The document discusses steady-state error in control systems. It defines steady-state error as the difference between the input and output of a system as time approaches infinity. It describes how to evaluate steady-state error for different test inputs like step, ramp, and parabolic inputs. The steady-state error depends on factors like system configuration and number/type of poles in the closed-loop transfer function. Systems with integrators have zero steady-state error for step inputs, while more poles may be needed at the origin to achieve zero error for ramps and parabolas. The document also introduces static error constants which characterize the steady-state error.

1. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Control Systems
B.Madhuri

2. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Second order system
H(s)
G(s)
R(s) C(s)+
-
𝐺 𝑠 =
𝜔 𝑛
2
𝑠(𝑠 + 2𝜉𝜔 𝑛) 22
2
2)(
)(
nn
n
sssR
sC




If H(s)=1

3. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Steady-State Error Definition
Steady-state error is the
difference between the input and
the output for a prescribed test
input as time approaches infinity

4. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Steady-State Error - Test Input
1.Step input
2.Ramp input
3.Parabolic input

5. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Steady-State Error – Introduction

6. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Evaluating Steady-State Error
Steady-state error analysis only
applicable when the system
response is stable. Unstable
system cannot be analyzed for
steady-state error

7. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Evaluating Steady-State Error – Step
input
Error = 0
Error = constant

8. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Evaluating Steady-State Error – Ramp
input
Error = 0
Error = constant
Error = ∞

9. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Evaluating Steady-State Error – General
Block Diagram
E(s) = R(s) – C(s)
E(s) = R(s) – C(s)
General Representation (unity and non-unity feedback)
Unity feedback System
Error Calculation

10. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Evaluating Steady-State Error –
Sources of Error
1.Non-linear elements
2.System configuration
3.Type of applied input

11. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Evaluating Steady-State Error – Sources of
Error by System Configuration
Since K is nonzero and
positive, e(t) will never be
zero for a step input
K
te
K
te
KsK
s
K
sR
sE
sRsKEsE
sKEsC
sCsRsE
t














1
1
)(
1
1
)(
)1(
1
1
1
1
)(
)(
)()()(
)()(
)()()(
The pure gain configuration
will never produce zero
steady-state error.

12. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Evaluating Steady-State Error – Sources of
Error by System Configuration
Since K is nonzero and
positive, e(t) will always
approach zero for a step
input.
The integrator
configuration will
always produce zero
steady-state error.
0)(
)(
1
1
)(
1
)(
)(
)()()(
)()(
)()()(















t
Kt
te
ete
Ks
s
Ks
s
s
Ks
sR
s
K
sR
sE
sRsE
s
K
sE
sE
s
K
sC
sCsRsE

13. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Steady-State Error for Unity Feedback System
Using Closed Loop Transfer Function
 )(1)()(
)()()()(
)()()(
)()()(
sTsRsE
sTsRsRsE
sTsRsC
sCsRsE




Since we are interested to
find the steady-state error
(not the transient error), we
can use final value theorem

14. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Steady-State Error for Unity Feedback System
Using Closed Loop Transfer Function
)(lim)(lim)(
0
ssEtee
st 

Final Value Theorem
 )(1)(lim)(lim)(
00
sTssRssEe
ss



15. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Steady-State Error for Unity Feedback System
Using Closed Loop Transfer Function - Example
s
sR
ss
sT
1
)(
107
5
)( 2



Find steady-state error?
 
5.0
107
5
1)
1
(lim)(
)(1)()(lim)(
20
0











sss
se
sTssRssEe
s
s

16. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Steady-State Error for Unity Feedback System Using
Forward Path Transfer Function
)(1
)(
)(
)()()()(
)()()(
)()()(
sG
sR
sE
sRsGsEsE
sGsEsC
sCsRsE





By final value theorem,
)(1
)(
lim)(lim)(
00 sG
ssR
ssEe
ss 


No need to
evaluate T(s)

17. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Steady-State Error for Unity Feedback System – Step
Input
)(lim1
1
)(1
1
lim)(lim)(
)(1
)(
lim)(lim)(
0
00
00
sGsG
s
s
ssEe
sG
ssR
ssEe
s
ss
ss









In order to have zero steady-
state error,


)(lim
0
sG
s
1)(lim
))((
))((
)(
0
21
21





nsG
pspss
zszs
sG
s
n


In order to have zero steady-state
error, n must be equal or greater
than one. There must exist at least
one pole at origin. If n = 0, we
have




21
21
0
21
21
)(lim
))((
))((
)(
pp
zz
sG
psps
zszs
sG
s





There is
finite
steady-
state error

18. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Steady-State Error for Unity Feedback System – Ramp
Input
)(lim
1
)(lim
1
)(lim1
/1
)(1
1
lim
)(1
)(
lim)(lim)(
000
2
000 ssGssGssG
s
sG
s
s
sG
ssR
ssEe
sss
sss











In order to have zero steady-state error,


)(lim
0
ssG
s
2)(lim
))((
))((
)(
0
21
21





nssG
pspss
zszs
sG
s
n


In order to have zero steady-state error, n must
be equal or greater than two. There must exist
at least two poles at origin.

19. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Steady-State Error for Unity Feedback System – Ramp
Input
If n = 1, we have






21
21
21
21
00
21
21
))((
))((
lim)(lim
))((
))((
)(
pp
zz
pspss
zszss
ssG
pspss
zszs
sG
ss








There is finite steady-state error
If n = 0, we have
0
))((
))((
lim)(lim
))((
))((
)(
21
21
00
21
21







 



psps
zszss
ssG
psps
zszs
sG
ss
There is infinite
steady-state error
)(lim
1
)(
0
ssG
e
s


20. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Steady-State Error for Unity Feedback System –
Parabolic Input
)(lim
1
)(lim
1
)(1
1
lim
)(1
)(
lim)(lim)( 2
0
2
0
2
3
000 sGssGsssG
s
s
sG
ssR
ssEe
ss
sss









In order to have zero steady-state error, 

)(lim 2
0
sGs
s
3)(lim
))((
))((
)(
2
0
21
21





nsGs
pspss
zszs
sG
s
n

 In order to have zero steady-state
error, n must be equal or greater
than three. There must exist at
least three poles at origin.
If n = 2, then finite error If n = 1 or less, then infinite error

21. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Static Error Constants
)(lim1
1
)(
0
sG
e
s
step



Step input
)(lim
1
)(
0
ssG
e
s
ramp


Ramp input
)(lim
1
)( 2
0
sGs
e
s
parabola


Parabolic input
Static Error ConstantSteady-state error
)(lim
0
sGK
s
p


Position error constant
Velocity error constant
Acceleration error constant
)(lim 2
0
sGsK
s
a


)(lim
0
ssGK
s
v


22 February 2015

22. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Static Error Constants
22
)(lim
0
sGK
s
p


The static error constants can
assume three values:
1.Zero
2.Finite constant
3.Infinity
)(lim 2
0
sGsK
s
a

)(lim
0
ssGK
s
v


The value of steady-state error
decreases as the value of static
error constant increases
p
step
K
e


1
1
)(
v
ramp
K
e
1
)( 
a
parabola
K
e
1
)( 
22 February 2015

23. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION22 February 2015
Static Error Constants - Example
Evaluate steady-state error
constants and the steady
state errors for step, ramp
and parabolic inputs?
208.5
)12)(10)(8(
)5)(2(500
)12)(10)(8(
)5)(2(500
lim)(lim
00




 sss
ss
sGK
ss
p
161.0
208.51
1
1
1
)( 




p
step
K
e
0
)12)(10)(8(
)5)(2(500
lim)(lim
00




 sss
ss
sssGK
ss
v
0
)12)(10)(8(
)5)(2(500
lim)(lim 2
0
2
0




 sss
ss
ssGsK
ss
a

0
11
)(
a
parabola
K
e

0
11
)(
v
ramp
K
e
Step Input

24. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Static Error Constants - Example
Evaluate steady-state error
constants and the steady
state errors for step, ramp
and parabolic inputs?




 )12)(10)(8)(0(
)6)(5)(2(500
)12)(10)(8(
)6)(5)(2(500
lim)(lim
00 ssss
sss
sGK
ss
p
0
1
1
1
1
)( 




p
step
K
e
25.31
)12)(10)(8(
)6)(5)(2(500
lim)(lim
00




 ssss
sss
sssGK
ss
v
0
)12)(10)(8(
)6)(5)(2(500
lim)(lim 2
0
2
0




 ssss
sss
ssGsK
ss
a 
0
11
)(
a
parabola
K
e
032.0
25.31
11
)( 
v
ramp
K
e

25. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Static Error Constants - Example
Evaluate steady-state error
constants and the steady
state errors for step, ramp
and parabolic inputs?




 )12)(10)(8(
)7)(6)(5)(4)(2(500
lim)(lim 200 ssss
sssss
sGK
ss
p
0
1
1
1
1
)( 




p
step
K
e




 )12)(10)(8(
)7)(6)(5)(4)(2(500
lim)(lim 200 ssss
sssss
sssGK
ss
v
875
)12)(10)(8(
)7)(6)(5)(4)(2(500
lim)(lim 2
2
0
2
0




 ssss
sssss
ssGsK
ss
a
1014.1
875
11
)( 
a
parabola
K
e
0
11
)( 


v
ramp
K
e
22 February 2015

26. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Steady-State Error for Unity Feedback System –
Example
Find steady-state error for input
5u(t), 5tu(t) and 5t2u(t).

27. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Steady-State Error for Unity Feedback System –
Example
Find steady-state error for input
5u(t), 5tu(t) and 5t2u(t).

28. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
H(s)
G(s)
R(s) C(s)+
-
Steady state error for non-unity feedback
system
𝑒𝑠𝑠 = lim
𝑠→0
𝑠𝑅(𝑠)
1 + 𝐺 𝑠 𝐻(𝑠)

29. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
22 February 2015
System Types
System type is the value of n.
n = 0, System type 0
n = 1, System type 1
n = 2, system type 2
Etc.

30. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
System Types

31. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
System Types - Example
Evaluate steady-state error constants
and the steady state errors for step,
ramp and parabolic inputs?
Check for stability first
)9)(7(
)8(1000
)(



ss
s
sG
2
s
1
s
0
s
1 8063
1016 0
8063
No sign changes.
Therefore, all poles are in the left-half plane.
Therefore, the closed-loop system is stable.