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Automatic Control Systems (FCS)
Lecture-6Time Domain Analysis of Control systems
Introduction
• In time-domain analysis the response of a dynamic systemto an input is expressed as a function of time.
• It is possible to compute the time response of a system if thenature of input and the mathematical model of the systemare known.
• Usually, the input signals to control systems are not knownfully ahead of time.
• For example, in a radar tracking system, the position and thespeed of the target to be tracked may vary in a randomfashion.
• It is therefore difficult to express the actual input signalsmathematically by simple equations.
Standard Test Signals
• The characteristics of actual input signals are asudden shock, a sudden change, a constantvelocity, and constant acceleration.
• The dynamic behavior of a system is thereforejudged and compared under application ofstandard test signals – an impulse, a step, aconstant velocity, and constant acceleration.
• Another standard signal of great importance is asinusoidal signal.
Standard Test Signals
• Impulse signal
– The impulse signal imitate thesudden shock characteristic ofactual input signal.
– If A=1, the impulse signal iscalled unit impulse signal.
0 t
δ(t)
A
00
0
t
tAt
)(
Standard Test Signals
• Impulse signal
Source: English Wikipedia. Iain. Original image: [1]
Standard Test Signals
• Step signal
– The step signal imitatethe sudden changecharacteristic of actualinput signal.
– If A=1, the step signal iscalled unit step signal
00
0
t
tAtu
)( 0 t
u(t)
A
Standard Test Signals
• Ramp signal– The ramp signal imitate
the constant velocitycharacteristic of actualinput signal.
– If A=1, the ramp signalis called unit rampsignal
00
0
t
tAttr
)(
0 t
r(t)
r(t)
unit ramp signal
r(t)
ramp signal with slope A
Standard Test Signals
• Parabolic signal
– The parabolic signalimitate the constantacceleration characteristicof actual input signal.
– If A=1, the parabolicsignal is called unitparabolic signal.
00
02
2
t
tAt
tp
)(
0 t
p(t)
parabolic signal with slope A
p(t)
Unit parabolic signal
p(t)
Relation between standard Test Signals
• Impulse
• Step
• Ramp
• Parabolic
00
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t
tAt
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00
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t
tAtu
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tAttr
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t
tAt
tp
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dt
d
dt
d
dt
d
Laplace Transform of Test Signals
• Impulse
• Step
00
0
t
tAt
)(
AstL )()}({
00
0
t
tAtu
)(
S
AsUtuL )()}({
Laplace Transform of Test Signals
• Ramp
• Parabolic
2s
AsRtrL )()}({
3
2
S
AsPtpL )()}({
00
0
t
tAttr
)(
00
02
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t
tAt
tp
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Time Response of Control Systems
System
• The time response of any system has two components
• Transient response
• Steady-state response.
• Time response of a dynamic system is response to an inputexpressed as a function of time.
Time Response of Control Systems
• When the response of the system is changed form rest orequilibrium it takes some time to settle down.
• Transient response is the response of a system from rest orequilibrium to steady state.
0 2 4 6 8 10 12 14 16 18 200
1
2
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6x 10
-3
Step Response
Time (sec)
Am
plit
ude Response
Step Input
Transient Response
Stea
dy
Stat
e R
esp
on
se
• The response of thesystem after the transientresponse is called steadystate response.
Time Response of Control Systems
• Transient response dependents upon the system poles only andnot on the type of input.
• It is therefore sufficient to analyze the transient response using astep input.
• The steady-state response depends on system dynamics and theinput quantity.
• It is then examined using different test signals by final valuetheorem.