TELKOM
NIKA
, Vol.13, No
.2, June 20
15
, pp. 518 ~ 5
2
7
ISSN: 1693-6
930,
accredited
A
by DIKTI, De
cree No: 58/DIK
T
I/Kep/2013
DOI
:
10.12928/TELKOMNIKA.v13i2.875
518
Re
cei
v
ed O
c
t
ober 2
6
, 201
4; Revi
se
d March 15, 201
5
;
Accepte
d
April 2, 2015
Guaranteed Cost Control for Uncertain Neutral Systems
with a Minimal Order Observer
Er
w
i
n Susanto
1
, Junar
t
ho
Halomoan
2
, Mitsua
ki Ishitobi
3
1,2
School of Electrical En
gin
e
e
rin
g
, T
e
lkom
Univers
i
t
y
,
3
Department o
f
Mechanic
a
l S
y
stems En
gin
e
e
rin
g
, Kumamo
to Univers
i
t
y
e-mail: er
w
i
nelektro@telk
o
muniversit
y
.
ac.id
1
, junarth
o@telk
omun
iversit
y
.a
c.id
2
,
mishi@k
u
mam
o
to-u.ac.jp
3
A
b
st
r
a
ct
T
h
is pa
per
pr
esents
a d
e
si
gn sch
e
m
e
of a
mi
n
i
mal
or
der o
b
serv
er-b
ased
gu
arant
e
ed cost
controller for uncertain neut
ral
system
s, in which some state vari
ables can not be m
e
asur
ed. The
uncerta
inties
a
r
e assu
med to
be n
o
r
m
-bo
u
n
ded. T
h
e in
itial
state is ass
u
med u
n
kn
ow
n b
u
t their
me
an
a
n
d
covari
ance
are
assu
med k
n
o
w
n. A sufficient conditi
on
for
robust stab
ility
ana
lysis a
nd r
obust stab
ili
z
a
t
i
o
n
are der
ived vi
a lin
ear matrix
ineq
ual
ities (
L
MIs). T
o
show
the advanta
ge of
the pro
pose
d
metho
d
,
a
nu
meric
a
l ex
a
m
p
l
e is g
i
ven.
Ke
y
w
ords
: gu
arante
ed cost
control, unc
er
tain n
eutral sys
tems, a mini
mal ord
e
r obser
ver, line
a
r matri
x
ine
qua
lities (
L
MIs)
1. Introduc
tion
Due to the p
r
esen
ce of th
e uncertai
n
tie
s
may ca
use
instability, bad pe
rform
a
n
c
e an
d
lack of the exact model for the controlle
d syst
em
s, then con
s
id
era
b
le attention has be
en dra
w
n
to the proble
m
of robust stability and stabilization
for
systems with
param
eter uncertaintie
s
[1].
Moreover, it
has been devoted to fi
nd a
controll
er which
guarante
es robust stability.
Especi
a
lly
in a
re
al pla
n
t co
ntrol, it i
s
also
de
sirabl
e to
d
e
si
gn
a
control
syste
m
which
n
o
t
only a
c
hieve
s
the
stability but also guarantees an a
dequat
e level of performance [2
].
The guaranteed cost control
is on
e app
ro
a
c
h to
solve thi
s
ki
nd of p
r
ob
lem be
cau
s
e
it has a
n
adv
antage i
n
pro
v
iding an u
p
p
e
r
boun
d on the
quad
ratic
co
st functio
n
of
the cl
o
s
e
d
l
oop
system.
Lyapun
ov method via lin
ear
matrix inequality is often
used
to proof the stability criteria
[3],[4]. Some research papers
on
guaranteed cost contro
ller
desi
gn can be found in [5],[6
] and references therein.
In practi
cal u
s
e, it is difficult to measure
all system states that are neede
d in controlle
r
desi
gn be
ca
u
s
e of some
reason
s su
ch
as p
oor
pl
ant
kno
w
led
ge,
sen
s
o
r
avaibi
lity, etc. On the
other h
and, the ob
serve
r
b
a
se
d co
ntrol
may be
better than a state feedb
ack on
e becau
se it may
not be p
o
ssib
le to mea
s
u
r
e all sy
stem
states.
T
he o
b
se
rver
co
uld
be emb
edd
e
d
in the
syste
m
s
for either
systems with
out
all available
states o
r
sy
stem
s with p
a
rtially availa
ble state
s
. A full
orde
r o
b
serv
er a
nd a
mini
mal orde
r ob
serve
r
a
r
e
ap
plied to
re
co
nstru
c
t the
system state
s
[7],
[8]. However,
a mi
nimal
order ob
se
rver
and
red
u
c
ed orde
r ob
serv
er are
fewer
i
n
vestigate
d
t
han
a full orde
r ob
serve
r
ba
se
d guarantee
d cost co
ntrol re
sea
r
ch [9].
This pa
pe
r wi
ll con
s
ide
r
a minimal orde
r obse
r
ver b
a
s
ed
control to
develop gu
a
r
antee
d
co
st control for un
certai
n neutra
l
syste
m
s. The neut
ral syste
m
is system that depe
nd
s on the
delay of the state and it
s derivat
ive. Th
is type ca
n b
e
found in m
any fields of
engin
eeri
ng
and
techn
o
logy a
pplication su
ch as
in
ch
emical
p
r
o
c
e
ss, networke
d
control system
s,
ro
boti
c
impleme
n
tation etc [10].
The un
ce
rtai
nties a
r
e a
s
sumed to be
norm
-
bo
und
e
d
with un
kn
o
w
n
initial state but their mean
and covari
an
ce are
kno
w
n
.
Since the inverse relation
s are ap
pea
red
in LMI solutio
n
, an iterative algorithm i
s
use
d
[11].
Not
a
tion
s
. T
h
rou
gho
ut th
e pap
er, th
e
su
perscript
s ”
T
” an
d ”
−
1
”
s
t
and for matrix
transpo
se an
d inverse,
n
de
notes the n
-
d
i
mensi
onal E
u
clid
ean spa
c
e,
X >
Y
or
X
≥
Y
means
that
X
−
Y
is
positive d
e
finite or semi-p
ositive definit
e,
I
is an id
entity matrix with ap
pro
p
ri
ate
dimen
s
ion
s
, and * re
pre
s
e
n
ts the symm
etric ele
m
ent
s in a symme
tric matrix.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
1693-6
930
Guarantee
d Co
st Control for Un
ce
rtain
Neutral
Sys
t
ems
with a Minimal ..
.. (Erwin Sus
anto)
519
2. Problem Statements
Con
s
id
er an
uncertain n
e
u
t
ral system
wi
th
(1)
(2)
,
(3)
whe
r
e
,
,
an
d
are the
gi
ven consta
nt
time del
ay in the
states
and their de
rivatives,
is the state vector,
is the control inp
u
t vector,
is the me
asu
r
ed o
u
tput ve
ctor,
are kno
w
n
con
s
tant real-valu
ed m
a
trice
s
and
is
res
t
ricted to the form of
. The uncertai
n
ties in
the states a
r
e
written in
,
,
,
whic
h sat
i
sf
y
,
,
,
(4)
w
h
er
e
are
con
s
tant
re
al
-valued
kno
w
n matri
c
e
s
with ap
propri
a
te
dimen
s
ion
s
, and
are
re
al time-va
r
ying u
n
kno
w
n
co
ntinuo
us an
d
determi
nisti
c
matrices.
We fu
rthe
r a
s
sume th
at t
he initial
stat
e varia
b
le
is
unkno
wn, b
u
t their mea
n
and
covari
an
ce a
r
e kno
w
n, eq
u
i
valently
(5)
(6)
(7)
whe
r
e
is the expecte
d value ope
rato
r.
The problem
determi
ned h
e
re is to de
si
gn a minimal
orde
r ob
se
rver
(8)
(9)
and a controll
er
(10
)
with
,
,
,
,
,
,
,
to achieve a
n
upper b
oun
d
of the following qua
drati
c
perfo
rman
ce i
ndex
(11
)
asso
ciated
wi
th the sy
stem
(1)
-(2
) wh
er
e
are given sy
mmetric
posit
ive definite matrice
s
.
3. Main Result
In this se
ction
,
a sufficient condition i
s
est
ablished for t
he existen
c
e
of a minimal orde
r
observe
r-b
ased gua
rante
e
d
co
st cont
rol
l
er fo
r the un
certain sy
stem
(1) an
d (2
).
The gain of controlle
r is formulated in:
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 16
93-6
930
TELKOM
NIKA
Vol. 13, No. 2, June 20
15 : 518 – 52
7
520
(12
)
whe
r
e
is a symmetric
posit
ive definite matrix.
The main result of this study is given by Theorem 1.
Theo
rem
1.
If the following
matrix inequ
a
lities optimiza
t
ion probl
em;
s
ubjec
t to
1
2
1
1
2
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
U
U
U
0
diag
diag
x
x
(13
)
,
,
,
(14
)
0
2
2
2
1
2
1
4
S
Yv
Yv
S
Yv
Y
v
Y
v
Y
v
m
T
T
m
T
T
T
T
(15
)
0
*
1
1
inv
x
H
E
M
(16
)
0
*
2
2
inv
x
H
F
M
(17
)
0
*
1
3
inv
x
U
G
M
(18
)
0
*
2
4
inv
x
U
H
M
(19
)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
1693-6
930
Guarantee
d Co
st Control for Un
ce
rtain
Neutral
Sys
t
ems
with a Minimal ..
.. (Erwin Sus
anto)
521
W
h
er
e
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
has a
set of solution
s
satisfy the inverse rel
a
tion
s
then the
mini
mal orde
r ob
serve
r
la
w
(8
)-(10
)
is
a
gu
arante
ed
co
st controller with a minim
u
m
expecte
d value of guarant
eed cost
(20
)
whe
r
e
a
nd
is the e
s
ti
mated e
r
ror
of the minim
a
l orde
r
observe
r.
Rem
a
r
k
1
. Si
nce
(13)-(19) have
co
nst
r
aints i
n
inve
rse
rel
a
tions,
an ite
r
ative L
M
I app
roa
c
h
is
applie
d to sol
v
e [11].
Before giving
a proof of Th
eore
m
1, a ke
y lemma is introdu
ce
d ([12]
).
L
e
mma
1
. Let
a
nd
b
e
matrices of
a
ppro
p
ri
ate di
mensi
o
n
s
, a
n
d
be
a m
a
trix function
sat
i
sf
y
i
ng
. Then for any p
o
sitive scala
r
, the following
inequ
ality holds
(21
)
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ISSN: 16
93-6
930
TELKOM
NIKA
Vol. 13, No. 2, June 20
15 : 518 – 52
7
522
Proof of Theo
rem
1
The clo
s
e
d
lo
op syste
m
is
obtaine
d by mathem
ati
c
al
sub
s
titution of (1)-(2
) and
(8)-(10)
)
(
)
(
))
(
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
2
1
2
1
t
t
x
t
h
t
x
t
x
BKP
T
D
TA
TA
BK
T
A
T
KP
t
B
t
A
t
A
K
t
B
t
A
t
t
x
(22
)
Define a
can
d
idate of Lya
punov fun
c
tio
n
(23
)
whe
r
e
Then, the time derivative o
f
(23) alo
ng to (22
)
is calculated a
s
(24
)
whe
r
e
,
2
2
9
8
7
6
5
4
3
2
1
0
)
1
(
0
0
0
0
0
0
0
0
0
)
(
U
d
H
t
,
,
,
,
,
,
,
,
,
,
Und
e
r conditi
on
(25
)
equatio
n (24
)
leads to
(26
)
for any
and the clo
s
e
d
loo
p
system i
s
a
s
ymptotically
stable.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
1693-6
930
Guarantee
d Co
st Control for Un
ce
rtain
Neutral
Sys
t
ems
with a Minimal ..
.. (Erwin Sus
anto)
523
Here, con
d
ition (24
)
is inv
e
stigate
d
by applying
l
e
mma
1
as
follows
(27
)
(28
)
(29
)
(30
)
(31
)
(32
)
(33
)
(34
)
(35
)
(36
)
(37
)
(38
)
(39
)
(40
)
(41
)
(42)
(43
)
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 16
93-6
930
TELKOM
NIKA
Vol. 13, No. 2, June 20
15 : 518 – 52
7
524
(44
)
Applying lem
m
a 1., denoting
, and usin
g schu
r co
mple
ment lead to (13)
Furthe
r, integ
r
ating (26) fro
m
0 to T infinity, we have
(45
)
Con
s
id
er the
optimal expe
cted val
ue of
the guarante
ed co
st, we h
a
ve
(46
)
A relation bet
wee
n
mean
and covaria
n
c
e
of the exp
e
cted valu
e is
(47
)
Substituting (47) into (46),
we obtai
n
(48
)
It is easily se
en that
(49
)
(50
)
(51
)
(52
)
(53
)
Here, we cons
ider pos
i
tive s
c
alars
, (54)
, (55)
(56
)
(57
)
(58
)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
1693-6
930
Guarantee
d Co
st Control for Un
ce
rtain
Neutral
Sys
t
ems
with a Minimal ..
.. (Erwin Sus
anto)
525
(59
)
(60
)
(61
)
(62
)
Minimizi
ng
results in mi
n
. By recalling
, (54)-(5
7) lea
d
to (14) a
nd
(59
)
-(62
) to (16)-(1
9).
By denoting
, (58
)
is calcul
ated as
(63
)
Schu
r com
p
le
ment derive
s
(15
)
from (63)
4. Numerical
example
Con
s
id
er sy
st
em (1)
-
(
2
)
wit
h
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
Applying the
minimal-order observer-bas
ed
app
roac
h [8],[9], we obtain the
s
o
lutions
of
controlle
r and
obse
r
ver g
a
i
n
:
,
,
guarantee
d cost value
an
d other
set of solutio
n
s a
s
follows
,
,
,
,
,
,
,
,
.
Figure 1 sh
o
w
s the traje
c
tory
of states with initial c
ondition for
nominal a
nd
neutral
sy
st
em
s
,
,
. It is sh
own th
at th
e state
s
conve
r
ge
to t
he
stable
co
n
d
ition.
He
nce
the controlle
r an
d o
b
serv
er g
a
in a
r
e
d
e
sig
ned
well
fo
r
the uncertain
neutral
syste
m
s with a min
i
mal ord
e
r ob
serve
r
.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 16
93-6
930
TELKOM
NIKA
Vol. 13, No. 2, June 20
15 : 518 – 52
7
526
0
10
20
30
40
50
60
70
80
90
10
0
-1
.
5
-1
-0.
5
0
0.5
1
Ti
m
e
(
s
e
c
)
St
at
es
Figure 1. Traj
ectory of stat
es
(
) and
(- -
)
5. Conclusio
n
This pa
per di
scusse
s a gu
arante
ed cost
cont
roll
er de
sign for u
n
ce
rtain neutral system
s
with a mini
m
a
l order
observer. T
he
stability on
the LMIs forms
is determined by deriving a
sufficie
n
t con
d
ition for the
existen
c
e of
state
feedb
ack controll
e
r
. To
illust
rat
e
the propo
sed
method, a n
u
meri
cal si
m
u
lated exam
ple is pr
ese
n
t
ed. The pro
b
lem of the guarantee
d cost
controlle
r for
uncertain
sy
stems i
s
still o
pen
acco
rdin
g to the
adva
n
tage
of this
method. In
th
e
future work, we will investi
gate t
he probl
e
m of discret
e
system
s.
Ackn
o
w
l
e
dg
ements
This
re
sea
r
ch is fun
ded
b
y
dire
ctorate
gene
ral of
hi
gher edu
cati
on (DIKTI), Indon
esi
a
ministry
of ed
ucatio
n an
d
cultur
e
(Deci
s
i
on of the
Di
re
ctor
of
Resea
r
ch
an
d
Com
m
unity Service
Numb
er: 02
6
3
/E5/2014
).
Referen
ces
[1]
Che
n
Y, Ma G, Lin S, Gao J.
Adaptiv
e fuzz
y
and ro
bust
compens
ation co
ntrol for uncert
a
in ro
bot.
Internatio
na
l Journ
a
l of Ro
bo
tics and Auto
mation
. 20
13; 2(
4): 174-1
8
8
[2]
Cha
ng SSL, P
eng T
K
C. Adaptive gu
ara
n
te
ed cost
contro
l of s
y
stems
w
i
t
h
unc
ertai
n
parameters.
IEEE Trans. on Automatic Contr
. 1972; 17(
4
)
: 474-48
3.
[3]
W
ang L, L
i
Z
.
Nonl
in
ear ro
bu
st control for s
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