6. steady state error

simple how to find S.S.E

1. Steady State Error

2. Introduction
•Steady-stateerrorisdefinedasthedifferencebetweentheinput
(command)andtheoutputofasysteminthelimitastimegoestoinfinity
(i.e.whentheresponsehasreachedsteadystate).
•Anyphysicalcontrolsysteminherentlysufferssteady-stateerrorin
responsetocertaintypesofinputs.
•Asystemmayhavenosteady-stateerrortoastepinput,butthesame
systemmayexhibitnonzerosteady-stateerrortoarampinput.
•Whetheragivensystemwillexhibitsteady-stateerrorforagiventypeof
inputdependsonthetypeofopen-looptransferfunctionofthesystem.
•Controlsystemsmaybeclassifiedaccordingtotheirabilitytofollowstep
inputs,rampinputs,parabolicinputs,andsoon.
•Themagnitudesofthesteady-stateerrorsduetotheseindividualinputs
areindicativeofthegoodnessofthesystem.

3. Classification of Control Systems
•Considertheunity-feedbackcontrolsystemwiththefollowingopen-loop
transferfunction
•Itinvolvestheterms
N
inthedenominator,
representingNpolesattheorigin.
•Asystemiscalledtype0,type1,type2,...,if
N=0,N=1,N=2,...,respectively.

4. Classification of Control Systems
•Asthetypenumberisincreased,accuracyisimproved.
•However,increasingthetypenumberaggravatesthestabilityproblem.
•Acompromisebetweensteady-stateaccuracyandrelativestabilityis
alwaysnecessary.

5. Steady State Error of Unity Feedback Systems
•Consider the system shown in following figure.
•The closed-loop transfer function is

6. Steady State Error of Unity Feedback Systems
•ThetransferfunctionbetweentheerrorsignalE(s)andthe
inputsignalR(s)is)()(
)(
sGsR
sE


1
1
•Thefinal-valuetheoremprovidesaconvenientwaytofind
thesteady-stateperformanceofastablesystem.
•SinceE(s)is
•Thesteadystateerroris

7. Static Error Constants
•Thestaticerrorconstantsarefiguresofmeritofcontrolsystems.Thehigher
theconstants,thesmallerthesteady-stateerror.
•Inagivensystem,theoutputmaybetheposition,velocity,pressure,
temperature,orthelike.
•Therefore,inwhatfollows,weshallcalltheoutput“position,”therateof
changeoftheoutput“velocity,”andsoon.
•Thismeansthatinatemperaturecontrolsystem“position”representsthe
outputtemperature,“velocity”representstherateofchangeoftheoutput
temperature,andsoon.

8. Static Position Error Constant (K
p)
•The steady-state error of the system for a unit-step input is
•The static position error constant K
pis defined by
•Thus, the steady-state error in terms of the static position
error constant K
pis given by

9. Static Position Error Constant (K
p)
•For a Type 0system
•For Type 1or higher systems
•For a unit step input the steady state error e
ssis

10. Static Velocity Error Constant (K
v)
•The steady-state error of the system for a unit-ramp input is
•The static position error constant K
vis defined by
•Thus, the steady-state error in terms of the static velocity
error constant K
vis given by

11. Static Velocity Error Constant (K
v)
•For a Type 0system
•For Type 1systems
•For type 2 or higher systems

12. Static Velocity Error Constant (K
v)
•For a ramp input the steady state error e
ssis

13. Static Acceleration Error Constant (K
a)
•The steady-state error of the system for parabolic input is
•The static acceleration error constant K
ais defined by
•Thus,thesteady-stateerrorintermsofthestaticaccelerationerror
constantK
aisgivenby

14. Static Acceleration Error Constant (K
a)
•For a Type 0system
•For Type 1systems
•For type 2systems
•For type 3or higher systems

15. Static Acceleration Error Constant (K
a)
•For a parabolic input the steady state error e
ssis

16. Summary

17. Example
•Forthesystemshowninfigurebelowevaluatethestatic
errorconstantsandfindtheexpectedsteadystateerrors
forthestandardstep,rampandparabolicinputs.
C(S)R(S)
-))((
))((
128
52100
2


sss
ss

18. Example (Steady Sate Errors)
p
K 
v
K 410.
a
K 0 0 090.

19. Example (evaluation of Static Error Constants)))((
))((
)(
128
52100
2



sss
ss
sG )(limsGK
s
p
0
 










 ))((
))((
lim
128
52100
2
0 sss
ss
K
s
p 
p
K )(limssGK
s
v
0
 










 ))((
))((
lim
128
52100
2
0 sss
sss
K
s
v 
v
K )(lim sGsK
s
a
2
0
 










 ))((
))((
lim
128
52100
2
2
0 sss
sss
K
s
a 410
12080
5020100
.
))((
))((












a
K