6Steady State Characteristics

8/9/2019 6Steady State Characteristics

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ME2142 Feedback Control Systems1

Steady-State Characteristics

Steady-State Characteristics

ME2142 Feedback Control Systems




Consider the unity-feedback system

Consider the unity-feedback system

System Type

RG

C

-

+ E

With … (2-1)

The parameter N associated with the term S N in the

denominator represents the “Type” of the system.Example Type ! if "#!$ Type 1 if "#1 and so on.

% free “s” term in the denominator represents aninte&ration. The hi&her the type n'mer$ the etter thesteady-state acc'racy of the closed-loop control system.

owe*er$ the hi&her the system type$ the &reater theprolem with system staility.

)1()1)(1()1()1)(1()(

21 +++

+++=

sT sT sT s

sT sT sT K sG p

N

mba




Examples of System Types

RG

C

-

+ E

Type 1 systems ! !

)1()(+

= s s

K

sG τ )2()( 22

nn s s s

K sG ω ζω ++

=

Type " systems ! !

)1(

)(

+

=

s

K sG

τ ))((

)(

21 p s p s

K sG

++

=

22 2)(

nn s s

K sG

ω ζω ++=

Type 2 systems ! !

)1()(

2+

= s s

K sG

τ ))(()(

21

2 p s p s s

K sG

++=




The unity-feedback system

The unity-feedback system

Steady-state Errors # Static Error Constants

RG

C

-

+ E R

C R

R

E −=

R

C −= 1

G

G

+

−=

11

G

GG

+

−+=

1

)1(

G R E

+=1

1

Error Transfer f'nction

Th's )()(1

1)( s R

sG s E

+=

and the steady-state error is)(lim t ee t

ss∞→=

)(lim0

s sE s→

=

)(1

)(lim

0 sG

s sRe

s ss

+=

→





)(1

)(lim

0 sG

s sRe

s

ss

+

=→

+or a 'nit-step inp't

+or a 'nit-step inp't s

s R 1)( = and

s sG

se

s ss

1

)(1lim

0 +=

→ )0(1

1

G+

=

,tatic osition Error onstant$ K p is defined as

)0()(lim0

G sG K s

p ==

→

and

p

ss K

e+

=1

1

with $

)1()1)(1(

)1()1)(1()(

21 +++

+++=

sT sT sT s

sT sT sT K sG

p

N

mba

for Type ! systems K K p =

for Type 1 or hi&her systems

K e ss

+=1

1

∞= p K 0= sse





)(1

)(lim

0

sG

s sRe

s ss

+

=→

+or a 'nit-ramp inp't

+or a 'nit-ramp inp't2

1)(

s s R = and

20

1

)(1lim

s sG

se

s ss

+=

→ )(

1lim

0 s sG s→=

,tatic /elocity Error onstant$ K v is defined as

and)(lim0 s sG K sv

→=

v

ss K

e 1=

with $

)1()1)(1(

)1()1)(1()(

21 +++

+++=

sT sT sT s

sT sT sT K sG

p

N

mba

for Type ! systems

for Type 1 systems

0= sse

0=v

K

for Type 2 or hi&her systems

K K v = K

e ss1

=

∞= sse

∞=v K


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Steady-state Errors

,'mmary

of steady-state errors

,'mmaryof steady-state errors

Step Input

1=r

Ramp Inputt r =

Accel. Input2/2t r =

Type 0 system K +1

1 ∞ ∞

Type 1 system 0 K

1 ∞

Type 2 system 0 0 K

1

Type ! systems ha*e finite steady-state errors for step inp'ts andcannot follow ramp inp'ts.

Type 1 systems ha*e ero steady-state errors for step inp'ts$ finiteerrors for ramp inp'ts$ and cannot follow acceleration inp'ts.

Type 2 systems are needed to follow step inp't and ramp inp'tswith ero steady-state errors.

n &eneral$ the hi&her the static &ain of the open-loop transferf'nction$ 3(s)$ the smaller the steady-state errors. owe*er$hi&her &ains normally lead to staility prolems.


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6Steady State Characteristics

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