time response 1
TRANSCRIPT
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Time Response Analysis, Design
Indices
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System
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a s e me response o a con rosystem?
Time response:
d ifferentia l e uation.
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a s e me response o a con rosystem?
me response c ons s s o :
2.Forced response
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a s e me response o a con rosystem?
Natural res onse is caused b the s stem itself.
Also called transient response or homogenous
solution
Forced response is caused by the input to the
-.or particular solution.
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a s e me response o a con rosystem?
Time response PlotNatural res onse or
Steady-stateTime solution to
Transient responseresponse
differential equation
Caused by the
system itself
Caused bythe input
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How to obtain time response?
Schematic Time ResponseWhyrans er
Function?
M)(t f
x M &&
Kx Transfer Function
Differential Equation
K MssF s X sG
+
==21
)()()(
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)(t f Kx x M =+&& )(2sF KX MXs =+
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ow o o a n me response romtransfer function?
Given:
,concept of poles and zeros
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ow o o a n me response romtransfer function?
What are oles ?First rule:
The values of Laplace transform variable,s, a cause e rans er unc on o
become infinite.
Any roots of the denominator of thetransfer function that are common to the
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ow o o a n me response romtransfer function?
Transfer FunctionFirst rule:
)3)(2)(5(
)3()(
+++
+=
sss
ssG
When s = -5, or s = -2 or s = -3, then,G(s)=infinity! Therefore, the poles
of the transfer function G(s) are -5,-2 and -3.
Second rule:
)3)(2)(5()(
+++=
ssssG Although the term (s+3) can be
cancelled out, the value -3 is stillthe poles of the transfer function
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G(s).
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ow o o a n me response romtransfer function?
What are zeros ?First rule:
The values of Laplace transform variable,s, a cause e rans er unc on o
become zero.
Any roots of the numerator of the transfer function that are common to roots of the
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ow o o a n me response romtransfer function?
Transfer FunctionFirst rule:
)3)(2)(5(
)3()(
+++
+=
sss
ssG
When s = -3, then, G(s)= 0.Therefore, the zero of the transfer
function G(s) is -3.
Second rule:
)3)(2)(5()(
+++=
ssssG Although the term (s+3) can be
cancelled out, the value -3 is stillthe zero of the transfer function
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G(s).
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ow o o a n me response romtransfer function?
Given:
How to obtain the time response?1. Find C(s)
2. Expan e rans er unc on us ng par a rac onexpansion technique
3. Perform inverse La lace transform
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ow o o a n me response romtransfer function? –Find C(s)
Given:
)(
)()(
s R
sC
Input
Output sG ==
)()()( s RsGsC =
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ow o o a n me response romtransfer function? –Find C(s)
What is R s ?
Step input: At r =)(S
As R =)(Laplace Transform
Ramp input: t t r =2
)(S
s R =Laplace Transform
Sine input: )sin( t r(t) ω = 22)( =
As RLaplace Transform
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ow o o a n me response romtransfer function? –Find C(s)
Given:
R(s) unit step input implies A = 1
)()()( = s RsGsC
)2(1)(
+=s
sC
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ow o o a n me response romtransfer function? – Expand C(s)
Ex and C s usin Partial Fraction Ex ansionTechnique
21 +s
)5( +=ss
s
)5()2(
5)(
+
+=
++=
sss
s B
s AsC
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ow o o a n me response romtransfer function? – Expand C(s)
)2( +==s B A How to find A
)5(5 ++ ssss and B?
5)5(0
=+
=
→ss
s A
53
52
5
3
)(
)2(=
+=s
s B
5+ss
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−
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How to obtain time response from transferfunction? – Inverse Laplace Transform
32
5++=ss
sC
2 Inverse Laplace 2
sTransform 5
3 3 −
t et c
532 −+=
5+s
Transform e5 55
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a s me response o a con rosystem?
Input pole generatesforce res onse
System pole generates
natural response
System pole generatesnatural response in theform of e-αt .
Both zero and polegenerates theam litude of time
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response
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a s me response o a con rosystem? An Example
)(sC s R1
)( =What are poles and
zeros?
)5)(4)(2( +++ sssPoles = -2, -4 and -5Zero = -3
By inspection:
)( 4321+++≡K K K K
sC
Force
on time response
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response
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a s me response o a con rosystem? An Example
)(sC s R1
)( =
)5)(4)(2( +++ sss
By inspection:)5()4()2(
)( 4321
++
++
++≡
s
K
s
K
s
K
s
K sC
−−−
Transform eeec 4321+++≡
–
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a s me response o a con rosystem? An Example
)( 4321+++≡K K K K
sC How tossss
t t t eK eK eK K t c
5
4
4
3
2
21)( −−−
+++≡
eva uate 1,K2, K3 and K4?
)5)(4)(2(
)3()(
+++
+=
ssss
ssC
11
3
)5)(4)(2(
)3(
)5)(4)(2(
)3()(
00
1 =+++
+=
+++
+==
→→ sssss
s
ssss
ssK ssC
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a s me response o a con rosystem? An Example
330)3()3()( 1 =
+=
+=
+==
sssK ssC
00 →→ ss
12
1
)52)(42(2
)32(
)5)(4(
)3(
)5)(4)(2(
)3)(2()()2(
222 −
=
+−+−−
+−=
++
+=
+++
++==+
−→−→ ss sss
s
ssss
ssK sC s
8
1
)54)(24)(4(
)34(
)5)(2(
)3(
)5)(4)(2(
)3)(4()()4(
44
1
−=
+−+−−
+−=
++
+=
+++
++==+
−→−→ sssss
s
ssss
ssK sC s
15
2
)45)(25(5
)35(
)4)(2(
)3(
)5)(4)(2(
)3)(5()()5( 1
−
−=
+−+−−
+−=
++
+=
+++
++==+
−−sss
s
ssss
ssK sC s
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a s me response o a con rosystem? An Example
t t t eK eK eK K t c
54
43
221)( −−−
+++≡
t t t eeet c 542 2113−−−
+−−≡3 1581240
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a s me response o a con rosystem? An Example
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ys e
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What is the first order of a control system?
One Pole Time Response
at et c−
−=1)(
Solution Using Inspection
)(ass
asC
+=
1)(1 =====aasa
ssC K
)()( 21
as
K
s
K sC
++=
00
++
→→
aasassss
1)(
)()( −===+
=+=aaaas
sC asK
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at eK K t c−
+= 21)( −+−→−→asass asas
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What is the first order of a control system? –Performance Parameters
Time constant
63% ofFinal
T c1
=
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a s e rs or er o a con ro sys em –Performance Parameters
Rise Time
me o r sefrom 0.1 to 0.9
of final va lue
T T T r
11.031.2
%10%90 −=
T
aa
r
r
2.2=
−=
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a
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What is the first order of a control system? –Performance Parameters
Settling Time
Time to reach
2% of finalvalue
T s4=
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What is the first order of a control system? –Performance Parameters
Time constant
Rise Time
aT c =
o na
value
aT r
2.2=
Time to rise from0.1 to 0.9 of finalvalue
Settling Time
4Time to reach
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as =2% of final value
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What is the first order of a control system? –An Example
Time constant50 1
Rise Time
sec50
==a
T c50+s
s
sec50
2.22.2==a
T r
pole: pole = a = -50
By inspection the solution of
Settling Time
44t et c
501
−−=
unit step input is:
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50as
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What is the first order of a control system? –An Example
By experiment we get this
Find transfer function?
Step 1: Find the poleTime constant is the time
to reach 63% of final value
Final value = 0.72
63% Final value = 0.63X0.72=0.45Time constant = 0.13 sec
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What is the first order of a control system? –An Example
Step 1: Find the pole
me cons an = . sec
1=c
a
T
7.713.0
11===
cT a
at K −−
General solution
a54.5)7.7(72.072.072.0 ===→= aK
a
K
54.5K
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7.7+=
+=
sass
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What is the first order of a control system? –An Example
)(=sG
at are t e t me constant, r se t me, sett ng t meand steady-state value?
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General Response of Sec ond
Order System
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control system?
m wi h w l
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control system?
Under-damped Time Response
a
acbb
2
42
2,1
−±−=σ
81
2322
)1(2)9)(1(422
2
2,1 j
+−=
±−=
−±−=
σ
σ
σ − t d
General solution
σ ±−= 81
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812 j−−=σ −− d
nd jω σ σ ±−=2,1
,
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me response o a secon or ercontrol system?
Un-damped Time Response
a
acbb
2
42
2,1
−±−=σ
2
36
)1(2
)9)(1(400 2
2,1
j±=
−±−=σ )cos(1)( φ ω −−= t t c
General solution
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3
2
1
j
j
−=
+=
σ
σ
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me response o secon or ercontrol system? - Summary
Over-damped Un-damped
Poles are realand distinct
Poles areimaginary
Under-damped Critic ally-damped
Poles arereal andimaginary
Poles are realand repeated
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me response o secon or ercontrol system? - Summary
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me response o secon or ercontrol system? - Example
)cos()( φ ω σ
−=−
d t d Aet cGeneral
acbb 42
2,1
−±−=
σ
under-damped )23.13cos()( 5 φ −=− t Aet c
)1(2
)200)(1(41010 2
2,1
−±−=σ
23.1352,
2
)175(410
1
2,1
j
j
±−=
±−
=
σ
σ
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The poles are real and imaginary.Therefore UNDER-DAMPED.
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me response o secon or ercontrol system? - Example
What are poles and zeroes?-
Write the general expression of the step response?
State the nature of response?
5
)6)(3( ++ ss
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me response o secon or ercontrol system? - Example
What are poles and zeroes?-
Write the general expression of the step response?
State the nature of response?
)7(10 +s
)20)(10( ++ ss
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me response o secon or ercontrol system? - Example
What are poles and zeroes?-
Write the general expression of the step response?
State the nature of response?
20
14462 ++ ss
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me response o secon or ercontrol system? - Example
What are poles and zeroes?-
Write the general expression of the step response?
State the nature of response?
2+s
92 +s
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me response o secon or ercontrol system? - Example
What are poles and zeroes?-
Write the general expression of the step response?
State the nature of response?
5+=s
2)10( +s
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Time Response of Second Order
Natural Frequency
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me response o secon or ercontrol system?
So far, we based on the poles of the
second order system
a
ac
22,1
−−=σ
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me response o secon or ercontrol system?
We are going to learn two
quantities that will help us analyze
second order systemNatural
Fre uenc
Damping
Ratio
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me response o secon or ercontrol system? Natural Frequency
Natural
Frequency n
a ura requency o a secon
order system is the frequency of
osc a on o e sys em w oudamping
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me response o secon or ercontrol system? Damping Ratio
Damping FrequencyDecaylExponentia=
Ratio (rad/sec)FrequencyNatural
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