Steady State Error1 1

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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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21

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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22

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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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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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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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



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)

Steady State Error1 1

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