A STUDYONCONTROL OF ACCUMULATORS IN CONTINUOUS
WEB PROCESSING LINES
By
lNDERPAL SINGH
Bachelor of Science
Thapar Institute of Engineering & Technology
Punjab, 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
December, 2002
A STUDY ON CONTROL OF ACCUMULATORS IN CONTINUOUS
WEB PROCESSING LINES
I I
Thesis Approved:
me \ no
I·
, ( un'
Thesis Ad isor
th 1
\
~tJ~.~
Dean of the Graduate College
11
ACKNOWLEDGMENTS
I wish to express my sincerest appreciation to my major advisor, Dr. Prabhakar R.
Pagilla for his mtelligent supervision, constructive guidance, inspiration, and friendship
throughout the development ofthis investigation.
I would like to extend my warmest thanks to my masters committee members: Dr.
Eduardo A. Misawa and Dr. Gary E. Young for their support and suggestions in completion
of this research. TheIr guidance and encouragement made the development ofthis thesis a
positive learning experience.
I would also like to thank my colleagues at Oklahoma State Universijy Ramamurthy V.
Dwivedula, SachinGupta, GimJdIuan N~ and yopg Liang Zfu for all the help they have
provided.
,II
. \, ('
..or) •
111
TABLE OF CONTENTS
Chapter
Linean"f~t1
Page
1 INTRODUCTION
1.1 Background.
nil ri (II:
1.2 Literature Review
• I
1.3 Thesis Contributions
1.4 Thesis Outline . . . .
2 DYNAMIC MODELLING
')' .....
2.1 Strip Tension and Carriage Dynamics ofWeb in Accumulator Spans
2.1.1 Speed Changes During Rewind Roll Change .. , ,
2.2 Hydraulic System ,' .
, \ " I
2.3 Full Model Dynamics Including Exit and Entry Roller.
2.4 System with Carriage Sway Dynamics
2.5 Average Dynamic Model
2.6 Synopsis .
3 DYNAMIC BEHAVIOR OF SYSTEM
3.1 System Parameter Calculations . . . . . . . . . . . . . . . . . . . . . . .. 24
3.1.1 Required Reference Pressure to Maintain Carriage Position and
Reference Tension in Web Spans. . . . . . . . . . . . . . . . . " 24
3.1.2 Calculation of Spool Valve Position for any Forced Equilibrium Point 25
3.1.3 Calculations for Weight Change of Web Spans during Carriage
Motion " 26
IV
3.2 Study of Dynamic Behavior of System . . . . . . . . . . . . . . . . . . . . 29
3.2.1 Reachability and Observability Analysis for the Accumulator System 30
3.2.2 Feedback Linearization " 34
3.2.3 Controllability and Observability Analysis for Jacobi Linearized
. . System . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3 System Dynamics Including Exit and Entry Roller Dynamics. 38
3.3.1 Analysis ofthe Accumulator System Ignoring Nonlinearities . 40
3.3.2 Analysis ofthe Accumulator System. . . . . . . 42
3.3.3 Jacobi Linearization ofthe Accumulator System 43
3.4 Synopsis.......................... 44
4 CONTROLLER AND OBSERVER DESIGN 46
4.1 Controller Design . . . . 46
4.1.1 Simulation Study 50
4.2 Controller Design Incorporating Adaptation Law for Friction Coefficient. 68
4.3 Simulation Study Considering Web Span Weight Acting on the Carriage 71
4.4 Synopsis.................................. 73
5 CONCLUSIONS AND FUTURE RESEARCH
5.1 Future Research .
BIBLIOGRAPHY
v
82
. . . . . . . . . . . . . . . .. 83
84
LIST OF TABLES
Table
4.1 Parameters of the accumulator.
Page
. 51
" :.
I ,
~ I
. I
• T. '1" .' 'iJ
t d II'
vi
7
LIST OFFIGURES rb 59
Figure r Page
2.1 Typical process line layout and terminology..
2.2 Sketch of an exit accumulator.
2.3 Sketch of an accumulator span. . ....
, II ' l
2.4 Exit and carriage speed profiles during rollchange.
r
2.5 Sketch of carriage and accumulator hydraulic system.
.~ .
2.6 Exit accumulator with processside and exitside driven roller.
I I
2.7 Illustration of carriage during sway. .
5
7
8
10
13
15
16
T ••
2.8 Equivalent of exit accumulator for average model, Le., with one span. 19
I .
3.1 Change in web weight during carriage motioI\ for 0.01 jp.che I thick aluminium
web. 27
3.2 Change in web weight during carriage motion for 0.125 iI).ches thick alu
3.4 Uncompensated system states incorporating friction term..
minium web. . .
3.3 Uncompensated system states ignoring friction term.
9·,9 28
30
31
3.5 Uncompensated system states incorporating friction term and sinusoidal
disturbance .
3.6 Compensated system with friction term and PI controller.
3.7 Exit accumulator with,process~ide and e~itside drive~lo11e.r.•
4.1 Desired exit speed, carriage speed and carriage position during rewind rollchange.
. .
4.2 Three cases of sinusoidal disturbances..
vii
32
33
39
52
54
.~
I
!
4.3 State errors of the industrial controllec Disturbance 1. 56
4.4 State errors of the proposed controller: Disturbance 1. . '57
4.5 Control inputs for the industrial controller: Disturbance I. 58
4.6 Control inputs for the proposed controller: Disturbance 1. . 59
4.7 State errors ofthe industrial controller: Disturbance 2. 60
4.8 State errors of the proposed controller: Disturbance 2. . 61
4.9 Control inputs for the industrial controller: Disturbance 2. 62
4.10 Control inputs for the proposed controller: Disturbance 2. . 63
4.11 State errors of the industrial controller: Disturbance 3. 64
4.12 State errors ofthe proposed controller: Disturbance 3. . 65
4.13 Control inputs for the industrial controller: Disturbance 3. 66
4.14 Control inputs for the proposed controller: Disturbance 3. . 67
4.15 State errors of the proposed controller with adaptation law: Disturbance 1. 70
4.16 Control inputs for the proposed controller with adaptation law: Disturbance 1. 71
4.17 Viscous friction coefficient estimation for the proposed controller: Disturbance
1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 72
4.18 State errors of the industrial controller: No compensation for varying mass. 73
4.19 State errors of the proposed controller: No compensation for varying mass. 74
4.20 Control inputs for the industrial controller: No compensation for varying
mass. . 75
4.21 Control inputs for the proposed controller: No compensation for varying
mass. . 76
4.22 State errors of the industrial controller with compensation for varying mass. 77
4.23 State errors of the proposed controller with compensation for varying mass. 78
4.24 Control inputs for the industrial controller with compensation for varying
mass. . 79
V111
4.25 Control inputs for the proposed controller with compen ation for varying
mass. 80
perpellldlcuJ
ril th roll r
~ It i I
( t t
.. "l! • I
.. T'l1 I 'II r t. t:>r i"_
r 1\ 
• I
, . t
IX
NOMENCLATURE
A = area of crosssection of the strip perpendicular to web travel direction
Acyl rod side cylinder area
Bj  viscous friction coefficient at the jth roller
l
em  coefficient ofinternal leakage of the cylinder
E  modulus of elasticity of the strip
Fd disturbance force
Fh  controlled force
9  acceleration due to gravity
Ie  mass moment of inertia of the carriage
Jj moment of inertia of the jth roller
Ke,Kp  motor constants
Kq flow gain coefficient
k  spring constant of the web span( ~:)
Lj  span length ofjth web span
ld  center to center distance between adjacent rollers
Me  carriage mass
N  number of spans
Pc cylinder pressure
Pr return pressure
Ps supply pressure
R· radius of the jth roller J
tc  average strip tension in N spans
tj tension in jth web span
tr downstream and upstream tension of accumulator
x
Va  total control volume in the rod. side cylinder chamber
Wj  angular velocity of the jth roller
Vc  carriage velocity
Xc  carriage position
X r ram displacement
() angular rotation ofcarriage in vertical plane
(3 bulk modulous of hydraulic fluid.
p = density of the web,mSiterial , d, I J
= strain in web span ... I
r t Ito l '11" {, l" ."
I. I I
, I
'I
, "
xi
CHAPTERl
INTRODUCTION .
1.1 Background
Any continuous material whose width is significantly less than its length and whose thickness
is significantly less than its width can be described as a web. Plastic wrap, paper,
photographic film, and aluminum strips are all examples of web. It is very important that
the tension in a web span be maintained within a' close tolerance band during processing of
a web. For example, if the tension in the web changes during printing/perforating processes,
the print/perforation gets skewed. Further, excessive tension variations may cause wrinkles
or may even tear the web: Thus, a good tension control system is an'important requirement
in a web handling system since any disturbance such as uneven roller Or variations in web
speed or roll size affect the tension.
As the demand for higher productivity and better performance from the web processing
industry increases, better models and more accurate control algoritJuns for the processes
have to be developed. Tension control plays a key role in improving the quality of the
finished web. It is essential to keep the web in the process at a required preset tension,
which could change throughout the process by many conditions. Poor tension control leads
to dishing, coning, and telescoping of the rewind roll, which are undesirable.
A continuous web processing line is a largescale complex interconnected dynamic
system with numerous control zones to transport the web while processing it. A continuous
web processing line typically consists of an entry section, a process section, and an exit
section. The entry section consists of an unwind stand, a tension leveller, and an entry
accumulator. Operations such as wash, coat, and quench on the web are performed in
1
the process section in the case of aluminum and steel webs, and printing perforating, and
laminating in the case of other consumer products. The exit ection con ist of an e'xi:,
accumulator and a rewind stand. Accumulators are primarily used to allow for rewind
or unwind core change while the process continues at a constant velocity. Dynamic of
the accumulator directly affects the behavior of web tension in the entire proc~ line.
Tension disturbance propagation both upstream and downstr16UD of the accumulator ha
been noticed due to motion of the accumulator carriage.
1.2 Literature Review
"
In [1], a mathematical model for longitudinal dynamics ofa web span between two pairs of
pinch rolls based on Hooke's law, which are driven by two motors, is given; this model does
not predict tension transfer and does not c~nsider tension in the ~ntering span. A modified
I model that considers tension in the entering span was developed in [3]. The moving web
is assumed to be equivalent to a moving continuum in [4] and the general methods of
continuum mechanics such as conservation of mass and conservation of momentum were
used in the development of a model. A large web process line consists of many rollers for
transporting the web, that are driven by electrical motors CDCIAC), while various different
operations being performed on the web. The entire system is interconnected through the
web. In [5], the authors proposed a decentralized method to decouple an interconnected
system. The decentralized decoupling is able to successfully separate the coupling factors
between the subsystems, which consists of the roller and the span upstream of the roller.
In this study the only subsystem considered is an accumulator. Although sufficient amount
of work has been done [1,2, 3, 4, 6, 7, 8] and is currently being done in tension control of
a web but very little published research exists in modelling and control of accumulators in
web processing lines. An overview of the lateral and longitudinal behavior and control of
moving webs was presented in [6]. A review of the problems in tension control of webs
can be found in [7]. Discussions on tension control versus strain control and torque control
2
versus velocity control were given in [8]. n.
Accumulators in web processing lines constitute an important element in all of the web
I
handling machines. Functional importance of these in web processing lines is quite ubs13ntial
as they are primarily responsible for continuous operation ofweb processing lines.
A preliminarY study on modelling and. control of accumula ors is given in 1[9]; characteristics
of an accumulator and its bperation are explained; throughout the study, discussions
are carried out to gain more insight into the dynamic behavior of the accumulator carriage.
web spans, and the current methods used in controlling the carriage. A dynamic model for
accumulator spans that consider the ~evarying nature of the span length was d~vel9ped
in [10]. '. t ,
lQ this study, control of the acew:nulator carriage in conjunction with control of the
driven roller~ both upstream and downstream ofthe accUIJlulator is. considered. '(he average
dynamic model de.veloped in [9] is further simplified base~ on practical observations and
is used fer controller design. The design of the control algorithm is carried out based
on Lyapunov's second method. An observer for estimating the average web tension in
accumulator web spans is also developed in the process. Simulation results on an industrial
continuous web process line for a typical operation of an accumulator are conducted using
the proposed controller and observer; these results are compared with the simulation results
of the current control techniques that are used in the industry.
1.3 Thesis Contributions
The following are the thesis contributions.
• Development of average dynamic model for tension in the web spans in the accumulator
carriage, processside driven roller and exitside driven roller.
• Analysis ofaccumulator dynamics along with processside and exitside driven roller
dynamics and a systematic study to analyze tension variation problem in accumula
3
tors considering different factors (friction, carriage sway etc.) and disturbances.
• Design and investigation of a controller for the accumulator system and it comparison
with a currently used industrial controller,
• Development of necessary simulation software to study the performance of an accuI
mulator system and tension variations in the web spans.
11
1.4 Thesis Outline
! ,. J t) M r
The rest of the report is organized as follows. Chapte 2 discusses the modeling of the
accumulator systein consisting ofweb dynamics~ roller dynamics and hydraulic 'system dynamics.
Chapter 3 describes the methodology to analyze and perform different calculations
for accumulator system. CHapte~ 4 explains 'the' process of control scheme design for the
system in investigation and explains the comparison oftbis proposed controller with industrial
control scheme using simulations. Chapter 5 gives conclusions of this research and
gives directions for future research.
'.
4
I
t·
I. '
5
11 t
CHAPTER 2
u
DYNAMIC MODELLING
A line sketch of a typical continuous strip pJ;ocess line layout is given in Figure 2.1. It conI
sists of an entry section that unwinds unprocessed s,trip, an entry accumulator that releases
web into the process section when the entry section (unwind roller) is stopped, a process
•
section where strip processing is J?erfonned, an exit accumulatqr that stores web when the
exit section (rewind roller) is stopped for a rev:ind changeover, and an exit section that
winds the processed web into rolls. Bridles shown in the figure are driven rollers and are
____~~ exit Sectlon
ExK Accumulalor t
!: Process Sectton
Entry Accumulator :...;
Entry Section
Figure 2.1: Typical process line layout and tenninology.
driven by either AC or DC drives. Bridle rollers facilitates the transportation of the web in
the web processing line. Both accumulator carriages are controlled by hydraulic systems,
( recently AC motors are used for this purpose) that provide regulation of tension in the
strip when the carriage is in motion. This study assumes that a hydraulic system provides
control action. Tills chapter is organized as follows. Section 2.1 presents the dynamics of
carriage and web span. Section 2.2 gives the governing equations of the hydraulic system.
5
Section 2.3 explains equations for entry and exit rollers. Section 2.4 describes the pIoces
of incorporating carriage sway dynamics in accumulator system dynamics and Section 2.5
introduces the accumulator dynamics in an average form.
2.1 Strip Tension a.nd Carriage Dynamics of Web lD Accumulator Spans
In accumulators, length of each web span varies with the motion of the carriage as shown
in Figure 2.2. Dynamics of the fixed length span are taken and the length of the span is
made timevarying according to the carriage motion to incorporate variable length of th.e
spans in the accumulator dynamics. The dynamics of this variable length is given by the
accumulator carriage dynamics by taking the summation of all the forces acting on the
carriage. The accumulator carriage dynamics is given by:
; I l
(2.1)
(2.2)
The controlled force, Fh(t), is generated by a hydraulic system. Figure 2.5 shows a
schematic ofthe hydraulic system. The disturbance force, Fd(t), includes the friction in the
hydraulic cylinder and the rod seals, friction in the carriage guides and the force due to carriage
chain elasticity. The torque shaft shown in Figure 2.2 is included in the accumulator
design to synchronize the side to side lifting action so that only vertical motion needs to be
considered in the control system design. The number of rollers on. the carriage is N/2. The
number of rollers in. the accumulator is N + 1. To derive the tension dynamics, consider
the sketch of a single roller and two web spans as shown in Figure 2.3. From [10], the law
of conservation of mass for a control volume in the first span of Figure 2.3 gives: : [lX2
(t) P(X,t)A(X,t)dX] = Pl(t)A1(t)Vl(t)  P2(t)A2(t)V2(t),
t Xl(t)
where Xl and X2 denote the coordinates of roller 1 and roller 2, respectively, from a fixed
reference frame. Notice that for the accumulator case, roller 1 is fixed (Xl (t) = 0) and
roller 2 moves along with the carriage (X2(t) = xc(t», where xc(t) denotes the variable
length of the span.
6
Tor ue Shaft
tFd
, ~~aE~ ~ _
I
I
I,
._._._._,_._._._.._. . ._._..
I
I
I
I
________________________________ J
fIXed
V e • vp "
Strip to Rewind Strip from Process
Figure 2.2: Sketch of an exit accumulator.
If we consider an infinitesimal element of the strip in the direction of web travel, the
geometric relations between unstretched and stretched el~~ent are given by:
(2.3)
(2.4)
(2.5)
where subscript u indicates the unstretched state of the element, wand h denote the width
and thickness of the web, respectively. The elemental mass, dm, in the unstretched and
stretched state is equal, and hence:
(2.6)
Combining equations (2.3)(2.6), we obtain
p(x, t)A(x, t)
Pu(x, t)Au(x, t)
1
(2.7)
7
r ( 1)
.I. ,_ _C_ .a.r_r_i I ~J~~. _.. __ ..,
~ ~_.__....
2 :,,,
 ._._ ._.__ ..... ~
xc
Figure 2.3: Sketch of an accumulator span.
(2.8)
Substituting (2.7) into (2.2), we obtain
~ [lX2(t~ Pu(x, t)A(x, t) dX] = Plu(X, t), A1u(x, t)VI(t)
dt XI(t) l+tx(x,t) 1 + ExI(X,t)
P2u(X, t), A2u(x, t)V2(t)
1 + ex2(X, t)
Assuming the density (p) and the modulus of elasticity (E) of the web in the unstretched
state are constant over the crosssection, (2.8) can be written as:
~ [lX2
(t) 1 dX] _ Vl(t) _ V2(t) (2.9)
dt xI(t) 1 + Ex(X, t)  1 + Exl (x, t) 1 + Ex 2(X, t)'
Assuming that the strain is very small, ex «< 1, we can neglect higher order terms and
(2.10)
write 1/(1 + ex) ~ (1  Ex). Then, (2.9) can be written as:
d [lX2
d (t) (lex(x,t))dx] =vI(t)[1 exl(x,t)]V2(t)[l ex2(X,t)].
t XI (t)
Assuming that the strain does not vary with x, i.e. ex(X, t) ~ cx(t), the lefthandside of
(2.10) can be written as:
d [lX2
(t) ] [lX2
d (1  ex(t))dx = (t) dx] dd (1  ex(t))
t Xl(t) XI(t) t
d [lX2
+ (1  Ex(t)) d (t) dx] .
t Xl (t)
(2.11)
8
Notice that the second tenn in the righthandside 0 (2.11) is the differentiation of an
integral with variable limits of integration. Hence, the integral can be differentiated using
Leibnitz rulel of differentiating an integral. For simplicity, taking the accumulator c e
given]:)y Figure 2.3, i.e., Xl (t) = 0: and X2(t) = xc(t); and applying Leibnitz ruteior (tu 1)
gives:
d [l;1:C(t) ] [lXC(t)] d
dt 0 ~~  cx(t))dx.. 0 dx dt (1  c~(t))
d [lXC
(t) ]
.' I. + (1  cx(t)) dt 0 'd:JJ. (2.12)
Substituting (2.12) into (2.10) and using Hooke's law, i.e., t2(t) = AEfx(t), gives the strip
. I I
tension in the j th accumulator span as: , . . "
Also the dynamics ofjth roller is given by
2.1.1 Speed Changes During Rewind Roll Change
(2.14)
A continuous aluminum strip processing line consists ofan entry section, a process section,
and an exit section. The entry section consists of an unwind stand and an entry accumulator.
The process section consists of several stations where washing, coating, and drying
operations are perfonned on the strip. The exit section consists of an exit accumulator and
a rewind stand. The process speed and exit speed are relative to the exit accumulator as
shown in Figure 2.2.
I Leibnitz rule
d
dt [ rt/J(t) f(x, t)dX] = r"'(t} af~, t) dx  ~~ f(¢(t), t) + ~~ f(1/J(t), t)
J"'(t} J""t)
9
A typical scenario of the exit speed and the carriage speed during a rewind roll change
is depicted in Figure 2.4. The following steps describe a rewind roll changeover scenario
when the strip velocity in the process section is maintained at a constant value: (i) AB 
velocity of the strip in the rewind side is decelerated to zero, as a result ofthis the accumulator
starts collecting strip and the carriage accelerates upwards; (li) Be  rewind stops and
the carriage i~ moving up with constant velocity; (iii) Cp  after rewind roll change, exit
side is accelerated up to the process speed, in this period the carriage is moving up while
decelerating; (iv) DE  exi.t side is accelerated up to a speed above the process speed; (v)
EF  exit speed is maintained at a constant speed; (vi) fG  exit speed is reduced to the PTO
" l
cess speed. The carriage after this cyqle returns to its original position. Notice that carriage
returns to its original position when the area under the carriage speed graph of Figure 2.4 is
zero. Since there are N spans in the accumulator, carriage velocity can be obtained based
>r.__\:A
1l v
Po.
CIl ....
";<
v
B c
D
Er ,.F
G
time
u >
g
v
Po.
CIl
(1)
bIl "'E"
u'"
A
B c
D
E F
G
time
Figure 2.4: Exit and carriage speed profiles during rollchange.
on the process speed and the exit speed, under the assumption that there is no web slippage
on the rollers. The relationship between the process speed, vp, the exit speed, Ve , and the
10
carriage speed, 110 , in an ideal situation is given by the following equation
V p  Ve
Vc = tV' '
vp  JVc
Wj = R
(2.15)
(2.16)
It is assumed that the process speed is the speed ofthe strip in the span just upstream ofthe
exit accumulator, as shown in Figure 2.2, and the exit speed is the speed of the strip in the
span just downstream of the accumulator. The carriage speed as given by equation'(2.15),
depends on the speed variation between the process section and the exit section. Generally,
the process speed is' kept constant and the exit speed varies according to the sketch shown
in Figure 2.4. When the strip is stationary at the exit side of the accumulator, Ve = 0, then
equations (2.15) and (2.16) become:
Vc Wj = R (N  J") .
(2.17)
(2.18)
Using this notion of speed changes at the exit side, some observations can be drawn from
the dynamics of the carriage given by equation (2.1). Assume that the external disturbance
force, Fd(t), is zero. If the web is stationary at the exit side then the carriage is moving up
with a constant velocity Vc thus, carriage acceleration ~ is zero, then (2.1) becomes:
N
FhO(t) = NIcg + L: tj(t).
j=1
Further, it can be seen from (2.18) that the angular acceleration ~ of each roller is zero.
Thus equation (2.14) gives tj = tj +1 for all j '= 1 : N. This is true under the assumption
that the roller bearing friction, Bj , is very small and can be ignored (Bj ~ 0). Thus, when
the exit side is stationary, the tension in all the spans is equal and the controlled force is
required to overcome the weight ofthe carriage, sum of the tensions in the spans, and other
force disturbances. Now consider the case of constant acceleration or deceleration of the
11
exit side and constant process speed. The carriage dynamics (2.1), can be rewritten a :
(2.19)
where Fhd(t) is the control force required when the exit side is accelerating or decelerating.
It should be observed that during deceleration of the exit side, the carriage is accelerating
up, i.e., dvc/dt is positive. Hence, the controlled force, Fh(t), in this case should be larger
than the controlled force in the case of stationary exit side. However, in the event of acceleration
of the exit side, dvc/dt is negative, and hence the controlled force Fh(t) in this case
is smaller than the one required for the case of stationary exit side. To sum up,
F/C;CC(t) > FhO(t)
Ftec(t) <: FhO(t)
..
Therefore, variations in the hydraulic force are required to account for acceleration/deceleration
ofthe carriage, as well as for compensating the friction loss between the ram and the cylinder.
The friction between the ram and the cylinder during carriage motion plays a major
role in the strip tension change through the accumulator. Friction for up and down motion
ofthe ram is different. Seal design and rod lubrication are big factors affecting the friction
coefficients.
2.2 Hydraulic System
A schematic of the accumulator hydraulic system and the carriage is shown in Figure 2.5.
It consists of two symmetrically placed cylinders, a directional proportional valve, and
a pressure compensated pump. The pressure in each cylinder is regulated by the same
directional proportional valve. Even though the pipe length from the proportional valve
(point A) to the entry point of the cylinder (point Pc) is generally large, pressure drop due
to friction in the pipe length and bends is comparatively small, as discussed in [9]. The
12
Proportional Directional VIV
(Rexroth 4WRD 16E)
1.5" sch80
x
Carriage
63GPM
1200 PSI
75 HP
ReliefVN
1350 PSI
Figure 2.5: Sketch of carriage and accumulator hydraulic system.
cylinder pressure dynamics for the setup shown in Figure 2.5, in a simplified way, is given
by:
Vjaj dP;e([t)t = Qe(t)  Cm(Pe(t )  Pr)  Acy/x.r(t ), (2.20)
where Qe(t) is the flow rate to and from the cylinder, which is related to the proportional
spool valve displacement, Xv, by:
Qe(t) = KqxvVPtJ  Pe(t) for Xv > 0,
Qe(t) = KqxvVPe(t)  Pr for Xv < 0.
(2.21)
(2.22)
The total cylinder control volume is Vo(t) = Yin + Acy/xr(t), where \lin is the initial clearance
volume of the cylinder when the ram is fully extended, i.e., when Xr = O. Due to the
13
pulley system, as shown in Figure 2.5, the ram displacement is exactly half of the carriage
displacement, i.e., xr(t) = O.5xc (t). The position ofthe ram, xr(t), is positive downwards
as shown in Figure 2.5, whereas the position of the carriage, xc(t), is positive upwards.
2.3 Full Model Dynamics Including Exit and Entry Roller
Instead of taking a defined function for Ve and vp, exit and entry roller dynamics is also
considered in the system. The dynamics of the complete system is given by:
• Dynamics of exitside driven roller:
where Ke is motor constant and U e is torque input to the exitside driven roller motor.
• Span dynamics ofjth spa~ in the accumulator:
• Accumulator carriage dynamics:
Xc = vc(t),
1 N
Vc = M (2::= tj(t) + AcylPc  Fd(t))  9,
C j=l
• Hydraulic system dynamics:
I '
(2,25)
(2,26)
• Dynamics of processside driven roller:
Vp = ~ ( Bjpvp(t) + R;(tl (t)  tr ) + RpKpup) (2.28)
p
where Kp is motor constant and Up is torque input to the processside driven roller
motor.
14
Strip 10 rewind
to
L.:l
N·2
Strip fronl pnlCCSI t,
Figure 2.6: Exit accumulator with processside and exitside driven roller.
Accumulator can be represented as shown in Figure 2.6. The system has three control
inputs, i.e., two motor inputs to the exit and entry rollers and the third is spool valve displacement
to the hydraulic system.
2.4 System with Carriage Sway Dynamics
Even though it is required that the carriage slots and the guide ways mate properly to give
smooth motion, there will invariably be sway and cocking reaction due to inherent limitations
on manufacturing processes. The manufacturing tolerances, which are necessary
to compensate for manufacturing process capabilities will generate clearance in the guide
ways. In general, this sway will be in both planes; plane parallel to the travel of the web
and plane perpendicular to the travel of the web and will affect tension in the web spans.
The effect of carriage sway in the vertical plane is incorporated in the following analysis.
For simplicity even number of spans are considered in this analysis. The web strand length
15

o
..,..... '
n
.
r
Vp
Strip from process
;(
   
Ve
Strip to rewind
\'
Figure 2.7: Illustration of carriage during sway.
(b) due to carriage rotation is given by:
o
for J" = t{  3 t{  4
2 ' 2
for J" = N  1 t{  2 2 , 2
fior J" = 2N ' 2N + 1
liJ for j = If + 2, ~ + 3
2ldO for j = ~ + 4, ~ + 5
16

Thus, (xc +OJ) will replace Xc and Xc+8j
win replace Xc in all the equations given by (2.23)
through (2.28). Incorporating carriage sway, the complete system equations are given oy:
Ve = Je
(Bfeve(t) + R~(tr  tN(t)) + ReKeue ) , (2.29)
dtj(t) AER R
d = () [) (Wj(t)  Wjl(t)) + () 0 [tjl(t)Wj_l(t)  tj (t)Wj (t)]
t Xr t + j Xc t + j
AE . 1 .
+ (Xc(t) + OJ) (Xc(t) +OJ)  (Xc(t) + OJ) tj(t)(Xc(t) +6j ), (2.30)
Xc = Vc(t) + 8j ,
1 N
Vc = M (L tj(t) + Acy/Pc  Fd(t))  9,
c j=l
V13o~dPc(t) = Qc (t )  Cm(P(c,t)  Pr)  AcylX.r(t ),
Vp = ~ (BfIVp(t) + R~(tl(t)  tr) + RpKpup).
p
• I
\ .
(2.31)
(2.32)
(2.33)
(2.34)
There will be one more equation for carriage dynamics due to rotational moment. This
equation can be written in ~olIowing form
where
(2.35)
(tj + koj )(2ld + r)
(tj + kOj )(2ld  r)
(tj + kOj)(ld + r)
(tj + kOj)(ld  r)
(tj + koj )(r)
(tj + k15j )(r)
(tj +koj)(ld  r)
(tj + koj)(ld + r)
(tj + koj )(2ld  r)
17
for j = If  4
for j = If  3
for j = If  2
for j = ~ 1
for j = If
for j = If + 1
for j = If + 2
for j = ~ + 3
for j = If + 4

and AcylPe = Fr +Fp ; Fr and Fp are forces acting on the two ends of the carriage. The k8j
term is there to account for elastic force due to web span. The effect ofllie carriage rotation
will be a important factor in designing a controller to control tension in accumulator web
spans.
2.5 Average Dynamic Model
Analysis and controller design for large scale nonlinearly interconnected systems is challenging
task even with the help of design/analysis techniques available at present. A comprehensive
study is required to implement these controllers. The web handling process is
a large scale multiinput multioutput system. Analysis of the models discussed in Sections
2.1 or Section 2.3 requires knowledge of all the states for each and every web span
in the accumulator. The motivation for using average dynamics is that practically it is not
feasible to control tension variations in each and every span as variati,on in one span affects
tension in next span and vice versa. Also, installing load cells on all the rollers is not a
feasible approach. The motivation for averaging the tension dynamics of N web spans in
the accumulator is to obtain a simpler dynamic model for all the accumulator spans in an
average sense. With aggregation of dynamics, the system is equivalent to a dynamic system
with one roller on the carriage, as shown in Figure 2.8. Therefore, a simpler dynamic
model for all the accumulator spans in an average sense can be obtained and implementation
of control scheme requires, perhaps only two load cells at each end of the accumulator,
i.e., first (oth) and last roller (Nth) of the accumulator. Consider a new variable te, which
denotes an average sum of the tensions in the accumulator web spans and is given by:
1 N
te(t) = N L tj(t).
j=l
18
(2.36)

Strip 10 rewind
tc tc
Strip from proces
~ ~
Ve
N 0
Figure 2.8: Equivalent of exit accumulator for average model, i.e., with one span.
Taking the sum from j = 1 to j = N of both sides of equation (2.13) and dividing by N
results in
Evaluating the sum on the righthandside results in
dtc(t) AER 1 1 R
~ = xc(t) N (WN(t)  wo(t)) + N xc(t) (to(t)wo(t)  tN(t)WN(t))
, v ' , v '
term) term2
AE 1 + ()Xc(t)  ()tc(t)Xc(t).
Xc t Xc t
~' V' '
term3 term4
(2.38)
Using this approach, the average dynamics in terms of tc(t) simplifies the dynamics of N
spans given by N equations of (2.13). Notice that the average dynamics, (2.38), simply
19

looks like the strip tension dynamics of an aocumulator with a single miler on its carriage,
which is illustrated in Figure 2.8. Also, notice that the last summation <in the carnage
dynamics in equation (2.1), can be replaced with Ntc(t). We can further simplify the
average tension dynamics given by equation (2.38), under the assumption that:the ratio of
the desired web tension to AE is very small; this is generally true for all web materials
processed in web processing machines. Under this assumption the tenn 2 and term 4 can
be ignored from the average tension dynamic model given by equation (2.38), resulting in
the following simplified average tension I;lynamic model:
dtc(t) AER 1 AE .
d = () N (WN(t)  wo(t)) + ()xc(t).
t Xc: t Xc t
(2.39)
In the average dynamics given by equation (2.39), the angular velocities ofthe Oth and nth
stationary rollers of the accumulator appear explicitly. Assuming that there is no slip on
the entry and exit rollers of the accumulator, the process velocity vp and the exit velocity Ve
are given by vp(t) = Rwo(t) and ve(t) = RwN(t). Therefore, the average tension dynamics
given by equation (2.39) becomes
dtc(t) AE 1 AE .
d:t = xc(t) N (ve(t)  vp(t)) + xc(t) xc(t). (2.40)
Notice that the average tension dynamics (2.40) clearly reflects variations in the process and
exit velocities on the average tension in the accumulator spans. The driven roller angular
dynamics at the process side and the exit side of the accumulator are given by the following
equations:
JeWe(t) = Bfewe(t) + Re(te(t)  te(t)) + Keue(t),
Jpwp(t) = Bfpwp(t) + Rp(tc(t)  tp(t)) + Kpup(t) ,
(2.41)
(2.42)
where We (t) and wp (t) are the exit side and process side driven roller angular velocities,
respectively, Bfe and B fp are the viscous friction coefficients in the exit side roller and
process side roller, respectively, te(t) and tp(t) are the web tension in the span downstream
of the exitside roller and in the span upstream of the processside roller, K e and Kp are
20
positive gains, and 'Ue (t) and Up(t) are exit side and process side driven roUer control inputs.
Notice that teet) and tp(t) are given by the tension dynamics downstream of the exitside
driven roller and upstream of the processside driven roller; since in this work we are
interested in the web tension behavior in the accumulator region, we assume that teet)
and tp(t) are maintained close to the desired web tension, that is, teet) = tT + oe(t) and
tp(t) = tT + op(t), where tr is the desired web tension in the process line, and oe(t) and
op(t) are disturbances. Assuming that there is sufficient web wrap on the driven rollers
which will ensure that there is no web slip, we have ve(t) = Rewe(t) and vp(t) = Rpwp(t),
the equations for exitside and processside web velocities are:
Ve(t) = Je (Bjeve(t) + R~(tr  teet)) + ReKeue(t) + R~oe(t)) ,
vp(t) = ~ (Bjpvp(t) + R;(te(t)  tT ) + RpKpup(t)  R;8p(t)) .
p
The carriage dynamics, (2.1), can be rewritten as:
(2.43)
(2.44)
(2.45)
Together, equations (2.40) and (2.45) characterize the dynamics of the accumulator carriage
and its spans in an average sense. This dynamic model is suitable for design of the
controller force, Fh (t), to provide a desired regulation of the average tension in the accumulator
spans. The control force Fh(t) is generated by a hydraulic system. Considering the
pressure (Pe(t» in each cylinder to be the same, the force on the carriage is obtained from
21
Using this expression for controlled force, equations (2.40), (2.45) and (2.20) can be combined
into the state space form by choosing the state variables and input as follows:
6(t) = teet)
6(t) = xe(t),
6(t) = ve(t),
~4(t) = Pe(t),
~5(t) = Veet)',
~6(t) = Vp(t),
u=xv ·
...
Then the state space form of the complete system dynamics in a simplified form is
· AE (1 ) ~l (t) = 6(t) 6(t) + N (~5(t)  ~6(t)) ,
~2(t) = ~3(t),
· 1
6(t) = Me (N~l (t)  Ff (6(t)) + ACYl~4(t))  g,
· A l
~4(t) = a(6)( :; 6  Cm(~4  Pr)) + h(~4, u)u,
~·5(t) = J1(2 2 )
e
Bfe~5(t) + Re(tr  6(t)) + ReKeue(t) + Reoe(t) ,
~·6(t) = Y1 (Bfp~6(t) + Rp2(~l (t)  t 2 ) r ) + RpKpup(t)  Rpop(t) ,
p
where
f3
a(6) = (Vin + Acyl6) ,
h(~4' u) = ~q (1 + sgn(u))JPa  ~4 + ~q (1  sgn(U))J~4  Pr ,
22
(2.46)
(2.47)
(2.48)
(2.49)
(2.50)
(2.51 )

and Vin is the initial volume of the cylinder when the ram is fully extended. The function
sgn is defined as:
+1 ifu> 0
sgn(u) = 0 ifu = 0
1 ifu < 0
2.6 Synopsis
In this chapter the dynamics of a web span, carriage and driven roller are developed. The
effect of carriage sway is also considered and a mathematical equation is developeq for the
same. To simplify the dynamics averaging is perfonned. This transfers a system with N
web spans into a system with one web span. This average model is used for further analysis
in the following ch"lpters.
23

CHAPTER 3
DYNAMIC BEHAVIOR OF SYSTEM
This chapter concentrates on the analysis ofdynamic behavior ofaccumulator system under
different approaches. Such systematic study of accumulator system is of utmost use in
designing a controller. The remainder of the chapter is organized as follows. Section 3.1
describes the process of calculating initial states of system parameters. This section also
explains the fact that with movement of the carriage, the weight of web spans that is being
acted on the carriage will be in significant proportion to the weight of the carriage for
certain web materials. Section 3.2 explains the open loop behavior of tension variations
in the system. Section 3.3 analyze the carriage dynamics along with dynamics of driven
rollers on exit and processside to establish groundwork for controller design.
3.1 System Parameter Calculations
3.1.1 Required Reference Pressure to Maintain Carriage Position and Reference
Tension in Web Spans
This section explains the calculation for the reference pressure to maintain carriage position
and reference tension in web spans throughout the operating range of the system, irrespective
of the case whether the carriage is moving or not. The calculations are performed for
an ALCOA continuous processing line (CPL). The following numerical values are used in
24

the calculations.
Reference tension to be maintained:
Number of spans:
Rod side cylinder an~a:
Weight of the carriage:
Accumulator carriage dynamics is given by:
tref = 1165 lbs.
N =34.
We = 16134lbs.
When carriage is not in motion
So
= 779.96 psi.
This is the reference pressure the hydraulic system has to maintain irrespective of the motion
of the carriage. To cqmpensate for acceleration or deceleration of carriage, pressure
boost as discussed in [9] is required.
3.1.2 Calculation of Spool Valve Position for any Forced Equilibrium Point
This section explains the calculations perfonned to calculate the position of spool valve
to maintain reference pressure calculated in previous section. The hydraulic dynamics 1S
given by (2.49). Calculations are perfonned for an ALCOA continuous process line (CPL),
25

for which hydraulic parameters are:
(3 = 100000 psi, em = 0.01,
Pr = 200 psi, Ps = 1200 psi,
Vin = ain2
, Kq = 0.7.
The lowest point of the carriage is taken as the forced equilibrium point for calculation. At
this point
6 = 71 in,
~4 = apsi.
In this scenario the spool valve will always be moving in the positive direction. The value
of functions a([2) and h([4, u) at a given forced equilibrium point are:
h([4, u) = 14.346.
From equation (2.49)
which is the reference value of spool valve position u and is 7.9 in. This is the reference
value of spool valve position to maintain reference pressure which in tum will compensate
for weight of the carriage and reference tension in all the accumulator web spans.
3.1.3 Calculations for Weight Change of Web Spans during Carriage Motion
This section explains the fact that, when carriage is in motion, the length of each web span
in the accumulator changes and thus the weight of the web material acting on the carriage
changes. The hydraulic system has to compensate for this changing weight of the web
spans. If the web thickness is quite sign ificant, in the range of 1/8th to 114th of an inch,
the weight of the web material is comparable to that ofthe carriage as shown in Figure 3.2.
Figure 3.1 shows the weight change for the aluminium strip ofthickness 0.01 inches, which
is the case used for the simulation study in the next chapter.
26

700r,,,,,.,.,
600
500
,e
1:
Cl
~ 400
..c
~
300
200
Web thickness = 0.01 inches
Web width" =='SO"jrlcities .
100 ' .1... .1... .1... .1... 1. .1... 1. _
o 50 100 150 200 250 300 350 400
time sec
Figure 3.1: Change in web weight during carriage motion for 0.01 inches thick aluminium
web.
27
9000 r,,,,,,......,
8000
7000
6000
:f1
~
0>
~ 5000
LJ
Q)
~
4000
3000
2000 .
Web thickness::'118 inch
Web wldth= 50 inches
200 250 300 350 400
time sec
50 100 150
1000 L L.__JL.__...l ...l .l .l L J
o
Figure 3.2: Change in web weight during carnage motion for 0.125 inches thick aluminium
web.
28
3.2 Study of Dynamic Behavior of System
This section considers the study of the dynamic behavior of the system using some simplifying
approximations. To simplify the analysis, initially the hydraulic dynamics is not
considered in the simulation process. Also the effect of viscous and coulomb friction is
ignored. It is assumed that there is no pressure dynamics, therefore hydraulic force is considered
as input to the system. Equations in the following form are used:
. AE6 AE 1
~J = ~2 6 + z; N (ve(t)  vp(t)),
6 =~3,
. 1
6 = Me (N6 + uc)  g.
The input Uc is chosen to consist of feed forward terms only, i.e.,
(3.1)
(3.2)
(3.3)
The results obtained from the simulation of this simplified system are shown in Figure 3.3.
A defined pattern of exit velocity as shown in Figure 2.4 is used to govern the motion ofthe
carriage. The carriage movement is confined, with a lower limit of 71 inches and a upper
limit of 400 inches. After each cycle the carriage will return back to its original position if
the area under the carriage velocity curve is zero. Large variations in the web tension can
be noticed and the reference tension value returns to 1165 lb after every cycle and stays
there if the carriage is not in motion.
In the second step, friction term is added in (2.48) and simulations are performed. The
variations in average tension are higher than the previous case as shown in Figure 3.4.
In the third step, a small sinusoidal disturbance is added in equation (2.48) and simulation
is performed. Figure 3.5 shows that in the absence of feedback control, this sinusoidal
disturbance will generate large variations in the tension.
In the fourth step, a compensator (a PI controller) is incorporated by using the error
signal, because of the difference in reference position and the actual position, as feedback.
29
.. , .. ... ..
50 100 150 200 250 300 350 400
1170 r,,r,,,...,...~
£ 1165
c:: o
'<ij
c::
{!!. 1160
40 r,,.,.,....~
4::'
<Ii 30
8.
~20
co
'EJ 101. __
50 100 150 200 250 300 350 400
OL__.....I..__I. ...L__L .L__.L__l__J
o
200 250 300 350 400
time sec
50 100 150
c::
~
10
4i
>
<Il
~ 0
'E
<0 u 10 L __.L__l__~~==::::::t:===:t::====t::=!...__I _l
o
20r,====:;.,r,,,
Figure 3.3: Uncompensated system states ignoring friction tenn.
The results from simulation is shown in Figure 3.6.
This study shows that variations in the hydraulic force are required to account for acceleration/
deacceleration ofthe carriage, as well as for compensating the friction loss between
the ram and the cylinder, The model described in the previous chapter is highly nonlinear.
Besides the friction, the nonlinearities present in the hydraulic system make controller design
a challenging task. A detailed analysis of the system is required to design a controller
for the system.
3.2.1 Reachability and Observability Analysis for the Accumulator System
The system given by (2.46) through (2.51) is a nonlinear system. Before designing a contro
ler one must first analyze the question ofreachability and observability. For this analysis
30
1200
1180 .......... ...........
:B
c:
0 1160 ....:............:........ "in '" ...... : ...........
c:
Q) ..... 1140 ......... .
1120
0 50 100 150 200 250 300 350 400
40
;::
eli 30 ........ :.
8.
~20
III 'E
III 10 .
()
0
0 50 100 150 200 250 300 350 400
20
c
'E
~ 10 ,. ,....
'ai
>
III
01 0 ... III
'E
ttl
0
10
0 50 100 150 200 250 300 350 400
time sec
Figure 3.4: Uncompensated system states incorporating friction term.
dynamics of 6 ,~2 and 6 are considered. Among these three states the last two states can
be measured and thus are available for feedback. These three state equations can be written
in the following form as those are linear in the input, i.e.,
( = /(0 + g(~)u, (3.4)
where f(~) and geE) are smooth vector functions of ~(t) and are given by:
(A~~{1 )(~3 + 1/(ve (t)  vp(t»)
f(~) = 6
_1 (N~1  F/(6)  9 Me
o
g(~) = 0
1
Me
(3.5)
Also the output equation can be written as:
(3.6)
31
1400
~ 1200
0
Vi c~
1000
800
0 50 100 150 200 250 300 350 400
40
;:::
<Ii 30 ...........
8.
~20
til
'E
~ 10 . ,," ...... o.
0
0 50 100 150 200 250 300 350 400
20
c
'E
¢!
10 _... , ..... . ......... .  ....... ... .... ..........
Q)
>
Q)
r0a1 0 ., .....   '. . ...... . ...
'E
ra
()
10
0 50 100 150 200 250 300 350 400
time sec
Figure 3.5: Uncompensated system states incorporating friction teml and sinusoidal disturbance.
A system given by (3.4), is said to be locally reachable around a state Xo E X ifthere exists
a neighborhood U ofXo such that for each xfEU, 3 T > aand 3 u(t), t E [0, T], such that
if the system starts in Xo at timet = 0, then it reaches the state x f at time t = T. Define
where adjg is the ith Lie bracket. The nonlinear system given by (3.4) is locally reachable
if ( has 3 linearly independent columns. ( is evaluated as:
__1 AE{l __1 AE:z{l (C + VeVp ) + _1 AE{l (~ _ !L)
Me 6 Me ~2 <,,3 N Me ~2 ~2 Me a
(= a 1
Me
__1 !L
MeMo
1
Me
_1 !L
Me Me
32
1167
f! 1166 . . . . . . . . . .. '..
c: rJ . I
.~ 1165 _ ... 
c: 1: Q)
I 1164 ....... '.' ..... ","
1163 ;
0 50 100 150 200 250 300 350 400
40
¢::
<Ii 30 ...... ...
8.
~20 . . . . . . . . .......... ... .
111
'E t3 10 ., ................  ........
0
0 50 100 150 200 250 300 350 400
20
c:
'E
ii:! 10 ..
(jj
>
Q)
CD 111 0
'E
111 u
10
0 50 100 150 200 250 300 350 400
time sec
Figure 3.6: Compensated system with friction tenn and PI controller.
The column rank of this matrix is three. Therefore the system is locally reachable. After
this the local observability of the system is checked. The system given by (3.4) and (3.6) is
said to be locally observable at Xo E X if there exists a deleted neighborhood N of Xo such
that each x E N is distinguishable from xo. The system is locally observable if it is locally
observable at each Xo EX. IfG denotes the set of all finite linear combinations of the Lie
derivatives of hI and h2 with respect to f and dG denotes the set ofall their gradients, then
the system is said to be locally observable if one can find 3 linearly independent vectors
within dG. In the present case ofthe accumulator system, G and dG are given by:
G= 6 ~JN6  F/(6»  g
_1 (NCI _ F
f
(C3» _g _N(AE~d(~ + VeVp) _ :!.LCN~lF[  g)
Me ':. ':. 6Mc 3 N Me Me
33
and
0 1 0 0 0 1
dG= 0 0 1 N  Me 0 it
N 0 _!!L K(1..(6 + ve;/P) + !!L) ~e A~~{l (6 + v"i/P ) _KAE{l + (i/;)2  Me Me Me ~2 Me Me ~2 e
The rank of dG is three. Therefore the system is locally observable.
3.2.2 Feedback Linearization
In this section, the dynamics of the web spans and the carriage are considered for analysis.
Only the position and velocity of the carriage are assumed to be measured. Measurement
of tension in each accumulator span is practically not feasible. Among the states of the
system given by (2.46), (2.47), and (2.48), only position of the carriage and velocity of the
carriage are directly available as measured signals. Full knowledge of the mathematical
model of system can be used to estimate the unknown state. For this purpose the feedback
linearization is used. Displacement of the carriage is considered as output of the system.
The output equation can be written as:
(3.7)
The derivatives of the output equation are given by:
Hence, the system has relative degree of 2 in R3
• Now let
This gives the transformed, linearized system as:
ii = up'
34
(3.8)
(3.9)
Now let
AE6 (~. + tietip) {2 3 N a
f(O = 6 p(~) = a .. (3.10)
_1(N6  F/(6))  9 1
Me Mr.
Then
'l./Jl = h(~) = ~2,
8'I./Jl
'l./J2 = 8~ f(~) = ~3'
The third function required to complete the transformation should satisfy
8ep
8~P(O = 0,
which gives
a¢
86 =0.
The state ¢ is independent of ~3 and it is a function of ~l and 6 only. To complete the
transformation let
The associated state transformation is given by:
After transformation the final state space fonn is given by:
. AE
</> = AE1jJz + iV(ve(t)  vp(t»),
VJl = 'l./J2,
VJ2 = Up.
35
(3.11)
(3.12)
(3.13)
(3.14)
(3.15)
(3.16)
This is the transformed state space model of the system. Thi sy tem is equivalent to a
system with a unit mass and a controller can be designed for this system. The control input
'Up can be obtained from this analysis, which will further govern the hydraulic force profile
'lIe. In equation (3.16), the control input up is being multiplied by Me to amplify its effect.
With this exercise of state transformation, the tension control problem is converted into a
position and velocity tracking problem.
Observability and Controllability of Feedback Linearized System
The question ofcontrollability and observability for the transfonned system given by equations
(3.12) through (3.16) is briefly considered in this section. [n matrix fonn the system
can be written as:
if> 0 0 AE ¢ 0
'l/Jl 0 0 1 'l/Jl + 0 up, (3.17)
'l/J2 0 0 0 'l/J2 1
and output equation can be written as:
[:] [: :]
if>
1
 'l/Jl (3.18)
0
'l/J2
The controllability matrix is given by
0 AE 0
We = [B AB A2B] = 0 1 0 (3.19)
1 0 0
Rank of this matrix is 2. From this a conclusion can be made that the system is not controllable.
Similarly the observability matrix is given by
0 0 0 0 0 0
Wo = [G' A'G' A'2G']  1 0 1 0 1 0 (3.20)
0 1 0 0 0 0
36
Again the rank is 2. All the states of the system are not observable. The first state i.e
product of position and tension in the web spans, is not observable.
3.2.3 ControUability and Observability Analysis for Jacobi Linearized System
In this section Jacobi linearization is performed on the system considered in the previous
section. The linearized system will be in the form
ox = Aox + Bou,
y = C6x.
In this particular system there are forced equilibrium points. Linearization is performed
about the point defined by:
~1 = 0,
6 = tref = 1165,
6 = ~~q = 71,
6 = 6 = 0,
veq = veq e p = 130,
Matrices A, Band C are computed as:
A= o
N
 Me
o
o
1
0
B= 0 and C=
1
Me
37
The controllability matrix is given by:
wc = [B AB A2B]
0 _1 AE$l _.2.AE2~1(6 + tietip) _ ...!..~AE~l
Me ~2 Me ~2 N Me Me ~2
0 1 _...!..~
Me Me Me
1 __1 ~ _J:L_1 AE{l + (~)2...!..
Me Me Me Me Me ~2 Me Me
Rank of this matrix is 2. Therefore the system is not controllable. Similarly the Observability
matrix is given by:
c
a 1 0
0 0 1
0 0 1
N 0 ~  Me Me
N 0 ~
Me Me
J:L 1.. (6 + VeVp ) + ~..!i ..!i. AE2{1 (~, + tietip) _..!i. AE{l + (~)2
Me {2 N Me Me Me ~2 3 N Me 6 Me
The rank of this matrix is 3. Therefore the system is observable as concluded in previous
section.
3.3 System Dynamics Including Exit and Entry Roller Dynamics
In the previous two sections, only the dynamics of ~b 6 and 6 are considered and Ve , vp
are taken as known perturbations to the system. In this section dynamics of exit and entry
38
NI 0)
N N·2
Strip from proe:c:s.
t
Figure 3.7: Exit accumulator with processside and exitside driven roller.
roller to accumulator are also considered for analysis. The final system is given by:
Ve = ~e (Bfeve(t) + R;(tr  te(t)) + ReKeue),
. _ AE  te(t) ( () (Ve(t)  Vp(t)))
te  Xe(t) Ve t + N '
Xc = Ve(t),
. 1
Vc = Me (Nte(t) + Uc  Fd(t))  g,
Up = J. (BflVp(t) + R;(tc(t)  tr)+ RpKpup).
p
(3.21)
(3.22)
(3.23)
(3.24)
(3.25)
Accumulator can be represented as shown in Figure 3.7, i.e. a system with three control
inputs. An assumption has been made here that both entry and exit spans to accumulator
39
have tension tco With this assumption, in matrix form the system can be written as:
Ve ~ R2 J e e 0 0 0 Ve ReKe 0 0
tc
AEte 0 0 AEte AEte t c 0 0 0 Ue NZe N NZe
Xc 0 0 0 1 0 Xc + 0 0 a Uc I (3.26)
Vc 0 MNe 0 0 0 Vc 0 1 0 Up
vp 0 R2 0 0 _!!h. vp 0 0 RpK, p J" p
V
J , ...,. "
A B
and the output matrix is given by
Ve
Yl 1 0 0 0 0
t c
Y2 0 0 1 0 0
 XC
Y3 0 0 0 1 0
VC
Y4 0 0 0 0 1 ,
'V' " vp c
3.3.1 Analysis of the Accumulator System Ignoring Nonlinearities
In this section the system is written in following form for analysis:
X = Ax + Bu + f(x),
y=Cx.
(3.27)
(3.28)
Considering the equations (2.46), (2.47),(2.48), (2.50), and (2.51), the matrices A, B, C,
and the term f(x) are given by:
0 0 a a a 0 0 0
a 0 1 a a a 0 0
A=  MNe a _!!L a 0 B= 0 1 0 Me
R2 a 0 ~ 0 ReKe 0 a e J e
Rp2 a 0 0 _!!h. 0 a RpKp J"
40
1 0 0 0 0
(~~6)(6 + ~ (ve(t)  vp(t))
0
0 0 1 0 0 c= f(x) = 0
0 0 0 1 0
0
0 0 0 0 1
0
Using these matrices the controllability and observability matrices can be calculated as per:
We = [B AB A2B A3B A4 B] ,
We = [C' A'C' A'2C' A'3C' A'4C'].
Due to construction of the matrix A, the column rank of We and We can not be five. Therefore
the system is not controllable as well as not observable. In the next step, state equations
are written using a new state vector defined as:
66 X2 I
X= 6 (3.29)
This transformation is equivalent to feedback linearized system along with exitside and
processside driven roller dynamics. The system can be rewritten in terms of new state
variables as:
(3.30)
(3.31 )
(3.32)
(3.33)
(3.34)
41
The new matrices A, B, C, and the tenn f(x) are given by:
_!!k J 0 0 0 0 ReKe 0 a e
AE 0 0 AE AE IT IT 0 0 a
A= 0 0 0 1 0 B= 0 0 0
0 0 0 _!!M.Le 0 0 1 0 Me
0 0 0 0 _!!.J.1u' . 0 0 RvKp
R2~
1 0 0 0 0 e X3
~(~ ~)
0 a 1 0 0 x3 N N
C= f(x) = 0
0 0 0 1 0
_.!i..~
0 0 0 0 1 Me X3
It;~
xa
Using these matrices the controllability matrix (Wc)and observability matrix CWo) are calculated
again and rank ofthese matrices is less than five. Thus the system is not controllable
as well as not observable.
3.3.2 Analysis of the Accumulator System
Considering the nonlinearities along with the fact that the position of the carriage and
reference average tension in the web spans is always a positive number, the controllability
and observability matrices are calculated using matrix form system given by (3.26). Rank
of the controllability as well as observability matrix comes out to be 5, which makes the
system controllable and observable. As discussed in a earlier section, the exit velocity is
zero for some part of the core change cycle. When the exit velocity is zero, the first control
42
input is lost. Therefore, matrix B is given by: .,
0 0 0
° 0 0
B= 0 0 0 (3.35)
0 1 0
0 0 RpKp
and matrix C is given by
0 0 1 0 a
c= ° 0 0 1 0 (3.36)
0 0 ° 0 1
Again the matrices We and Wo have rank 5, which keeps the system controllable and observable.
The system is controllable and observable if considered in its nonlinear form,
without neglecting any nonlinear terms. A nonlinear controller has to be designed for this
system as linear control schemes may not be feasible.
3.3.3 Jacobi Linearization of the Accumulator System
To analyze the system about some equilibrium point Jacobi linearization can be very helpful.
In this particular system there are many different forced equilibrium points. Linearization
is performed about the point, when the carriage is at its lowest position and it is not in
motion. At this equilibrium point following identities will be true,
ie = 0,
Vp = Ve = V = 130,
t c = tref = 1165,
Xc = Xeq = 71,
eq _ Ntrcf + Meg
Ue  A '
cyl
43
From matrices defined in previous section, this system in linear form can be r pre ented
as:
_!!.I.!. =R2 0 0 0 ~ 0 0 J e Je Je
AE 0 a AEtret AE 0 0 0  NXeq Xeq NXeq
o'x= a 0 0 1 0 ox+ 0 0 0 Oll.
0  M1'1e 0 0 0 0 1 0 Me
0 _!!i 0 0 _!!n O 0 !!P..!!.J? J p J p J p
This final representation is in the fonn of
x = Ax+ Bu.
The roller bearing friction for entry and exit roller, Bj , is very small and can be ignored
(Bj ~ 0). For that case, the eigen values of this matrix A, using parameters for ALCOA
CPL, are given by:
'\15 = 0, ± 69.1197t, 0, O.
Also the rank of Jacobi matrix A is 3 in this case. All the eigen values are at origin or at the
imaginary axis. So linearization about this forced equilibrium point does not suggest any
thing new in understanding the system. This system is a real time system and practically it
is stable. So it can be concluded that because of the nature of the process of linearization
about a single point cannot guarantee the same characteristics of the system, in the whole
space.
3.4 Synopsis
In all the previous sections, the system given by equations (2.46) through (2.51), is analyzed
after performing different state transformations. The system given by equations
(2.46) through (2.48) along with exitside driven roller dynamics (2.50) and processside
driven roller (2.51), is controllable and observable. The system loses its controllability and
observability ifany transformation is performed ,therefore, a linear control scheme can not
44
be designed. The nonlinearities in the system are responsible for the controllability and
observability of this system. This nonlinear system is used for further analysis in following
chapters. A nonlinear controller scheme is designed and investigated for this system.
45
CHAPTER 4
CONTROLLER AND OBSERVER DESIGN
4.1 Controller Design
This chapter considers the design of control algorithms for the accumulator carriage, the
exitside driven roller and the processside driven roller such that the average web tension,
the carriage position, the carriage velocity, the exitside web velocity and processside
web velocity track their desired trajectories. It is assumed that all the state variables are
measurable except for the average web tension, 6(t). An observer will be designed to
estimate the average web tension.
Consider the following error variables: el(t) = 6(t)  ~f, e2(t) = ~2(t)  (~(t),
e3(t) = 6(t)  ~g(t), ee(t) = ~5(t)  ~t(t),and ep(t) = ~6(t)  ~g(t), where ~t is the
desired web tension, ~~(t) and ~g(t) are the desired accumulator position and velocity,
respectively and ~t(t) and ~g(t) are desired exit and processside velocities, respectively.
Choose the following control inputs for the accumulator carriage, exitside driven roller,
and the processside driven roller:
(4.1)
(4.2)
(4.3)
where ucn.(t), uea(t), and Upa(t) are auxiliary control inputs that win be designed later.
Substituting the control input and using the error definitions, the error dynamics is given
46
by:
. AE AE
el(t) = 6(t) e3(t) + N6(t) (ee(t)  ep(t)) ,
e2(t) = e3(t),
. IV vf
e3(t) =  M el(t)  M e3(t) + Uca(t),
c c
ee(t) = ~ (Bjeee(t)  R;el(t) + R;8e (t») + uea(t),
e
ep(t) = ~ (Bfpep(t) + ~el(t)  R;A,(t») + upa(t).
p
Consider the following observer for estimating the average tension dynamics:
(4.4)
(4.5)
(4.6)
(4.7)
(4.8)
;... AE AE ~ ~ ~
~l (t) = 6(t) e3(t) + N~2(t) (ee(t)  ep(t» + ff,l' ~1(O) = ~1O, (4.9)
where ~ (t) is the estimate of~I (t), and hI will be chosen later during the stability analysis.
Define the observation error as el(t) = 6(t)  fl(t). Also, define el(t) = fl{t)  ~t.
Therefore, the observer error dynamics is
(4.10)
Notice that we have used the fact that ~g(t) = (~g(t)  ~g(t»/N, that is, the difference
between the desired process velocity and the desired exit velocity divided by the number
of spans gives the desired carriage velocity.
Consider the following Lyapunov function candidate for the accumulator carriage systern:
(4.11)
The derivative of the Lyapunov function candidate along the trajectories of the error dynarnics
is given by:
(4.12)
47
Choose the following auxiliary control input for the oarriage:
(4.13)
where 1'3 is a positive gain.
Substituting (4.13) into (4.12), the derivative of the Lyapunov function candidate becomes
Consider the following Lyapunov function candidate for the exit and the processside roller
dynamics:
(4.15)
The derivative of~p(t) along the trajectories of the error dynamics, (4.7) and (4.8), is
(4.16)
Consider the following Lyapunov function candidate for the combined system ofthe accumulator
carriage, exitside driven roller, and the processside driven roller:
Vet) = Ve(t) + Vep(t).
The derivative ofV(t) is:
(4.17)
Vet) = Ve(t) + Vep(t)
( Vf) 2 AE ) (AE N ) ~ J.'" ~
=  1'3 + Me e3 + N6(t) (ee  ep el + ~  Me ele3  ~lel
+ ~e (Bfee~  R~elee + R~<5e(t)ee) + eeUea
+ ~ (BJpe; + R;ele.,;,  R;<5p(t)ep) + epupa' (4.18)
p
48
Assuming that oe(t) and op(t) are bounded by some known constants, that is, IOe(t)1 ~ J
and IOp(t)1 ~ Jp, we choose the following auxiliary control inputs:
(4.19)
(4.20)
where "fe and "fp are positive gains. As all the rollers are of same size and inertia in real
industries, therefore J e = Jp = J and Re = Rp = R is used in further analysis.
Using these auxiliary control inputs and rearranging tenns, we obtain
(4.21)
Choosing
we get
V.(t) ~  ("fVa + Mfe) ea2  ( "fe + JBfc ) ee2  ( "fp + JBfP) ep2 '
(4.22)
(4.23)
Therefore, V(t) ~ 0 is a non increasing function of time for all t ~ O. Hence, V(t) E .coo
and limtooo V(t) = Voo < 00. Also, el(t), el(t), e2(t), e3(t), ee(t), ep(t) E .cry:, and
e2(t), e3(t), ee(t), ep(t) E .c2. From the error dynamics, (4.5}(4.8), e2(t), e3(t) I ee(t) I ep(t) E
.coo' Therefore, using Barbalat's lemma, we have e2(t), ea(t), ee(t), ep(t) + 0 as t + 00.
The following theorem summarizes the results of this section.
Theorem 4.1.1 For the dynamics ofthe accumulator carriage and the driven rollers upstream
and downstream ofthe accumulator given by equations (2.46) through (2.51), the
49
following control inputs
( ) (
'd( ) vI d N d AE ~ N
U c t =Mc~3 t +g+ Mc~3(t)+ MC~l 6(t)e1(t)e2 (t) + Mce;(t)')'ses(t)),
(4.24)
J ( (AE R
2
) ~ R2
ue(t) = RK  B .) e 'eee(t)  N6(t)  J e] (t)  J§e!Jgn(e ) + .r~t(t) + ~t(t) ,
(4.25)
J ( (AE R
2
) R
2
up(t) = RK  Bf .)
p
,pep(t) + N6(t)  J €let)  J§pSgn(ep ) + JP ~g(t) + ~g(t) ,
(4.26)
and the average tension observer
will result in the signals e] (t), el(t), e2(t), e3(t), ee(t), ep(t) being bounded andfurther, the
signals e3(t), ee(t), ep(t) asymptotically converge to zero.
Equations (4.24), (4.25), (4.26), and (4.27) give the accumulator carriage input, exitside
driven roller input, processside driven roller input, and the observer dynamics, respectively.
The proof of the theorem follows from the prior analysis. Notice that the control
inputs given by equations (4.24), (4.25) and (4.26) are dependent on e], which in tum depends
upon ee and ep . Therefore, in the proposed control scheme all the control algorithms
are coupled, i.e., it is centralized control as compared to what is presently used in industry,
which is decentralized control.
4.1.1 Simulation Study
In this section, the proposed control scheme is investigated by conducting simulations on an
industrial continuous web process line. The simulations are perfOlmed using the parameters
of an ALCOA continuous process line (CPL) and its exit accumulator. Different values of
the parameters of the accwnulator used in the simulations are given in Table 4.1. The
50
Description Symbol Value
Mass ofthe carriage Mc 501.06 slugs
Crosssection area ofweb A 0.5 in2
Modulus of elasticity E 107psi
Number of web spans N 34
Viscous friction coefficient I vf 20000Ib/(in/sec)
Radius of exit and processside roller R 6 in
Moment of inertia J 228.8448 slugsin2
Bearing friction coefficient Bf 0.02
Table 4.1: Parameters of the accumulator.
desired tension in the web spans is 1165 Ibs. The desired process speed is 650 fpm. A
typical case ofthe exit speed and the carriage speed during a rewind roU change is depicted
in Figure 4.1. The rewind roll changeover scenario, when the web velocity in the process
section is maintained at a constant value, is described in the following steps by referring
to Figure 4.1: (i) AB  velocity of the web in the rewind side is decelerated to zero from
a value of 650 fpm, as a result of this the accumulator starts collecting the web and the
carriage accelerates upwards; (ii) Be  rewind stops and the carriage is moving up with
constant velocity; (iii) CD  after rewind roll change, exit side is accelerated up to the
process speed, in this period the carriage is moving up while decelerating; (iv) DE  exit
side is accelerated up to a speed above the process speed, 950 fpm in this case; (v) EF  exit
speed is maintained at this constant speed; (vi) FG  exit speed is reduced to the process
speed. The desired profile for carriage velocity is given by:
From this velocity profile the desired carriage position profile is calculated. After each
cycle of rewind rollchange operation the carriage will return to its original position; this
means that the area under the carriage velocity curve is zero. The goal is to track the
51
F
350 400
G
100 150 200 250 300
1000
c
"E A
¢:!
Qi 500
>
"x w
0
0 50
20
c
"E
¢:!
10 " "
Qi >
G)
01 «l 0 " "
"E
tV
D
10
0 50
40
¢::
c ,g 30 ,"
w0
Cl. 20
Q)
01 B
«l
E 10 "A
tV
D
0
0 50
Figure 4.1: Desired exit speed, carriage speed and carriage position during rewind rollchange.
desired profiles of the carriage position and speed, exit velocity and process velocity while
maintaining the desired level of the average web tension. The simulations are conducted
using the system model given by equations (2.46)(2.51) and the control algorithms given
by (4.24), (4.25), (4.26), and the observer given by equation (4.27).
Two types ofcontrollers are considered for comparative study via simulations: (l) Currently
used industrial controller, and (2) the controller proposed in this paper. Currently,
the industrial controller uses only feed forward of the desired profile for the accumulator
carriage and PI controllers for the exit and processside driven rollers. The industrial controllers
for the accumulator carriage, exitside driven roller, and processside driven roller
52
are given by:
(4.28)
(4.29)
(4.30)
where kpe and kpp, and kie and ~p are the proportional and integral gains, respectively. Notice
that ifsame control gains are considered for proposed control scheme and for industrial
controller, the proposed controller expressions g~ven by equations (4.24), (4.25) and (4.26)
have few additional terms, as auxiliary input, in comparison to expressions for industrial
controllers given by equations (4.28), (4.29) and (4.30). Later with simulations, it is shown
that with this little extra effort at each instant, the tension variations can be controlled more
effectively.
The disturbances be, present on the exitside driven roller tension system, and 6p , present
on the processside driven roller tension system, considered in equations (2.50) and (2.51),
are of sinusoidal nature with a frequency of 0.2 Hz and amplitude 10 lbs. For comparing
the results of the two types of controllers, three types of sinusoidal disturbances are introduced
into the accumulator carriage dynamics, that is, into equation (2.48). The amplitude
ofthese disturbances is 10 in/sec2 and the frequency is 0.5 Hz. Low frequency disturbances
are used because they are typical disturbances on the accumulator carriage; the accumulator
carriage does not have the ability to respond to high frequency disturbances due to
its large mass. We consider the three time profiles shown in Figure 4.2 for introducing
sinusoidal disturbances into the accumulator carriage. In the first case the sinusoidal disturbance
is introduced throughout the time duration of 400 seconds. In the second case the
sinusoidal disturbance is introduced at specific time intervals of 20 to 30 seconds, 106 to
126 seconds, and then from 318 to 328 seconds. In the third case the disturbance is introduced
at specific time intervals of 2030 seconds, 4060 seconds, 96116 seconds, 136156
seconds, and 308328 seconds. The time durations in the second and third case are picked
53
10
2f c:
CeIl 0
::J
U;
0
10
0 50 100 150 200 250 300 350 400
N 10 ... '." ..... . ........
~
c:
CIl € 0
::J
U;
0
10
0 50 100 150 200 250 300 350 400
'" 10
ill
0c:
CIl € 0 .a(/)
0
10 . '" ........ , .........
0 50 100 150 200 250 300 350 400
time sec
Figure 4.2: Three cases of sinusoidal disturbances.
54
to reflect observations made on an industrial process line that periodic disturbances occur
during the initiation of carriage motion from its stationary state and when the exit velocity
starts to accelerate or decelerate. The simulations are performed for these three cases of
disturbances and for both types of controllers. All the gains that are common to both controllers
are chosen to be the same. The results are shown in Figures 4.3 through 4.14. The
three cases of disturbances shown in Figure 4.2 are labelled as Disturbance I, Disturbance
2, and Disturbance 3, respectively. For each disturbance the following errors are shown:
web tension error (el), carriage position error (e2), carriage velocity error (es), exit velocity
error (ee), and process velocity error (ep ). The control signals of accumulator carriage,
exitside driven roller, and processside driven roller for both controllers are also shown.
Figures 4.3 and 4.4 show the errors for the industrial controller and the proposed controller,
respectively. From the results we can gather that the proposed controller does a
much better job at the sinusoidal disturbances. As tension variations propagate both upstream
and downstream of the accumulator, these variations in tension and process speed
will affect the web in the entire process line, which is undesirable. The magnitude of all
the error signals for the proposed control scheme is very small compared to the industrial
controller. Figures 4.5 and 4.6 show the plots for control effort required for the two
schemes. The control effort required for accumulator carriage is similar in both schemes
but the control effort for both driven rollers for the proposed controller is smaller than the
industrial controller. Overall, the performance ofthe proposed controller is much improved
compared to the industrial controller with no additional control effort.
Figures 4.7 through 4.10 and Figures 4.11 through 4.14 show results corresponding to
Disturbance 2 and Disturbance 3, respectively. Similar observations as made in the case of
Disturbance 1 can also be made for Disturbance 2 and Disturbance 3.
55
:Q 200
Gl~_20:
:_0:055:0";.~ ...';~~i'.~1:~'M~"::!O" ..3::0 ..'jOO
",,,·r"o"w.l\\'NNH'··.. .: :.
0.1
!"00:
0.05'· o 50 100 150 200 250 300 350 400
(:
~o 0
0.5o 50 100 150 200 250 300 350 400
time sec
Figure 4.3: State errors of the industrial controller: Disturbance 1.
56
:~:I P:·: ::. !.J
<:5:··· ';o;r ·~r 2j"T'T .. T
o 50 100 150 200 250 300 350 400 (::o 50 100 150 200 250 300 350 400
(:~~
(:: r o 50 100 150 200 250 300 350 400
time sec
Figure 4.4: State errors of the proposed controller: Disturbance 1.
57
58
X 104
15 ,.,r..,~__._~
:::l"
100 150 200 250 300 350 400
OL_~L.......L.LL..L.l..J o 50
100 r~,...,___r_.______.__,_~
50
50 100
50
100 '__L L ' .l I__!_...:..._L.. 1
o
.,
:::l
:B 0
40 r,,...,__.rr,.
20
.!:
:B 0
a.
:::l
20
200 250 300 350 400
time sec
50 100 150
40 '__L ' L... ..l l__I I.. 1
o
Figure 4.6: Control inputs for the proposed controller: Disturbance 1.
59
400
400
350
350
! Iii: I
300
300 350 400
300
300 350 400
250
200 250
200
time sec
50 100 150
~_~Pt: :I' ; :
o 50 100 150 200 250
~"_:::h:~! ; .!
o 50 100 150 200 250 (J=i ii, i i : o 50 100 150
(:~.~.:,....~..~,..t..,~'1j
o 50 100 150 200 250 300 350 400 (:E!,: ,*=~: ~f'±!~:1
0.5   o
Figure 4.7: State errors of the industrial controller: Disturbance 2.
60
~~:I ~ : :::! ]
~~::E:o . '~;r . 2~oT 'r ,'r ....T
o 50 100 150 200 250 300 350 400
~:::~~
o 50 100 150 200 250 300 350 400
.~ 0: . ':, ",,: :. . .
~ . ,
0,2o 50 100 150 200 250 300 350 400
(,) 0.05
~.oJ~~~Q~TT!~~ij~~~ijij~~
o 50 100 150 200 250 300 350 400
time sec
Figure 4.8: State errors of the proposed controller: Disturbance 2.
61
X 104
15
10 ,. . , • •••• o •
f!
u
::>
5 .. ,'" . , .........
0
0 50 100 150 200 250 300 350 400
200
100 : ......... . ......
.£
f! 0
III
::>
100 ..... ", ".' .. . ... ','
200
0 50 100 150 200 250 300 350 400
100
50 •••• f ......
.£
f! 0
0
::>
SO . . . . .. . . . .
100
0 50 100 150 200 250 300 350 400
time sec
Figure 4.9: Control inputs for the industrial controller: Disturbance 2.
62
X 104
15
......
:9
:J" ......... . ... . .. .. . .. .. . ... ........
0
0 50 100 150 200 250 300 350 400
100
50 ...........
.~
:9., 0
:J
50
100
0 50 100 150 200 250 300 350 400
40
20 . . . . . . . . .
.~
:9 0
a.
:::J
20
I ...... , . ',' ......... ... ........ , ....
40
0 50 100 150 200 250 300 350 400
time sec
Figure 4.10: Control inputs for the proposed controller: Disturbance 2.
63
Figure 4.11: State errors of the industrial controller: Disturbance 3.
64
~1 ~~':~'~:.~:.~:1
o 50 100 150 200 250 300 350 400
:"_::E; ;/:njn!!'!,n nn]
o 50 100 150 200 250 300 350 400
(::~~.~~
o 50 100 150 200 250 300 350 400
~.o:nw~~~
0.2· o 50 100 150 200 250 300 350 400
o O.~ . . . .
~o. 0 " ' ." ".. '. . '
Q) ,. I
0.05o 50 100 150 200 250 300 350 400
time sec
Figure 4.12: State errors of the proposed controller: Disturbance 3.
65
X 104
15
10 .... . . . ........... . ..... ......  ..... 
f!
;:,"
5 ' . ..  ...... .... . .........
0
0 50 100 150 200 250 300 350 400
200
100 . ,', ..........
.5
f!.. 0 ;:,
100 ",' . '." . . . . '.' .. '.'
200
0 50 100 150 200 250 300 350 400
100
50
.5
f! 0
Q.
;:,
SO ....
100
0 50 100 150 200 2SO 300 350 400
time sec
Figure 4.13: Control inputs for the industrial controller: Disturbance 3.
66
f .'
X 10·
15
10
::Q
u
::;, ... ~. .. .
0
0 50 100 150 200 2.50 300 350 400
100
.... ,................
£
::Q
III
::;,
100
0 50 100 150 200 250 300 350 400
40
20
£
::Q 0
a.
:;:l
20 .... ", ...
40
0 50 100 150 200 250 300 350 400
time sec
Figure 4.14: Control inputs for the proposed controller: Disturbance 3.
67
4.2 Controller Design Incorporating Adaptation Law for Friction Coefficient
Friction is a complicated phenomenon, which exists in all mechanical systems. Knowledge
of the friction. coefficients is essential for a stable controller design, which may not
be practically feasible. High accuracy control cannot be achieved if friction effect is not
considered properly. Therefore, an adaptation law is designed for friction coefficient along
with a stable closed loop controller. The coefficient ofviscous friction will be estimated online.
Considering the same error definitions and control inputs Ue and up given by equations
(4.2) and (4.3), as in section 4.1 and Uc as follows:
(4.31)
gives the same error dynamics for e}, e2, ee, ep and new error dynamics for e3:
(4.32)
By adding and subtracting the tenn it~t from equation (4.32) and rearranging tenn this
equation can be written as:
(4.33)
where vf = vf  Vf' Considering the same observer dynamics, the new Lyapunov function
candidate will be:
(4.34)
where Q is a positive gain. The new derivative of the Lyapunov function candidate along
the trajectories of the error dynamics is given by:
(4.35)
Following the same process to choose Uco.(t) , Uca(t) , upa(i), as per equations (4.13), (4.19),
(4.20) and if we let
(4.36)
68
•
then the adaptation law is given as follows:
(4.37)
Since actual parameter vI is not varying, t;I = t;I. SO above equation can be rewritten as:
(4.38)
~
Using these auxiliary control inputs and the term ifI as per equation (4.22), the same conclusion
can be made as in section 4.1. Therefore, Vmod(t) ~ 0 is a non increasing function
of time for all t ~ O. Hence, Vmod(t) E .coo and limt.oo Vmod(t) = Voo < 00. Also,
el (t), el (t), e2 (t), e3(t), ee(t), ep(t), viet) E .coo and e2(t), e3(t), ee(t), ep(t) E .c2. From
the error dynamics, (4.5}(4.8), e2(t), e3(t), ee(t), ep(t) E £'00' Therefore, using Barbalat's
lemma, we have e2(t), e3(t), ee(t), ep(t)  0 as t  00.
The following theorem summarizes the results ofthis section.
Theorem 4.2.1 For the dynamics ofthe accumulator carriage and the driven rollers upstream
and downstream of the accumulator given by equations (2.46) through (2.51), the
following control inputs,
"d 0(t) d N d AE __ ) N __ ( ) ( ))
uc(t)=Mc(E3(t)+g+ Me ~3(t)+ Mc~l E2(t)e l(t)e2(t + M
e
el t 'Y3e3 t ,
(4.39)
J ( (AE R
2
)__ R
2
_ BIe ue(t) = RK d) 'd())
e
'Yeee(t)  N6(t)  J el(t)  jOesgn(ec ) + J~4 (t + e4 t ,
(4.40)
J ( (AE R
2
) ~ R
2
up(t) = RK  ) Bfp d() 'd())
p
'Ypep(t) + N6(t)  j ('l(t)  JDpsgn(€p + Jes t +~5 t, .
(4.41)
the average tension observer and the adaptation law
~ (0) = ~o (4.42)
(4.43)
69
will result in the signals el (t), el (t), e2(t), e3(t), ee(t), ep(t), vI(t) being bounded andfurther,
the signals e3(t), ee(t), ep(t) asymptotically con.verge to zero.
The simulations are conducted using the system model given by equations (2.46)(2.51)
and the control algorithms given by (4.39), (4.40), (4.41), the observer given by equation
(4.42) and adaptation law given by equation (4.43). The simulations are performed for
first case of disturbance. The results are shown in Figures 4.15 through 4.17. For this
disturbance the following errors are shown: web tension error (el), carriage position error
(e2), carriage velocity error (e3), exit velocity error (ee), and process velocity error (ep).
The control signals of accumulator carriage, exitside driven roller, and processside driven
roller for both controllers are also shown.
~~t ·tti~,;!:..;I
o 50 100 150 200 250 300 350 400 <:E;;;/:'~:'~l
o 50 100 150 200 250 300 350 400 !.oo:
001
. 0 50 100 150 200 250 300 350 400
() 0.2 .:
~. .
!·o:~
. 0 50 100 150 200 250 300 350 400
~oo:
005
. 0 50 1'00 150 200 250 300 350 400
time sec
Figure 4.15: State errors of the proposed controller with adaptation law: Disturbance 1.
70
X 104
15
.... .. ....... . .... . . ...... . .... ... ... 
:9
0
::J
. . . . . . . . . . . . ........ ..
0
0 50 100 150 200 250 300 350 400
100
50
.!;
:9
CD
:::J
100
0 50 100 150 200 250 300 350 400
40
20 .. ....... .
.!;
:9 0
a.
:::J
20 ..... .. .' .
40
0 50 100 150 200 250 300 350 400
time sec
Figure 4.16: Control inputs for the proposed controller with adaptation law: Disturbance
1.
All the state errors are very low. The results are similar to what is earlier shown. Figure
4.17 shows how the estimate of viscous friction coefficient is behaving.
4.3 Simulation Study Considering Web Span Weight Acting on the Carriage
In section 3.1.3, the change in web span weight acting on carriage is discussed, as the carriage
is in motion. In metal industry, if the thickness ofweb material is in the range of 1/8th
of an inch, the change in web span weight acting on carriage is considerable as shown in
Figure 3.2. In this section the effect ofthis weight change is studied with simulations for
the proposed controller and for industrial control scheme. The simulations are performed
71
X 104
2.5r,,r,,r,,
2
1.5
0
OJ
.!!!
§,
£!
.E
>
0.5
................ ., ', . ,'" .
'if:
........: .
200 250 300 350 400
time sec
50 100 150
0.5 '__' ' L ' .l....__'L.__I. .J
o
Figure 4.17: Viscous friction coefficient estimation for the proposed controller: Disturbance
1.
for first case of disturbance used in section 4.1. Figures 4. I8 through 4.21 are corresponding
to case when in the actual model the effect of varying mass is considered by replacing
Ale with Me +N A6p and there is no compensation for this varying mass in the controller.
In case of industrial controller the variations in tension are very high. If no compensation
is being used for varying mass the tension is dropping to a low value, i.e., a part of control
effort is being used to compensate for weight change and there is not enough effort to
account for average total tension required in all the web spans. The proposed controller is
performing well in this case also. Even though there is no compensation for weight change
in controller, the control algorithm is accounting for weight change as the control effort
72
~~ 10:t"IlIIlIll....IIJMI;"WlliM~·~~··.·~ ·"'·~ ..~.ili'.MMIW.WIlIlUW~WWII\WMII'.W~W ......W",..j Q)","""",,"ffln'h,·l"l., _'»WYJff 1111.11.'."'" """,,,,,,,£",,,4""',",,,,,",,,,,, , II , nIft'r11 ft,n ,'I '" ,
1000 '' :"J 5:0 100 '~ H2~0 2~0 3;0~0 1"
o 50 100 150 200 250 300 350 400
~ O:t·..*..IMMM :,.""I/II" ::::IIlIl!&lIU~MWN\WWI'M III'_ 'Il'IlI,~ , j M 94" inA. ;mh\n,Unrdlf mlmm"," .' • • .... "mV 4. J;'#"'{iJ. IVK An (ViMa••
~ '.' .
0.2 ' '' ' L .l. ...L ....l.. _
o 2
0
t
5~n 10~~0I\ll.lll1llllll1l~15OIIIIIIIIliI .. 20~? 2~50 300: 35
r
O 4~00
~ rtN,Wli/tNI·.WN~W...fA?II~fHHIbM"4WMmm.:~I\WNMWMr~~wJm..u.rNt••__.M...... '.. u .
~._: ....~ "'HH_HH~H~H_"""..h ..n",~_
o 50 100 150 200 250 300 350 400
~ :~.~ii'"..';,;,~:,,;,_.':.iI"'";';O ...Ii'Iiw._ __j~~
cD'" r ·"'"..."~I"'M"""j~"llll"'"hJrJjn~ 1 r 2 ' ' J J .l. .L _
o 50 100 150 200 250 300 350 400
time sec
Figure 4.18: State errors of the industrial controller: No compensation for varying mass.
required in the case of proposed controller is much higher than the industrial controller.
Figures 4.23 through 4.25 are for the case when compensation for varying mass is considered
in the process of controller design. Performance is improved in case of both type of
controllers. Overall the proposed controller is showing more promising results.
4.4 Synopsis
In this chapter, design of a control algorithm for web tension regulation in an accumulator,
in web processing lines, is considered. A feedback controller together with an observer
for web tension is proposed. It is shown that the proposed feedback controller results in a
stable closedloop system. Simulation results on an industrial continuous web processing
line are given and discussed for the proposed controller/observer and compared with a
73
jq : ]
300 350 400 j, : J
~lq2 : : : o 50 100 150 200 250 <:E: ;/:: j
o 50 100 150 200 250 300 350 400
(::~
o 50 100 150 200 250 300 350 400
~ 0:. • :. .. .. ...:
OJ'" ::.
0.2 I o 50 100 150 200 250 3"00 350 400
~O 0:
0.05o 50 100 150 200 250 300 350 400
time sec
Figure 4.19: State errors of the proposed controller: No compensation for varying mass.
74
Figure 4.20: Control inputs for the industrial controller: No compensation for varying
mass.
75
X 105
2....r..,.
1.5 ~.,~~~~,
0.5
.D
" 1
::::l
50 100 150 200 250 300 350 400
0'1''''1'' o
100 ,___r,...,,.~r____r_,__,
50
.5
:9 0
::::l'"
50
50 100 150 200 250 300 350 400
100 L.....__I. ' ....l... ' L.....__I. ' '
o
40,___rr...,..,.,.,,
20
.5
:9 0
C.
::::l
20
200 250 300 350 400
time sec
50 100 150
40 L.....__I. L ....L ' L.....__' ' '
o
Figure 4.21: Control inputs for the proposed controller: No compensation for varying mass.
76
500
:8 :. ":.
~ 0
Ql .. . .... .
500· .
0,1
05 5
;0 1~0~10~ 2oo~ 2~5O 30~0 35
1
0 4jOO
C' 1M1YNi.·MMW..W..•.........·........·..·w·..·· .. " :
.NO, ':  ~. . . :ff~L eft>, ~l~ ..l~~'!'~~!,~~.w:..~.~ ~.~::: .'. :·~\Wo""".Ili"'''.w~ _/\1\1I\\ .
Q) _.:ronUhM1U.tmIllU"U.I1f.U.uim "n· :.:•
0.1 o 50 100 150 200 250 300 350 400
.CUl~ 0'0
1t\lllUWI4~lJ.II'~~~~~:WI"'''''''IIII'M~:'MvyWtIMllIII~ iM,mrhwhlm ,,"" """,,'U""" """",mnHAJ
ar' :. . ~ :
0.1 L.. '__' L '__' L...__J
o 50 100 150 200 250 300 350 400
o 1 .. *. . .
~~:
o 50 100 150 200 250 300 350 400
0
1
• 3l' ...: .
E 0
Q)a. : • . . . • :. .
1
o 50 100 150 200 250 300 350 400
time sec
Figure 4.22: State errors of the industrial controller with compensation for varying mass.
77
~~r ...~..... Ii . j. ·il ·1
o 50 100 150 200 250 300 350 400 <:E: ;/; , : :'!l o 50 100 150 200 250 300 350 400 (::o 50 100 150 200 250 300 350 400
1ilO'2~~~. ~_0_.hi~dn,"''M''yU . . . . . . . . . , .. . ' . '.
4) ':: :
~ . .
0.2o 50 100 150 200 250 300 350 400
1il 0.05 .... .
..!!!:!:.Co 0.. ... " ' ,
~ '..
0.05a 50 100 150 200 250 300 350 400
lime sec
Figure 4.23: State errors of the proposed controller with compensation for varying mass.
78
X 105
2
1.5
f! 1 ..:............:. .. ....... . .........
0
..... ..
~
0.5 ...... . . ~ . . . . . . , . . . . .... . , .. , .. ., ..... .... '" ....
0
0 50 100 150 200 250 300 350 400
400
.!: 200
f!
CD
~ 0
200
0 50 100 150 200 250 300 350 400
400
200
.!:
f! 0
n
~
200
400
0 50 100 150 200 250 300 350 400
time sec
Figure 4.24: Control inputs for the industrial controller with compensation for varying
mass.
79
X 105
2
1.5
f!
<> 1 ...
::::J
0.5
0
0 50 100 150 200 250 300 350 400
100
.!:
f!
CD
::::J
100
0 50 100 150 200 250 300 350 400
40
20
.!:
f! 0
Co
::I
20
40a 50 100 150 200 250 300 350 400
time sec
Figure 4.25: Control inputs for the proposed controller with compensation for varying
mass.
80
•
currently used industrial controller. The effect of change in web spans weight acting on the
carriage and adaptation law for viscous friction coefficient in case of uncertainties is al 0
considered. Simulations are performed for these cases also.
81
I,
CHAPTERS
CONCLUSIONS AND FUTURE RESEARCH
Feedback control algoritluns for the accumulator carriage, and for the upstream and downstream
driven rollers to the accumulator, are designed for tracking the desired exit and
process web velocities, and to maintain the web tension at the desired level. It is common
in the web handling industry to just apply a desired force on the caniage using a hydraulic
system in opposition to the carriage weight and the force required to produce desired tension
in all the accumulator web spans; thus, ignoring the dynamics ofthe caniage motion.
This strategy often leads to large tension variations not only in the accumulator web spans
but also in web spans in the entire process line due to tension disturbance propagation both
upstream and downstream of the accumulator. Simulation results comparing the proposed
control algorithm with the currently used control scheme in industry show that the proposed
control algorithm results in much less web tension variations.
Friction is a complicated phenomenon, which exists in all mechanical systems. Knowledge
of the friction coefficients is essential for a stable controller design, which may not
be practically feasible. High accuracy control cannot be achieved if friction effect is not
considered properly. Therefore, an adaptation law is designed for friction coefficient along
with a stable closed loop controller. The control scheme along with this adaptation law
shows the same performance. Estimating the friction coefficient is a more practical approach,
which has been attained in this dissertation.
The weight of the web spans acting on the carriage changes with the carriage motion.
The effect of the weight of the web on the accumulator caniage can effect the tension in the
web spans in a substantial way. The weight of the web on the carriage can be substantial in
82
metal process lines where the thickness of the web can be as high as 0.2 inches. The effect
ofthis timevarying weight is also investigated in this dissertation. The proposed controller
is accounting for this weight change, with minimal variations in tension in web spans.
5.1 Future Research
This research also provides a solid background for some further advanced study on accumulators.
Instead of assuming that the force on the carriage is directly accessible as an
input, future work should include the dynamics of the actuator, either electrohydraulic or
electromechanical, coupled with the dynamics ofthe carriage and the web spans. Although
a friction model that includes linear viscous friction is proposed, better friction modeling
may be required for accumulators as these have some ofthe peculiar friction characteristics.
There is a need to conduct experiments on a processing line with a well designed experimental
procedure. This can better validate the concepts developed and provide directions
to iterate on these concepts to better model and control the dynamic behavior.
83
BIBLIOGRAPHY
[I] D.P. Campbell, Dynamic Behavior of the Production Process, Process Dynamics.
John Wiley and Sons. Inc., New York, 1958.
[2] K.P. Grenfell, "Tension control on papermaking and converting machinery," Proc. of
the 9th IEEE Annual Con! on Electrical Engineering in the Pulp and Paper Industry,
Boston, MA, June, 1963.
[3] D. King, "The mathematical model of a newspaper press," Newspaper Techniques,
December, 1969.
[4] G. Brandenburg, "New mathematical models for web tension and register error," Proc.
ofthe 3rd IntI. IFAC Con! on Instrumentation and Automation in the Paper, Rubber,
and Plastics Industry, vol. I, pp. 41143,1977.
[5] W. Wolfermann, & D. Schroder, "New Decentralized Control in Processing Machines
with Continuous Moving Webs," Proc. of the Second IntI. Con! on Web Handling,
June 69, 1993.
[6] G.E. Young and K.N. Reid, "Lateral and longitudinal dynamic behavior and control
of moving webs," ASME Journal of Dynamic Systems. Measurement, and Control,
vol. 115,pp.309317,June, 1993.
[7] W. Wolferrnann, "Tension control of webs  A review the problems and solutions in
the present and future," Proc. of the Third Inti. Conf. on Web Handling, Stillwater,
OK, June, 1995.
84
[8J 1.1. Shelton, "Limitations to sensing of web tension by means of roller reaction
forces," Proc. ofthe Fifth Inti. Con! on Web Handling, Stillwater, OK, June, 1999.
[9] P.R. Pagilla, 8.S. Garimella, L.H. Dreinhoefer, and E.O. King, «Dynamics and control
of accumulators in continuous strip processing lines," IEEE Transactions on Industry
Applications, vol. 37, no. 3, pp. 934940, 2001.
[10] P.R. Pagilla, E.G. King, L.H. Dreinhoefer, and S.S. Garimella, "Robust observerbased
control of an aluminium strip processing line," IEEE Transactions on Industry
Applications, vol. 36, no. 3, pp. 835840, 2000.
85
86
VITA
Inderpal Singh
Candidate for the Degree of
Master of Science
Thesis: A STUDY ON CONTROL OF ACCUMULATORS IN CONTINUOUS
WEB PROCESSING LINES
Major Field: Mechanical Engineering
Biographical:
Personal Data: Born in India, on April 3, 1977, the son ofBalwant Singh and Jaswant
Kaur.
Education: Received the B.S. degree from Thapar Institute of Engineering & Technology,
Punjab, India, in 1998, in Mechanical Engineering; Completed the requirements
for the Master of Science degree with a major in Mechanical Engineering
at Oklahoma State University in December, 2002.
Experience: Research Assistant at Oklahoma State University from August 2000 to
December 2002; Teaching Assistant at Oklahoma State University from August
2000 to August 2002; Design Engineer at ISGEC, Haryana, India from July
1998 to July 2000.
Professional Memberships: American Society of Mechanical Engineers, Honorary
Society of Phi Kappa Phi.