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Steady-state error
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Why Worry About Steady
State Error?Control systems are used to control some physical variable. That variable may be
temperature somewhere, the attitude of an aircraft or a frequency in a
communication system. Whatever the variable, it is important to control the variable
accurately.
If you are designing a control system, how accurately the system performs is
important. If it is desired to have the variable under control take on a particular value
you will want the variable to get as close to the desired value as possible. Certainly,
you will want to measure how accurately you can control the variable. Beyond thatyou will want to be able to predict how accurately you can control the variable.
To be able to measure and predict accuracy in a control system, a standard
measure of performance is widely used. That measure of performance is steady state
error - SSE - and steady state error is a concept that assumes the following:
The system under test is stimulated with some standard input. Typically, the test inpuis a step function of time, but it can also be a ramp or other polynomial kinds of
inputs.
The system comes to a steady state, and the difference between the input and the
output is measured.
The difference between the input - the desired response - and the output - the actualresponse is referred to as the error.
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Error and steady-state error
( )R s ( )Y s( )G s
( )r t ( )y t
( )R s ( )Y s( )G s
( )r t
( )E s
( )y t( )e t
Open-loop control system Closed-loop control system
Error:
Steady-state error:
Utilizing the final value theorem:
( ) ( ) ( )e t r t y t
lim ( )sst
e e t
0
lim ( ) lim ( )t s
f t sF s
0lim ( ) lim ( )sst s
e e t sE s
Assuming r(t)=1(t) is a unit-step input, according to the above
definition, could you calculate the steady-state error of the open-loop and closed-loop control systems?
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( )R s ( )Y s( )G s
( )r t ( )y t
( )R s ( )Y s( )G s
( )r t
( )E s
( )y t( )e t
Open-loop control system Closed-loop control system
( ) ( ) ( )
( ) ( ) ( )[1 ( )] ( )
E s R s Y s
R s G s R sG s R s
0
0
0
lim ( )
1lim [1 ( )]
lim[1 ( )]
1 (0)
sss
s
s
e sE s
s G ss
G s
G
Unit-step input r(t)=1(t),1
( )R ss
( ) ( ) ( )
( )( ) ( )1 ( )
1( )
1 ( )
E s R s Y s
G sR s R sG s
R sG s
0 0
0
1 1lim ( ) lim1 ( )
1 1lim
1 ( ) 1 (0)
sss s
s
e sE s sG s s
G s G
Feedback has the effect to reduce steady-state error.
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First order system (SSE)
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Steady state error :unit ramp input
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Type 0
1
10
s
)(sY)(sR )(sE
11
1
11
1lim
1
101
1
lim][lim)]([lim
0
0)()(1
)(
00
s
s
s
ss
ssEe
s
ssHsG
ssR
ssss
ssR
1)(
pK1
1
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3
1)(
ssR
)11(
1lim
1
101
1
lim][lim)]([lim
20
3
0)()(1
)(
00
ss
s
s
ss
ssEe
s
ssHsG
ssR
ssss
2
1
)( ssR
)11(
1lim
1
101
1
lim][lim)]([lim
0
2
0)()(1
)(
00
ss
s
s
ss
ssEe
s
ssHsG
ssR
ssss
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Automatic control by Meiling CHEN 23
Type 1
23
5
s
)(sY)(sR )(sE
s
1
010)1(
)1(lim
1
1011
1
lim][lim)]([lim
0
0)()(1
)(
00
ss
ss
ss
ss
ssEe
s
ssHsG
ssR
ssss
ssR
1)(
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Automatic control by Meiling CHEN 24
1.010)1(
)1(lim
1
1011
1
lim][lim)]([lim
0
2
0)()(1
)(
00
ss
s
ss
ss
ssEe
s
ssHsG
ssR
ssss
21)(
ssR
)10)1(()1(lim
1
1011
1
lim][lim)]([lim
0
3
0)()(1
)(
00
ssss
ss
ss
ssEe
s
ssHsG
ssR
ssss
3
1)(
s
sR
vK
1
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Automatic control by Meiling CHEN 25
1
10
s
)(sY)(sR )(sE
2
1
s
Type 2
010)1(
)1(lim
11011
1
lim][lim)]([lim
2
2
0
2
0)()(1
)(
00
ss
ss
ss
ss
ssEe
s
ssHsG
ssR
ssss
ssR
1)(
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Automatic control by Meiling CHEN 26
010)1(
)1(lim
11011
1
lim][lim)]([lim
20
2
2
0)()(1
)(
00
ss
ss
ss
ss
ssEe
s
ssHsG
ssR
ssss
2
1)(
ssR
1.010)1(
)1(lim
1
1011
1
lim][lim)]([lim
20
2
3
0)()(1
)(
00
ss
s
ss
ss
ssEe
s
ssHsG
ssR
ssss
3
1)(
s
sR
aK
1
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0 0
00
1lim ( ) lim ( )
1 ( )ss
s se sE s s R s
kG s
s
=0, type 0 system
1
1sse
k
sse
sse
Steady-state errorexists and
is finite.
Unstable
Unstable
Step input:
1( ) 1( ) ( )r t t R s
s
2
Ramp input:
1( ) ( )r t t R s
s
2
3
Parabolic input:
1 1( ) ( )
2r t t R s
s
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0 0
01
1lim ( ) lim ( )
1 ( )
sss s
e sE s s R sk
G ss
=1, type 1 system
10
1sse
1sse
k
sse
No steady-state error
Steady-stateerror exists
Unstable
Step input:
1( ) 1( ) ( )r t t R s
s
2
Ramp input:
1( ) ( )r t t R s
s
2
3
Parabolic input:
1 1( ) ( )
2r t t R s
s
Type-1 system can track step signal accurately.
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0 0
02
1lim ( ) lim ( )
1 ( )ss
s se sE s s R s
kG s
s
=2, type 2 system
10
1sse
0sse
1sse k
No steady-state error
No steady-state error
Steady-stateerror exists
Step input:
1( ) 1( ) ( )r t t R s
s
2
Ramp input:
1( ) ( )r t t R s
s
2
3
Parabolic input:
1 1( ) ( )
2r t t R s
s
Type-2 system can track step and ramp signals accurately.
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with step input - step-error constant
with ramp input- ramp-error constant
with parabolic input
- parabolic-error constant
0lim ( )ps
k G s
0lim ( )s
k sG s
2
0lim ( )as
k s G s
Steady-state error constants
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Type of
System
Error
constantsSteady-state error
j
0
)(1)( 0 tRtr tVtr 0)( 2)( 2
0tAtr
k
R
1
0
k
V0
k
A0
0
00
0
00k
k
k
pk vk ak
sse
Summary of steady-state error and error constantsfor unit feedback systems (H(s)=1)