloop feedback is very essential for satisfactory
lateral control of web. Equation (4.24) is our desired velocity estimator that lateral
motion of the web is digitally controlled. Open loop dynamic response of the web for
a step, impulse and a sinusoidal disturbance are conducted which show that the web
system can follow the disturbance after a short time period as shown in Figures 4.7,
4.8 and 4.9. The open loop dynamics of the system also appears to be characterized
by a low pass filter.
87
Experiments having been conducted using both Fife A9 PI controller and the
digital PI controller using the computer. Results show that the computer control
is very similar to the Fife A9' analog controller. T\\'o solutions were offered for the
purpose of stabilizing the system in the absence of motor tachometer:
• Finite difference of edge sensor signal to obtain the lateral velocity of the web.
• Estimating the velocity of the web using minimumorder observer design.
The results using the finite difference method is found to be unsatisfactory since the
finite difference velocity signal depends on the noise level in the position signal, which
makes it unusable at some instances when the lateral position signal noise is high.
On the other hand, the results using velocity observer are found to be more effective
since the estimated velocity feedback in the innerloop can provide similar closedloop
performance as that of the system with tachometer feedback.
7.2 Future Work
Investigation of the effect of different kinds of controllers (like adaptive control, robust
control) on the lateral control of web will be undertaken in the future. The study will
involve use of ultrasonic sensors and an offset pivoted guide (displacement guide) will
replace the remotely pivoted guide.
Little research has been done for the area of web due to uneven flIp loading.
However, research has been done for the rubber covered roBs. Foreman assumed that
the web achieves the velocity of the rubber rolls in the contact region. The velocity in
this region should be greater than the velocity of the covering away from the contact
zone. Foreman formulated the following equation for change in velocity per unit
88
velocity.
6.V RO 501 5  =0.35 o' .
V t 2 (7.1 )
An accurate model and basis for this greater velocity could provide a more accurate
strain.
An investigation ofmodul us could also be useful. Research could also be performed
on the sman deflection assumption of the rubber covered roll. Being able to determine
the point where this assumption does not apply could be useful in predicting the
lateral deflection of the web. This could give a more accurate description of lateral
web movement. The effect of friction between rollers and the web would also be
interesting. Investigation into this area could show a relationship between frictiul1
and lateral web movement. Once the equations for lateral deflection and effective
nip load have been refined, they could be then used for web guiding. By using an
edge sensor and a feedback controller, the nip setup could be used as a web guidance
system.
89
Bibliography
[1] J .J. Shelton, Lateral Dynamics ofa l'vIo,rjng Web, Ph.D. thesis. Oklahoma State
University, 1968.
[2] J.J. Shelton and K.N. Reid, Lateral Dynamics of a Real Moving vVeb, ASME
Journal of Dynamic Systems, Measurement, and Control, vol. 93, no. 3, pp.
180192, 1971.
[3] J.J. Shelton and K.N. Reid, Lateral Dynamics ofan Ideallvloving Web, ASME
Journal of Dynamic Systems, Measurement, and Control, vol. 93, no. 3, pp.
187186, 1971.
[4] K.I. Hopcus, Unwind and Rewind Guiding, Proceedings of the Second International
Conference on Web Handling, June 69, 1993.
[5] G.E. Young and K.N. Reid, Lateral and Longitudinal DY'namic Behavior and
Control of Moving Webs, ASME Journal of Dynamic Systems, Measurement,
and Control, vol. 115, no. 2B, pp. 309317, 1993.
[6] G.E. Young, J.J. Shelton and C.E. Kardamilas, Modeling and Control of Multiple
Web Spans Using State Estimation, ASME Journal of Dynamic Systems,
Measurement, and Control, vol. Ill, pp. 505510, 1989.
90
[7] C.E. Kardamilas and G.E. Young, Stochastic Modeling of Lateral Web Dynamics,
Proceedings of the American Control Conference, San Diego, CA,
May 2325, 1990.
[8] C.E. Kardamilas, Stochastic Modeling and Control of Lateral "Veb Dynamics
Ph.d. thesis, Oklahoma State University, 1990.
[9] G.E. Young, J.J. Shelton and B. Fang, Interaction of Web Spans: Part 1Statics,
ASME Journal of Dynamic Systems, Measurement, and Control, vol.
111, pp. 490496, 1989.
[10] G.E. Young, J.J. Shelton and B. Fang, Interaction of Web Spans: Part IIDynamics,
ASME Journal of Dynamic Systems, Measurement, and Control,
vol. 111, pp. 497504, 1989.
[11] P.B. Lindley, LoadCompression Relationships of Rubber Units, Journal of
Strain Analysis, vol. 1, no. 3, pp. 190195.
[12] A.R. Foreman, Application of Rubber Covered Rolls to Pinch Rolls and Bridles,
Iron and Steel Engineer, pp. 111120, 1964.
[13] J.J. Shelton, Informal notes on web deformation due to nonuniform nip pressure,
1994.
[14] M.N. Ahmad, Lateral Deflection of a Web due to a Diffcn~I1tjalrv lOaded Nip,
Masters Thesis, Oklahoma State University, 1995.
[15] J.J. Shelton, Effects of Web Camber on Handling, Proceedings of the Fourth
International Conference on \Neb Handling, June 14, 1997.
91
[161 P.R. Pagilla, K.N. Reid and K. Hopcus, Modeli.ng and Advanced Control
of Web Handling Systems, Project Number: AR928021, Contract Number:
5401, January 2000.
[17] K. Ogata, DiscreteTime Control Systems, Prentice Hall, Englewoods Cliffs,
New Jersey, 1995.
92
..
Appendix A
Derivation of Velocity Observer
A typical control system with estimated state feedback is shown in Fig. A.l
~
x=Ax + Bu
y =ex
! u
"x MinimumOrder y
Siale Observer
Figure A.l: Typical control system with estimated state feedback
The transfer function of web guiding system, which is a remotely pivoted steering
guide is given by the following equation:
YL(S)
Z(s)
2 12(1{L) h(J{ L)
s + S + 2.,'
T T Xl
3 2 + h(KL) s + h(KL)
T T 2
93
(A.l)
r
A state space model of the above system is given by:
The state variables Xl, X2, X3 and X4 correspond to motor angle, motor angular velocity,
web lateral position, web lateral velocity, respectively. For our web system, the
web lateral position X3, is measured by the optical sensor. Thus, the minimumorder
observer is designed to estimate the state variables XI, X2 and X4'
The procedure for deriving the observer is summarized below.
(1) Choose a matrix C'* such that
C. = [~ ~ ~ ~]
Consider the following similarity transformation matrix,
1 0 0 0
yl = [~ ]
0 1 0 0
0 0 0 1
0 0 1 0
Therefore,
1 0 0 0
0 1 0 0
T=
0 0 0 1
0 0 1 0
94
(2) Apply similarity transformation to matrices A, B, and C.
A= T1AT
Therfore,
1 0 0 0 0 1 0 0 1 0 0 0
0 1 0 0 0 am 0 0 0 1 0 0
A=
0 0 0 1 a a 0 1 0 0 0 1
0 0 1 0 13 0 al G2 0 0 1 0
a 1 0 0
a am 0 0 _ [ All A12
A= ]
13 0 a2 Gl A21 A22
a 1 1 0
where
Also,
.'121 = [0 0 1]; A22 = [ 0]
B = T1B
Therefore,
1 0 0 0 0 0
a 1 0 0 km km
= [ ~: B= ]
a 0 a 1 0 a
a 0 1 a 0 0
where
B1 =
[ k~ ] ;
B2 = [ 0 ]
95
1 0 0 0
6 = CT = [ 0 o] 0 1 0 0
0 1
0 0 0 1
0 0 1 0
Therefore,
6=[0001]
(3) Define the following matrix,
E = All  LA21
T
where L is the observer feedback gain matrix which is given by [i l l2 l31
The observer gains, ll, i 2 and 13 are obtained by choosing the desired poles of the
observer matrix E. The matrix E is
The characteristic polynomial of E is given by
det[sI  E] = 8
3 + (0'2 + l3 + am)s2 + (am Q'2 + aml3 + i1l3)8 + (l2{J + [[(Jam)
(A.2)
Let 1l.J , fl2 ,J13 be the desired poles of the observer, i.e. of matrix E. Then the
following characteristic equation gives the desired observer poles:
det[sI  E] = 8
3 + (/11 + fl2 + /13)8
2 + (/11fl2 + fl2fl3 + fl3J1ds + (fllfL2/13) = 0
(A.3)
96
•
Comparing coefficients of (A.2) and (A.3L we obtain
Solving the above equations we obtain the observer gain matrix, L.
(4) Now define matrices J,D,R,S such that,
D = B1  LB2
On computation the matrices J, D, R, 5 are
D = [ k~ ]
1 0 0 II
0 1 0 l2 R= 5=
0 0 1 l3
0 0 0 1
1 0 0 0 1 o 0 l1 1 0 0 II
T[R 5]=
0 1 0 0 0 1 0 12 0 1 0 l2
0 0 0 1 0 0 1 13 0 0 0 1
a 0 1 0 0 0 0 1 0 0 1 lJ
The estimated state vector is given by
1 0 0 I}
x(t) = 0 1 0 12 [ i(t) ] (A.4)
0 0 0 1 y( t)
0 0 1 13
97
where z is given by the observer error dynamics,
i(t} = Ez(t) + Du(t) + Jy(t)
Taking Laplace transform of the above equation, we obtain
(s1  E)Z(s) = DU(s) + JY(s)
(A.5)
The above equation can be divided into three parts:
(A.6)
(A.7)
(A.8)
Solving equations (A.6), (A.7) and (A.8) we get
, kms2 + (kmCf2+ kml3)s + l]kmfJ )U(s)
8 3 + (am + a2 + l3)82 + (a2am + l3am + lIfJ).'> + (llamfJ + l213
+ J28
2+ (,ha 2+ J2[3  J1 l2)S + (ll·lzl3  JI12(3) P'(S)
8 3 + (am + 0:2 + l3)82+ (a2 am + [3am + ll(3)s + (lIUm!J + l2(3
To obtain the transfer function relating the estimated motor velocity, ,\"2 (s) to U(s)
and Y(s), we use equation (A.4). From (A.4),
Therefore,
98
where
"12 am + G:'2 + l3
"11 G:'2am + l3am + ltf3
"10 ll am(3 + l2/3
(33 12
(32 l2am + G:'2 l2+ 1213 + J2
(31 J2 G:'2 + J2l 3  J3 l2 + (}:2am l2 + l3am l2 + hlz(3
(30 II J2f3  Jl l2(J + II aml2(3 + (3l2 2
A2 km
Al km CX2 + km l3
AO llkm (3
99
Appendix B
Matlab Script File
Cm=O.012; am=53; Km=1413*Cm; Kml=1413;
xl=88; %instant center (inch)
L=46; % in
v=400*12/60; %in/sec
Tension=10; %Ib
T=L/v;
K=sqrt«Tensionv2*1.7*10(8»/(10800*(1+0.008658*Tension»);
KL=K*L;
fffl=KL2*(cosh(KL)1)/(KL*sinh(KL)2*(cosh(KL)1»
fff2=KL*(KL*cosh(KL)sinh(KL»/(KL*sinh(KL)2*(cosh(KL)1»
fff3=KL*(sinh(KL)KL)/(KL*sinh(KL)2*(cosh(KL)1»;
a=T2/fffl b=fff2*T/fff1 c=4/3*fff2/fff1
Kpl=10; Kp2=40; Kp3=80; Kp4=4; Kd=O; Ki=O.l; Kv=O.744;
100
beta=(cl)/a; alphal=b/a; alpha2=1/a;
mul=50; mu2=50; mu3=50;
M=[O 0 l;beta 0 am;beta*am beta 0]; N=[mul+mu2+mu3amalpha2
mul*mu2+mu2*mu3+mu3*mulam*alpha2
mul*mu2*mu3] ;
LL=inv(M)*N;
11=LL(1); 12=LL(2); 13=LL(3);
J1=1211*13; J2=am*1212*13; J3=beta*11alpha2*1313*13alpha1;
beta3=12; beta2=12*(a1pha2+13+am)+J2;
betal=12*(am* (alpha2+13)+beta*11)+J2* (a1pha2+13)J3*12;
bataO= (J2*beta*11Jl*beta*12) +12* (aIlJ*beta*l1 +12*beta) ;
gammaO=am*beta*11+12*beta; gammal=am*(alpha2+13)+beta*11;
gamma2=a1pha2+13+am;
1ambdaO=Km*beta*11; 1ambdal=(a1pha2+13)*Km; 1ambda2=Km;
gvu=tf([lambda2,lambdal,lambdaO] ,[l,gamma2,gammal,gammaO]);
gvy=tf ([beta3, beta2, beta!, betaO] , [1, gamma2 ,gamma1, gammaO] ) ;
101
%********************* Coef2.m*******************
f1=(K2)*(cosh(KL)1)!(KL*sinh(KL)2*(cosh(KL)1));
f2=K* (KL*cosh (KL) sinh(KL)) / (KL*sinh (KL) 2* (cosh (KL) 1) ) ;
f3=K* (sinh (KL) KL) / (KL*sinh(KL) 2* (cosh (KL) 1)) ;
nurnz= [1, v*f2, v2*f2/x1] ;
denz=[1,v*f2,v2*f1] ;
gl=tf(numz,denz);
num3R=[O,v*f3,v2*f1] ;
den3R= [1 ,v*f2, v2*f 1] ;
g3=tf(num3R,den3R);
gm=tf([O 0 Km] ,[1 arn+Km 0]);
gcl=tf([Kp1 Ki],[l 0]);
gc2=tf([Kp2 Ki],[l 0]);
gc3=tf([Kp3 Ki] ,[1
0]) ;
g_close1=g3/(1+gcl*gm*gl) ;
g_close2=g3/(1+gc2*gm*gl) ;
g_close3=g3/(1+gc3*gm*gl) ;
102
close all;
%open loop response (Impulse)
t=O: 0.01: 5;
%[yl,t]=impulse(tf([l], [l]),t);
%plot(t,yl,'');
hold on;
[y2,t]=impulse(g3,t);
plot(t,y2, '');
title('Response at guide roller to impulse disturbance (open
loop)');
xlabel('Time (Sec.)');
ylabel ( , Amplitude') ;
hold off;
zoom on;
% open loop response (step)
figure(2);
t=O: 0.01: 5;
y2=ones(size(t»;
[yl,t]=step(g3,t) ;
plot(t,y2,'b' ,t,yl, 'b');
legend( 'Disturbance' ,'Response') ;
103
title('Response at guide roller to unit step disturbance (open
loop)');
xlabel('Time (Sec.)');
ylabel('Amplitude');
zoom on;
% open loop response (sin)
figure(3);
t=O:O.Ol:10;%open loop response (Impulse)
yl=sin(t); sinl=sin(t);
[y2,t]=lsim(g3,sinl,t) ;
plot(t,yl, 'b',t,y2,'b');
legend('Disturbance', 'Response');
title('Response at guide roller to sinusoidal disturbance (open
loop)') ;
xlabel('Time (Sec.) I);
ylabel('Amplitude');
zoom on;
%%%%%%%%%%%%closedloop response%%%%%%%%%%%%%%%%
%close loop response (Impulse)
figure(4);
t=O:O.Ol:3;
/'[yl,t]=impulse(tf([l], [l]),t);
1Q4
._  
%plot(t,y1,'');
[y1,t]=impulse(g_close1,t);
[y2,t]=impulse(g_close2,t);
[y3,t]=impulse(g_close3,t);
plot (t , y1 , ,b' ,t, y2, , b: ' , t , y3 , ,b. ' ) ;
legend('Kp=10,Ki=O.1','Kp=40,Ki=O.1','Kp=80,Ki=0.1');
titIe ( ,Response at guide' roller to impulse disturbance
(closedloop)');
xlabel('Time (Sec.)');
ylabel('YL Amplitude');
zoom on;
%close loop response (unit step)
figure(5);
t=O: 0.01: 10;
y4=ones(size(t»;
[y1,tJ=step(g_close1,t);
[y2,tJ=step(g_close2,t);
[y3,tJ=step(g_close3,t);
plot(t,y4, 'b' ,t,y1, 'b' ,t,y2, 'b:' ,t,y3, 'b.');
legend( 'Disturbance' , ' Kp=10, Ki=O. l' , ' Kp=40 ,Ki=O. l' , 'Kp=80 ,Ki=O. 1') ;
title('Response at guide roller to unit step disturbance
(closedloop) ');
xlabel('Time (Sec.)');
ylabel('YL Amplitude');
hold off;
105
zoom on;
%close loop response (sin step)
figure(6);
t=O:O.Ol:10;
y4=sin(t);
[yl,t]=lsim(g_close1,y4,t);
[y2,t]=lsim(g_close2,y4,t);
[y3,tJ=lsim(g_close3,y4,t) ;
plot (t, y4, ' b ) , t , y1 , )b' , t , y2, , b: ' , t , y3, 'b . ' ) ;
legend ( 'Disturbance) , )Kp=10 ,.Ki=O. P , , Kp=40, Ki=O.l' , 'Kp=80 ,Ki=O .1') ;
title('Response at guide roller to sinusoidal disturbance
(closedloop)');
xlabel ('Time (Sec.)');
ylabel('YL Amplitude');
zoom on;
%%%%%%%%%%%%closedloop response with velocity observer%%%%%%%%"!.%%"!.'!.%%%
g_obsl=g3*(1+gvu)!(1+gvu+gm*gl*(gcl+gvy»);
g_obs2=g3*(1+gvu)!(1+gvu+gm*gl*(gc2+gvy»);
g_obs3=g3*(1+gvu)/(1+gvu+gm*gl*(gc3+gvy»);
g_obs_ref=gm*gcl*gl/(l+gvu+grn*gl*(gcl+gvy));
%closedloop response with velocity observer (Impulse)
106
figure (7) ;
t=O:O.01:3;
%[yl,t]=impulse(tf([l],[l]),t);
%plot(t,yl,'');
[y1, t] =impulse (g_obs1, t) ;,
[y2,tJ=impulse(g_obs2,t);
[y3,tJ=impulse(g_obs3,t);
plot(t,yl, 'b' ,t,y2, 'b:' ,t,y3, 'b.');
legend('Kp=10,Ki=O.l','Kp=40,Ki=O.1' ,'Kp=80,Ki=O.1');
title('Response at guide roller to impulse disturbance (using
estimated motor velocity feedback)');
xlabelC'Time (Sec.)');
ylabel('YL Amplitude');
hold off;
zoom on;.
%closedloop response with velocity observer (unit step)
figure(8); t=O:O.Ol:10;
y4=ones(size(t»);
[y1,tJ=step(g_obs1,t) ;
[y2,t]=step(g_obs2,t);
[y3,t]=stepCg_obs3,t) ;
plot (t ,y4, , b' , t ,y1, ' b' , t ,y2 , ' b: ' , t ,y3, ' b . ') ;
legend('Disturbance','Kp=10,Ki=O.l', 'Kp=40,Ki=O.l' ,'Kp=80,Ki=O.1');
title('Response at guide roller to unit step disturbance (using
estimated motor velocity feedback)');
107
xlabel('Time (Sec.)');
ylabel('YL Amplitude');
zoom on;
%closedloop response with velocity observer (sin)
figure(9);
t=O:O.Ol:10;
y4=sin(t);
[yl,t]=lsim(g_obsl,y4,t);
[y2,t]=lsim(g_obs2,y4,t);
[y3,t]=lsim(g_obs3,y4,t);
plot(t,y4, 'b' ,t,yl,. 'b' ,t,y2, 'b:' ,t,y3, 'b. ');
legend('Disturbance' ,'Kp=10,Ki=O.1','Kp=40,Ki=O.1' l 'Kp=80,Ki=O,1');
title('Response at guide roller to sinusoidal disturbance (using
estimated motor velocity feedback)');
xlabel('Time (Sec,)');
ylabel('YL Amplitude');
zoom on;
%responses to impulse disturbance using tachometer and estimated velocity feedba
%%%% impulse response
figure(10); t=O:O.Ol:3;
%[yl,t]=impulse(tf([l], [1]) ,t);
%plot(t,yl,'') ;
[yl,t]=impulse(g_close_ref,t) ; %with tach
108
•
'%with estimated
plot(t,yl, 'b' ,t,y2, 'b:'); legend('Observor', 'Tachometer');
title('Comparsion of impulse responses between using tachometer
and estimated velocity feedback');
xlabel('Time (Sec.)');
ylabel(' YL Amplitude');
hold off;
zoom on;
%%%% unit step response
figure (11) ;
t=O:O.Ol:10;
y4=ones(size(t»;
[yl,t]=step(g_close_ref,t) ;
[y2,t]=step(g_obs_ref,t);
%with tach
%withestimated
plot(t,y4,. 'b',t,yl, 'b' ,t,y2, 'b:');
legend('Reference','Observor','Tachometer');
title('Comparsion of unit step responses between using tachometer
and estimated velocity feedback');
xlabel('Time (Sec.)');
ylabel('YL Amplitude');
zoom on;
%sin response
figure(12) ;
109
•
t=O:O.Ol:10;
y4=sin(t);
[yl,t]=lsim(g_obs_ref,y4,t);
[y2,t]=lsim(g_close_ref,y4,t) ;
plot (t •y4, 'b , , t, yl, 'b' , t , y2 , , b: ' ) ;
legend('Reference','Observor' . 'Tachometer');
title('Comparsion of sinusoidal responses between using tachometer
and estimated velocity feedback');
xlabel('Time (Sec.)');
ylabel('YL Amplitude');
zoom on;
110 ..
1...,...
VITA
Shaibal Sailaza Mandai
Candidate for the Degree of
Master of Science
Thesis: LATERAL CONTROL OF A WEB USING ESTIMTED VELOCITY
FEEDBACK
Major Field: Mechanical Engineering
Biographical:
Education: Received Bachelor of Engineering degree in Mechanical
Engineering from Universjty ofMumbai, Maharashtra, India in May 1998.
Completed requirements for the Master of Science degree at Oklahoma
State University in July, 2000.
Professjonal Experience: Graduate Research Assistant, Web Handling Research
Center, School of Mechanical and Aerospace Engineering, Oklahoma
State University, March 1999 to July 2000. Trainee Engineer in Bhabha
Atomic Research Center, Trombay, India:, August 1997 to April] 998.
