Rochester Institute of Technology
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Thesis/Dissertation Collections
1990
Simulation and analysis of a digital scanning system
Eugene A. Rogalski
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Recommended Citation
Simulation and Analysis of a Digital Scanning System
by
Eugene A. Rogalski, Jr.
A Thesis Submitted
in Partial Fulfillment
of the
Requirements for the Degree of
MASTER OF SCIENCE
in Mechanical Engineering
Approved by:
Prof.
Joseph S. Torok
(Thesis advisor)
Prof.
Prof.
P Venkatar
Prof.
Wayne W. Walter
Mr.
Thaddeus Kciuk
Prof.
B.
Karlekar
(Department Head)
Department of Mechanical Engineering
College of Engineering
Rochester Institute of Technology
Rochester, New York
Simulation
and
Analysis
of
a
Digital
Scanning
System
I,
Eugene
A.
Rogalski,
Jr.
_{hereby}
grantpermissionto the
Wallace
Memorial
Library,
ofthe
Rochester Institute
of_{Technology}
to
reproduce_{my}
thesis
in
wholeorpart.
_{Any}
reproduction willnotbe for
commercial use or profit.Acknowledgment
This
work could nothave
been
possible withoutthe
supportandguidance ofseveral
_{individuals.}
_{Its}
_{success}
_{is due}
largely
to their
encouragementandsuggestions
andI
extend_{my}
deepest
appreciation:To Xerox
Corporation
for
the
useof_{computing}
facilities.
To
Dr. Wayne
_{Walter,}
Dr.
P.
_{Venkataraman,}
andDr.
_{Hany}
Ghoneim for
taking
time
outoftheir
_{busy}
schedulesto
read and critique_{my}
work.To Michael Boyack
whodesigned
the
control system which was usedin
this
analysis
and who providedthe
controlshardware
and software.His
enthusiasm
for
the
design
and analysisof controlsystems wasaninspiration.
To
Thaddeus
Kciuk
who providednumerous suggestions and guidance.To Dr. Joseph
_{Torok,}
Rochester
Institute
ofTechnology,
Department
ofMechanical
_{Engineering}
for
his
very
functional
guidance andencouragement.And
_{lastly,}
though
notatall_{last,}
to
my
wifeMarlene
whosuppliedAbstract
The
analysis
and simulation ofthe
motion of adigitallycontrolled digital
imaging
systemis
presented.The
dynamic
system consists ofa_{scanning}
carriage,
powertransmission
elements,
a prime_{mover,}
andacontrolsubsystem.
The
prime moveris
apermanent magnetDC
motorthat
is
positioned_{by}
adirect
digital
control system.The
scan carriage motionis
mathematically
modeledand simulated_{using ACSL}
andDADS
simulationsoftware.
T
he
simulationresultsare comparedto
empiricaldata.
It is
shownthat
the
dynamic
response ofthe
actualscan system canbe
predicted quitewell
_{using}
suchsimulations._{Furthermore,}
these
simulations can aidin
the
Table
of
Contents
Page
List
ofTables
viList
ofFigures
viiList
ofSymbols
ix
I
Introduction
1
II
Control System
Preliminaries
11
III
Description
ofthe
Physical
System
24
IV
Mathematical
_{Modeling}
28
V
Simulations
42
VI
Results
44
VII
Conclusions
55
VIII
References
58
List
of
Tables
Page
1.
_{Full Model}
_{Step}
_{Response}
_{Comparison}
_{47}
2.
_{Physical Constants}
_{6970}
List
of
Figures
Page
1.
_{Building}
Blocksof
aDigital
_{Scanning}
Device
2
2.
_{Imaging}
Definitions
3
3.
Contrast
_{Sensitivity}
ofthe
Human Eye
to
Square Wave Gratings
5
4.
_{Perceptibility}
Threshold
ofEdge
Raggedness
6
5.
Motion Error
_{Detectability}
ThresholdDesign Goal
8
6.
_{Velocity}
Error Design Goal
9
7
Basic Components
of aClosed
_{Loop}
Control System
11
8.
Nonlinear
Element
12
9.
_{Linearity}
Improvement
13
10.
Block Diagram
Transfer Function
15
11.
Block Diagram
Reduction
17
12.
Digital
Control
System
19
13.
Uniform
Sampler
19
14.
Sampler
withZero Order Hold
20
15.
Physical
System
24
16.
Control
System
/Mechanical
Block
Diagram
25
List
of
Figures
(continued)
Page
18.
Controls
Block Diagram
29
19.
Motor
Circuit
31
20.
Simple Mechanical
Model
Block Diagram
35
21.
Lumped Parameter
Mathematical Model (Translational
DOF)
38
22.
_{Linearized Lumped}
_{Parameter Model}
(all
_{DOF)}
39
23.
_{Eigenvalues,}
Eigenvectors
andMode
shapesfor
3 DOF
model41
24.
Simple
Model Motor
_{Step}
Response
ACSL Simulation
45
25.
Simple
Model Motor
_{Step}
Response
DADS Simulation
45
26.
Full Model Motor
_{Step}
Response
ACSL Simulation
46
27.
Motor
_{Step}
Response
Experimental Result
46
28.
Motor/ Carriage
_{Ramp}
Response
ACSLSimulation
4849
29.
Motor/ Carriage
_{Ramp}
Response
Experimental Result
50
30.
Motor
_{Ramp}
Response Increased
_{Sampling}
Period
52
ACSLSimulation
31.
Motor
_{Ramp}
Response Increased
_{Sampling}
Period
52
Experimental
Result
32.
Mechanical
_{Frequency}
Response
53
List
of
Symbols
a;
Linear
_{acceleration, ith}
elementB
Viscous
_{damping}
coefficient
c
Command
positionCj
Viscous
_{damping}
coefficientek
discrete
errorkth
sample
f
Position
feedback
Ffj
Linear
friction
G
Plant Transfer Function
H
Feedback Transfer Function
ia
Armature
currentli
Inertia
lm
System
inertia
Kd
Derivative
gain constantKe
Generated
back
EMF
constantK,
Spring
constant,
ith
elementKp
Proportional
gain constantKj
Motor
torque
constantLa
Motor
armatureinductance
Mj
Mass,
ith
elementNenc
Encoder
line
_{density}
R
ResultOuput
Ri
Radius
ofthe
ith
elementRc
Capstan
radiusRm
Motor
armature resistances
Laplace Variable
Tfj
Torsional
friction
TmotorTorque
produced_{by}
motorTres
Resultant
Torque
Ts
Sampling
time
List
of
Symbols
(continued)
Va
Applied
motor voltagevi
Linear velocity
Vb
Generated
back
EMF
voltageVm
Motor
armature
voltagexj
Linear
positionz
Z Transform Variable
9i
Angular
positionai
Angular
acceleration8(t)
Unit Impulse Function
Qi
Angular
velocity
$
Magnetic
field
flux
I
INTRODUCTION
Digital
scanners
are
_{taking}
on an everincreasing
role asinput
devices
to
computerbased
systems.
As
typical
opticaldisk
storage
capacitiesexceed
1
Gigabyte
and
_{desktop}
publishing
equipment
rivals print_{shop quality,}
digital
scanners are
_{becoming}
a popular_{computing accessory for}
data
storage,
andimage
manipulation
purposes.
_{Acting}
as
alink between
the
paper andelectronic
_{worlds, the}
input
scannercreates
an electronicimage
ofthe
paperoriginal which
is
then
processed
_{according}
to
the
user's requirements.Downstream
_{processing}
_{may include intelligent}
character recognitionfor
text
input
and subsequent
word_{processing}
_{applications,}
enhancementfor image
input,
orfile
compression
for
archival storage purposes.The
operation of_{any digital scanning}
device
requiresthree
basic
_{building}
blocks
as
shownin
Figure
1
.The
first
and
mostimportant block
containsthe
image
sensor and optical system.Together
these
elements convertthe
light
anddark
portionsofthe
originalimage into
individual picture elementsor pixels.Charged
coupled semiconductordevices
(CCD's)
are
_{typically}
usedas
image
sensorsin
_{many}
scanners onthe
markettoday.
The
second
_{building}
block
assures
that
the image
sensor_{physically}
scansthe
input
to
create atwo
dimensional
_{map}
ofthe
original.This
motion canbe
achieved
_{by}
_{moving}
the
image
sensororthe
input document
relativeto the
other_{using}
a_{variety}
ofdrive
transmission
schemes.The
last step is
then to
processthe
two
dimensional
pixelbitmap
into
usefuldata.
_{Depending}
uponthe
downstream
Original Image
(Side
View)"2 Drive Subsystem Relative Motion between Original
andImage
_{Sensing}
SubsystemChargedCoupled Device CCDimage sensor
Signal
_{Processing}
Electronics
Optics
1 Image
Sensing
Subsystem3 Image
Processing
Subsystem
FIGURE 1
:_{Building}
Blocks
ofa
Digital
_{Scanning}
Device
The
two
dimensional
_{map}
ofpixels
formed
_{by}
scanning
the
image
_{may}
be
conveniently
thought
ofas
a_{group}
ofparallelbut
separate
scanlines
eachcontaining
a
line
of pixelsimaged
at aninstant
in
time.
The
lines
oriented
parallel
to
a scanline
are
saidto
lie
in
the
fastscan
_{direction,}
because
in this
direction
the
pixels
areimaged
more
rapidly.This is
the
direction
perpendicular
to
the
motion ofthe
_{imaging}
carriage
or paper processdirection.
The
slowscandirection is
defined
as
that
whichis
perpendicular
to
the
fastscan direction
and
is
in the
direction
of motion ofthe
carriage.In
the
_{scanning}
_{lines}
of_{a cathode}
ray
tube.
The
above
definitions
are
illustrated
in
Figure
2
and are
_{further described}
_{in}
_{HI.}
>?
SlowScan Direction
FastScan Direction
Original
_{being}
scannedRelativemotion of sensor and original image scan carriage motion or process
_{direction)}
!*"rrr. m
^
individualScan lines
(
orrasterlines)are represented_{by}
columnsof thismatrix
FIGURE
2:
_{Imaging}
Definitions
Image distortion
can occurwhen eitherthe
fast
or slowscan ratesare
not constant.Perturbations
ofscanrates
in
either_{direction}
cause pixelsto
be
imaged
_{incorrectly}
and
as a result createlocal
variationsin
the
reflectance
ofthe image.
Random
fluctuations
in
onedirection
cause
onedimensional
disturbances in
the
recordedimage. This
phenomenonis
termed
"banding"and
_{typically}
refersto
image
distortion
caused_{by}
disturbances
which occurin
the
processdirection.
_{Banding}
in
scannedimages
refers
to
the
fluctuations
output
image.
_{Thus,}
to
_{faithfully}
reproduce
aninput
and
keep
customerperceptions
of_{scanning devices}
_{favorable,}
vibrationsin
the
slowscandirection
must
be kept below
perceptible
limits
[23]
.To
avoid such errors
in the
_{image,}
the
main objective of scan mechanismdesign
must
be
to
provide a
_{smooth, i.e.}
_{constant,}
slowscan velocity.However,
friction,
systemdynamics
and
otherdisturbances
cause_{velocity}
fluctuations
and
vibrationswhichmust
be
attenuatedif
image distortions
are
to
be
minimized.
The
useoffeedback
to
control slowscan motionis
_{usually}
anecessity,
but
evenwiththe
added
robustnessofa controlsystem,
vibrationand
image
motion problems can stillbe
present._{Additionally,}
downstream
processing
andimage
outputdevices
can either_{magnify}
or attenuatethese
errors and
like
consideration mustbe
givento these
additionaldevices.
Thus
some
questionsremain, namely,
how
to
determine
the
perceptiblelevel
ofslowscan motion error
and
whatis
an acceptablelevel
of vibration?Much
ofthe
published work_{relating to perceptibility}
and motionrequirements
is
concerned with_{printing}
oroutputdevices,
_{specifically}
xerographic
laser
printersMl.
However,
the
psycophysical experiments usedto
determine
human
vision_{sensitivity to image}
defects
are
still applicableto
input
_{scanning devices.}
Visual
perceptual phenomenaare
highly
dependent
upon avariety
offactors.
Variations
in
exposureduration,
illumination,
viewing
distance,
image
significantly
shift
_{perceptibility}
thresholds.
A
full
discussion
ofthese
parameters,
the
eye's
_{structure,}
andthe
_{psychology}
of perceptionis beyond
the
scope
ofthis
work.
The
reader
is
directed
to the
appropriate references[471
for
more
_{detailed information}
onthese
subjects.
When
_{imaging}
aspecific
_{pattern, if}
groups
ofscanlines are
imaged
more
closely
together
orfarther
_{apart, the}
average
reflectance ofthat
regionincreases
or_{decreases,}
_{respectively.}
_{As}
_{demonstrated}
_{by}
_{Hilzfs],}
_{the}
resulting
AL
cd
5
4
IL
r
1.5
(.

_{1.0}
3
2
(L*ALl
luminance
of linesL
luminance
ofbackground
 HOcd/m*

32
_{tL}
0.5
I 1
_{}
'_{'"I}
_{I} r"'
_{I ""I}
_{1}
0002
0.005 0 01
0.02
0.05
0.1
0.2
cycles/minuteofarc
FIGURE
3: Contrast
_{Sensitivity}
ofthe Human
Eye
to
Square Wave Gratings
8
periodiconedimensional
brightness
fluctuations
and contrast
_{sensitivity}
ofthe
human
vision system relatedto
local
_{frequency}
is
depicted
in Figure
3.
Hilz's
work presentsthe
eye as most sensitive
(with
respect
to
background
1
.2cyc/mm
(assuming
a_{300mm viewing}
_{distance).}
Bestenreiner
et
alIU
reinforce
the
supposition
that the
_{detectability}
ofline
lattice
interference
structures
in
recorded
images
relies
uponthe
contrast
ofthe
lattice,
the
luminosity
ofthe
background
and
the
spatial_{frequency}
ofthe
lattice.
The
work of_{Hamerly}
and
SpringerOl
(see
Figure
_{4),}
in
regardsto
edge
100
rThreshold
Amplitude
microns_{pp}
80
60
40
20
0
8
10
Spatial
_{Freguency}
(cycles/mm)
FIGURE
4:
_{Perceptibility}
Threshold
of
Edge Raggedness
8
detection
anddegree
ofraggedness,
alsosupports
the
presence
of_{frequency}
select channels
in
the
eye
which mightbe
appropriate
for
_{describing}
the
detectability
andthresholds
ofedge
raggedness.Figure
4
depicts
the
frequency
based
upon a
_{viewing distance}
of400mm. This
result
reinforcesHilz'
findings
and
_{describes}
the
eye as most sensitive
to
errors
ofspatialfrequencies
in
the
0.5
1
.2
cyc/mm
range.The
previous
findingsM9]
and
Figures
3
and4
reinforce
the
inclination
to
develop
a
steadystate
motion specification_{for input scanning}
devices
withthe
spatial
ortemporal
_{frequency}
ofthe
motionas
the
independent
variableand
the threshold
amplitude
in
units ofdisplacement
asthe
dependent
variable.
_{Replotting}
Figure
4
on a_{loglog}
scale
(see
Figure
_{5),}
severalregions can
be
identified
and a
linear
approximation canbe
usedto
delineate
the
threshold
values ofperceptible
error._{Furthermore,}
since
the
curveindicates
threshold
values(i.e.
errorswithamplitudes above
the
curveare
noticible
to
an_{observer)}
,
dividing
the threshold
by
asafety factor
of2
yieldsaspecification which
_{may}
be
anappropriate
design
goal.Realizing
that the
measurement
of motionis
aninevitable
_{task,}
it
is
appropriate
to
expressthe
_{perceptibility}
threshold
in terms
of_{velocity}
error(as
shownin Figure
_{6)}
to
accountfor
different
sensortypes.
Refer
to
Appendix
A
for
the
conversionfrom
adisplacementbased perceptibility
threshold
to
a velocitybased_{perceptibility}
threshold.
The
motion requirementsandthresholds
for
_{many}
commercial_{scanning}
devices
reflectthe
_{scanning}
resolutionand
oftentake
into
consideration
the
anticipated outputor
_{hardcopy}
devices.
For example,
a
xerographiclaser
Threshold Amplitude microns pkpk
^ 3 1
001 0.1
Spatial
_{Frequency}
(cycles/mm)
FIGURE
5: Motion Error
_{Detectability}
Threshold
Design Goal
requirements
are
_{typically}
_{very}
stringent
to
allowfor
this
uncertainty.It
should
be
notedhere
that
the
derivation
ofthe
specification curves
is
100
Velocity
Error%0pk
yI
Sfe_{^^^^fM^^fg^}
_{Jig}
_{gMfi}
,4lferfftFtrHt:'nT3:n^l1i:>r
;
r
Mist
*A~h:l\i\:
>!:
;. ;Design Goal '.!2^Ip.
t
4t&j*
wi'it 2D" TTTT XXt#tSE!;CI
13x1_{CAiErfejtSitip}
lit * Jtjt."fci~ f'
;
"
tyrt.{.jrtiS;'*.f;''* r.
Silt rH'~T
irn'tl'" t^^ffi^rr?
7* _{::':^:::i} ;:4...:_.:..ai;i4;. . pifp Si}:!a::::r r+f7'Trj '
0.1 10
Spatial
_{Frequency}
(cycles/mm)
FIGURE
6:
_{Velocity}
Error Design Goal
In
light
ofthe
abovediscussion,
the
characterization ofthe
_{scanning}
systemand measurementof motion
are
the
maingoals
in
an
analysis anddesign
ofa
scanning
system.The
motion goalsdefined
in
Figures 5
and6
are
the
specifications.
Mathematical
modeling
and simulationtools
canbe
usedto
predict
the behavoirof
a
systembefore
_{any}
hardware
is
constructed.These
analysis.
Experimental
verification
of_{any}
simulationis
prudentand
oftena
necessity.
The
goal ofthis
workthen
is
to
discuss
the
basic
operation ofaparticular
_{scanning}
_{system,}
_{perform}_{the}
_{analysis and}
simulationsto
predictthe
characteristics
ofthat
systemand
_{finally,}
_{to verify}
these
predictions withII
CONTROL
SYSTEM
PRELIMINARIES
It is
appropriate
to
engage
in
a generaldiscussion
ofthe
mathematicsand
basic
principles
of_{control systems}
_{that}
_{will}_{be}
_{required}_{in the}
_{simulation}process.
Control
systems are
usedin
a wide_{variety}
ofapplications
to
enablea
desired
performance
of adynamic
system under_{changing}
conditionsand are
generally lumped
into
two
categoriesopen and closed
_{loop}
systems.Open
loop
(nonfeedback)
systems are
less
robustand are_{easily}
affected_{by}
externaldisturbances.
A
closed
_{loop}
_{system,}
onthe
otherhand,
usesfeedback
to
monitor
the
output and
is
able
to
compensatefor
externaldisturbances.
Feedback
can alsobe
usedto
improve
the
_{linearity}
ofasystem and enablesit
to
be
more
robustin
_{rejecting}
externaldisturbances.
Figure
7
shows
the
basic
componentsofa closed_{loop}
control system.The
CONTnni sic;mai <;
PLANT
ACTUAL OUTPUT
COMPENSATION
UNIT
REQUIRED
'
'
OUTPUT
FEEDBACK
SENS ORS
Figure 7:Basic Components
of aClosed
_{Loop}
Control System
plant
defines
the
processor system whichis
to
be
controlledand
_{may}
containany
number ofcontrollable variables.To
obtain optimalperformance
from
adjusted accordingly.
Feedback
is
provided
_{by}
sensors
which monitorthe
variables
that
affect
the
performance
ofthe
plant.The
compensation unitmakes
acomparison
between
the
required output and
feedback from
the
plant,
determines
if
the
error
between
them
is
withinacceptable
limits
and
then
adjusts
the
control signals
to
the
plant
to
achieve
the
actual output.To
examine
how
closed
_{loop}
controlsystems
canimprove
the
_{linearity}
of asystemtiO],
consider
the
configurations
shown inFigure 8.
The inputoutput
c
G
R
NPUT OUTP
Figure
8a: Nonlinear Element
INPUT i
o
fc
G
J
' f
H
OUTPUT
Figure 8b: Nonlinear Element
withFeedback
relationship
for
nonlinear elementG
in
Figure 8a
is
R
= C2Figure
_{8b}
_{shows}
the
same nonlinear element with a_{unity}
gainfeedback
_{loop}
element
H
(H
=1).
The
equationfor
the
output_{R,}
in
terms
ofthe
input
C
for
the
closedloop
system ofFigure
8b
canbe developed
by
_{expressing}
E
and
F
in
terms
ofC
E
=CF
E
=R/G
F
=HR
(2.1)
(2.2)
(2.3)
Substituting
2.2
and
2.3
for
E
and
F
in
equation
2.1
and
_{rearranging}
yields:R
=CG
_{(2.4)}
(1
+_{GH)}
Figure
9
plots
both
the
open
_{loop}
and closed_{loop}
outputs versesthe
input
C.
The
output
R
ofthe
closed
_{loop}
caseis
more
linear
and
follows
the
input
C
more
_{closely}
due
to
the
feedback
loop.
10
6
4
2
0
R
w/ofeedback
R
w/feedback
Figure
_{9:Linearity}
Improvement
Another
feature
of_{using}
closed_{loop}
controlis
that
if
designed
properly, the
system
_{may be}
less
sensitiveto
plant variations.If the term GH
in
the
denominator
of equation2.4
is
madeto
be
suchthat GH>
_{>1,}
then the
R~
CG_
(2.5)
GH
and
further:
R
~C_
(2.6)
H
If
GH>>1,then
variations
in
G
will_{have}
little
or no affect onthe
_{ability}
ofthe
control system
to
_{effectively}
controlthe
processG
as shown_{by}
2.6.
The
Laplace
Transform
and
Transfer
Functions
Another
tool
that
is
used when_{analyzing}
and
_{modeling linear}
control systemsis
the
Laplace
transform.
This
is the
main mathematicaltool
ofthe
controlsystem
designer
because
it
simplifies
the
manipulation oftransfer
functions.
The Laplace
transform
relates afunction
oftime
_{f(t)}
to
afunction
_{F(s)}
wherethe
variable,
s,
is
in
general a complex quantity.The
transform
is
defined
_{by}
the
definite
integral
F(s)
=/.
{
f
(t)
>
= 0/f(t)estdt(2.7)
The
mainadvantage
ofthe
transform
is that
differentiation
andintegration
operations
are
changedinto
simple
algebraic operations.Consider
the
block
diagram
in
the
_{top}
portion ofFigure
10.
The
generic
function
H
can represent_{any}
mathemathical_{relationship}
between
the
input
and
output.The
function
H
couldbe
a
constant value_{indicating}
a
linear
relationship,
a
firstorder
pole ora
complexnonlinearfunction.
_{Applying}
the Laplace
transform
to the
relationsdepicted
by
this
block diagram
wex(t)
H
y(t)
INPUT _{OUTPUT}
\
Laplace Transform
x(s)
TRANSFORMED EXCITATION
H(s)
_{y(s)}
SYSTEM TRANSFER FUNCTION rRANSFORMED RESPONSE
Figure
10: Block Diagram
Transfer Function
y(s)
=_{H(s)x(s)}
(2.8)
Once
the
transfer
function
is
manipulated
to
the
properform
in
the
s_{domain,}
the
response
_{y(t)}
ofthe
systemto
an excitationx(t)
may
be
recovered
_{by}
the
inverse
Laplace Transformation
of_{y(s)}
represented
by:
y(t)
=Z.1
{y(s)}
=M
{
H(s)
_{x(s)}
}
(2.9)
Equation
2.9
canbe
usedto
derive
the
response
of_{any linear}
timeinvariant
system subjected
to
an_{arbitrary}
excitation.The Laplace
transform
ofthe
unit
impulse, 8(t),
is
8(s)
=J8(t)estdt
=
estt
=00/8(t)dt=
1
(2.10)
where
the
_{meanvalue}
_{theorem}
_{has been}
_{used.}_{Inserting}
_{x(s)}
=8(s)
=1and
y(t)
=h(t)
_{into}
2.10,
we obtain
h(t)
=M
{H(s)}
(2.12)
Thus
the
impulse
response
_{function, h(t),}
of a constant coefficientlinear
system
is
equal
to the
inverse
Laplace
transformation
ofthe
transfer
function.
Furthermore,
the
unitimpulse
andtransfer
function
representaLaplace
transform
pair.Tables
oftransform
_{pairs,}
applicable_{fundamentals,}
andtheorems
ofLaplace
transform
_{theory}
canbe
found
in References 101 3
and
will
not
be
covered
here.
Block Diagram
Reduction
CDH
G,G2"3
1 + G,G,H_{2u3n2}
Minor
Loop
reducedtosingleblockFigure 11b: Block Diagram Reduction
Block
diagrams
are
_{very}
useful when_{modeling linear}
systems,
however
the
analysis can getcumbersome
for
a
system with_{many blocks.}
The
reduction
ofa
large
systemis
accomplished_{using}
equation2.4.
The
minorfeedback
_{loop}
c +
a
y~*Gi
\^J^_{G*}
*_{G'}
i
1
H2
*K
1
H,
* 1
Figure
11a: Minor
_{Loop}
Feedback
shown
in
Figure
11b.
Further
reduction
ofthe
_{remaining}
_{loop}
in Figure 11b
is
represented
by:
R
=_{Gi G2G3}
1
+_{G2G3H2}
+_{G1G2G3H1}
(2.13)
Control Laws
The
controllogic
elements
incorporated into
the
compensation unitare
designed
to
act uponthe
errorsignalto
produce
the
control orcompensation
signal.
The
algorithmthat
carries outthe
production ofthe
control signalis
the
controllaw
or compensation action.The
main controlobjectives are
to
minimize
_{steady}
state
_{error,}
minimize_{settling}
time
andachieve
othertransient
specifications such as_{minimizing}
the
maximumovershoot.
Two
control
laws
that
form
the
basis
of_{many}
control systemsare
twoposition
and
proportional control.
As
the
name_{implies,}
twoposition
controlis
either
_{fully}
on or
_{fully}
_{off,}
_{depending}
uponthe
magnitude ofthe
error.
Proportional
control
implements
a
control signal proportionalto the
errorsignal
to
correct
Other
control
law
features
are possible
_{by}
further
mathematical manipulationof
the
error signal.
_{By}
_{using}
the
derivative
and/orintegration
ofthe
errorsignal
in
conjunction
withproportional
_{control,}
_{many}
othertypes
of controlare possible.
The
transfer
function
_{X(s)}
for
aproportionalintegralderivative
(PID)
controller
is
given
_{by}
X(s)
=M(sl
=_{KP}
_{+}_{Ki}
_{+}KDs
(2.14)
E(s)
swhere
_{Kp, K,}
and
_{Kq}
are
the
_{proportional,integral,}
and
derivative
constants
respectively.
The
constants
_{Kp,}
_{K,}
and_{KD}
mustbe
chosen_{carefully}
to
avoidinstability
and
to
yield an optimal response.References
10
and12
explainthe
advantages
of_{using}
varioustypes
of controlfor
first
and
second ordersystems.
DIGITAL
CONTROL SYSTEMS
Digital
controlsystems are
_{fundamentally}
different
from
continuoussystems
in
that
adigital
computeris
usedto
performthe
controllogic
operationsthat
are
requiredto
act
uponthe
error signal.Thus sampling
of acontinuous
variable
and
the
generation ofa
digital
compensation signalrepresent a
fundamental
departure
from
the
continuous systemanalysis
discussed
so
far.
The
respectiveeffects
upon systemdynamics
require
carefulattention.
The
sampling
period and
type
of control mustbe
selected_{properly}
to
prevent
aliasing
ofthe
variablesand
possibleinstability.
The
control systemdiagram
changes_{slightly}
withthe
introduction
ofthe
change
_{may}
take
place.
Consider
the
digital
control system shownin Figure
Interface
Input +
Interface
<;
Output
^V
A/D Computer _{D/A} PlantJ^
>
Sensor
Figure
12: Digital Control System
12.
The interface
atthe
input
ofthe
computeris
an analogtodigital_{(A/D)}
converter,
and
is
required
to
convertthe
continuous error signalto
aform
that
is
readable
_{by}
the
computer.At
the
outputend
is
a_{digitaltoanalog}
(D/A)
converterthat
convertsthe
digital
compensation signalinto
aform
necessary
to
drive
the
plant.The
maindifference
from
a continuous systemis
that
some
portion ofthe
_{loop,}
_{namely}
the
controllogic,
is implemented
digitally.
Discrete
Time
Systems
Sampling
a continuoussignal canbe
represented_{by}
the
_{closing}
and
_{opening}
ofa switch
as
shownin Figure 13. The
continuous signal_{y}
(t)
is
sampled
uniformly
every T
secondsto
yieldthe
discrete
time
representation y*(t).The
sampled sequence
y*(t)
may
be
representedby
atrain
of unitimpulse
functions
_{occuring}
at_{sampling}
time
kT,
eachhaving
a
strength equalto the
o
o^Y
o
oContinuous
signal
y(t)
Discrete
signal y*(t)
Figure
13:
Uniform Sampler
withPeriod T
y*(t)
=_{y(0)8(t)}
_{+}_{y(T)8(t}
T)
+_{y(2T)8(t2T)}
+=
E
y
(iT)
8(t
iT)
fori
=_{0,1,}
.,
N
(2.15)
The
sampler represented
in
Figure
1
3
does
notexist
in
real controlsystems
because
the
impulse
funtion
exists_{only for}
abrief instant
ofthe
_{sampling}
interval
T. For
the
remainderofthe
_{sampling}
interval
the
value ofthe
discrete
signal
_{y*(t)}
is
equalto
zero.In
real_{systems,}
adata
hold
is
usedto
make
anO
^ CYT
ZeroOrder
Hold
Continuous
Discrete
signal_{y*(t)}
Digital
signal_{y(t)}
Representation
Y(t)
y(t)
_{I}
y*(t)_{f}
Y(t)

.,
tin,
"t
0
T
2T
3T
0
T
2T
3T
Figure 14: Sampler
withZero
Order Hold
iT
<
t
<
(i
+1)T.
A
zeroorder
hold
is
the
most
common althoughfirst
andfractionalorder
data holds
are also
used.A
samplerand
zeroorderhold
andthe
associated
outputs are shown
in
Figure
14.
The
Z
Transform
The Z
transform
is
usedin the
analysis
of_{discrete}
time
systemsfor
the
analogous reasons
that
the
Laplace
transform
is
usedin linear
continuoussystems
itmakes
the
analysis
easier._{By}
_{taking}
the
Lapace
transform
ofthe
impulse
representation
of asampled
signalas
given_{by}
equation2.1
5
wearrive
at:Y*(s)
=_{y(0)}
_{+} y(T)eTs _{+} y(2T)e2Ts +_{(2.16)}
In
equation_{2.16,}
the
_{shifting property}
ofthe
Laplace
transform
has
been
used.
Using
the
definition
z = eTs
(2.17)
Y*(s)
may
be
rewritten as afunction
ofz:Y*(z)
=_{y(0)}
+_{y(T)}
_{1}
+_{y(2T)}
_{1}
+. . .(2.18)
z z2
and
_{Y*(z)}
is
the Z
transform
ofthe
function
y*(t).As
shownin
equation
_{2.18,}
1/z
is
a
_{delay}
operatorand
represents atime
_{delay}
ofT. Thus the
powerseries
Y*(z)
=2 +3+6
2
+7 +..._{(2.19)}
corresponds
to
sampled values
_{y(t)}
=_{2, 3, 6,}
2,
7.
. .at
t
=0,
T, 2T, 3T,
4T.
.Tables
of
ztransform
pairs and
further discussion
ofthis
discrete
transform
can
_{be found}
in
references
10,13
and
14.
Difference
Equations
The
control algorithms
ofadigital
control system areimplemented
_{by}
discrete
time
approximations
oftheir
continuous
counterparts.This is
achieved
_{by}
using
difference
equationapproximations
for
the
derivative
and
integral
parts
ofthe
controllaw.
First
difference
approximations are
often usedto
represent
the
slope
_{(derivative)}
ofthe
error signal_{e(t)}
at
t
=kT:
forward
difference
backward
difference
_{(2.20)}
central
difference
2T
Similarily
the
secondderivative
ofthe
error signal_{e(t)}
_{may}
be
approximated
by
using
seconddifference
equations(see
Reference 14).
To
approximatethe
integral
term
/e
_{dt,}
the
last
approximation ofthe integral
up to time
(k1)T,
given as_{vki}
is
used.The
approximation_{vk}
_{up to}
some
time
kT
_{may}
be
written as:Vk
_Vk
_} +jek1
forward
rectangle rule
Vk
_Vk
^
+_{jek}
backward
rectangle rule(2.21)
vk
=_{vu}
_{+T(eki}
+ ek)/2trapezoidal
ruleek
11
ekT
ek
ik1
T
Any
difference
equation control algorithm
_{is essentially acting}
as a
digital
filter.
It
is
quite
different
from
an
_{analog}
_{filter,}
since
it is
implemented using
a
set
ofdigital
computer
instructions
ratherthan
a
physical circuit.As
anexample, the ouput,
Uk,
ofa
PD
_{(proportionalderivative)}
digital filter
_{using}
backward
difference
is
given_{by}
Uk
=_{KPek}
_{+}_{Kn(ek}
_{eki)}
(2.22)
T
For
further
information
on controlsystems
anddigital
control consultIll
_{DESCRIPTION}
_{OF}
_{THE PHYSICAL}
_{SYSTEM}
The
physical system under
investigation
is
shown_{schematically}
in
Figure
1
5.
An
original
is
placed
onthe
platen glass positioned at a
fixed height
in
the
zdirection
above
the
carriage.
The
carriage provides
the
relative
_{scanning}
motion
in
the
_{xy}
plane
_{between}
_{the}
_{original}_{and}
_{the}
_{image}
_{sensor.}_{The}
motion
is
achieved
_{by}
asinglecapstan
dualcable
system(Figure
15).
The
Drive Cable
IdlerPulhes
Capstan
Motor/Encoder
FIGURE
15: Physical System
carriage
rides
ontwo
precisionground carbon steelshafts
whichare
fixed
to
the
machine
base.
Five
bearing
surfaces supportthe
carriage
onthe
shafts.
This
configuration prevents rotation ofthe
carriageabout
the y
and
xaxes.
from
the
drive
motor
(4:
1
drive
ratio).
The
feedback device
is
a
1000
lines/rev
incremental encoder.
Two
encoder
_{outputs,}
90
degrees
out
of_{phase,}
yield aquadrature
output
ofposition
_{and an effective resolution}
of4000
lines/rev.
A
4pole
_{permanentmagnet}
_{DC}
_{motor}
_{is}
_{the}
_{prime mover.}
_{Specifications}
_{for}
the
drive
components
and
_{manufacturer's}
_{literature}
_{on}_{the}
_{motor/encoder}
package are given
in Appendix
B.
Figure
16
shows a
block diagram
ofthe
control andmechanical
systeminteractions.
The
control system produces
the
_{torque,}
_{Tmot0r,}
whichsets
the
mechanical system
in
motion.
The
dynamics
ofthe
mechanical system affect
the
motor position
whichis
the
_{primary feedback}
variable.The
controlMicroprocessor
Position Xcarriage
Vcarriage
Acarriage Position _{Error}
Command
^^^ Digital
i _{PWM}
Filter/ i _{amplifier/}
DtoA
j
Motor Converter iCircuit
'_{motor}
Motor/Capstan
Carriage Dynamics
+s
. _ _ _ J
^motor
Sack EMF
Feedback
Position
^motor
Encoder
hardware
is
_{based}
onthe
Intel
80286
microprocessor
which residesin
a
personal
computer.
A
_{prototype expansion}
_{board handles}
decoding
ofthe
encoder
_{output,}
_{digital}
_{to}
analog
conversion,
control
_{loop}
_{timing,}
andassigns
communication addresses.
_{Initialization}
ofvariables and
generation ofthe
position and error commands are encoded
in
software
(C
language). Since
the
position and error commands reside
_{in software,}
aninherent
_{flexibility}
in
customizing
and
_{optimizing}
these
portions
ofthe
control systemis
allowed.An
_{external pulsewidth modulated amplifier}
circuit,
based
uponthe
L298
integrated
circuit,
amplifies
the
errorcommand
to the
motor.Operationally,
the
motoris
commanded
to
ramp up
as shownin
Figure
17,
Velocity
(m/s)
Velocity
requiredfor 50% magnificationNominal
_{Velocity}
Velocity
requiredfor 200%magnificationImagesensor
activated
Time
(seconds)
and must
then
settle
to
anacceptable
constant_{velocity}
before
the
image
sensor
is
activated.
The
nominal
_{velocity}
ofthe
carriage
depends
uponthe
speed and resolutionof
the
_{imaging}
sensor.The
nominal
slowscan_{velocity}
forthe
_{scanning}
device
is 208
mm/s.This
correspondsto
a motor_{velocity}
of51
rad/s.To
provide
for 50
200%image
magnification, the
carriage mustbe
ableto
scanone
_{half}
_{and}
_{twice}
_{the}
_{nominal}velocity
(104
and
416
mm/s respectively).The
motion requirements
_{during}
the
steadystateor_{imaging}
portion ofthe
IV
_{MATHEMATICAL}
MODELING
For
the
_{scanning}
system
under_{consideration, two}
distinct
subsystemscanbe
identified.
The
mechanical
subsystem
includes
several elements  scancarriage, cables,
drive belt
and motor.
The
dynamics
ofthese
lumped
elements are
_{readily}
described
_{by}
the
usuallaws
of classical mechanicsandthe
subsequent system
ofdifferential
equations.The
control subsystemelements,
however,
are considered
_{individually}
due
to the
nonlinearbehavior
ofsome elements.
Each
function
ofthe
control subsystemis
represented
_{by}
ablock.
_{By}
_{using}
the
appropriateassumptions,
a_{relationship}
between
the
input
and
ouput variables ofthat
block
canbe
determined.
The
reduction of
the
control systemto
a mathematical modeldemands
that
eachelement and
its
behaviour be
_{thoroughly}
understood.This
sectiondiscusses
in
detail
how
the
control and mechanicalsubsytems are modelled.Controls
Model
The
initial step in mathematically
modeling
the
control nucleus ofthe
scanning
systemis
a
more specificdefinition
ofits individual
controlelementsMO14].
Figure 18
elaborates onthe
singleblock representation ofthe
control system_{initially}
givenin Figure 16.
Each
block
representsa
mathematical
_{relationship}
between input
and
outputvariables.All
variablesdepicted
are timevarying.
This
systemhas direct digital
control ofthe
motorposition,
01,
and providesthe
_{forcing}
function
torque
necessary
for
input
to
the
system ofdifferential
equationsthat
describe
the
dynamics
ofthe
mechanical system.
In
turn,
the
dynamics
ofthe
mechanical systemaffect
the
the
_{secondary}
_{loop}
_{variable}
_{of}_{motor shaft}
velocity,G)i
.The
control systemPC
based
Microprocessor1 Position 1 _{Command} ^ e
C>
i i i i up U PD Control LawKd
kp
v3
vm
Motor Circuit 1. 1 ,
h
DAC PWM_{(!)}
+
\
*
'
Ts
1
+"
vb
BackEMF!
f KaCO,

Position ! Feedback Encoder ' _{from} motorUi6
s4Nenc
e,
\
inFigure 18:Controls
Block Diagram
consists of
a
digital
section and an_{analog}
section.The
microprocessorsamples
the
motorencoder_{every}
_{Ts}
seconds_{(nominally}
_{600psec)}
to
obtainthe
position_{feedback,}
_{f,}
and comparesit
to
a positioncommand,
c.The
resulting
discrete
error,
e,
given_{by}
e =
fc
(4.1)
isthen
_{digitally}
filtered
using
a proportional/derivative
(PD)
controllaw.
The
discrete
corrected or compensatederror, up,
whichis
outputto the
Digital to
_{Analog}
Converter,
(DAC),
canbe
expressed
_{using}
the
present
valueThe
result
is
expressed
as
up
=Kpek
+_Kd_[ekeki]
(4.2)
Ts
where
_{Kp}
is
the
proportional
gainconstant and
_{Kd}
is
the
derivative
gainconstant.
This
portion
ofthe
control system
is
encodedin
software, thus
the
derivative
and proportional
gainconstants
canbe
adjustedto
optimize systemperformance.
The
discrete
compensated
_{error,}
_{up,}
computed
_{by}
the
microprocessor_{every}
_{Ts}
seconds
is
then
converted
to
an_{analog}
_{signal, u,}
_{by}
the
D
to
A
converter,
DAC.
The
modelled
DAC
has
aneightbit
resolutionand
maximum output voltageof
3.3
volts.Since
it is
adiscrete
_{device,}
the
DAC
can_{only}
recognize
alimited
range of
digital input
valuesbeyond
whichits
outputbecomes
saturated.
Thus
the
range ofdiscrete
compensated error enabled_{by}
the
DAC is
28/2
or128
encoder counts.A
valueabove
128or
below128
willsaturatethe
DAC
and
limit
the
output voltageto
the
amplifier.Since
the
D to A
converter
must calculate an
_{analog}
voltage,
a
computational_{delay}
of10
microseconds
was
also
included in
the
representation ofthis
control element.Amplification
ofthe
_{analog}
error_{signal, u,}
is
accomplished
_{by}
aPulse Width
Modulated
amplifier(PWM)
witha
_{switching}
_{frequency}
of20khz.
This
type
of