Steady-State Analysis
DNT 354 - CONTROL PRINCIPLE
CONTENTS
Introduction
 Steady-State Error for Unity Feedback System
 Static Error Constants and System Type
 Steady-State Error for Non-Unity Feedback
Systems

INTRODUCTION

Steady-state error, ess: The difference between the input and
the output for a prescribed test input as time, t approaches ∞.
Step Input
INTRODUCTION

Steady-state error, ess: The difference between the input and
the output for a prescribed test input as time, t approaches ∞.
Ramp Input
TEST INPUTS


Test Inputs: Used for steady-state error analysis and design.
Step Input:



Ramp Input:


Represent a constant position.
Useful in determining the ability of the control system to position itself
with respect to a stationary target.
Represent constant velocity input to a position control system by their
linearly increasing amplitude.
Parabolic Input:


Represent constant acceleration inputs to position control.
Used to represent accelerating targets.
TEST INPUTS
UNITY FEEDBACK SYSTEMS

To determine the steady-state error, we apply the Final Value
Theorem:
f ()  lim sF ( s )
s 0


The following system has an open-loop gain, G(s) and a unity
feedback since H(s) is 1. Thus to find E(s),
Substituting the (2) into (1) yields,
R( s)
E ( s) 
1  G ( s)
E ( s)  R( s)  C ( s)
…(1)
C ( s)  R( s)G ( s)
…(2)
UNITY FEEDBACK SYSTEMS

By applying the Final Value Theorem, we have:
e()  lim sE ( s)
s 0
sR ( s)
 lim
s 0 1  G ( s )

This allows the steady-state error to be determined for a given
test input, R(s) and the transfer function, G(s) of the system.
UNITY FEEDBACK SYSTEMS

For a unit step input:
s (1 / s )
s 0 1  G ( s )
1

1  lim G ( s )
estep ()  lim
s 0

The term:
lim G ( s )
s 0


The dc gain of the forward transfer function, as the frequency
variable, s approaches zero.
To have zero steady-state error,
lim G ( s )  
s 0
UNITY FEEDBACK SYSTEMS

For a unit ramp input:
s (1 / s 2 )
eramp ()  lim
s 0 1  G ( s )
1

s  lim sG ( s )
s 0

1
lim sG ( s )
s 0

To have zero steady-state error,
lim sG ( s )  
s 0

If there are no integration in the forward path:
lim sG ( s )  0
s 0
Then, the steady state error will be infinite.
UNITY FEEDBACK SYSTEMS

For a unit parabolic input:
s (1 / s 3 )
e parabola()  lim
s 0 1  G ( s )
1
 2
s  lim s 2G ( s )
s 0

1
lim s 2G ( s )
s 0

To have zero steady-state error,
lim s 2G(s)  
s 0

If there are one or no integration in the forward path:
lim s 2G( s)  0
s 0
Then, the steady state error will be infinite.
UNITY FEEDBACK SYSTEMS

Example: Find the steady-state errors for inputs of 5u(t), 5tu(t), and
5t2u(t).
estep () 
5
5

1  lim G( s) 21
eramp () 
s 0
eramp () 
5

lim sG ( s)
s 0
5

2
lim s G( s)
s 0
SYSTEM TYPE


System Type: The value of n in the denominator or, the number
of pure integrations in the forward path.
Therefore,
i.
ii.
iii.
If n = 0, system is Type 0
If n = 1, system is Type 1
If n = 2, system is Type 2
SYSTEM TYPE


Example:
i.
Gs  
K s  2
s  1s  3
Type 0
ii.
Gs  
K 0.5s  1
ss  12s  1 s 2  s  1
Type 1
iii.
G s  
K 2 s  1
s3
Type 3

Problem: Determine the system type.

STATIC ERROR CONSTANT


Static Error Constants: Limits that determine the steady-state
errors.
Position constant:
K p  lim G ( s )
s 0

Velocity constant:
K v  lim sG ( s )
s 0

Acceleration constant:
K a  lim s 2G(s)
s 0
POSITION ERROR CONSTANT, KP

Steady-state error for step function input, R(s):
Rs   R s
 sR s  
R
ess  lim 

s 0 1  G s  

 1  lim G ( s )
s 0

Position error constant:

Thus,
R
ess 
1 K p
K p  lim G ( s )
s 0
VELOCITY ERROR CONSTANT, KV

Steady-state error for step function input, R(s):
Rs   R s
2
 sR s  
R
ess  lim 

s 0 1  G s  

 lim sG ( s )
s 0

Position error constant:

Thus,
ess 
R
Kv
K v  lim sG ( s )
s 0
ACCELERATION ERROR CONSTANT, KA

Steady-state error for step function input, R(s):
Rs   R s
3
 sR s  
R
ess  lim 

2
s 0 1  G s  
lim
s
G ( s)


s 0

Position error constant:

Thus,
ess 
R
Ka
K a  lim s 2G(s)
s 0
STATIC ERROR CONSTANT & SYSTEM TYPE

Relationships between input, system type, static error
constants, and steady-state errors:
ANALYSIS VIA STATIC ERROR CONSTANT

Example: Find the steady-state errors for inputs of 5u(t), 5tu(t), and
5t2u(t) by first evaluating the static error constants.
K p  20, K v  0, K a  0
estep () 
R
5

1  K p 21
eramp () 
eramp () 
R

Ka
R

Kv
NON-UNITY FEEDBACK SYSTEMS

Example: Calculate the error constants and determine ess for a
unit step, ramp and parabolic functions response of the
following system.
1
5s  1
G s  
; H s  
ss  12
s5
NON-UNITY FEEDBACK SYSTEMS

Example: Calculate the error constants and determine ess for a
unit step, ramp and parabolic functions response of the
following system.

For step input,
K p  lim G ( s ) H ( s)
s 0

5s  1 
 lim 
s 0 s s  12 s  5 




sR s  
ess  lim 
s 0 1  G s H s  






s 1 s 
 lim 

s 0


5
s

1
1 

 s s  12 s  5 
 1 
 lim 
0
s 0 1   


NON-UNITY FEEDBACK SYSTEMS

Example: Calculate the error constants and determine ess for a
unit step, ramp and parabolic functions response of the
following system.

For ramp input,
K v  lim sG s H ( s )
s 0
 5s  1 
 lim 
s  0 s  12 s  5  


1

12

sR s  
ess  lim 
s 0 1  G s H s  




2


s1 s
 lim 

s 0


5
s

1
1 

 s s  12 s  5 
 
 12
NON-UNITY FEEDBACK SYSTEMS

Example: Calculate the error constants and determine ess for a
unit step, ramp and parabolic functions response of the
following system.

For parabolic input,


K a  lim s 2G s H s 
s 0
 5s s  1 
 lim 
s  0 s  12 s  5  


0

sR s  
ess  lim 
s 0 1  G s H s  






s 1 s3
 lim 

s 0


5
s

1
1 

 s s  12 s  5 
 

NON-UNITY FEEDBACK SYSTEMS

Problem: Calculate the error constants and determine ess for a
unit step, ramp and parabolic functions response of the
following system.