7/24/2019 4.1 Time Response Analysis
1/33
Time Response Analysis
7/24/2019 4.1 Time Response Analysis
2/33
Introduction
Infuence o Poles on Time Response
Transient Response o First-OrderSystem
Transient Response o Second-OrderSystem
7/24/2019 4.1 Time Response Analysis
3/33
The concept o poles andzeros, undamental to theanalysis o and design o control system, simpliesthe e!aluation o system response"
The poleso a transer unction are#
i" $alues o the %aplace Transorm !aria&les s, thatcause the transer unction to &ecome innite"
ii" Any roots o the denominator o the transer unctionthat are common to roots o the numerator"
The zeroso a transer unction are#
i" The !alues o the %aplace Transorm !aria&le s, thatcause the transer unction to &ecome zero"
ii" Any roots o the numerator o the transer unctionthat are common to roots o the denominator"
7/24/2019 4.1 Time Response Analysis
4/33
The output response o a system is a sum oi" Forcedresponse
ii" 'aturalresponse
a( System sho)ing an input and anoutput
&( Pole-zero plot o the system
7/24/2019 4.1 Time Response Analysis
5/33
c( *!olution o a system response" Follo)the &lue arro)s to see the e!olution osystem component generated &y the
pole or zero
7/24/2019 4.1 Time Response Analysis
6/33
a( First-order system
&( Pole plot o thes stem
*+ect o a real-ais pole upon transientresponse
7/24/2019 4.1 Time Response Analysis
7/33
eneral orm#
Pro&lem# .eri!e the transer unction or the
ollo)ing circuit
1)(
)()(
+==
s
K
sR
sCsG
1
1)(
+=RCs
sG
7/24/2019 4.1 Time Response Analysis
8/33
Transient Response# radual change o output rominitial to the desired condition"
/loc0 diagram representation#
/y denition itsel, the input to the system should &ea step unction )hich is gi!en &y the ollo)ing#
C(s)R(s) 1+s
K
ssR
1)( =
1here,
2 # ain # Time constant
7/24/2019 4.1 Time Response Analysis
9/33
eneral orm#
Output response#
1)(
)()(
+==
s
K
sR
sCsG
1
1
1)(
++=
+
=
s
B
s
A
s
K
ssC
teB
Atc +=)(
)()()( sRsGsC =
7/24/2019 4.1 Time Response Analysis
10/33
Pro&lem# Find the orced and natural responses orthe ollo)ing systems
7/24/2019 4.1 Time Response Analysis
11/33
First-order system response to a unit step
7/24/2019 4.1 Time Response Analysis
12/33
Time constant, The time or e-atto decay 345 o its
initial !alue"
Rise time, tr The time or the )a!eorm to go
rom 6"7 to 6"8 o its nal !alue"
Settling time, ts The time or the response to reach,
and stay )ithin 95 o its nal !alue"
a
1=
atr
2.2=
ats
4=
7/24/2019 4.1 Time Response Analysis
13/33
Pro&lem# For a system )ith the transer unctionsho)n &elo), nd the rele!ant responsespecications
i" Time constant,
ii" Settling time, ts
iii" Rise time, tr
50
50)(
+=s
sG
7/24/2019 4.1 Time Response Analysis
14/33
eneral orm#
Roots o denominator#
( )22
2
2nn
n
ss
KsG
++=
1here,2 # ain: # .amping ration # ;ndamped natural
re
7/24/2019 4.1 Time Response Analysis
15/33
'atural re
7/24/2019 4.1 Time Response Analysis
16/33
Pro&lem# Find the step response or the ollo)ingtranser unction
Ans)er#
( )22530
2252 ++
=ss
sG
( ) tt teetc 1515 151 =
7/24/2019 4.1 Time Response Analysis
17/33
Pro&lem# For each o the transer unction, nd the!alues o : and n, as )ell as characterize the nature
o the response"
a(
&(
c(
d(
( )40012
4002 ++
=ss
sG
( )000
002 ++
=ss
sG
( )
22530
2252
++
=
ss
sG
( )!25
!252 +
=s
sG
7/24/2019 4.1 Time Response Analysis
18/33
7/24/2019 4.1 Time Response Analysis
19/33
7/24/2019 4.1 Time Response Analysis
20/33
Step responses or second-order system dampingcases
7/24/2019 4.1 Time Response Analysis
21/33
Pole plot or the underdamped second-order system
7/24/2019 4.1 Time Response Analysis
22/33
Second-order response as a unction o dampingratio
7/24/2019 4.1 Time Response Analysis
23/33
Second-order response as a unction o dampingratio
7/24/2019 4.1 Time Response Analysis
24/33
1hen 6 > : > 7, the transer unction is gi!en &y theollo)ing"
Pole position#
( )( ) ( )dndn
n
jsjs
KsG
+++=
2 1here,2
1 =nd
7/24/2019 4.1 Time Response Analysis
25/33
Second-order response components generated &ycomple poles
7/24/2019 4.1 Time Response Analysis
26/33
Second-order underdamped responses or dampingratio !alue
7/24/2019 4.1 Time Response Analysis
27/33
Second-order underdamped response specications
7/24/2019 4.1 Time Response Analysis
28/33
Rise time, Tr The time or the )a!eorm to go rom 6"7 to 6"8 o its
nal !alue"
Pea0 time, Tp The time re
7/24/2019 4.1 Time Response Analysis
29/33
Percent o!ershoot, 5OS The amount that the )a!eorm o!ershoots the steady-
state, or nal !alue at pea0 time, epressed as apercentage o the steady-state !alue"
"100" )1/(2
= eOS
)100/("ln
)100/ln("
22 OS
OS
+
=
7/24/2019 4.1 Time Response Analysis
30/33
Percent o!ershoot !ersus damping ratio
7/24/2019 4.1 Time Response Analysis
31/33
%ines o constant pea0 time Tp, settling time Tsandpercent o!ershoot 5OS
Ts9> Ts7Tp9> Tp7
5OS7> 5OS9
7/24/2019 4.1 Time Response Analysis
32/33
Step responses o second-order underdampedsystems as poles mo!e
a( 1ith constantreal part
&( 1ith constantimaginary
part
7/24/2019 4.1 Time Response Analysis
33/33
Step responses o second-order underdampedsystems as poles mo!e
c( 1ith constant dampingratio