Control Systems
Steady state error
ch.7
R(s ) 
G (s )

Lecturer: Musaab Zaroug
2023
C (s )
Outline
• Definition of Steady-State Error.
• Steady-state error for unity feedback systems.
• Steady-state error for non-unity feedback systems.
• System types.
• Steady-state error for disturbances
• Sensitivity
• Section 7.8 is not included
Introduction
• control systems analysis and design focus on three specifications: (1)
transient response, (2) stability, and (3) steady-state errors
Elevator Location (floor)
Definition of the Steady-State error
Input command
4
Transient
response
Steady-state
responce
1
Time
Steady-state
error
Input test signals
r (t )
• Unit Step
fwe want to test
t
constant position
using Unit step
r (t )
• Unit Ramp
for velocity
t
If
r (t )
• Unit Parabola
ace
t
S.S. Error in terms of G(s)
Oo
Change between input and output
if
we know
input Rli and Tf G
we can calculate the
steadystate ero
openLoop Tf
sing if we want
teady stateerror in time domain
S.S. Error for unity feedback systems
R(s) 

E (s)
G (s)
C (s)
whenwe have closeloop T f
R(s)

 E (s)
G ( s)
T ( s) 
1  G(s) C (s)
close
loop TF
closeloopTf
• Example: Find the S.S. Error for a unity feedback
system with
• Answer
step
43
R'd Le
In
Li
Tin
Tin
ligy.gg
1
I
S.S. Error in terms of G(s)
O
Dy
Example
• Find the S.S. Error for the following system for inputs
R(s ) 

E (s )
p
GI
100( s  2)( s  6)
s ( s  3)( s  4)
C (s )
TyPI
f
544
es
5
tutti
fr
Ga
Iya
s
Ey
of
Ey
o
SE ult
I
so
a
os
Error constants
• Position constant
• Velocity constant
• Acceleration constan
The system type
R(s ) 
• For the unity feedback
system shown above the
type of the system is
defined as (n).

E (s )
K ( s  z1 )( s  z2 )  C (s )
n
sO
( s  p1 )( s  p2 ) 
n o
n l
n2
É
O
This type Is
input give us
go to
a
the table
and
see
value for steady state
using Ramp Kult
what the
type of
error
e
a
0.1 54
esta
6 78
IT
46722
another
input
85
s
detain
wt
witnatmyth
Tron
RIn
Eastin
that the steady-state error produced by a step
ashows
disturbance can be reduced by increasing the dc gain of
FinTisdeftbecuatthe
step G1(s) or decreasing the dc gain of G2(s).
m
Ms
ofsystem exit in theCys
 The degree to which changes in system parameters affect system transfer functions, and hence performance, is called
sensitivity.
 A system with zero sensitivity (that is, changes in the system parameters have no effect on the transfer function) is ideal.
 The greater the sensitivity, the less desirable the effect of a parameter change.
T
Ts
Sta
f
I
East
y
Its
S.S. Error for Non-unity feedback
systems
R(s) +
-
G
C (s )
R(s) +
G
- - H(s)
H(s)
(a)
-1
R(s) +
-
G( s)
1  G( s) H ( s)  G( s)
(c)
C (s )
(b)
C (s )
Tutorial
• For the system shown in the following figure, find the system type,
the appropriate error constant, and the steady state error for a unit
step input. Assume input and output units are the same.
R(s) +
-
100
s ( s  10)
1
( s  5)
C (s )