tr tn

tc sG1 sG2

sH tb

Two definitions of error

)()()( sCsRsE

)()()( sBsRsE

or

The steady state error of control system

)}({)( )(lim 1 sEteteet

ss

L

The steady state error

Here we define:

Final Value Theorem

Suppose has the Laplace transform and the

Limits and exist. Then the final value

Is Hence

The steady state error

)()()( sBsRsE

tr tn

tc sG1 sG2

sH tb

)(

1)(

1

1

)(

21

2

21

sNsHsGsG

sHsGsR

sHsGsG

sNssRssE ener

)(tf )(sF

)(lim tft

)(lim0

ssFs

)(lim)(lim0

ssFtfst

ssEteest

ss0

limlim

sR E sC

sB

sG

sH

Relationship between system and steady-state error

only caused by input signal

gain loopopen 121

12122

22

K

sTsTsTs

sssKsHsG

kkki

v

lllj

Open-loop transfer function

In terms of to the number of the poles of G(s)H(s) at s=0

(corresponding with the number of the integral elements in the

open loop systems), we name:

=0 type 0 system

=1 type system

=2 type system

Define: )()()( sBsRsE

121

121

22

22

sTsTsTs

sssKsHsG

kkkiv

lllj

0 0

0

1

0

1lim ( ) lim ( )

1 ( ) ( )

1 lim

1

lim

sss s

s

s

e sE s s R sG s H s

s R sK s

sR s

s K

Relationship between system and steady-state error

only caused by input signal

0 0

1lim ( ) lim ( )

1 ( ) ( )ss

s se sE s s R s

G s H s

1) Step input

R

R ss Ps

ssK

R

sHsG

Re

1)()(1lim

0

The position error constant: )()(lim0

sHsGKs

P

sR E sC

sB

sG

sH

Relationship between system and steady-state error

only caused by input signal

)(1)( tRtr

0 0

1lim ( ) lim ( )

1 ( ) ( )ss

s se sE s s R s

G s H s

2) Ramp input

Vs

ssK

V

sHssG

Ve

)()(lim

0

The velocity error constant: )()(lim0

sHssGKs

V

sR E sC

sB

sG

sH

2s

VsR

Relationship between system and steady-state error

only caused by input signal

tVtr )(

0 0

1lim ( ) lim ( )

1 ( ) ( )ss R

s se sE s s R s

G s H s

3) Parabolic input

3A

R ss

Relationship between system and steady-state error

only caused by input signal

The parabolic error constant: )()(lim2

0sHsGsK

sa

sR E sC

sB

sG

sH

as

ssK

A

sHsGs

Ae

)()(lim

20

3

2

1)( tAtr

Summy of the steady-state errors

Type of

system

Error constant Steady-state error

Step input

Ramp Input

Parabolic Input

0 0 0

1 0

2

3

0sse

0sse 0sse

p

ssK

Re

1

V

ssK

Ve

a

ssK

Ae

0sse 0sse 0sse

sse

pK VK aK

K

K

K

)(1)( tRtr tVtr )( 2

2

1)( tAtr

By using the method described, the steady-state error of any linear

closed-loop system subject to an input with order higher than the

parabolic function can also be derived if necessary.

Summy of the steady-state errors

We emphasize often enough that, for these table results to be valid,

the closed-loop system must be stable.

As a summary of the error analysis, Table shows the relations among

the error constants, the types of systems, and the input types with

reference to the following conditions:

)()()( sBsRsE The error defination

sR E sC

sB

sG

sH

The system configuration

121

121

22

22

sTsTsTs

sssKsHsG

kkkiv

lllj

Open-loop transfer function

Summy of the steady-state errors

As a summary, the following points should be noted when applying

the error-constant analysis just presented.

1. The step-, ramp-, or parabolic-error constants are significant for the

error analysis only when the input signal is a step function, ramp

function, or parabolic function, espectively.

2. Because the error constants are defined with respect to the

forward-path transfer function G(s)H(s), the method is applicable to

only the system configuration shown in the above figure . Because the

error analysis relies on the use of the final value theorem of the

Laplace transform, it is important to check first to see if sE(s) has any

poles on the j-axis or in the right-half s-plane.

3. The steady-state error properties summarized in the Table are for

systems with the above figure and the error defination only.

4. The steady-state error of a system with an input that is a linear

combination of the three basic types of inputs can be determined by

superimposing the errors due to each input component.

5. When the system configuration and the error defination differ from the

above , we can either simplify the system to the form of above figure or

establish the error signal and apply the final-value theorem. The error

constants defined here may or may not apply, depending on the dividual

situation.

6. When the steady-state error is infinite, that is, when the error increases

continuously with time, the error-constant method does not indicate how

the error varies with time. This is one of the disadvantages of the error-

constant method.

7.The error-constant method also does not apply to systems with inputs

that are sinusoidal, since the final-value theorem cannot be applied.

Summy of the steady-state errors

tr

tn

tc

1G 2G

e

)(1

lim21

2

0SN

GG

Gse

sssn

ssnssree

sNGG

Gslim

GG

sRslim

ssElimssElim

ssElime

ss

Ns

Rs

sss

)(11

)(

)()(

)(

21

2

021

0

00

0

When R(s)=0 , N(s)=1/s

Steady-state error only caused by

the disturbance signals

)(1

)(21

2 sNGG

GsEN

s

KsGKsGset 2211 )( , )(

121

2

00

11lim)(lim

KsKKs

sKssEe

sN

sssn

tr

tn

tc

1G 2G

e

)(1

lim21

2

0SN

GG

Gse

sssn

When R(s)=0 , N(s)=1/s

Steady-state error only caused by

the disturbance signals

)(1

)(21

2 sNGG

GsEN

s

KsGset 22 )(

)(

1lim

1

)(lim)(lim

10

21

2

00 sGsKsGs

sKssEe

ssN

sssn

0,0,0 )1(

)( 211

1

KKs

sKsGif Stability and essn=0