NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Steady State error of a System
Dr Bishakh Bhattacharya
Dr.
Professor, Department of Mechanical Engineering
IIT Kanpur
Joint Initiative of IITs and IISc - Funded by MHRD
Module 2- Lecture 11
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
This Lecture Contains
Steady state error of a system
 Position, Velocity and Acceleration Constants
Position Velocity and Acceleration Constants
Optimal Steady State Error
Joint Initiative of IITs and IISc - Funded by MHRD
Module 2- Lecture 11
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Module 2- Lecture 11
Steady
y state error - Introduction
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Steady state error refers to the long-term behavior of a dynamic system.
The Type of a system is significant to predict the nature of this error.
A system having no pole at the origin is referred as Type-0 system.
Thus, Type-1, refers to one pole at the origin and so on.
It will be shown in this lecture that
that, it is the type of a system which can directly
determine whether a particular command will be followed by a system or not.
We will consider three common commands: namely, step, ramp and parabolic
ramp and find out the steady state response/error of a system to follow these
commands.
d
A closed loop control system shows remarkable performance in reducing the
steady state error of a system.
t Initiative of IITs and IISc ‐ Funded by MHRD
3
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Module 2- Lecture 11
Steady state error of a system
• Error in a system: E(s) = U(s) /(1+G(s))
sU ( s )
s 0 1  G ( s )
lim e(t )  ess  lim
t 
• For a step input
A
e ss 
1  G (0)
• Plant Transfer function G(s) is defined as
i 1(s  z i )
K ΠM
G (s) 
sk ΠN
j 1 (s  p j )
Joint Initiative of IITs and IISc ‐ Funded by MHRD
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Module 2- Lecture 11
Error Constants
•
Position error constant
K p  limG ( s )
s 0
ess = A/(1+Kp)
•
St d state
Steady
t t error off a step
t iinputt off magnitude
it d A is
i
•
Steady state error will be zero for system with type greater than or equal to 1
Joint Initiative of IITs and IISc ‐ Funded by MHRD
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
•
For ramp input
•
Define velocity constant as
A
ess  lim
s 0 sG ( s )
K v  lim sG (s)
s 0
A/Kv
•
Hence steady state error is
•
Error will be zero for k g
greater than or equal
q
to 2
Joint Initiative of IITs and IISc ‐ Funded by MHRD
Module 2- Lecture 11
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Module 2- Lecture 11
Summary of Steady State Errors
Summary of Steady State Errors
Type
Step
(A/s)
0
Ess= A/(1+Kp) Inf
Parabolic
Ramp
(A/s3)
Inf
1
0
A/Kv
Inf
2
0
0
A/Ka
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Ramp
(A/s2)
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Module 2- Lecture 11
Design
g of Optimal
p
systems
y
You can chose optimal gain such that the error constant could be minimized. Consider the following system:
Joint Initiative of IITs and IISc ‐ Funded by MHRD
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Module 2- Lecture 11
Optimized
p
Gain
2  ks
T (s )  2
s  3s  2
This is a Type zero system. Hence, yp
y
,
consider step input.
E (s ) 
A
k  2 k 1
(1  Te (s))  A (

)
s
s 1 s  2
2
A
J ( k )   e 2 ( t )dt 
[ k 2  6 k  11]
12
0

Note that the Performance index is a quadratic function of gain k, which can q
g
,
be minimized to obtain k.
Joint Initiative of IITs and IISc ‐ Funded by MHRD
dJ/dk 0
dJ/dk=0 ‐‐‐
k 3
k = ‐3
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Module 2- Lecture 11
Special
p
References for this lecture
 Feedback Control of Dynamic Systems: Frankline,
Frankline Powell and Emami-Naeini
Emami-Naeini,
Pearson Publisher
 Control Systems Engineering: Norman S Nise
Nise, John Wiley & Sons
 Systems Dynamics and Response: S. Graham Kelly, Thomson Publisher
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