Indonesian
Journal
of
Electrical
Engineering
and
Computer
Science
V
ol.
2,
No
.
3,
J
une
2016,
pp
.
486
500
DOI:
10.11591/ijeecs
.v2.i3.pp486-500
486
Routh
Appr
o
ximation:
An
Appr
oac
h
of
Model
Or
der
Reduction
in
SISO
and
MIMO
Systems
D
.
K.
Sambariy
a*
and
Omveer
Sharma
Depar
tment
of
Electr
ical
Engg.,
Rajasthan
T
echnical
Univ
ersity
Ra
w
atbhata
Road,
K
ota,
324010,
India
*Corresponding
author
,
e-mail:
dsambar
iy
a
2003@y
ahoo
.com
Abstract
In
this
paper
the
Routh
Appro
ximation
method
is
e
xplored
f
o
r
getting
the
reduced
order
model
of
a
higher
order
model.
The
reduced
order
modeling
of
a
large
system
is
necessar
y
to
ease
the
analysis
of
the
system.
The
approach
is
e
xamined
and
compared
to
single-input
single-output
(SISO)
and
m
ulti-input
m
ulti-
output
(MIMO)
systems
.
The
response
compar
ison
is
considered
in
ter
ms
of
step
response
par
ameters
and
g
r
aphical
compar
isons
.
It
is
repor
ted
that
the
reduced
order
model
using
proposed
Routh
Appro
ximation
(RA)
method
is
almost
similar
in
beha
vior
to
that
of
with
or
iginal
systems
.
K
e
yw
or
ds:
Model
order
reduction
(MOR),
Single-input
single-output
(SISO),
Multi-input
m
ulti-outp
ut
(MIMO),
Routh
Appro
ximation
method.
Cop
yright
c
2016
Institute
of
Ad
v
anced
Engineering
and
Science
.
All
rights
reser
v
ed.
1.
Intr
oduction
The
analysis
of
high
order
systems
(HOS)
is
gener
ally
v
er
y
m
uch
complicated
and
costly
.
On
other
hand
it
became
easy
to
analysis
of
lo
w
er
order
system
[1,
2].
The
reduced
models
f
or
the
or
iginal
high
order
system
is
achie
v
ed
b
y
using
mathematical
optimization
procedures
or
simplification
procedures
based
on
ph
ysical
consider
ations
[3].
Thus
analysis
,
synthesis
and
sim
ulation
of
reduced
lo
w
order
systems
is
easier
and
pr
acticab
le
as
compared
to
it’
s
high
order
systems
[4].
An
approach
to
get
reduced
order
model
of
a
higher
order
system
using
time-
moments
method
is
presented
b
y
[5].
The
reduced
model
ha
v
e
a
prob
lem
of
stability
because
of
mathematical
appro
ximation
in
model
reduction
technique
.
The
reduced
model
m
a
y
be
unstab
le
e
v
en
though
the
high
order
system
is
stab
le
[6].
The
instability
prob
lem
of
reduced
models
w
as
studied
b
y
Hutton
[7],
Shamash
[8],
Gut-
man
et.
al.
[9]
and
W
an
[10].
Some
method
based
on
stability
cr
iter
ion
and
other
not
based
on
stability
cr
iter
ion
b
ut
the
reduced
model
f
or
a
stab
le
high
order
system
(HOS)
is
alw
a
ys
stab
le
[11,
12].
Diff
erent
methods
giv
e
diff
erent
approach
some
giv
es
batter
result
in
r
ise
time
,
some
giv
es
batter
results
in
settling
time
[13].
The
combin
ation
of
these
methods
giv
es
batter
results
.
The
reduced
model
using
combination
of
methods
is
near
ly
wit
h
its
higher
order
system.
The
com-
bination
of
Routh
appro
ximation
and
par
ticle
s
w
ar
m
optimization
(PSO)
is
presented
in
[14].
The
concept
of
preser
v
ation
of
stability
is
presented
in
[15].
The
diff
erentia
tion
method
f
or
reduction
of
systems
is
prese
nted
in
[16].
The
diff
erentiation
method
is
used
to
der
iv
e
reduced
order
model
of
single
machine
infinite
b
us
po
w
er
system
in
[17].
The
application
of
Routh
stability
algor
ithm
is
presented
in
[18,
19].
The
application
of
soft
computing
techniques
ha
v
e
been
presented
in
liter
ature
in
the
field
of
model
order
reduction
[20].
The
concept
used
is
minimization
of
integ
r
al
squared
err
or
using
bat
algor
ithm
[20].
The
application
of
fire
fly
algor
ithm
in
model
order
reduction
is
presented
in
[21].
The
application
of
par
ticle
s
w
ar
m
optimization
(PSO)
is
pr
esented
in
[22].
The
application
of
Routh
appro
ximation
with
Cuc
k
oo
search
algor
ithm
f
or
model
order
reduction
is
presented
in
[23].
The
h
ybr
id
application
of
stability
equation
method
with
self-adaptiv
e
bat
algor
ithm
to
reduce
po
w
er
system
to
a
reduced
model
is
presented
in
[24].
Receiv
ed
J
an
uar
y
1,
2016;
Re
vised
Apr
il
24,
2016;
Accepted
Ma
y
10,
2016
Evaluation Warning : The document was created with Spire.PDF for Python.
IJEECS
ISSN:
2502-4752
487
In
this
paper
the
application
of
Routh
appro
ximation
method
is
presented
f
or
der
iving
reduced
order
model
of
the
higher
order
L
TI
systems
which
include
s
bechmar
k
prob
lems
.
The
statement
of
prob
lem
is
presented
in
section
2..
The
detailed
procedur
al
steps
on
Routh
appro
x-
imation
method
are
included
in
section
3..
The
systems
under
consid
er
ation
and
their
reduced
order
models
are
presented
in
section
4..
The
results
of
or
iginal
system
and
reduced
models
are
subjected
to
step
input
and
compared
in
this
section.
Finally
the
man
uscr
ipt
is
concluded
in
section
5.
and
f
ollo
w
ed
b
y
ref
erences
.
2.
Pr
ob
lem
Form
ulation
Consider
a
high
order
tr
ansf
er
function
of
a
system
represented
as
in
Eq.
1.
G
(
s
)
=
n
1
P
i
=0
b
i
s
i
n
P
i
=0
a
i
s
i
(1)
where
,
the
G
(
s
)
represents
a
high
order
sy
st
em
with
the
order
of
n
.
The
p
ur
pose
of
man
uscr
ipt
is
to
reduce
the
order
of
such
high
order
system
to
r
.
The
reduced
order
model
ma
y
be
represented
as
in
Eq.
2.
R
(
s
)
=
r
1
P
j
=0
d
j
s
j
r
P
j
=0
c
j
s
j
(2)
where
,
a
i
,
b
i
,
c
j
and
d
j
are
the
scalar
constants
of
or
iginal
high
order
system
and
the
reduced
order
system.
The
obje
ctiv
e
is
to
find
a
reduced
r
th
order
system
model
R
(
s
)
such
that
it
retains
the
impor
tant
proper
ties
of
G
(
s
)
f
or
the
same
types
of
inputs
.
3.
Re
vie
w
on
Routh
appr
o
ximation
This
method
n
umber
of
useful
proper
ties
lik
e
if
or
iginal
system
is
stab
le
then
reduce
model
will
be
stab
le
,con
v
erge
monotonically
of
or
iginal
system
in
ter
ms
of
step
and
impulse
response
.
By
increase
order
of
appro
ximation
poles
and
z
eros
of
the
appro
ximants
mo
v
e
to
w
ards
the
poles
and
z
eros
of
the
or
igina
l.
In
this
method
Routh
T
ab
le
f
or
or
iginal
system
is
use
to
constr
uct
the
appro
ximate
in
a
manner
that
it
will
stab
le
f
or
stab
le
or
iginal
system
[22].
3.1.
Description
of
Method
G
(
s
)
=
b
n
s
(
n
1)
+
b
n
s
(
n
2)
+
:
:
:
+
b
1
a
n
s
n
+
a
(
n
1)
s
(
n
1)
+
:
:
:
+
a
0
(3)
By
taking
reciprocal
of
Eq.
3
and
sho
wn
in
Eq.
4
^
G
(
s
)
=
1
s
G
1
s
=
b
1
s
(
n
1)
+
:
:
:
+
b
n
a
0
s
n
+
a
1
s
(
n
1)
+
:
:
:
+
a
n
(4)
If
s
i
,
represents
the
i
th
pole/z
eros
of
the
or
iginal
system
then
1
=s
i
,
the
i
th
poles/z
eros
of
the
reciprocal
system.
Routh
Appro
ximation:
An
Approach
of
Model
Order
Reduction
...
(D
.
K.
Sambar
iy
a)
Evaluation Warning : The document was created with Spire.PDF for Python.
488
ISSN:
2502-4752
3.2.
Alpha-Beta
e
xpansion
The
tr
ansf
er
function
of
Eq.
4
can
be
e
xpanded
in
the
canonical
f
or
m
as
presented
in
Eq.
5.
^
G(s)
=
1
F
1
(s)+
2
F
1
(s)F
2
(s)+
3
F
1
(s)F
2
(s)F
3
(s)
+
:
:
:
+
n
F
1
(s)F
2
(s)F
3
(s)
:
:
:::
F
n
(s)
=
n
P
i=1
i
Q
i
i
=1
F
j
(
s
)
(5)
The
F
i
(
s
)
can
be
defined
b
y
the
contin
ued
fr
action
e
xpansions
as
sho
wn
in
Eq.
6.
F
i
(
s
)
=
1
i
s
+
1
i
+1
s
+
1
i
+2
s
+
.
.
.
n
1
s
+
1
n
s
(6)
In
Routh
T
ab
le
1,
the
first
tw
o
ro
ws
of
tab
le
are
f
or
med
b
y
coefficients
of
the
denominator
of
function
^
G
(
s
)
and
taking
assumption
that
the
entr
ies
of
a
0
J
=
a
I
(
J
1)
=
0
f
or
j
>
n
.
a
i
+1
0
=
a
i
1
2
i
a
i
2
a
i
+1
2
=
a
i
1
4
i
a
i
4
.
.
.
a
i
+1
n
i
1
=
a
i
1
n
i
i
a
i
n
i
(7)
where
,
Eq.
7
stands
f
or
i
=
1
;
2
;
3
;
:
:
:
;
n
1
.
If
the
v
alue
of
n
i
as
odd,
the
last
ter
m
in
Eq.
7
is
replaced
b
y
as
sho
wn
in
Eq.
8.
a
i
+1
n
i
1
=
a
i
1
n
i
1
(8)
F
or
i
=
1
;
2
;
3
;
:
:
:
;
n
,
the
marginal
entr
ies
f
or
i
are
calculated
as
in
Eq.
9.
i
=
a
i
1
0
a
i
0
(9)
The
i
coefficients
of
the
canonical
f
or
m
Routh
tab
le
are
deter
mined
using
coefficients
of
the
n
umer
ator
of
^
G
(
s
)
and
is
sho
wn
in
Eq.
10.
i
=
b
i
0
a
i
0
(10)
b
i
+2
j
2
=
b
i
j
i
a
i
j
(11)
The
Routh
T
ab
le
1
is
equiv
alent
to
constr
uction
of
f
ollo
wing
finite
contin
ued
fr
action
e
xpansion
as
sho
wn
in
Eq.
12.
^
D
(
s
)
=
1
s
+
1
2
s
+
1
3
s
+
.
.
.
n
1
s
+
1
n
s
(12)
It
could
be
easy
to
sa
y
that
the
system
wit
h
all
par
ameters
being
positiv
e
ref
ers
to
an
asymptot-
ically
stab
le
system
[1].
IJEECS
V
ol.
2,
No
.
3,
J
une
2016
:
486
500
Evaluation Warning : The document was created with Spire.PDF for Python.
IJEECS
ISSN:
2502-4752
489
T
ab
le
1.
Alpha
tab
le
I
st
ro
w
a
0
0
=
a
0
a
0
2
=
a
2
a
0
4
=
a
4
2
nd
ro
w
a
1
0
=
a
0
a
1
2
=
a
3
a
1
4
=
a
5
1
=
a
0
0
a
1
0
a
2
0
=
a
0
2
1
a
1
2
a
2
2
=
a
0
4
1
a
1
4
a
2
4
=
a
0
2
1
a
1
6
2
=
a
1
0
a
2
0
a
3
0
=
a
1
2
2
a
2
2
a
3
2
=
a
1
4
2
a
2
4
:
:
:
3
=
a
2
0
a
3
0
a
4
0
=
a
2
2
3
a
3
2
a
4
2
=
a
2
4
3
a
3
4
:
:
:
4
=
a
3
0
a
4
0
a
5
0
=
a
3
2
4
a
4
2
:
:
:
:
:
:
5
=
a
4
0
a
5
0
a
6
0
=
a
4
2
5
a
5
2
:
:
:
:
:
:
6
=
a
5
0
a
6
0
:
:
:
:
:
:
:
:
:
T
ab
le
2.
Beta
tab
le
I
st
ro
w
b
1
0
=
b
1
b
1
2
=
b
3
b
1
4
=
b
5
2
nd
ro
w
b
2
0
=
b
2
b
2
2
=
b
4
b
2
4
=
b
6
1
=
b
1
0
a
1
0
b
3
0
=
b
1
2
1
a
1
2
b
3
2
=
b
1
4
1
a
1
4
:
:
:
2
=
b
2
0
a
2
0
b
4
0
=
b
2
2
2
a
2
2
b
4
2
=
b
2
4
1
a
2
4
:
:
:
3
=
b
3
0
a
3
0
b
5
0
=
b
3
2
3
a
3
2
:
:
:
:
:
:
4
=
b
4
0
a
4
0
b
6
0
=
b
4
2
4
a
4
2
:
:
:
:
:
:
5
=
b
5
0
a
5
0
:
:
:
:
:
:
:
:
:
3.3.
Routh
Con
ver
g
ent
The
reduced
k
th
order
tr
ansf
er
function
as
^
R
k
(
s
)
f
or
an
or
iginal
tr
ansf
er
function
G
(
s
)
is
der
iv
ed
b
y
tr
uncating
the
e
xpansion
and
r
ational
arr
angement
of
the
results
.
The
ter
ms
appear
ing
k
+1
;
:
:
:
;
n
and
k
+1
;
:
:
:
;
n
are
eliminated
using
e
xpansion.
In
this
w
a
y
the
the
resultant
is
dependent
on
the
first
k-ter
ms
[7,
25].
Assuming
a
set
of
k-functions
,
which
are
defined
b
y
G
i;k
f
or
i
=
2
;
3
;
:
:
:
;
k
and
is
repre-
sented
as
in
f
ollo
wing
Eq.
13
[25].
G
i;k
(
s
)
=
1
i
s
+
1
i
+1
s
+
1
i
+2
s
+
.
.
.
k
1
s
+
1
k
s
(13)
The
abo
v
e
method
possess
slight
modification
f
or
i
=
1
.
The
I
st
ter
m
in
the
contin
ued
fr
action
e
xpansion
is
1
+
1
s
instead
of
1
s
.
In
this
w
a
y
,
the
k
th
con
v
ergent
ma
y
be
giv
en
b
y
as
in
Eq.
14
[1,
25].
^
R
k
(s)
=
1
G
1
;
k
(s)+
2
G
1
;
k
(s)G
2
;
k
(s)+
+
k
G
1
;
k
(s)G
2
;
k
(s)
G
k
;
k
(s)
=
k
P
i
=1
i
I
Q
i
=1
G
i;k
(
s
)
(14)
The
A
k
(
s
)
is
the
denominator
of
the
k
th
con
v
ergent
while
B
k
(
s
)
represents
the
n
umer
ator
of
it.
In
Routh
Appro
ximation:
An
Approach
of
Model
Order
Reduction
...
(D
.
K.
Sambar
iy
a)
Evaluation Warning : The document was created with Spire.PDF for Python.
490
ISSN:
2502-4752
this
w
a
y
,
the
k
th
con
v
ergent
ma
y
be
represented
as
in
f
ollo
wing
Eq.
15
[26].
A
1
(
s
)
=
1
s
+
1
B
1
(
s
)
=
1
A
2
(
s
)
=
1
2
s
2
+
2
s
+
1
B
2
(
s
)
=
2
1
s
+
2
A
3
(
s
)
=
1
2
3
s
3
+
2
3
s
2
+
(
1
+
3
)
s
+
1
B
3
(
s
)
=
2
3
1
s
2
+
3
2
s
+
(
1
+
3
)
A
k
(
s
)
=
k
sA
k
1
(
s
)
+
A
k
2
(
s
)
B
k
(
s
)
=
k
sB
k
1
(
s
)
+
B
k
2
(
s
)
+
k
A
1
(
s
)
=
1
;
B
1
(
s
)
=
0
A
0
(
s
)
=
1
;
B
0
(
s
)
=
0
(15)
The
^
R
k
(
s
)
represents
the
appro
ximation
of
^
G
(
s
)
with
preser
ving
the
frequency
beha
viour
.
The
k
th
appro
ximate
can
be
der
iv
ed
b
y
consider
ing
the
reciprocal
of
^
R
k
(
s
)
as
sho
wn
in
Eq.
16
[25].
R
k
(
s
)
=
1
s
^
R
k
1
s
(16)
3.4.
Algorithm
of
Routh
appr
o
ximation
The
f
ollo
wing
steps
can
be
f
ollo
w
ed
f
or
deter
mining
the
reduced
order
of
a
high
order
system.
(i)
Initially
deter
mine
the
reciprocal
(
^
G
(
s
)
)
of
the
full
order
system
G
(
s
)
(ii)
Der
iv
e
the
elements
(iii)
Deter
mine
k
th
con
v
ergent
using
^
R
k
(
s
)
=
B
k
(
s
)
A
k
(
s
)
(iv)
Reciprocate
^
R
k
(
s
)
f
or
k
th
order
Routh
appro
ximation
R
k
(
s
)
.
4.
Results
and
Discussions
4.1.
Example-1:
SISO
Consider
ing
the
8
th
order
system
presented
in
Shamash,
1975
[8]
and
presented
in
Eq.
17.
G
(
s
)
=
18
s
7
+
514
s
6
+
5982
s
5
+
36380
s
4
+
122664
s
3
+
222088
s
2
+
185760
s
+
40320
s
8
+
36
s
7
+
546
s
6
+
4536
s
5
+
22449
s
4
+67284
s
3
+
118124
s
2
+
109584
s
+
40320
(17)
The
reduced
2
nd
order
and
3
r
d
order
models
are
presented
in
Eq.
18
and
Eq.
19,
respectiv
ely
using
Routh
Appro
ximation
method.
R
2
(
s
)
=
1
:
990
s
+
0
:
432
s
2
+
1
:
174
s
+
0
:
432
(18)
R
3
(
s
)
=
4
:
968
s
2
+
4
:
331
s
+
0
:
940
s
3
+
2
:
545
s
2
+
2
:
555
s
+
0
:
940
(19)
The
step
response
compar
ison
of
the
or
iginal
system
[
8]
and
it’
s
reduced
2
nd
and
3
r
d
order
models
are
g
r
aphically
compared
in
Fig.
1.
It
can
be
obser
v
ed
that
the
stability
of
the
system
that
of
with
reduced
models
are
retain
ed
e
xcept
slight
v
ar
iation
in
r
ise
time
,
settling
time
,
peak
v
alue
and
peak
time
as
included
in
T
ab
le
3.
Since
,
the
impor
tant
proper
ties
of
the
higher
order
system
are
preser
v
ed
in
it’
s
reduced
(
2
nd
order)
system,
consequently
the
mathematical
ease
is
increased
g
reatly
.
IJEECS
V
ol.
2,
No
.
3,
J
une
2016
:
486
500
Evaluation Warning : The document was created with Spire.PDF for Python.
IJEECS
ISSN:
2502-4752
491
0
5
10
15
20
0
0.5
1
1.5
2
2.5
Time (s)
Amplitude
Step response
Original (Shamash, 1975)
RA−MOR: 2
nd
order (Proposed)
RA−MOR: 3
rd
order (Proposed)
Figure
1
.
Step
response
of
the
or
iginal
8
th
order
system
[8]
and
its
reduced
model
R
2
(
s
)
(Eq.
18)
and
R
3
(
s
)
(Eq.
19)
using
the
Routh
appro
ximation
method
T
ab
le
3.
Step
response
compar
ision
of
or
iginal
system
in
Example-1,
with
reduced
models
using
Routh
Appro
ximation
T
r
ansf
er
Rise
Settling
P
eak
P
eak
Function
Time
Time
V
alue
Time
Or
iginal:
G
8
(
s
)
[8]
0.0569
4.8201
2.2035
0.4493
MOR-RA:
R
2
(
s
)
0.5514
8.7327
1.5717
2.3235
MOR-RA:
R
3
(
s
)
0.1973
7.0765
2.1128
1.1637
4.2.
Example-2:
SISO
Consider
ing
the
4
th
order
system
presented
in
Hw
ang,1996
[27]
and
presented
in
Eq.
20.
G
(
s
)
=
10
s
4
+
82
:s
3
+
264
s
2
+
396
s
+
156
2
s
5
+
21
s
4
+
84
:s
3
+
173
s
2
+
148
s
+
40
(20)
The
reduced
2
nd
order
and
3
r
d
order
models
are
presented
in
Eq.
21
and
Eq.
22,
respectiv
ely
using
Routh
Appro
ximation
method.
R
2
(
s
)
=
1
:
990
s
+
0
:
432
s
2
+
1
:
174
s
+
0
:
432
(21)
R
3
(
s
)
=
4
:
968
s
2
+
4
:
331
s
+
0
:
940
s
3
+
2
:
545
s
2
+
2
:
555
s
+
0
:
940
(22)
The
step
response
compar
ison
of
the
or
iginal
system
[27]
and
it
’
s
reduced
2
nd
and
3
r
d
order
models
are
g
r
aphically
compared
in
Fig.
2.
It
can
be
obser
v
ed
that
the
stability
of
the
system
that
of
with
reduced
models
are
retain
ed
e
xcept
slight
v
ar
iation
in
r
ise
time
,
settling
time
,
peak
v
alue
and
peak
time
as
included
in
T
ab
le
4.
In
this
case
the
r
ise-time
of
the
or
iginal,
2
nd
and
3
r
d
order
reduced
models
are
2.7456,
2.6830
and
2.5549
seconds
,
respectiv
ely
.
The
diff
erence
in
the
r
ise
times
is
minimal
and
is
enough
to
pro
v
e
similar
ity
of
the
or
iginal
and
reduced
models
.
The
other
step
response
data
are
enlisted
in
T
ab
le
4.
4.3.
Example-3:
SISO
Consider
ing
the
7
th
order
system
presented
in
J
amshidi,
1983
[28]
and
presented
in
state-space
f
or
m
b
y
Eq.
23
-
24
and
in
tr
ansf
er
function
b
y
Eq.
25.
It
represents
the
SMIB
po
w
er
Routh
Appro
ximation:
An
Approach
of
Model
Order
Reduction
...
(D
.
K.
Sambar
iy
a)
Evaluation Warning : The document was created with Spire.PDF for Python.
492
ISSN:
2502-4752
0
5
10
15
20
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Time (s)
Amplitude
Step response
Original (Hwang, 1996)
RA−MOR: 2
nd
order (Proposed)
RA−MOR: 3
rd
order (Proposed)
Figure
2.
Step
response
of
the
or
iginal
4
th
order
system
[27]
and
its
reduced
model
R
2
(
s
)
(Eq.
21)
and
R
3
(
s
)
(Eq.
22)
using
the
Routh
appro
ximation
method
T
ab
le
4.
Step
response
compar
ision
of
or
iginal
system
in
Example-2,
with
reduced
models
using
Routh
Appro
ximation
T
r
ansf
er
Rise
Settling
P
eak
P
eak
Function
Time
Time
V
alue
Time
Or
iginal:
G
8
(
s
)
[27]
2.7456
5.4346
3.8944
10.4392
MOR-RA:
R
2
(
s
)
2.6830
3.9639
3.
9748
6.4974
MOR-RA:
R
3
(
s
)
2.5549
5.5932
3.
8992
15.6589
system
and
the
details
are
giv
en
in
[29].
_
x
(
t
)
=
2
6
6
6
6
6
6
6
6
4
0
:
58
0
0
0
:
269
0
0
:
2
0
0
1
0
0
0
1
0
0
0
5
2
:
12
0
0
0
0
0
0
0
377
0
0
0
:
141
0
0
:
141
0
:
2
0
:
28
0
0
0
0
0
0
0
0
:
0838
2
173
66
:
7
116
40
:
9
0
66
:
7
16
:
7
3
7
7
7
7
7
7
7
7
5
x
(
t
)
+
2
6
6
6
6
6
6
6
6
4
1
0
1
0
1
0
1
3
7
7
7
7
7
7
7
7
5
u
(
t
)
(23)
y
(
t
)
=
1
1
1
1
0
1
0
x
(
t
)
(24)
G
(
s
)
=
2
s
6
+
420
:
4
s
5
+
9435
s
4
+
1
:
39
10
5
s
3
+4
:
663
10
5
s
2
+
4
:
342
10
5
+
1
:
877
10
5
s
7
+
23
:
48
s
6
+
331
:
7
s
5
+
2640
s
4
+
1
:
757
10
4
s
3
+5
:
165
10
4
s
2
+
3
:
534
10
4
s
+
1
:
729
10
4
(25)
The
reduced
2
nd
order
and
3
r
d
order
models
are
presented
in
Eq.
26
and
Eq.
27,
respectiv
ely
using
Routh
Appro
ximation
method.
R
2
(
s
)
=
10
:
085
s
+
4
:
360
s
2
+
0
:
821
s
+
0
:
402
(26)
IJEECS
V
ol.
2,
No
.
3,
J
une
2016
:
486
500
Evaluation Warning : The document was created with Spire.PDF for Python.
IJEECS
ISSN:
2502-4752
493
T
ab
le
5.
Step
response
compar
ision
of
or
iginal
system
in
Example-3,
with
reduced
models
using
Routh
Appro
ximation
T
r
ansf
er
Rise
Settling
P
eak
P
eak
Function
Time
Time
V
alue
Time
Or
iginal:
G
8
(
s
)
[28,
29]
0.1126
5.9294
16.0310
0.4307
MOR-RA:
R
2
(
s
)
1.1998
7.4456
13.92
03
3.3225
MOR-RA:
R
3
(
s
)
0.5740
9.0915
13.22
69
2.1099
R
3
(
s
)
=
29
:
318
s
2
+
27
:
948
s
+
12
:
081
s
3
+
3
:
26
s
2
+
2
:
275
s
+
1
:
113
(27)
0
5
10
15
20
0
2
4
6
8
10
12
14
16
18
Time (s)
Amplitude
Step response
Original (Jamshidi, 1983)
RA−MOR: 2
nd
order (Proposed)
RA−MOR: 3
rd
order (Proposed)
Figure
3.
Step
response
of
the
or
iginal
7
th
order
system
[28,
29]
and
its
reduced
model
R
2
(
s
)
(Eq.
26)
and
R
3
(
s
)
(Eq.
27)
using
the
Routh
appro
ximation
method
In
this
e
xample
,
the
considered
system
is
from
the
po
w
er
system
engineer
ing.
The
or
ig-
inal
syste
m
and
it’
s
reduced
2
nd
and
3
r
d
order
models
are
subjected
to
step
signal
and
super-
imposed
to
compare
the
responses
in
Fig.
3.
It
can
be
seen
that
the
response
due
to
or
iginal
system
is
ha
ving
more
oscillations
as
compared
to
that
of
with
the
reduced
order
models
.
The
step
response
inf
or
mation
of
these
responses
are
enlisted
in
T
ab
le
5.
4.4.
Example-4:
SISO
Consider
ing
the
9
th
order
boiler
system
represented
in
tr
ansf
er
function
f
or
m
in
Eq.
28
as
presented
in
[26,
30].
The
reduced
2
nd
order
and
3
r
d
order
models
are
presented
in
Eq.
29
and
G
(
s
)
=
146
:
4
s
8
+
9
:
81
10
4
s
7
+
5
:
999
10
7
s
6
+
3
:
206
10
10
s
5
+
3
:
582
10
12
s
4
+1
:
113
10
14
s
3
+
1
:
154
10
15
s
2
+
3
:
971
10
15
s
+
3
:
063
10
15
s
9
+
659
:
8
s
8
+
4
:
136
10
5
s
7
+
2
:
13
10
8
s
6
+
2
:
422
10
10
s
5
+
8
:
737
10
11
s
4
+1
:
523
10
13
s
3
+
1
:
221
10
14
s
2
+
3
:
636
10
14
s
+
2
:
406
10
14
(28)
Routh
Appro
ximation:
An
Approach
of
Model
Order
Reduction
...
(D
.
K.
Sambar
iy
a)
Evaluation Warning : The document was created with Spire.PDF for Python.
494
ISSN:
2502-4752
T
ab
le
6.
Step
response
compar
ision
of
or
iginal
system
in
Example-4,
with
reduced
models
using
Routh
Appro
ximation
T
r
ansf
er
Rise
Settling
P
eak
P
eak
Function
Time
Time
V
alue
Time
Or
iginal:
G
9
(
s
)
[26,
30]
0.5432
2.2753
12.6986
4.5555
MOR-RA:
R
2
(
s
)
0.6375
2.9668
13.28
09
1.6504
MOR-RA:
R
3
(
s
)
0.2577
2.4749
12.69
20
4.5431
Eq.
30,
respectiv
ely
using
Routh
Appro
ximation
method.
R
2
(
s
)
=
35
:
448
s
+
27
:
343
s
2
+
3
:
246
s
+
2
:
148
(29)
R
3
(
s
)
=
90
:
835
s
2
+
319
:
054
s
+
246
:
1
s
3
+
9
:
662
s
2
+
29
:
214
s
+
19
:
331
(30)
The
considered
9
th
order
system
is
a
pr
actical
boiler
system
as
presented
in
[26,
30].
The
0
1
2
3
4
5
6
7
8
0
5
10
15
Time (s)
Amplitude
Step response
Original (Salim, 2009)
RA−MOR: 2
nd
order (Proposed)
RA−MOR: 3
rd
order (Proposed)
Figure
4.
Step
response
of
the
or
iginal
9
th
order
system
[26,
30]
and
its
reduced
model
R
2
(
s
)
(Eq.
29)
and
R
3
(
s
)
(Eq.
30)
using
the
Routh
appro
ximation
method
system
is
reduced
to
2
nd
and
3
r
d
order
models
using
Routh
appro
ximation
method.
The
or
iginal
and
reduced
models
are
subjected
to
step
input
and
the
g
r
aphical
compar
ison
is
presented
in
Fig.
4.
It
can
be
seen
that
the
or
iginal
and
main
proper
ties
of
the
or
iginal
higher
order
system
are
retained
in
it’
s
reduced
model
responses
with
compar
ativ
ely
reduced
o
v
ershoots
.
The
step
response
inf
or
mation
are
included
in
T
ab
le
6.
4.5.
Example-5:
MIMO
A
po
w
er
p
lant
system
can
be
classified
as
a
m
ultiv
ar
iab
le
large-scale
system.
Numerous
methods
of
analysis
and
synthesis
f
or
such
processes
ha
v
e
been
de
v
eloped,
b
ut
the
remar
kab
le
dimensions
of
the
model
str
ucture
mak
es
their
implemen
tation
v
er
y
difficult.
Consider
ab
le
atten-
tion
has
theref
ore
been
de
v
oted
to
the
prob
lem
of
der
iving
reduced-order
models
f
or
such
sys-
tems
.
The
siz
e
and
comple
xity
of
current
electr
ic
po
w
er
netw
or
ks
in
v
olv
es
methods
f
or
studying
appro
ximated
models
to
in
v
estigate
the
dynamic
beha
viour
of
such
system
types
in
a
more
suit-
ab
le
w
a
y;
the
methods
currently
used
f
or
deter
mining
reduced-order
dynamic
models
f
or
po
w
er
systems
in
m
ulti-b
us
,
m
ulti-machine
fr
ames
are
gener
ally
ref
erred
to
as
”dynamic
equiv
alents”.
IJEECS
V
ol.
2,
No
.
3,
J
une
2016
:
486
500
Evaluation Warning : The document was created with Spire.PDF for Python.
IJEECS
ISSN:
2502-4752
495
An
electr
ic
po
w
er
system
consisting
of
a
salient-pole
synchronous
gener
ator
connected
to
an
infinite
b
us-bar
is
considered.
T
aking
into
account
the
w
ell
kno
wn
perf
or
mance
equations
of
both
the
machine
and
the
tr
ansmission
line
,
a
v
er
y
accu
r
ate
non-linear
mathematical
model
in
the
state-space
f
or
m
has
been
der
iv
ed.
As
state
v
ar
iab
les
of
the
electr
ical
par
t
of
the
synchronous
machine
,
the
set
of
winding
currents
of
the
q
d
equiv
alent
circuit
has
been
chosen.
The
se
v
enth-
order
state
v
ector
of
the
or
iginal
sys
t
em
consists
of
the
stator
currents
i
d
,
i
q
,
the
field
circuit
current
i
f
d
,
tw
o
damping
circuit
currents
i
k
q
,
i
k
d
and
the
mechanical
quantities
and
!
.
The
input
v
ector
,
in
the
chosen
representation,
consists
of
tw
o
quantities
,
the
mechanical
torque
T
m
and
the
v
oltage
V
f
.
As
output
v
ar
iab
les
,
the
machine
v
oltage
VI
and
the
mechanical
state
v
ar
iab
les
and
!
ha
v
e
been
chosen
[3].
By
consider
ing
small
v
ar
iations
(
)
around
a
steady-state
oper
ating
point,
a
linear
model
has
been
der
iv
ed.
The
v
alues
of
the
par
ameters
,
steady-state
w
or
king
conditions
and
fur
ther
details
on
the
adopted
model
are
repor
ted
b
y
Ramamoor
ty
and
Ar
um
ugan
[31].
w
e
indicate
with:
Change
in
mechanical
torque
(
T
m
)
as
input
1
Change
in
field
v
oltage
(
V
f
)
as
input
2
Change
in
ter
minal
v
oltage
(
V
t
)
as
output
1
Change
in
po
w
er
angle
(
)
as
output
2
Change
in
speed
(
!
)
as
output
3
The
tr
ansf
er
function
of
m
ulti-input
m
ulti-output
(MIMO)
single-machine
infinite-b
us
(SM
IB)
po
w
er
system
can
be
represented
as
in
Eqn.
31.
The
tr
ansf
er
function
of
the
system
with
output
V
t
to
input
T
m
can
be
represented
b
y
G
11
(
s
)
=
g
11
(
s
)
=d
(
s
)
and
si,ilar
ly
f
or
others
.
The
considered
MIMO
SMIB
consists
of
six
diff
erent
tr
ansf
er
function
with
diff
erent
sets
of
input
and
output
sig-
nals
.
The
denominator
of
these
systems
is
common
and
represen
ted
b
y
d
(
s
)
in
Eqn.
32.
The
polynomials
presented
in
Eqn.
33
-
Eqn.
38,
are
the
n
umer
ators
of
diff
erent
tr
ansf
er
functions
due
to
diff
erent
sets
of
input
and
output
signals
.
G
(
s
)
=
2
4
g
11
(
s
)
g
21
(
s
)
g
12
(
s
)
g
22
(
s
)
g
13
(
s
)
g
23
(
s
)
3
5
d
(
s
)
(31)
d
(
s
)
=
8
>
>
<
>
>
:
s
7
+
258
:
7
s
6
+
4
:
31
10
5
s
5
+4
:
835
10
7
s
4
+
1
:
853
10
9
s
3
+2
:
54
10
10
s
2
+
5
:
973
10
10
s
+1
:
886
10
10
(32)
g
11
(
s
)
=
8
<
:
12
:
41
s
4
+
1
:
213
10
4
s
3
2
:
866
10
6
s
2
3
:
325
10
8
s
6
:
404
10
9
(33)
g
12
(
s
)
=
8
<
:
12
:
41
s
5
+
1
:
213
10
4
s
4
2
:
866
10
6
s
3
3
:
325
10
8
s
2
6
:
404
10
9
s
+
0
:
0006087
(34)
g
13
(
s
)
=
8
>
>
<
>
>
:
0
:
2005
s
6
+
47
:
88
s
5
+3
:
928
10
4
s
4
+
5
:
122
10
6
s
3
+2
:
288
10
8
s
2
+
3
:
434
10
9
s
+5
:
492
10
9
(35)
Routh
Appro
ximation:
An
Approach
of
Model
Order
Reduction
...
(D
.
K.
Sambar
iy
a)
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