TELK
OMNIKA
,
V
ol.
12,
No
.
3,
September
2014,
pp
.
525
532
ISSN:
1693-6930,
accredited
A
b
y
DIKTI,
Decree
No:
58/DIKTI/K
ep/2013
DOI:
10.12928/telk
omnika.v12.i3.125
525
Quantiz
ed
Feedbac
k
Contr
ol
of
Netw
ork
Empo
werment
Amm
unition
With
Data-Rate
Limitations
F
ang
Jin
1
,
Zhi-hua
Y
uan*
1
,
Qing-Quan
Liu
1,2,3
,
and
Zhang
Xing
4
1
College
of
Inf
or
mation
Science
and
Engineer
ing,
Shen
y
ang
Ligong
Univ
ersity
No
.
6,
Nan
Ping
Zhong
Road,
Hun
Nan
Xin
Distr
ict,
Shen
y
ang,
110159,
Chin
a
2
Shen
y
ang
Institute
of
A
utomation,
Chinese
Academ
y
of
Sciences
No
.
114,
Nanta
Street,
Shenhe
Distr
ict,
Shen
y
ang,
110016,
China
3
Allwin
T
elecomm
unication
CO
.,
L
TD
No
.
6,
Gaoge
Road,
Hun
Nan
Xin
Distr
ict,
Shen
y
ang,
110179,
China
4
Ag
r
icultur
al
De
v
elopment
Bank
of
China
br
anch
in
Liaoning
pro
vince
China
*Corresponding
author
,
e-mail:
lqqneu@163.com
Abstract
This
paper
in
v
estigates
quantiz
ed
f
eedbac
k
control
prob
lems
f
or
netw
or
k
empo
w
er
ment
amm
uni-
tion,
where
the
sensors
and
the
controller
are
conne
cted
b
y
a
digital
comm
unication
netw
or
k
with
data-r
ate
limitations
.
Diff
erent
from
the
e
xisting
ones
,
a
ne
w
bit-allocation
algor
ithm
on
the
basis
of
the
singular
v
alues
of
the
plant
matr
ix
is
proposed
to
encode
the
plant
states
.
A
lo
w
er
bound
on
the
data
r
ate
is
presented
to
ensure
stabilization
of
the
unstab
le
plant.
It
is
sho
wn
in
our
results
that,
the
algor
ithm
can
be
emplo
y
ed
f
or
the
more
gener
al
case
.
An
illustr
a
tiv
e
e
xample
is
giv
en
to
demonstr
ate
the
eff
ectiv
eness
of
the
proposed
algor
ithm.
K
e
yw
or
ds:
netw
or
k
empo
w
er
ment
amm
unition,
bit-allocation
algor
ithms
,
data-r
ate
limitations
,
quantiz
ed
control,
f
eedbac
k
stabilization
Cop
yright
c
2014
Univer
sitas
Ahmad
Dahlan.
All
rights
reser
ved.
1.
Intr
oduction
Netw
or
k
ed
control
systems
ha
v
e
attr
acted
g
reat
interests
in
recent
y
ears
[1-3].
In
this
paper
,
w
e
study
quantiz
ed
f
eedbac
k
control
prob
lems
f
or
netw
or
k
empo
w
er
ment
amm
unition
with
limited
inf
or
mation
about
the
plant
states
.
This
prob
lem
ar
ises
when
t
he
state
measurements
are
to
be
tr
ansmitted
to
the
controller
via
a
limited
capacity
comm
unication
channel.
Issues
of
the
type
discussed
are
motiv
ated
b
y
se
v
er
al
pie
ces
of
w
or
k
in
the
recent
liter-
ature
.
The
research
on
the
inter
pla
y
among
coding,
estimation,
and
control
w
as
initiated
b
y
[4].
A
high-w
ater
mar
k
in
the
study
of
quantiz
ed
f
eedbac
k
using
data
r
ate
limited
f
eedbac
k
channels
is
kno
wn
as
the
data
r
ate
theorem
that
states
the
larger
the
magnitud
e
of
the
unstab
le
poles
,
the
larger
the
required
data
r
ate
through
the
f
eedbac
k
loop
.
The
intuitiv
ely
appealing
result
w
as
pro
v
ed
[5-8],
indicating
that
it
quantifies
a
fundamental
relationship
betw
een
unstab
le
ph
ysical
sys-
tems
and
the
r
ate
at
which
inf
or
mation
m
ust
be
processed
in
order
to
stab
ly
control
them.
When
the
f
eedbac
k
channel
capacity
is
near
the
data
r
ate
limit,
control
designs
typically
e
xhibit
chaotic
instabilities
.
This
result
w
as
gener
aliz
ed
to
diff
erent
notions
of
stabilization
and
system
models
,
and
w
as
also
e
xtended
to
m
ulti-dimensional
systems
[9-12].
Liu
and
Y
ang
in
v
estigated
quantiz
ed
control
prob
lems
f
or
linear
time-in
v
ar
iant
systems
o
v
er
a
noiseless
comm
unication
netw
or
k
[13].
Fur
ther
more
,
Liu
addressed
coordinated
motion
control
of
autonomous
and
semiautonomous
mobile
age
nts
in
[14],
and
der
iv
ed
t
he
condition
on
stabilization
of
unmanned
air
v
ehicles
o
v
er
wireless
comm
unication
channels
in
[15].
F
or
the
m
ulti-state
case
,
one
needs
to
present
an
optimal
bit-allocation
algor
ithm
to
reg-
ulate
the
tr
ansmission
of
inf
or
mation
about
each
mode
such
that
stabilization
can
be
guar
anteed
f
or
all
modes
.
In
the
liter
ature
,
the
bit-allocation
algor
ithms
w
ere
on
the
basis
of
the
eigen
v
alues
of
Receiv
ed
Apr
il
25,
2014;
Re
vised
J
uly
4,
2014;
Accepted
J
uly
24,
2014
Evaluation Warning : The document was created with Spire.PDF for Python.
526
ISSN:
1693-6930
the
system
matr
ix
A
.
Namely
,
it
states
the
larger
the
magnitude
of
the
unstab
le
eigen
v
alues
,
the
larger
the
required
data
r
ate
through
the
f
eedbac
k
loop
.
Thus
,
it
needs
to
tr
ansf
or
m
the
system
matr
ix
A
b
y
a
real
tr
ansf
or
mation
matr
ix
H
2
R
n
n
to
a
diagonal
matr
ix
J
or
a
Jordan
canoni-
cal
f
or
m
J
(i.e
.,
J
=
H
AH
1
)
in
order
to
decouple
its
dynamical
modes
to
achie
v
e
an
optimal
bit-allocation
algor
ithm.
Ho
w
e
v
er
,
f
or
the
more
gener
al
matr
ix
A
,
modal
decomposition
might
not
be
possib
le
and
putting
the
system
matr
ix
into
Jordan
canonical
f
or
m
gener
ally
requires
a
tr
ans-
f
or
mation
matr
ix
with
comple
x
elements
such
that
the
e
xisting
bit-allocation
algor
ithms
do
not
w
or
k.
In
this
paper
,
w
e
present
a
ne
w
bit-allocation
algor
ithm
f
or
the
more
gener
al
matr
ix
A
.
The
algor
ithm
proposed
here
is
on
basis
of
not
the
eigen
v
alues
b
ut
the
singular
v
alues
of
the
system
matr
ix
A
.
In
par
ticular
,
w
e
quantiz
e
,
encode
the
plant
states
b
y
an
adaptiv
e
diff
erential
coding
str
ategy
.
The
rest
of
the
paper
is
organiz
ed
as
f
ollo
ws
.
In
Section
2,
the
prob
lem
f
or
m
ulation
is
pre-
sented.
Section
3
presents
the
bit
-allocation
algor
ithm
f
or
stabilization.
The
results
of
n
umer
ical
sim
ulation
are
presented
in
Section
4.
Conclusions
are
stated
in
Section
5.
2.
Pr
ob
lem
Form
ulation
Consider
the
control
system
of
netw
or
k
empo
w
er
ment
amm
unition
descr
ibed
b
y
the
state
equation
X
(
k
+
1)
=
AX
(
k
)
+
B
U
(
k
)
(1)
where
X
(
k
)
2
R
n
is
the
measur
ab
le
state
,
and
U
(
k
)
2
R
p
is
the
control
input.
A
and
B
are
kno
wn
constant
matr
ices
with
appropr
iate
dimensions
.
The
f
ollo
wing
is
assumed
to
hold:
Assumption-1:
The
pair
(
A;
B
)
is
controllab
le
,
and
the
plant
states
are
measur
ab
le;
Assumption-2:
The
initial
condition
X
(0)
is
a
r
andom
v
ector
,
satisfying
E
k
X
(0)
k
2
<
0
<
1
;
Assumption-3:
The
sensors
and
controllers
are
geog
r
aphically
separ
ated
and
connected
b
y
er-
ror
less
,
bandwidth-limited,
digital
comm
unication
channels
without
time
dela
y
.
The
channel
is
also
assumed
t
o
be
a
time-in
v
ar
iant,
memor
yless
channel.
Then,
the
encoder
and
de-
coder
ha
v
e
access
to
the
pre
vious
control
actions
.
Let
^
X
(
k
)
denote
the
decoder’
s
estimate
of
X
(
k
)
on
the
basis
of
the
channel
output.
Our
simplified
model
of
the
channel
neglects
the
eff
ects
of
netw
or
k-induced
dela
ys
and
data
dropout,
and
f
ocuses
on
the
bit-allocation
algor
ithm.
Then,
w
e
ma
y
implement
a
quantiz
ed
state
f
eedbac
k
control
la
w
of
the
f
or
m
U
(
k
)
=
K
^
X
(
k
)
:
(2)
As
in
[9],
the
system
(1)
is
said
to
be
asymptotically
MS-Stabilizab
le
via
quantiz
ed
f
eed-
bac
k
if
f
or
an
y
initial
states
X
(0)
,
there
e
xists
a
control
policy
relying
on
the
quantiz
ed
data
such
that
the
states
of
the
closed-loop
system
are
asymptotically
dr
iv
en
to
z
ero
in
the
mean
square
sense
,
namely
lim
sup
k
!1
E
k
X
(
k
)
k
2
=
0
:
(3)
Our
main
task
is
to
present
a
bit-allocation
algor
ithm
f
or
the
more
gener
al
matr
ix
A
,
and
to
der
iv
e
the
sufficient
condition
on
the
data
r
ate
f
or
stabilization
of
the
system
(1)
in
the
mean
square
sense
(3).
3.
The
Bit-Allocation
Algorithm
This
section
deals
with
the
stabilization
prob
lem
under
data
r
ate
constr
aints
,
and
presents
a
bit-allocation
algor
ithm
f
or
the
more
gener
al
matr
ix
A
.
Since
the
matr
ix
A
0
A
is
a
real
symmetr
ic
matr
ix,
there
e
xists
a
real
or
thogonal
matr
ix
H
2
R
n
n
that
diagonaliz
es
A
0
A
=
H
0
2
H
.
Here
,
w
e
define
:=
diag
[
1
;
;
n
]
where
i
denotes
the
i
th
singular
v
alue
of
A
(
i
=
1
;
;
n
).
TELK
OMNIKA
V
ol.
12,
No
.
3,
September
2014
:
525
532
Evaluation Warning : The document was created with Spire.PDF for Python.
TELK
OMNIKA
ISSN:
1693-6930
527
The
coding
technique
presented
in
this
paper
is
an
adaptiv
e
diff
erential
coding
str
ategy
.
Define
X
(
k
)
:=
H
X
(
k
)
;
~
X
(
k
)
:=
H
^
X
(
k
)
:
Then,
the
system
(1)
ma
y
be
re
wr
itten
as
X
(
k
+
1)
=
H
AH
0
X
(
k
)
+
H
B
K
H
0
~
X
(
k
)
:
Let
~
X
(
k
)
:=
H
(
A
+
B
K
)
H
0
~
X
(
k
1)
and
Z
(
k
)
:=
X
(
k
)
~
X
(
k
)
denote
the
prediction
v
alue
and
the
prediction
error
of
X
(
k
)
,
respectiv
ely
.
Here
,
w
e
set
~
X
(0)
=
0
.
Then,
Z
(0)
=
X
(0)
.
It
is
sho
wn
in
Assumption-3
that
the
encoder
and
decoder
ha
v
e
access
to
the
pre
vious
control
actions
,
update
their
estimator
and
scaling
in
the
same
manner
,
and
obtain
the
same
prediction
v
alue
.
Then,
w
e
ma
y
quantiz
e
,
encode
Z
(
k
)
,
and
tr
ansmit
the
inf
or
mation
of
Z
(
k
)
via
a
digital
channel.
Let
^
Z
(
k
)
and
V
(
k
)
denote
the
quantization
v
alue
and
the
quantization
error
of
Z
(
k
)
,
respectiv
ely
.
Then,
^
Z
(
k
)
=
Z
(
k
)
V
(
k
)
:
The
v
alue
of
^
Z
(
k
)
ma
y
be
computed
on
the
basis
of
the
channel
output
at
the
decoder
.
Then,
the
decoder’
s
estimate
is
defined
as
^
X
(
k
)
:=
H
0
(
~
X
(
k
)
+
^
Z
(
k
))
:
Thus
,
^
X
(
k
)
ma
y
be
obtained
b
y
the
controller
.
Let
Z
(
k
)
:=
[
z
1
(
k
)
z
2
(
k
)
z
n
(
k
)]
0
.
As
in
[16],
giv
en
a
positiv
e
integer
M
i
and
a
nonneg-
ativ
e
real
n
umber
i
(
k
)
(
i
=
1
;
;
n
),
define
the
quantiz
er
q
:
R
!
Z
with
sensitivity
i
(
k
)
and
satur
ation
v
alue
M
i
b
y
the
f
or
m
ula
q
(
z
i
(
k
))
=
8
<
:
M
+
;
if
z
i
(
k
)
>
(
M
i
+
1
=
2)
i
(
k
)
;
M
;
if
z
i
(
k
)
(
M
i
+
1
=
2)
i
(
k
)
;
b
z
i
(
k
)
i
(
k
)
+
1
2
c
;
if
(
M
i
+
1
=
2)
i
(
k
)
<
z
i
(
k
)
(
M
i
+
1
=
2)
i
(
k
)
(4)
where
define
b
z
c
:=
max
f
k
2
Z
:=
k
<
z
;
z
2
R
g
.
The
indeces
M
+
and
M
will
be
emplo
y
ed
if
the
quantiz
er
satur
ates
.
The
algor
ithm
to
be
used
here
is
based
on
the
h
ypothesis
that
it
is
possib
le
to
change
the
sensitivity
(b
ut
n
ot
the
satur
ation
v
alue)
of
the
quantiz
er
on
the
basis
of
a
v
ailab
le
quantiz
ed
measurements
.
Ho
w
e
v
er
,
it
is
unclear
in
[16]
ho
w
m
uch
each
M
i
should
at
least
be
in
order
to
guar
antee
stabilization
of
the
system
(1).
A
lo
w
er
bou
nd
of
each
M
i
f
or
stabilization
will
be
presented
in
our
results
.
Based
on
the
quantizatio
n
algor
ithm
abo
v
e
,
w
e
can
constr
uct
a
code
with
code
w
ord
length
r
i
(
i
=
1
;
;
n
).
Let
(
c
1
c
2
c
r
i
)
denote
the
code
w
ord
corresponding
to
z
i
(
k
)
.
Namely
,
z
i
(
k
)
is
quantiz
ed,
encoded,
and
tr
ansf
or
med
into
the
r
i
bits
f
or
tr
ansmission.
Then
w
e
ma
y
compute
c
j
2
f
0
;
1
g
(
j
=
1
;
;
r
i
)
(
c
1
c
2
c
r
i
1
)
=
arg
max
(
c
1
c
2
c
r
i
1
)
P
r
i
1
j
=1
c
j
2
j
1
(5)
subject
to
the
condition:
P
r
i
1
j
=1
c
j
2
j
1
jb
z
i
(
k
)
i
(
k
)
+
1
2
cj
:
Fur
ther
more
,
w
e
set
c
r
i
=
0
when
z
i
(
k
)
0
and
set
c
r
i
=
1
when
z
i
(
k
)
<
0
.
This
implies
that
r
i
=
log
2
M
i
+
1
:
Then
the
data
r
ate
of
the
channel
is
giv
en
b
y
R
=
P
n
i
=1
r
i
(bits/sample)
:
Our
main
result
is
the
f
ollo
wing
theorem:
Quantiz
ed
F
eedbac
k
Control
of
Netw
or
k
Empo
w
er
ment
Amm
unition
with
...
(F
ang
Jin)
Evaluation Warning : The document was created with Spire.PDF for Python.
528
ISSN:
1693-6930
Theorem
1:
Consider
the
system
(1).
Suppose
that
all
the
eigen
v
alues
of
A
+
B
K
lie
inside
the
unit
circle
.
Then,
f
or
an
y
giv
en
"
2
(0
;
1)
,
there
e
xist
a
control
la
w
of
the
f
or
m
(2),
a
quantization
algor
ithm
of
the
f
or
m
(4),
and
a
coding
algor
ithm
of
the
f
or
m
(5)
to
stabiliz
e
the
system
(1)
in
the
mean
square
sense
(3)
if
the
data
r
ate
of
the
channel
satisfies
the
f
ollo
wing
condition:
R
>
P
i
2
1
2
log
2
2
i
"
(bits/sample)
where
:=
f
i
2
f
1
;
2
;
;
n
g
:
2
i
"
>
1
g
.
Pr
oof:
Consider
the
closed-loop
system
X
(
k
+
1)
=
AX
(
k
)
+
B
K
^
X
(
k
)
which
is
equiv
alent
to
X
(
k
+
1)
=
H
AH
0
X
(
k
)
+
H
B
K
H
0
~
X
(
k
)
:
Fur
ther
more
,
notice
that
~
X
(
k
+
1)
=
H
(
A
+
B
K
)
H
0
~
X
(
k
)
:
By
the
definitions
abo
v
e
,
w
e
ha
v
e
^
X
(
k
)
:=
H
0
(
~
X
(
k
)
+
^
Z
(
k
))
,
Z
(
k
)
:=
X
(
k
)
~
X
(
k
)
,
^
Z
(
k
)
=
Z
(
k
)
V
(
k
)
,
and
~
X
(
k
)
:=
H
^
X
(
k
)
.
Thus
,
it
f
ollo
ws
that
Z
(
k
+
1)
=
X
(
k
+
1)
~
X
(
k
+
1)
=
H
AH
0
(
X
(
k
)
~
X
(
k
))
=
H
AH
0
[(
~
X
(
k
)
+
Z
(
k
))
(
~
X
(
k
)
+
^
Z
(
k
))]
=
H
AH
0
V
(
k
)
:
(6)
Clear
ly
,
the
prediction
error
Z
(
k
+
1)
is
deter
mined
b
y
the
pre
vious
quantization
error
V
(
k
)
,
and
is
independent
of
the
control
signals
applied.
Notice
that
X
(
k
)
=
H
0
X
(
k
)
=
H
0
(
~
X
(
k
)
+
Z
(
k
))
:
Thus
,
it
holds
that
E
k
X
(
k
)
k
2
=
E
k
X
(
k
)
k
2
=
E
k
~
X
(
k
)
k
2
+
E
k
Z
(
k
)
k
2
+
2
E
[
~
X
0
(
k
)
Z
(
k
)]
:
(7)
By
the
definition
abo
v
e
,
w
e
see
that
~
X
(
k
)
=
H
(
A
+
B
K
)
H
0
~
X
(
k
1)
:
Fur
ther
more
,
it
f
ollo
ws
from
(6)
that
Z
(
k
)
=
H
AH
0
V
(
k
1)
:
Thus
,
w
e
ha
v
e
E
[
~
X
0
(
k
)
Z
(
k
)]
=
E
[
~
X
0
(
k
1)
H
(
A
+
B
K
)
0
AH
0
V
(
k
1)]
=
E
[(
X
(
k
1)
V
(
k
1))
0
H
(
A
+
B
K
)
0
AH
0
V
(
k
1)]
:
(8)
Here
,
X
(
k
1)
and
V
(
k
1)
are
m
utually
independent
r
andom
v
ar
iab
les
.
This
implies
E
[
X
0
(
k
1)
V
(
k
1)]
=
0
:
(9)
Substitute
(9)
into
(8),
and
obtain
E
[
~
X
0
(
k
)
Z
(
k
)]
=
E
[
V
0
(
k
1)
H
(
A
+
B
K
)
0
AH
0
V
(
k
1)]
:
(10)
Thus
,
substituting
(10)
into
(7),
w
e
ma
y
obtain
E
k
X
(
k
)
k
2
=
E
k
~
X
(
k
)
k
2
+
E
k
Z
(
k
)
k
2
2
E
[
V
0
(
k
1)
H
(
A
+
B
K
)
0
AH
0
V
(
k
1)]
:
(11)
TELK
OMNIKA
V
ol.
12,
No
.
3,
September
2014
:
525
532
Evaluation Warning : The document was created with Spire.PDF for Python.
TELK
OMNIKA
ISSN:
1693-6930
529
F
or
an
y
giv
en
v
ector
X
,
w
e
define
X
:=
E
[
X
X
0
]
.
It
f
ollo
ws
from
(6)
that
E
k
Z
(
k
+
1)
k
2
=
tr
[
Z
(
k
+1)
]
=
tr
[(
H
AH
0
)
V
(
k
)
(
H
AH
0
)
0
]
=
tr
[(
H
AH
0
)
0
(
H
AH
0
)
V
(
k
)
]
=
tr
[(
H
A
0
AH
0
)
V
(
k
)
]
=
tr
[
2
V
(
k
)
]
(12)
Let
Z
(
k
)
:=
[
z
1
(
k
)
z
2
(
k
)
z
n
(
k
)]
0
and
V
(
k
)
:=
[
v
1
(
k
)
v
2
(
k
)
v
n
(
k
)]
0
.
If
there
e
xists
the
quanti-
zation
algor
ithm
of
the
f
or
m
(4)
such
that
the
f
ollo
wing
condition
holds:
"E
[
z
2
i
(
k
)]
>
2
i
E
[
v
2
i
(
k
)]
;
(
i
=
1
;
2
;
;
n
)
(13)
then
w
e
ha
v
e
E
[
z
2
i
(
k
+
1)]
<
"E
[
z
2
i
(
k
)]
(
i
=
1
;
2
;
;
n
)
:
This
means
that
E
[
z
2
i
(
k
)]
<
"
k
E
[
z
2
i
(0)]
=
"
k
E
[
x
2
i
(0)]
(
i
=
1
;
2
;
;
n
)
:
Thus
,
it
f
ollo
ws
that
lim
sup
k
!1
E
[
z
2
i
(
k
)]
=
0
(
i
=
1
;
2
;
;
n
)
:
Namely
,
lim
sup
k
!1
E
k
Z
(
k
)
k
2
=
0
:
(14)
Clear
ly
,
it
f
ollo
ws
from
(6)
and
(14)
that
lim
sup
k
!1
E
k
V
(
k
)
k
2
=
0
:
(15)
This
implies
lim
sup
k
!1
E
[
V
0
(
k
1)
H
(
A
+
B
K
)
0
AH
0
V
(
k
1)]
=
0
:
(16)
Notice
that
~
X
(
k
+
1)
=
H
(
A
+
B
K
)
H
0
~
X
(
k
)
=
H
(
A
+
B
K
)
H
0
~
X
(
k
)
+
H
(
A
+
B
K
)
H
0
^
Z
(
k
)
:
It
f
ollo
ws
from
(14)
and
(15)
that
lim
sup
k
!1
E
k
^
Z
(
k
)
k
2
=
0
:
Fur
ther
more
,
w
e
kno
w
that
all
eigen
v
alues
of
A
+
B
K
lie
inside
the
unit
circle
.
Thus
,
it
f
ollo
ws
that
lim
sup
k
!1
E
k
~
X
(
k
)
k
2
=
0
:
(17)
Thus
,
substitute
(14),
(16),
and
(17)
into
(11),
and
obtain
lim
sup
k
!1
E
k
X
(
k
)
k
2
=
0
:
This
means
that,
the
system
(1)
is
asymptotically
stabilizab
le
in
the
mean
square
sense
(3)
if
there
e
xists
the
quantization
algor
ithm
of
the
f
or
m
(4)
such
that
the
condition
(13)
holds
.
By
the
quantization
algor
ith
m
of
the
f
or
m
(4),
w
e
kno
w
that
the
quantization
error
V
(
k
)
ma
y
be
small
enough
to
mak
e
the
condition
(13)
hold
if
the
quantization
le
v
els
are
large
enough.
Notice
that
f
or
the
case
with
2
i
"
<
1
,
the
condition
(13)
m
ust
hold
though
w
e
do
not
quantiz
e
the
corresponding
z
i
(
k
)
,
and
do
not
tr
ansmit
its
inf
or
mation
to
the
controller
.
Thus
,
in
order
to
mak
e
the
condition
(13)
hold,
w
e
set
M
i
>
i
2
p
"
;
r
i
>
1
2
log
2
2
i
"
Quantiz
ed
F
eedbac
k
Control
of
Netw
or
k
Empo
w
er
ment
Amm
unition
with
...
(F
ang
Jin)
Evaluation Warning : The document was created with Spire.PDF for Python.
530
ISSN:
1693-6930
when
2
i
"
>
1
holds
.
Thus
,
it
f
ollo
ws
that
R
=
P
i
2
r
i
>
P
i
2
1
2
log
2
2
i
"
(bits/sample)
where
:=
f
i
2
f
1
;
2
;
;
n
g
:
2
i
"
>
1
g
.
Remark
2:
Theorem
1
states
that,
the
bit-allocation
algor
ithm
on
the
basis
of
not
the
eigen
v
alues
b
ut
the
singular
v
alu
es
of
the
system
matr
ix
A
can
still
guar
ant
ee
stabilization
of
the
sys
t
em
(1).
The
par
ameter
"
has
an
impor
tant
eff
ect
on
the
r
ate
of
con
v
ergence
.
F
or
the
case
with
"
=
1
,
w
e
ma
y
also
present
a
similar
argument.
Ho
w
e
v
er
,
w
e
stress
that,
the
system
(1)
ma
y
be
not
asymptotically
stabilizab
le
,
b
ut
be
boundab
le
if
the
data
r
ate
satisfies
the
condition:
R
>
P
i
2
log
2
j
i
j
(bits/sample)
where
:=
f
i
2
f
1
;
2
;
;
n
g
:
j
i
j
>
1
g
.
If
w
e
set
"
=
1
and
assume
that
all
the
singular
v
alues
of
the
system
matr
ix
A
are
larger
than
1,
it
f
ollo
ws
from
Theorem
1
that
R
>
P
n
i
=1
log
2
j
i
j
=
log
2
j
det(
A
)
j
(bits/sample)
:
Namely
,
our
result
ma
y
reduce
to
the
w
ell
kno
w
data
r
ate
theorem
[9,11].
4.
Numerical
Example
In
this
section,
w
e
present
a
n
umer
ical
e
xample
f
or
netw
or
k
empo
w
er
ment
amm
unition
to
illustr
ate
the
eff
ectiv
eness
of
the
bit-allocation
algor
ithm
giv
en
in
our
results
.
Let
us
con
s
i
der
a
netw
or
k
ed
control
system
which
e
v
olv
es
in
discrete-time
according
to
X
(
t
+
1)
=
2
4
3
:
4331
17
:
3913
37
:
9582
12
:
4331
32
:
3913
33
:
9582
7
:
4331
17
:
3913
7
:
9582
3
5
X
(
t
)
+
2
4
1
:
4
1
:
7
2
:
1
3
5
U
(
t
)
:
A
control
la
w
is
giv
en
b
y
K
=
[11
:
32830
24
:
3046
21
:
9182]
0
.
The
initial
plant
state
is
giv
en
b
y
X
0
=
[2000
1000
200]
0
.
The
eigen
v
alues
of
the
system
matr
ix
A
are
7,
7
+
6
i
,
7
6
i
,
and
ha
v
e
magnitudes
of
7,
9.2195,
9.2195.
T
o
achie
v
e
the
minim
um
bit-r
ate
needed
to
stabiliz
e
the
unstab
le
plant,
one
ma
y
find
a
matr
ix
H
=
2
4
0
:
9464
0
:
8546
0
:
8546
0
:
2983
0
:
4037
0
:
2202
i
0
:
4037
+
0
:
2202
i
0
:
1234
0
:
0499
0
:
236
i
0
:
0499
+
0
:
236
i
3
5
which
diagonaliz
es
A
or
tr
ansf
or
ms
A
to
a
Jordan
canonical
f
or
m.
Then,
b
y
emplo
ying
coordinate
tr
ansf
or
m,
w
e
define
X
(
k
)
:=
H
X
(
k
)
;
and
quantiz
e
X
(
k
)
,
encode
X
(
k
)
,
and
tr
ansmit
the
inf
or
mation
of
X
(
k
)
to
the
controller
o
v
er
a
digital
comm
unication
channel.
Let
X
(
k
)
:=
[
x
1
(
k
)
x
2
(
k
)
x
3
(
k
)]
0
.
Clear
ly
,
x
2
(
k
)
and
x
3
(
k
)
are
comple
x
v
alues
,
b
ut
are
not
comple
x
conjugates
of
each
other
.
Thus
,
each
x
2
(
k
)
and
x
3
(
k
)
could
not
be
reconstr
ucted
from
only
its
real
par
t
or
its
comple
x
par
t.
Then,
one
has
to
quantiz
e
the
real
and
imaginar
y
par
ts
of
each
x
2
(
k
)
and
x
3
(
k
)
in
order
to
guar
antee
b
oundedness
of
each
x
i
(
k
)
.
Thus
,
the
minim
um
bit-r
ate
needed
to
stabiliz
e
the
unstab
le
plant
is
19
bits/sample
(since
the
bit
r
ate
m
ust
be
an
integer).
TELK
OMNIKA
V
ol.
12,
No
.
3,
September
2014
:
525
532
Evaluation Warning : The document was created with Spire.PDF for Python.
TELK
OMNIKA
ISSN:
1693-6930
531
Figure
1.
T
r
ajector
y
of
plant
states
.
T
o
reduce
the
conser
v
atism,
w
e
find
a
real
matr
ix
H
=
2
4
0
:
8802
0
:
4288
0
:
2034
0
:
4580
0
:
6549
0
:
6011
0
:
1246
0
:
6223
0
:
7728
3
5
which
can
diagonaliz
e
A
0
A
=
H
0
2
H
with
=
diag[0.5477,
15.4175,
65.5362].
It
f
ollo
ws
from
Theorem
1
that,
the
minim
um
bit-r
ate
needed
to
stab
iliz
e
the
unstab
le
plant
is
11
bits/sample
which
is
same
as
that
in
[9],
[11],
etc.
The
corresponding
sim
ulation
is
obtained
and
sho
wn
in
Fig.1.
Clear
ly
,
the
system
is
stabilizab
le
.
5.
Conc
lusion
In
this
paper
,
w
e
presented
a
ne
w
bit-allocation
algor
ithm
f
or
the
more
gener
al
system
ma-
tr
ix,
on
which
no
assumption
that
there
e
xists
a
real
tr
ansf
or
mat
ion
matr
ix
such
that
the
system
matr
ix
can
be
tr
ansf
or
med
to
a
diagonal
matr
ix
or
a
Jordan
canonical
f
or
m
w
as
made
.
Diff
erent
from
the
e
xisting
bit-allocation
algor
it
hms
,
the
algor
ithm
proposed
here
w
as
on
the
basis
of
not
the
eigen
v
alues
b
ut
the
singular
v
alues
of
the
system
matr
ix.
It
w
as
der
iv
ed
that
the
bit-allocation
algor
ithm
can
guar
antee
stabilization
of
the
system.
The
sim
ulation
results
on
netw
or
k
empo
w
er-
ment
amm
unition
ha
v
e
illustr
ated
the
eff
ectiv
eness
of
the
proposed
algor
ithm.
Ref
erences
[1]
K
otta
H
Z,
Rantelobo
K,
T
ena
S
,
Klau
G.
Wireless
Sensor
Netw
or
k
F
or
Landslide
Monitor-
ing
in
Nusa
T
enggar
a
Tim
ur
.
TELK
OMNIKA
T
elecomm
unication
Computing
Electronics
and
Control
.
2011;
9(1):9-18.
[2]
T
umir
an
C
,
Nandar
S
.
P
o
w
er
Oscillation
Damping
Control
Using
Rob
ust
Coordinated
Smar
t
De
vices
.
TELK
OMNIKA
T
elecomm
unication
Computing
Electronics
and
Control
.
2011;
9(1):65-72.
[3]
Atia
D
M,
F
ahm
y
F
H,
Ahmed
N
M,
Dorr
ah
H
T
.
A
Ne
w
Control
and
Design
of
PEM
Fuel
Cell
System
P
o
w
ered
Diffused
Air
Aer
ation
System.
TELK
OMNIKA
T
elecomm
unication
Comput-
ing
Electronics
and
Control
.
2012;
10(2):291-302.
[4]
W
ong
W
S
,
Broc
k
ett
R
W
.
Systems
With
Finite
Comm
unication
Bandwidth
Constr
aints
II:
Stabilization
With
Limited
Inf
or
mation
F
eedbac
k.
IEEE
T
r
ans
.
A
utomat.
Control
.
1999;
44(5):1049-1053.
[5]
Baillieul
J
.
F
eedbac
k
Designs
F
or
Controlling
De
vice
Arr
a
ys
With
Comm
unication
Channel
Bandwidth
Constr
aints
.
in
AR
O
W
or
kshop
on
Smar
t
Str
uctures
,
P
ennsylv
ania
State
Univ
,
A
ug.
1999.
Quantiz
ed
F
eedbac
k
Control
of
Netw
or
k
Empo
w
er
ment
Amm
unition
with
...
(F
ang
Jin)
Evaluation Warning : The document was created with Spire.PDF for Python.
532
ISSN:
1693-6930
[6]
Baillieul
J
.
F
eedbac
k
Designs
in
Inf
or
mation
Based
Control.
in
Stochastic
Theor
y
and
Control
Proceedings
of
a
W
or
kshop
Held
in
La
wrence
,
Kansas
.
B
.
P
asik-Duncan,
Ed.
Ne
w
Y
or
k:
Spr
inger-V
er
lag,
2001;
35-57.
[7]
Baillieul
J
.
F
eedbac
k
Coding
f
or
Inf
or
mation-Based
Control:
Oper
ating
Near
The
Data-Rate
Limit.
in
Proc.
41st
IEEE
Conf
erence
on
Decision
and
Control
,
Las
V
egas
,
Ne
v
ada
USA,
2002;
3229-3236.
[8]
Li
K,
Baillieul
J
.
Rob
ust
Quantization
F
or
Digital
Finite
Comm
unication
Bandwidth
(DFCB)
Control.
IEEE
T
r
ans
.
A
utomat.
Control
.
2004;
49(9):1573-1584.
[9]
Nair
G
N,
Ev
ans
R
J
.
Stabilizability
of
Stochastic
Linear
Systems
With
Finit
e
F
eedbac
k
Data
Rates
.
SIAM
J
.
Control
Optim.
.
2004;
43(2):413-436.
[10]
Elia
N.
When
Bode
Meets
Shannon:
Control-Or
iented
F
eedbac
k
Comm
unication
Schemes
.
IEEE
T
r
ans
.
A
utomat.
Control
.
2004;
49(9):1477-1488.
[11]
T
atik
onda
S
,
Mitter
S
K.
Control
Under
Comm
unication
Constr
aints
.
IEEE
T
r
ans
.
A
utomat.
Control
.
2004;
49(7):1056-1068.
[12]
T
atik
onda
S
,
Sahai
A,
Mitter
S
K.
Stochastic
Linear
Control
Ov
er
a
Comm
unication
Channel.
IEEE
T
r
ans
.
A
utomat.
Control
.
2004;
49(9):1549-1561.
[13]
Liu
Q,
Y
ang
G
H.
Quantiz
ed
F
eedbac
k
Control
F
or
Netw
or
k
ed
Control
Systems
Under
Inf
or-
mation
Limitation.
Inf
or
mation
T
echnology
And
Control
.
2011;
40(3):218-226.
[14]
Liu
Q.
Stabilization
of
Unmanned
Air
V
ehicles
Ov
er
Wireless
Comm
unication
Channels
.
TELK
OMNIKA
Indonesian
Jour
nal
of
Electr
ical
Engineer
ing
.
2012;
11(7):1701-1708.
[15]
Liu
Q.
Coordinated
Motion
Control
of
A
utonomous
and
Semiautonomous
Mobile
Agents
.
TELK
OMNIKA
Indonesian
Jour
nal
of
Electr
ical
Engineer
ing
.
2012;
10(8):1929-1935.
[16]
Liberz
on
D
,
Ne
s
i
c
D
.
Input-T
o-State
Stabilization
of
Linear
Systems
With
Quantiz
ed
State
Measurements
.
IEEE
T
r
ansactions
on
A
utomatic
Control
.
2007;
52(5):767-781.
TELK
OMNIKA
V
ol.
12,
No
.
3,
September
2014
:
525
532
Evaluation Warning : The document was created with Spire.PDF for Python.