time delay systems
TRANSCRIPT
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Control of
Time-Delay Systems
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Time-Delay Element
x(t) Time-DelayElement
x(t - Tl)
Tl= Time-DelayTime Response
x(t)
unit step1
0t
y(t)
1
0t
Tl
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Time-Delay SystemsExample 1
emperature Measurement of Heat-Exchanger Output
Heat
cold in
Sensor
a(t)
hot
out
d
b(t)
v
hot
out
Sensor
Heat
cold in
v = velocity
d = distance
b(t) = a(t Tl)
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Time-Delay SystemsExample 2
Thickness Measurement of Rolled Steel Plate
+
+
d
v
Thickness
gauge
Steel Plate
Time delay betweenchange in thickness
and measurement
v
dT
l=
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Difference Between Time-Lag and Time-Delay
(a)
(b)
(c)
!n case (c)" transfer function of the block = "T= time lag" Tl= delay ( ) sT
e
sG
lsT
+=
1
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Due to the #resence of time-delay" any corrective action from
the controller cannot be immediately a##lied to the #rocess$Thus during that time" #rocess may buildu# a deviation"
sometimes %uite large" due to any load disturbance or change
in set-#oint$
Problem due to Presence of Time-Delay
in a Process Control Loop
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Transfer Function of Time-Delay Element
x(t) Time-DelayElement y(t) = x(t - Tl)
Taking &a#lace Transform( )
( )
lsTe
sX
sY =
Transfer 'unction of a Time-Delay Element
ow"
( )
( )
1
2
1
2
21
21
2
21
21
2
12!
12!
l
l
l
lsT
lsT
sT
l
l
sTsT
e
e e sTsT
+
= =+ +
sing 'irst-*rder +ade ,##roimation
12
1
2
1
1
lsT l
l
sTe
sT
+
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Effect of Time-Delay on Process Loop Response
T$'$ of a +rocess with Time-Delay ( ) ( ) lsTesGsG =where G/(s) is the transfer function of the delay free #art of the
#rocess$
Case Study: Proportional Control of a First-Order Process
with Time-Delay
+ Kp
1
lsTe
sT
+
(s)!(s)
+ "
controllerprocess
Kp= +rocess .ain
T = +rocess Time Constant
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Case Study: Proportional Control of a First-Order Process
with Time-Delay(contd )
+ K p1
lsT
esT
+(s)!(s)
+"
controllerprocess
The Characteristic E%n$ of the C$&$ System 1 01
lsT
pK e
sT
+ =
+sing 'irst-*rder +ade ,##roimation
( )0
2
111
2
11
1 =
++
+
l
lp
sTsT
sTK
0
2
1
1
2
12
1
2
1
2
=
++
+
+l
p
l
lpl
TTK
TT
TKTT
ss
The standard form
2 2
2 0,n ns s + + =
( )2 1,
p
n
l
K
TT
+
=
The atural 're%$
( )
( )
1 12 2
2 1
l p l
p l
T T K T
K T
+
=
+ +
The Dam#ing /atio
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Case Study: Proportional Control of a First-Order Process
with Time-Delay(contd )
+ K p1
lsT
esT
+(s)!(s)
+"
controllerprocess
The dam#ing ratio becomes 0ero when
The system becomes unstable when
+=
l
l
p
T
TTK
2
+>
l
l
p
T
TTK
2
Conclusion:Time-delay may cause instability even in
first-order system for large values ofKp$
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Case Study: Proportional Control of a First-Order Process
with Time-Delay(contd )
+ Kp
1
lsTe
sT
+(s)!(s)
+"
controllerprocess
&oo# T$'$ of the C$ &$ SystemsT
eK lsTp
+
1
Root Locus
Plot
sing 'irst-*rder +ade ,##roimation
( )
21
2
1 21 1
2
plp
l
l
l
KsTsK
T T
sTsT s s
T T
=
+ + + +
Case ! 2l
T
T>K
p#K
c
$
K p # 0 K p # 0
s%plane
0
K p # K c
2
lT
1T
2
lT
1c Critical .ain
22
l
c
l
TT
K T
+
=
2 l
l
T T
T
+=
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Case Study: Proportional Control of a First-Order Process
with Time-Delay(contd )
+ Kp
1
lsTe
sT
+(s)!(s)
+"
controllerprocess
&oo# T$'$ of the C$ &$ SystemsT
eK lsTp
+
1
Root Locus
Plot
sing 'irst-*rder +ade ,##roimation
( )
21
2
1 21 1
2
plp
l
l
l
KsTsK
T T
sTsT s s
T T
=
+ + + +
Case !! 2l
T
T