Indonesian J
ournal of Ele
c
trical Engin
eering and
Computer Sci
e
nce
Vol. 1, No. 3,
March 20
16, pp. 575 ~ 5
8
2
DOI: 10.115
9
1
/ijeecs.v1.i3.pp57
5-5
8
2
575
Re
cei
v
ed
No
vem
ber 3
0
, 2015; Re
vi
sed
Febr
uary 10,
2016; Accept
ed Feb
r
ua
ry
22, 2016
Comparison Methods for Conv
erting a Spindle Plant to
Discrete System
M Khairudin
Dep
a
rtment of Electrical E
ngi
neer
ing, Un
iver
sitas Neg
e
ri Yo
g
y
akarta, Ind
o
nesi
a
Karan
g
mal
a
n
g
, Yog
y
akarta, 5
528
1, telp. +
6
2
-
274-
548
16
1
e-mail: mo
h_k
hair
udi
n@u
n
y
.
a
c.id
A
b
st
r
a
ct
T
h
is study
pres
ents co
mparis
o
n
meth
ods
in t
he c
onver
s
i
on
of a sp
in
dle
pl
a
n
t in
ord
e
r to
o
b
tain
an
accurate discr
ete system
when c
o
m
par
ed to a continuo
us system of s
p
indle. Th
e ac
curate
c
onv
ersion
results of
the continuous
sys
tem
into discr
ete form ar
e requir
ed for
implementing t
he
control system of
spin
dle.
Com
p
arison m
e
thods
that
w
ill
be c
onducted to convert the
c
o
ntinuous system of spi
n
dle
pl
ant
into discrete
sy
stem
t
h
rou
g
h
zero-or
der ho
ld (Z
OH),
fi
rst-order h
o
ld
(F
OH), impuls
e
invar
i
a
n
t dicr
etisatio
n,
tustin (bili
ne
ar)
,
and pol
e-
z
e
r
o
match
i
n
g
methods. T
he p
e
rformanc
es of each metho
d
in the conv
ersi
on
process h
a
ve
bee
n pres
ente
d
.
Convers
i
on
performanc
es
of continous
spin
dle p
l
a
n
t into discrete fo
r
m
usin
g F
O
H
me
thod, sh
ow
ed
mor
e
acc
u
rate
co
mpar
ed
to
other
methods.
Perfor
ma
nces
of the
c
onv
ers
i
o
n
accuracy
of F
O
H method
ha
ve be
en
eva
l
u
a
ted i
n
ter
m
s
o
f
transient r
e
sp
onses
an
alysis
that clos
ed s
i
mi
la
r
results with a c
ontinuous system
of
spindle plant. At the clos
ed similar tr
ansient resp
onses for the disc
ret
e
system
usi
n
g FOH method s
h
ow
the fi
nal
val
ue, ti
m
e
to peak,
percertages
oversh
oot and setlling
ti
me ar
e
0.863 v, 0.91
0 s, 0 % and 0.5
50 s respectiv
e
ly.
Ke
y
w
ords
: Co
mp
ariso
n
, conti
nous, di
scr
ete, convers
i
on,
me
thod
1. Introduc
tion
The digital si
gnal processi
ng wa
s beg
a
n
with
explo
s
ive gro
w
th i
n
the deca
d
e
of th
e
1960
s,
whe
n
re
se
arche
r
s
discovered
h
o
w to
u
s
e
re
cursive
digita
l filters for si
mulating
anal
og
filters [1]. The
improveme
n
t of conversio
n
techni
que
s
from co
ntinuo
us sy
stem int
o
discrete fo
rm
is
still requi
re
d to
su
ppo
rt t
he im
pleme
n
tation e
s
p
e
ci
a
lly whe
n
u
s
in
g comp
utatio
nal al
go
rithm
s
.
In the curren
t issu
e, almo
st all
control
system
s
u
s
in
g
digital cont
rol system b
a
se
d
computi
ng
algorith
m
. In
fact most of the system i
s
gene
ra
lly sh
a
ped in co
ntin
ous pla
n
t. It
will req
u
ire a
n
y
techni
que
s fo
r co
nverting i
n
to a discret
e
form. Thu
s
it would be
in harm
ony with the use of
digital co
ntrol
system
s.
Most ind
u
st
ri
al pro
c
e
s
se
s esp
e
cially
engin
eeri
ng
system
s a
r
e
con
s
tru
c
ted
using
discrete
mo
dels.
Chemi
c
al p
r
o
c
e
s
se
s an
d el
e
c
trical
system
s are typi
cal
example
s
.
The
recursive features of these disc
rete models and the re
cent availability of
high performance
low
co
st mi
cro
c
o
n
trolle
rs hav
e en
able
d
a
ne
w
con
s
id
eration
of
control
and
a
nalysi
s
of th
ese
system
s. Shieh and
Wan
g
[2] present
ed metho
d
s
for model
co
nversi
on
s of contin
uou
s-ti
me
state-spa
c
e e
quation
s
an
d disc
rete
-time
state-spa
c
e e
quation
s
, ma
ny well-d
e
vel
oped the
o
re
ms
and metho
d
s in either con
t
inuou
s or di
screte dom
ai
ns ca
n be effectively appli
ed to a suitable
model in eith
er dom
ain.
Melwin a
nd F
r
ey [3] descri
bed continu
o
u
s-tim
e
to discrete
-time co
nversi
on with
a novel
para
m
etri
sed
s-to-z-pla
ne
mapping. A
param
etri
se
d s-to
-z-pl
a
n
e
map was i
n
trodu
ce
d, where
the conventio
nal bili
nea
r m
ap a
nd
ba
ckward
and
forward Eul
e
r ru
les
app
ear a
s
spe
c
ial
cases.
With a
sim
p
l
e
techniq
ue
for ap
plying
this ma
p to
adaptively re
duce tru
n
cation e
r
ror i
n
t
he
contin
uou
s-ti
me to discret
e
-time conve
r
sio
n
probl
e
m
. In order to conve
r
t co
ntinou
s sign
a
l
to
discrete, Kell
er et al [4]
e
x
plored
a me
thod for
th
e discrete
-time simulatio
n
of contin
uou
s-ti
me
sigma
-
d
e
lta modulato
r
s. Via
the appli
c
ation
of
lifting
, co
rre
ction
value
s
for ea
ch state
vari
ab
le of
a mod
u
lator are
cal
c
ul
ated, whi
c
h
subsequ
ently
are
used to
calib
rate
on
line the
s
e
state
variable
s
du
ri
ng a discrete-time simulatio
n
of the conti
nuou
s-tim
e
system.
Korlin
cha
k
a
nd Coman
e
scu [5] explai
ned di
screte
time integration of ob
se
rv
ers with
contin
uou
s f
eedb
ack b
a
s
ed
on Tu
stin'
s
metho
d
with vari
able p
r
e
w
arping, imp
r
ov
ed
improvem
ent
to trape
zoid
al integration
the state
e
q
uation
s
of ob
serve
r
s are i
n
tegrate
d
u
s
i
ng a
discrete
-time
filter that is prewa
r
pe
d as
a
function of the drive'
s ope
rating frequ
en
cy.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 25
02-4
752
IJEECS
Vol.
1, No. 3, March 20
16 : 575 – 582
576
Cha
r
le
s [1] also expl
ain
ed that Hu
re
wicz
theo
ry
sho
w
e
d
ho
w a z-tran
sform coul
d
descri
b
e
a
sampled
data
tran
sfer fun
c
tion. O
n
th
e
othe
r h
and
Cag
a
tay et a
l
[6] propo
se
d a
definition of the
di
screte
fraction
al
F
o
urie
r
tr
an
sform that g
ene
ralises the
d
i
screte
Fou
r
i
e
r
transfo
rm i
n
the sa
me
se
nse th
at the
contin
uou
s fraction
al Fo
urier tra
n
sfo
r
m
gene
rali
se
s
the
contin
uou
s o
r
dina
ry Fo
uri
e
r tran
sform.
Xiao et
al
[
7
] de
sign
ed t
he filterin
g m
odule
with fil
t
er
usin
g
hig
h
preci
s
ion and wide dynami
c
resp
on
se
range,
can m
eet the requi
rements of sp
eed
and preci
s
io
n
of laser gyro demod
ulation
aero
s
pa
ce fi
elds.
Other
wi
se, several co
ntrol
method
s are
fo
rmul
ated
u
s
ing
continuo
us m
odel
s fo
r whi
c
h
several theori
e
s and p
r
a
c
tical metho
d
s
have been d
e
v
eloped. A large practi
cal
control metho
d
s
con
s
i
s
t of both continu
o
u
s-tim
e
and
discrete
-t
ime
sub-syste
m
s. For effecti
v
e analysis
and
synthe
sis of these com
p
o
s
ite system
s it is often nece
s
sary to
conv
ert a discrete
sub-syste
m
to
an equivale
n
t
continuo
us model. Gah
i
net and
Sh
ampine [8] p
r
esented the
framework for
modelin
g lin
ear time
-inva
r
iant (LTI)
system
s wi
th
de
la
ys
, th
e
de
la
ys
in
fe
edb
a
c
k
lo
op
s
ar
e
gene
ral
eno
u
gh for mo
st control
appli
c
a
t
ions, a
nd le
n
d
s it
self well
to com
pute
r
-aided
analy
s
i
s
.
Antonie and
George [9] d
e
velope
d the
first frame
w
ork
of system
approxim
atio
n that applie
s
to
both
di
screte and contin
uo
us syst
em
s b
y
developin
g
notion
s
of ap
proximate l
a
n
guag
e in
clu
s
ion,
approximate
simulatio
n
, a
nd a
pproxim
ate bi-sim
ulat
ion relation
s.
Othe
r way,
Indah
et al [
1
0
]
pre
s
ente
d
th
e co
mpa
r
iso
n
between t
he conventio
nal
pa
rticle f
ilter and
pa
rticle filter
with
Gau
ssi
an wei
ghting metho
d
s, the wei
g
h
t
was calc
ula
t
ed in each p
a
rticle, the re
main pa
rticle’
s
weig
ht wa
s calcul
ated u
s
in
g the Gau
ssi
an wei
ghting.
Ho
wever,
mo
st of the
pu
bl
ishe
d
work
o
n
buil
d
ing
filters to
co
nvert contio
nou
s
sign
al to
discrete
signal with limited possibility for co
m
pari
n
g several m
e
thods.
Moreover, not much
works on co
mpari
s
o
n
of contin
ou
s co
nversi
on
in
to
discrete fo
rm u
s
ing
with
out meth
od,
ZOH,
FOH, Impul
se-Invari
ant, Tustin App
r
oxi
m
ati
on and
Zero
-Pole M
a
tchin
g
Meth
ods. Ge
neral
y
it
can
be seen
whe
n
ZO
H
method a
r
e
use
d
by mo
st rese
arch
ers for co
nverti
ng a
contino
u
s
system into a
discrete sy
st
em. Includin
g
matlab by
default also u
s
e
s
the ZO
H method. But, it
is
necessa
ry to clarify that
wheth
e
r Z
O
H metho
d
is the most a
c
curate
meth
od to convert a
continuous system into a discrete
syst
em. This
study will examine the comparison of vari
ous
method
s i
n
o
r
der to o
b
tain t
he m
o
st
accu
rate m
e
thod
f
o
r
co
nversion
of
a co
n
t
in
uou
s
s
y
s
t
em in
to
a discrete system. This
is a
challe
ng
ing ta
sk for
finding the
close
s
t metho
d
com
p
a
r
ed
to
contin
uou
s
system is affe
cted
by seve
ral facto
r
s.
Th
is p
ape
r p
r
e
s
ents
com
p
a
r
i
s
on
metho
d
s for
conve
r
ting th
e spindl
e co
ntinou
s plant
into disc
rete
system usi
n
g several me
thods to find the
most a
c
curat
e
method. It is found that the seve
ral m
e
thod
s ha
s n
o
t been explo
r
ed to u
s
e in
the
same
pla
n
t to find
out th
e cl
osest th
e
spi
ndle
co
ntinou
s pla
n
t
with di
screte
system.
Usi
ng
without
meth
od, the
ZO
H, FO
H, Im
pulse-Inva
r
ia
nt, Tustin
A
pproxim
ation
and
Ze
ro
-Pole
Matchin
g
Met
hod
s to obtai
n the clo
s
e
s
t
discrete
sy
st
em co
mpa
r
e
d
int
o
co
nt
in
ous
sy
st
em.
For
performance assesment,
the
spi
ndle continous
plant
will compared into disc
rete
system in terms
of transie
nt resp
on
se
s an
alysis. The a
nalysi
s
re
sult
s sh
ow that b
e
tter system
perfo
rman
ce
and
clo
s
e
s
t
simila
r wit
h
c
ont
ino
u
s sy
st
em are achi
eved wi
th FOH meth
od.
2. Rese
arch
Metho
d
In this work
use
d
seve
ral
methods for conv
ertin
g
the co
ntinou
s of spindle p
l
ant into
discrete
syst
em. The met
hod
s that will
be used in
this work con
s
ist of ZOH, F
O
H, and Imp
u
lse
-
Invariant, Tustin Approxim
ation and Ze
ro-Pole Mat
c
h
i
ng Method
s. This work also pre
s
ente
d
the
conve
r
si
on re
sult witho
u
t any method.
In the ea
rly p
a
rt will
be
pre
s
ente
d
comp
arison
s b
e
tween the
meth
ods
of ZO
H,
FOH
and
Impulse
-Invariant. The
ZO
H, FO
H,
and
impulse-i
nv
ariant
meth
od
s prod
uce exact
discretizatio
n
s
in the tim
e
d
o
main
for:
(1
)
system
s
wit
hout ti
me
del
ays, (2) sy
stems with
tim
e
del
ays
on
the
inputs an
d o
u
tputs
only (no inte
rnal
d
e
lays). B
e
ca
use
of the
e
x
act mat
c
h, i
t
can
u
s
e th
ese
discreti
sation
method
s for
time-dom
ain
simulatio
n
s. I
n
this context, exact discre
tisation me
an
s
that the time resp
on
se
s of the
continu
o
u
s
an
d di
screti
sed m
odel
s
match exa
c
tly for the follo
wing
cla
s
ses
of in
put sig
nal
s: (1) Stairca
s
e i
nputs fo
r Z
O
H, (2
) Pie
c
e
w
ise line
a
r i
n
puts fo
r FO
H, (3)
Impuls
e
trains
for impulse IMP. For
s
y
stems
with
int
e
rnal
del
ays
(delay in
feed
back l
oop
s), t
h
e
ZOH
and F
O
H meth
ods re
sults i
n
ap
pro
x
imate discre
ti
sation
s. An i
n
ternal
delay
is illu
strated
in
the followin
g
Figure 1.
For such systems, the co
ntinou
s to discrete (c2
d
)
perfo
rms the
following a
c
tions to
comp
ute an
approximate
ZOH o
r
FO
H discret
ization [11]: (1)
De
comp
ose
the delay
as
Evaluation Warning : The document was created with Spire.PDF for Python.
IJEECS
ISSN:
2502-4
752
Com
pari
s
o
n
Method
s for
Con
v
e
r
ting a
Spindle Plant
to Discrete S
ystem
(
M
Khair
u
d
i
n)
577
s
kT
with
s
T
0
. (2) A
b
sorb
s th
e fra
c
t
i
onal
delay
into
H(
s)
. (3) Dis
c
r
e
tises
H(
s)
to
H(z
)
.
(4
) Rep
r
e
s
e
n
ts the i
n
teg
e
r p
o
rtion
of
the delay
s
kT
a
s
an inte
rn
al
discrete
-time
delay
k
z
. The final di
scretise
d model
app
ears in th
e fo
llowing
Figu
re 2. The im
p
u
lse
-
inva
rian
t
method do
es
not sup
port systems
with internal d
e
lay
s
.
Figure 1. An internal d
e
lay
Figure 2. The
final discretized model
Tustin
App
r
o
x
imation an
d
Zero-Pol
e
Matchin
g
M
e
thods.
Wh
en
discretisi
ng
a sy
stem
with time del
ays, the Tust
in and match
ed met
hod
s: (1)
Roun
d an
y time delay
to the neare
s
t
multiple of the sam
p
ling ti
me, (2) Ap
proximat
e the fractio
nal time
delay. Whe
n
discreti
sing
tf
and
zpk
model
s usin
g the Tu
stin or m
a
tch
ed metho
d
s,
c2d first ag
gregate
s
all in
put, output,
and input
-out
put (i/o) dela
y
s into
a sin
g
le input-out
put delay
TOT
for each cha
nnel
. The c2d
then a
p
p
r
oxi
m
ates
TOT
as a
Thira
n
filter
and
a
ch
ain
of unit
delay
s in
the
sam
e
way a
s
descri
bed fo
r each of the time del
ays in
statespa
ce m
odel
s [11].
2.1. The Spindle of La
th
e Machine
The
spi
ndle
o
f
lathe m
a
chi
ne in
this
work
wa
s
pre
s
e
n
t
ed in
Figu
re
3. The
ri
g
co
nsi
s
ts
of
three mai
n
p
a
rts: a
spin
dle, sen
s
o
r
s
an
d a pro
c
e
s
so
r. The spi
ndle
is rotate
d by the main m
o
to
r,
hold
s
the
cut
t
ing tool, whi
c
h
cuts the
work
pie
c
e, then the
cutting force
s
a
r
e
gene
rate
d which
effects the
sp
indle a
c
cura
cy directly. Th
e system
id
e
n
tification for
the spin
dle of
lathe machin
e
wa
s imple
m
e
n
ted withi
n
th
e Matlab a
n
d
Simulink e
n
vironm
ent on I
n
tel Pentium
1.80 G
H
z
an
d
2.00 GB RAM as detaile
d in M. Khairudin [12].
The data obtain
ed in the form of collectio
n of
variation
s
wit
h
the tacho
output voltag
e. The
tran
sfer functio
n
form is
obtai
ned by Matl
ab
prog
ram. Th
rough the ide
n
tification dat
a obtained
transfe
r functi
on. Experime
n
tal works were
con
d
u
c
ted u
s
ing the experi
m
ental rig fo
r
s
y
s
t
em identific
a
tion results.
Figure 3. The
Experimental
Setup of a Spindle
For the spindl
e without cutting pro
c
e
s
s,
the tran
sfer fu
nction
can b
e
writen a
s
:
H(
s)
S
e
k
z
H(
s)
S
e
T
a
il Stock
Bed
H
e
ad Stock
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 25
02-4
752
IJEECS
Vol.
1, No. 3, March 20
16 : 575 – 582
578
1.648e005
s
2.487e004
s
137.5
s
1.424e005
s
1794
s
9.925
2
3
2
)
s
(
G
1
(1)
3. Results a
nd Analy
s
is
The contino
u
s
sy
stem
of spindle plant can
be
co
nvert into discrete
system
with
out any
method. It mean
s the
con
v
erting p
r
o
c
e
ss
witho
u
t usi
ng any meth
od. The
conv
erting p
r
o
c
e
s
s in
this work
wil use th
e sam
p
ling time of
0.01 s. T
he conversion re
sult
of
di
screte
system
with
out
any method can be seen in
equation (2)
belo
w
:
0.2528
-
0.4324z
1.107z
-
z
01246
.
0
0.04855z
0.09911z
)
z
(
G
2
3
2
1
(2)
When
the in
put ste
p
is gi
ven to the
co
nt
inuou
s
syst
em an
d di
screte sy
stem in
equatio
n
(1) a
nd (2
) re
spe
c
tively, the respon
se
s
can b
e
se
en
at Figure
s
4 a
nd 5.
Figure 4. Step respon
se of
contino
u
s
spi
ndle
sy
st
em
Figure 5. Step respon
se of
discrete
syst
em
without meth
od
Figure 5 p
r
e
s
ents the
conv
ersi
on p
r
o
c
e
s
s into
di
scret
e
sig
nal u
s
ing
without any
method;
the discrete
signal is lag
g
in
g a few mom
ents compa
r
e
d
to contino
u
s
syste
m
. It
mean
s that in the
ca
se of
discrete sig
nal
h
a
ve delay ti
mes.
T
he
steady stat
e e
rro
r b
e
twe
e
n
co
ntinuou
s
and
discrete
si
gn
al ap
pea
rs
without
any
differen
c
e
s
.
To an
alyse
the tra
n
si
ent
respon
se
s
of the
discrete
syste
m
without me
thod ca
n be seen at Table
1.
To conve
r
t
th
e
contion
o
u
s
spin
dle plant into
di
screte
system, th
ere
are
several
method
s
can
be used. The methods
will be used i
n
this
work
consi
s
t of ZOH, F
O
H, and Impulse-
Invariant, Tu
stin App
r
oximation an
d
Zero
-Pole M
a
tchin
g
meth
ods. G
ene
ral
y
it can be
see
n
whe
n
ZO
H method a
r
e
use
d
by most rese
arche
r
s
for conve
r
tin
g
a co
ntinuo
us sy
stem in
to a
discrete
syst
em. Includin
g
matlab by default also
u
s
es the ZO
H
method. But, it is nece
s
sa
ry to
clarify th
at wh
ether ZO
H m
e
thod i
s
th
e
most
accu
rat
e
meth
od to
convert
a
conti
nuou
s
syste
m
to
a discrete
system. Thi
s
study ex
plo
r
ed
the co
mpa
r
ison of vari
ou
s
method
s in
order to
obtain
the
most a
c
curat
e
method for
conve
r
si
on of
a continu
o
u
s
system into
a discrete
system.
The contin
uo
us syste
m
in equatio
n
(1) will
b
e
conve
r
ted
by ZO
H method with sampli
ng
time of 0.01 s. The discrete
system will b
e
obtaine
d as equation (3)
belo
w
0.2528
-
0.4324z
1.107z
-
z
0.01246
0.04855z
0.09911z
)
z
(
G
2
3
2
1
(3)
0
0.
1
0.2
0.3
0.
4
0.5
0.6
0.
7
0.
8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
S
t
e
p
R
e
s
p
ons
e
ti
m
e
s
(
s
ec
)
am
pl
i
t
ude (
v
)
0
0.
2
0.
4
0.6
0.
8
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
S
t
e
p
R
e
sp
o
n
se
ti
m
e
s
(
s
e
c
)
am
pl
it
ude
(
v
)
co
n
t
i
n
u
e
w
i
tho
u
t m
e
tho
d
-
d
i
s
c
r
ete
Evaluation Warning : The document was created with Spire.PDF for Python.
IJEECS
ISSN:
2502-4
752
Com
pari
s
o
n
Method
s for
Con
v
e
r
ting a
Spindle Plant
to Discrete S
ystem
(
M
Khair
u
d
i
n)
579
It is obtained
the sam
e
di
screte e
quati
ons b
e
twe
e
n
using
ZO
H method
conv
ersi
on in
equatio
n (3
)
and the
conv
ersi
on u
s
in
g without any
method in
eq
uation (2). It has
been
pro
v
en
that the matl
ab by defa
u
lt with aut
oma
t
ically pr
oce
s
s for
co
nverti
ng into a
di
screte
form
using
the ZOH met
hod. Althoug
h when
con
d
u
cting the co
nversi
on u
s
in
g without an
y method, it
will
give the imp
a
ct that the result
in the
conversion p
r
oce
s
s is t
he same as whe
n
usin
g
the
Z
O
H
method. It also will p
r
ovid
e the sam
e
resp
on
se
s wh
en the discre
t
e system wit
h
ZOH m
e
th
od
given the ste
p
input. It is
noted the re
spon
se
s of
ZOH metho
d
is sam
e
as th
e respon
se
s
of
discrete
syste
m
s u
s
ing with
out any method.
Subse
que
ntly pre
s
e
n
ted
compa
r
ison u
s
ing
FO
H an
d impul
se
-inv
ariant m
e
tho
d
s. Th
e
contin
uou
s system
in equ
ation
(1) will be conve
r
ted
by
FO
H
met
hod with sam
p
ling
time
of 0.01
s. The di
scret
e
system i
s
o
b
tained in e
q
uation (4
) bel
ow.
0.2528
-
0.4324z
1.107z
-
z
0.008922
0.02254z
0.02557z
0.05107z
)
z
(
G
2
3
2
3
1
(4)
Otherway the
continuo
us
system in equ
ation (1
) whe
n
the conve
r
sion into the
discrete
form is co
nd
ucted by imp
u
lse
-
inva
riant
method with
samplin
g time is 0.01 s, it will
yield the
followin
g
syst
em in equatio
n (5).
0.2528
-
0.4324z
1.107z
-
z
016
-
1.503e
-
0.6768z
2.609z
9.925z
*
0.00949
)
z
(
G
2
3
2
3
1
(5)
Based
on
th
e equ
ation
s
(4) and
(5)
sho
w
the
different
equ
atio
ns
com
pared
to the
previou
s
eq
u
a
tions. It means
will give impact for the system
resp
on
se
s. Figure
s
6 an
d
7
pre
s
ente
d
the step respo
n
se
s of the d
i
screte
syste
m
with FOH
and impul
se
-i
nvariant met
hod
s
r
e
spec
tively.
Figure 6. Step respon
se of
discrete
syst
em
with FO
H method
Figure 7. Step respon
se of
discrete
syst
em with
impulse-i
nvariant method
Figure 6 sh
o
w
s the
conve
r
sio
n
re
sult
s of c
ontinu
o
u
s
spindl
e plant
using F
O
H
method
s.
The discrete
signal u
s
ing
FOH metho
d
provide
s
t
he sign
al located at the same po
sition
or
chime
d
in on
e line with a
continu
ous
sign
al. It me
ans that the
discrete si
g
nals u
s
ing F
O
H
method alm
o
st simulta
neo
usly at one time with
conti
nuou
s spindl
e plant. In other
words, th
e
discrete
sig
n
a
l usi
ng F
O
H method
s o
b
tained
a si
gna
l with a little
delay. This condition
provi
des
more
ide
a
l condition
s
fo
r a
time of
di
screte
sign
al appe
arin
g
al
most simulta
neou
sly with the
contin
ou
s o
r
i
g
inal
sign
al.
Certai
nly it wi
ll impa
ct
that the
re
sp
on
se
s
a
nalysi
s
wh
ich co
nsi
s
t
of the
final value (v), time to peak (s), pe
rce
n
tage
of ove
r
sh
oot and
settling time (s) on
a discrete
0
0.
2
0.
4
0.
6
0.
8
1
0
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
1
S
t
e
p
R
e
sp
o
n
se
ti
m
e
s
(
s
e
c
)
a
m
pl
i
t
ude
(
v
)
co
n
t
i
n
u
e
f
o
h-
di
s
c
r
e
t
e
0
0.
2
0.4
0.6
0.8
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
S
t
ep
R
e
s
pon
s
e
ti
m
e
s
(
s
e
c
)
am
pl
i
t
ud
e
(
v
)
c
onti
n
u
e
im
p
u
ls
e
-
d
i
s
c
r
e
t
e
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 25
02-4
752
IJEECS
Vol.
1, No. 3, March 20
16 : 575 – 582
580
system with
FOH method will
be almost the same
as the co
ntinuous spindle plant. For th
e
details respo
n
se
s an
alysi
s
of discrete sys
tem usin
g FOH
can b
e
see
n
at Table
1.
Figure 7
sho
w
s th
e re
sult
s of the
conv
ersi
on
of
con
t
inous
spi
ndl
e plant u
s
ing
impulse-
invariant that
the discrete
signal i
s
lea
d
ing
compa
r
ed to contin
ous
system.
This me
an
s the
discrete
sy
stem with
the i
m
pulse-i
nvari
ant meth
o
d
a
ppea
rs mo
re
faster than
continou
s
spin
dl
e
plant. Obvio
u
sly this
con
d
ition re
sulte
d
in the differen
c
e
s
b
e
twee
n the si
gnal di
screte
an
d
contin
uou
s si
gnal for respo
n
se
s an
alysi
s
.
The next chal
lenge
s meth
o
d
s that will b
e
used i
n
this
work a
r
e the
Tustin a
pproximation
and Ze
ro-P
ol
e Matchin
g
m
e
thod
s. The continuo
us
system in equ
ation (1
) ca
n b
e
conve
r
ted i
n
to
the discrete form by Tu
stin app
roximat
i
on metho
d
with sa
mplin
g time is 0.0
1
s. The di
screte
system
will be obtaine
d as equation (6)
belo
w
.
0.3921
-
0.7522z
1.289z
-
z
0.009689
0.01763z
0.02087z
0.04819z
)
z
(
G
2
3
2
3
1
(6)
Usi
ng the sa
me techni
que
, the continuo
us sy
stem in equatio
n (1)
can be
conve
r
ted into
the discrete f
o
rm by
Zero-Pole Matchin
g
metho
d
wit
h
sa
mplin
g time is
0.01
s. The di
scret
e
system
will be obtaine
d as equation (7)
belo
w
.
0.2528
-
0.4324z
1.107z
-
z
0.01748
0.06097z
0.1065z
)
z
(
G
2
3
2
1
(7)
Based
on th
e equ
ation
s
(6) and
(7
)
reveal
the different equ
atio
ns com
pared
to
the
previou
s
eq
u
a
tions. It means
will give impact fo
r the
system re
sp
onses. Figu
re
s 8 and 9
sh
ow
the step respon
se
s of the disc
rete system with
the
Tust
in
approximatio
n and Ze
ro
-Pole
Matchin
g
met
hod
s re
spe
c
ti
vely.
Figure 8. Step respon
se of
discrete
syst
em
with Tus
t
in method
Figure 9. Step respon
se of
discrete
syst
em with
matche
d met
hod
Figure 8
sh
o
w
s that the
conve
r
si
on
result
s
of co
n
t
inuou
s spind
l
e
plant usi
n
g
Tu
stin
method
re
sul
t
s. The
di
screte si
gnal
u
s
i
ng T
u
st
in
me
thod p
r
ovide
s
the
di
scret
e
si
gnal
like
the
FOH
method.
But the di
screte sig
nal
usi
ng Tu
stin
m
e
thods
obtain
e
d
a
sign
al wit
h
a d
e
lay. Th
e
time to peak of 0.920 is
more
slo
w
er
than t
he co
n
t
inous
syste
m
. For the d
e
tails respon
se
s
analysi
s
of di
screte
syste
m
using T
u
sti
n
method can
be see
n
at Table 1.
Figure 9 sho
w
s the
re
sult
s of the co
nversi
on
of cont
inou
s spi
ndle
plant usin
g Z
e
ro
-Pole
Matchin
g
me
thod that the
discrete si
g
nal is
lag
g
in
g a fe
w mo
ments
com
p
ared
to conti
nou
s
0
0.2
0.
4
0.
6
0.
8
1
0
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
1
S
t
ep R
e
s
pon
s
e
ti
m
e
s
(
s
e
c
)
am
pl
i
t
ude (
v
)
c
ont
i
n
ue
tus
t
i
n
-
d
i
s
c
r
ete
0
0.
2
0.
4
0.
6
0.
8
1
0
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
1
S
t
ep R
e
s
pons
e
ti
m
e
s
(
s
e
c
)
am
pl
i
t
ude (
v
)
c
ont
i
n
ue
m
a
t
c
h
ed-
di
s
c
r
e
t
e
Evaluation Warning : The document was created with Spire.PDF for Python.
IJEECS
ISSN:
2502-4
752
Com
pari
s
o
n
Method
s for
Con
v
e
r
ting a
Spindle Plant
to Discrete S
ystem
(
M
Khair
u
d
i
n)
581
system. It mean
s that in the ca
se
of discrete
sign
al
have delay times.
The
st
eady state e
r
ror
betwe
en cont
inuou
s an
d di
screte
signal
appe
ars with
out any differences.
To p
r
ovide
t
he exh
a
u
s
tive comp
ari
s
o
n
bet
wee
n
several m
e
th
ods that
pre
s
ente
d
previou
s
ly, Figure
10 p
r
e
s
ents the
com
pari
s
on fo
r
th
e re
spo
n
ses
of Tustin, zo
h, foh, match
e
d
and impul
se
method
s.
Figure 8. Co
mpari
s
o
n
ste
p
respon
se of
discrete
syst
em with seve
ral metho
d
s
To analyse the re
spo
n
se
s pe
rform
a
n
c
e of a
discrete system,
it is noted the be
st
perfo
rman
ce
is the most
similar
with
the orig
inal
system or t
he co
ntinuou
s syste
m
. It is
necessa
ry to find the most accu
rate o
f
conversion
method
com
par
e
d
to oth
e
r. The
r
efore
the
discrete
sy
stem respon
se
s n
eed to
be
analy
s
ed
by com
p
a
r
ing t
he p
e
rfo
r
man
c
e
s
of
re
sults fo
r
each method.
Table 1 presents t
he tra
n
sient respon
se
s analy
s
is for
all method
s mentione
d.
Tabel 1. Co
m
pari
s
on T
r
an
sient Re
spo
n
ses Analysi
s
o
f
Continue a
n
d
De
screte S
y
stem
No
Plant w
i
th meth
o
d
Final Value (v)
Time
to Peak (s)
% Ove
r
s
hoot
Settling time (s)
1 Continue
0.863
0.911
0
0.552
2 Tustin
0.863
0.920
0
0.550
3 ZOH
0.863
0.910
0
0.560
4 FOH
0.863
0.910
0
0.550
5 Matched
0.863
0.920
0
0.560
6 Impulse
0.863
0.930
0
0.560
7 Without
method
0.863
0.910
0
0.560
Table 1 de
scrib
e
s the transi
ent re
sp
onses
an
alysis, it can be
stated that the FOH
method
with
the final valu
e ap
pro
a
chin
g a
co
ntinuo
us
sig
nal. In
other
han
d a
time to p
e
a
k
is
almost th
e
sa
me a
s
the
co
ntinou
s sy
ste
m
. Also a
setlling time i
s
a
l
most e
qual
to the
contin
o
u
s
system. It is noted th
at the FO
H m
e
thod i
s
t
he
m
o
st a
c
curatel
y
method fo
r conve
r
ting t
he
contin
ou
s system into the discrete
syste
m
comp
are
d
to other meth
ods.
0
0.2
0.
4
0.
6
0.
8
1
0
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
1
S
t
e
p
R
e
s
pons
e
ti
m
e
s
(
s
e
c
)
am
pl
i
t
ud
e (
v
)
c
ont
i
n
ue
t
u
s
t
in
-
d
is
c
r
e
t
e
zo
h
-
d
i
scr
e
t
e
f
oh-
d
i
s
c
r
e
te
m
a
t
c
h
ed-
di
s
c
r
e
te
i
m
pul
s
e
-
d
i
s
c
r
et
e
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 25
02-4
752
IJEECS
Vol.
1, No. 3, March 20
16 : 575 – 582
582
4. Conclusio
n
The d
e
velop
m
ent of
con
v
erting a
co
ntinou
s spin
dle pla
n
t int
o
discrete
sy
stem
with
variou
s meth
ods
ha
s b
e
en p
r
e
s
ente
d
. A set of
linea
r mod
e
l of spindle
plant h
a
s
bee
n
develop
ed. T
he cl
osest di
screte
syste
m
is r
equi
red
to develop t
he control
sy
stem. Compa
r
iso
n
method
s h
a
ve be
en
con
d
u
cted
to
con
v
ert the
cont
i
nuou
s
syste
m
of spindl
e
plant into
discret
e
system th
rou
gh the meth
o
d
of ze
ro-ord
er hol
d (Z
OH), first-o
r
d
e
r
hold (FO
H
), i
m
pulse invari
ant
dicretisatio
n, Tustin
(bili
near
), an
d pole-ze
ro m
a
tchin
g
meth
ods
and al
so co
nverting
the
contin
ou
s sy
stem with
out
any method.
Conve
r
si
on
perfo
rman
ce
s of contin
uou
s spindle
pla
n
t
into discrete
system
usin
g
FOH m
e
tho
d
, sho
w
e
d
m
o
re a
c
cu
rate
comp
ared to
other m
e
thod
s.
Perform
a
n
c
e
s
of the con
v
ersio
n
accu
racy
of FOH
method have
been evalua
ted in terms
of
transi
ent
re
spon
se
s a
naly
s
is t
hat clo
s
ed simila
r re
sults
with
a
contin
uou
s system
of spi
ndle
plant.The respon
se
s anal
ysis na
mely final value,
time to pea
k, percerta
g
e
s
overshoot a
nd
setlling time.
Referen
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