TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol. 14, No. 2, May 2015, pp. 288 ~ 29
2
DOI: 10.115
9
1
/telkomni
ka.
v
14i2.764
4
288
Re
cei
v
ed Fe
brua
ry 15, 20
15; Re
vised
Ap
ril 11, 201
5; Acce
pted
April 27, 201
5
Temperature Control of Liquid Filled Tank System
Using Advance State Feedback Controller
Kunal Chakr
aborty
, Sankha Subhra G
hosh, Rah
u
l Dev
Basak, Indranil Ro
y
Electrical E
ngi
neer
ing D
e
p
a
rtment, IMPS Colleg
e
of Eng
i
ne
erin
g & T
e
chnolog
y,
Nit
yan
and
ap
ur, P.O. Chand
ip
ur (Kazigr
a
m), Mald
a
A
b
st
r
a
ct
In this paper
mo
de
lin
g of a temper
ature
me
asuri
ng tan
k
system has
been d
one a
nd then
a
Advanc
e state
feedb
ack co
ntroller
hav
e be
e
n
use
d
fo
r con
t
rollin
g the ste
p
resp
onses
of the system. T
h
e
proposed system extends to
a three tank
s
ystem
&
eac
h
tank has sa
me am
ount of liquid. The res
u
lt
s of
computer si
mul
a
tion for the sy
stem w
i
th Adva
nce state F
eed
back is show
n
.
Ke
y
w
ords
:
te
mp
eratur
e, tank system,
contr
o
l, non-
lin
ear c
ontrol,
SFB controll
er [7,8]
Copy
right
©
2015 In
stitu
t
e o
f
Ad
van
ced
En
g
i
n
eerin
g and
Scien
ce. All
rig
h
t
s reser
ve
d
1. Introducti
on
The Tem
pera
t
ure mea
s
u
r
e
m
ent of liquid in
a tank can be control
l
ed by cla
ssi
cal an
d
advan
ce cont
rol algo
rithm.
Here
we a
r
e con
s
id
erin
g
a three tan
k
non-inte
ra
cti
ng syste
m
. W
e
observed th
a
t
tank1 affe
cts the
dyna
mi
c be
haviou
r
of tank 2, si
milarly for ta
nk2
affects t
h
e
dynamic be
h
a
vior [2]
of ta
nk3
an
d vice
versa,
be
ca
u
s
e th
e flo
w
ra
te dep
end
s
o
n
the
differe
n
c
e
betwe
en liqui
d levels
h
1
an
d h
2
.Thu
s a
chang
e in the
inlet flow rate
affects the li
quid level in t
he
tank, which in
tern affect
s th
e tempe
r
ature of the li
quid
.
Basically it i
s
a the
r
mal p
r
ocess. Vari
o
u
s
type of temperature sensor RTD,
T/C, Thermis
t
or [1], [9-10].In th
at partic
u
lar
projec
t we used a
mercury t
hermometer a
s
sen
s
o
r
. Math
ematical
mo
d
e
ls
of three t
ank metho
d
give a thi
r
d
o
r
de
r
[6] equation.
Each tan
k
giv
e
a tran
sfe
r
functio
n
of
first orde
r sy
ste
m
. They ma
ke it easy to
check
wheth
e
r a pa
rticula
r
algo
rit
h
m is giving the req
u
isite result
s. A lot
of work ha
s been
carried
out
on the temp
eratu
r
e control in terms o
f
its
stabiliza
t
ion. Many attempts have
been mad
e
to
control the response of temperat
ure m
easuring
syst
em this meth
od i
s
utilized
to investigate [3]
global p
r
op
ert
i
es of the de
signed
controll
er.
2. Mechanic
al Cons
truc
tion
The sy
stem
comp
ri
se
s of
a mercu
r
y-i
n
-gla
ss the
r
mometer
pla
c
ed i
n
a liqu
i
d tank to
measure the
temperature
of t
he liqui
d
whi
c
h i
s
he
ated by st
ea
m throu
gh a
coil
system.
The
temperature
of the liquid (T
F
) varie
s
[5]
with time. T is the tempe
r
ature of the mercury in th
e well
of the therm
o
meter. T
he
followin
g
assumption
s a
r
e
made to d
e
termin
e the transfe
r fun
c
tion
relating the v
a
riation of the
thermomete
r (T) fo
r chan
g
e
in the temperatu
r
e of the
liquid (T
F
).
(1) Th
e exp
a
n
sio
n
o
r
co
ntractio
n
of the
gla
s
s walled
well
containi
ng m
e
rcu
r
y i
s
n
egligi
b
le
(that
mean
s the re
sista
n
ce offered by glas
s wall for heat tra
n
sfer i
s
negli
g
ible)
(2) T
he liquid
film surroun
di
ng the bulb i
s
the only resi
stan
ce to the heat tran
sfer.
(3) T
he me
rcury assum
e
s
isothe
rmal
co
ndition th
ro
ug
hout. The sy
stem is sh
own
in Figure 1.
3. Proposed
Mathem
atica
l
Model
Applying unsteady state h
eat balance for the
bulb,
we get input
heat rate- output heat
rate=Rate of heat accumul
a
tion:
0
(
1
)
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TELKOM
NIKA
ISSN:
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046
QCA and
CM
OS Nanote
c
h
nolog
y Based
Desi
gn an
d De
velo
pm
ent… (S. Deven
d
ra K. Verm
a)
289
Figure 1. Three tank
syste
m
where, A=surf
ace area of the bulb for hea
t transfer in m
2
M=Mass of mercury in the
bulb in kg
C
P
=Heat cap
a
city of the mercury in kj/kg k
U=Film heat transfer coefficient kw/m
2
k
At steady state, the
Equatio
n (1) can be rewritten as:
0
(
2
)
Subtracting Equation (2
) from Equation
(1):
Defining
the deviation variables,
and
and substituting in the
above
equation, we
get:
= MC
P
=
(
3
)
Defining time
constant
for the Thermom
e
ter,
Equation (3) can be rewritten as:
(
4
)
Taking Lapla
c
e transform, we get:
transfer fun
c
tion of Tank1
Similarly, for
tank2 & tank3 we ca
n abl
e to
get a first order syste
m
. So we ca
n able to
say that the e
n
tire system i
s
a third
ord
e
r
syst
em. He
re we
can
abl
e to con
s
truct
overall transf
e
r
function of
th
e three tank system is:
=
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02-4
046
TELKOM
NI
KA
Vol. 14, No. 2, May 2015 : 288 – 292
290
4
.
Transfer F
unction Mo
d
e
lling
As per our problem, let us assume:
= time constant for
tank1=0
.
5 minute
=
time constant for
tank2=1
.
2 minute
= time constant for
tank3=1
.
5 minute
=
= 0.25 min
=
= 0.30 min
=
0.35 min
.
.
.
.
This transfer
function is ca
ll
ed plant transfer function.
T
h
e
e
n
t
ir
e
e
x
p
e
r
ime
n
t
a
l
s
e
t
u
p
is
given in Figu
re 2.
Figure 2. Pro
posed Experi
m
ental Set up
5. Adv
a
nce State F
eedb
ack Co
ntr
o
ller Desig
n
Conditio
n
s:
Clo
s
ed lo
op system ha
s an
oversh
oot of 10% and settling time of 1 se
c.
Equations
:
Overshoot (M
P
)=e
-
ξπ
/
√
(1-
ξ
2)
=0.
1
Takin
g
natu
r
a
l
log on both
side
s
ln(e
-
ξπ
/
√
(1-
ξ
2)
)=ln(0.1
)
or, -
ξπ
/
√
(1-
ξ
2
)=
ln 0.1
Squari
ng bot
h side
s,
ξ
2
π
2
/(1-
ξ
2
)=(l
n 0.1)
2
or,
ξ
2
π
2
=(l
n
0.1)
2
(1
-
ξ
2
)
or,
ξ
2
π
2
+ (ln 0
.
1)
2
ξ
2
= (ln 0.
1)
2
or,
ξ
2
{
π
2
+ (l
n 0.1)
2
}
=
(ln 0.1
)
2
or,
ξ
2
=(ln 0.1)
2
/{
π
2
+ (ln 0.1)
2
}
Therefore,
ξ
=
ln 0.1/
√
π
2
+ (ln 0.1)
2
}
=
[-2.302
5/
√
9.8596
+5.3
018
)]
=
(-2.30
25/3.8
937)
=
(-0.59
132
8)
Therefore,
ξ
=-(-0.5
913
28
)
= 0.5913
28
t
s
=1 se
c
or, 4/
ξ
w
n
=1
or,
ξ
w
n
=4
or, w
n
=4/
ξ
=4/(0.5913
28)
=
6
.7644 rad/
s
e
c
The domi
nant
poles a
r
e at
-
ξ
w
n
jw
n
√
(1-
ξ
2
)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
QCA and
CM
OS Nanote
c
h
nolog
y Based
Desi
gn an
d De
velo
pm
ent… (S. Deven
d
ra K. Verm
a)
291
=-3.999
9
j*6.7644*
√
(1-0.3496
)
=-3.999
9
j 5.4553
1
≅
-4
j5.4553
1
The third p
o
l
e is pla
c
ed
10 times d
e
e
per into the
s-pl
ane tha
n
the domina
n
t poles.
Hen
c
e the d
e
s
ire
d
ch
aract
e
risti
cs e
quati
on is:
(s+ 39.99
9)(s + 3.9999
+ j 5.4553
1)(s
+ 3.9999 – j 5.4
5531
) =0
Or, (s+3
9.99
9){
(
s
+ 3.999
9)
2
+
5
.45531
2
} = 0
(s
2
+39.99
9)(s
2
+16
+ 8s+
29.760
4) = 0
s
3
+48
s
2
+36
5
.7604
s + 1
8
30.416
=0
(5)
Let k
=
[k
1
k
2
k
3
]
Now, [SI-A]=
S
100
010
001
-
01
0
00
1
1.111
3.5556
3.5000
=
1
0
0
1
1.111
3.5556
3.
5000
Clo
s
ed lo
op chara
c
te
risti
c
s equation:
S
3
+(3.50
0)s
2
+
(
3
.
55
5)
s
1
+1
.1111
=
0
(6)
Comp
ari
ng th
e coeffici
ent of E
quation (5) & (6). Th
erefore,
k =[
k
1
k
2
k
3
]
k = [(1
830.41
6-1.11
11
) (365.760
4-3.5
5
5
) (48
-
3.50
0)]
=[182
9.304
9 362.205
4 44
.5]
Similarly we t
a
ke
10 results to ob
se
rve
if the pole
s
a
r
e le
ss
or m
o
re de
epe
r in
s pla
ne
then wh
at ch
ange
s seen i
n
their
initial and ste
p
re
sp
onse.
6. Graphical
Resul
t
6.1. Step Re
spons
e of th
e Plant
w
i
th
SFB Con
t
roll
er
Figure 3. Re
spon
se of stat
e feedba
ck cont
rolle
r con
s
iderin
g step
condition in M
A
TLAB
6.2. Step Re
spons
e of th
e Plant
w
i
th Initial SFB Controller
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046
TELKOM
NI
KA
Vol. 14, No. 2, May 2015 : 288 – 292
292
Figure 4. Re
spon
se of stat
e
feedba
ck controlle
r con
s
iderin
g initial con
d
ition in
MATLAB
7. Conclusio
n
Modelin
g of t
h
ree
tan
k
te
mperature
m
easurin
g
syst
em
sho
w
s th
at syste
m
i
s
unsta
ble
for a ce
rtain range. Th
at’s
why we tri
ed
to desig
n a convention
a
l controlle
r st
rat
egy pro
c
e
s
s so
that we can
minimize the
steady
state error & ma
xi
mize the
settl
ing time. In future
we may
use
Geneti
c
Algorithm for desi
g
ning the controller.
Referen
ces
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idi
n
e
.
Process Ins
t
ruments an
d
Contro
l Ha
n
d
Book. 2
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a
w
Hi
ll Bo
ok
Comp
an
y. 19
7
4
.
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e
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r
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e
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ope
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