Stready state errors which refers to compensation between actual

Steady state

1. Steady State Errors
Dr ManduA. Jeffrey
. . . . .
Department of Electrical Engineering
Faculty of Engineering and Technology
University of Botswana
CONTROL SYSTEMS I

2. Learning Outcomes
After completing this Lecture, the student will be able to:
You will also continue with the case study -Antenna azimuth position control
system
Steady State Errors
Control Systems Engineering, 7
th
Edition by Norman S. Nise. Copyright © 2015. All rights reserved. 2
Find the steady-state error for a unity feedback system
Specify a system’s steady-state error performance
Design the gain of a closed-loop system to meet a steady-state error
specification
Determine the steady-state error for disturbance inputs
Find the steady-state error for non-unity feedback systems

3. Steady State Errors
Overview
Steady State Errors
Control Systems Engineering, 7
th
Edition by Norman S. Nise. Copyright © 2015. All rights reserved. 3
Recall that control systems analysis and design focus on three
specifications: -LE01
a)Transient response,
b)Stability, and
c)Steady-state errors, taking into account the robustness of the
design along with economic and social considerations.
Elements of transient analysis have been derived –LE04.
This section examines steady-state errors.
a)Start by defining the errors
b)Then derive methods of controlling them

4. Steady State Errors
Steady State Errors
Control Systems Engineering, 7
th
Edition by Norman S. Nise. Copyright © 2015. All rights reserved. 4
Definition:
Steady-state error is the difference between the input and the output
for a prescribed test input as t 
Ultimately: Find that control system design entails trade-offs between
a)desired transient response,
b)steady-state error, and the
c)requirement that the system be stable.
Test Inputs for Steady state errors
Step input
Ramp input
Parabolic input
.

5. Test Input Signals
Steady State Errors
Control Systems Engineering, 7
th
Edition by Norman S. Nise. Copyright © 2015. All rights reserved. 5
Test Inputs

6. Test Input Signals
Steady State Errors
Control Systems Engineering, 7
th
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Step
Step inputs represent constant positionand thus are useful in determining the
ability of the control system to position itself with respect to a stationarytarget,
such as a satellite in geostationary orbit
An antenna position control is an example of a system that can be tested for
accuracy using step inputs.
Ramp
Ramp inputs represent constant-velocityinputs to a position control system by
their linearly increasing amplitude. These waveforms can be used to test a
system’s ability to follow a linearly increasing input or, equivalently, to track a
constant-velocity target.
A position control system that tracks a satellite that moves across the sky at a
constant angular velocity is an example of a system that can be tested for
accuracy using ramp input

7. Test Inputs
Steady State Errors
Control Systems Engineering, 7
th
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Parabola Input
Parabolas second derivatives
are constant, and represent
constant accelerationinputs
to position control systems –
they can be used to
represent accelerating
targets.
Refer to figure for application of
all the test input signals

8. Steady State Errors
Steady State Errors
Control Systems Engineering, 7
th
Edition by Norman S. Nise. Copyright © 2015. All rights reserved. 8
Example -Step Input
Steady State Errors
Output 1 has zero steady
state error
Output 2 has a finite
steady state error e2()

9. Steady State Errors
Steady State Errors
Control Systems Engineering, 7
th
Edition by Norman S. Nise. Copyright © 2015. All rights reserved. 9
Example –Ramp Input
Steady State Errors
Output 1 has zero steady
state error
Output 2 has a finite
steady state error e2()
Output 3 ??

10. Unity Feedback Systems
Steady State Errors
Control Systems Engineering, 7
th
Edition by Norman S. Nise. Copyright © 2015. All rights reserved. 10
Derivation of Steady State Error –e() oress
E(s) = R(s) -C(s)
Also
C(s) = E(s)G(s)
E(s) = R(s) -E(s)G(s)

Applying the final value theorem:
E(s)
For Unity Feedback Systems

11. Unity Feedback Systems
Steady State Errors
Control Systems Engineering, 7
th
Edition by Norman S. Nise. Copyright © 2015. All rights reserved. 11
Derivation of Steady State Error -e() oress
Step input: R(s) = 1/s
For zero steady state error:
Ramp input: R(s) = 1/s
2
For zero steady state error:

12. Unity Feedback Systems
Steady State Errors
Control Systems Engineering, 7
th
Edition by Norman S. Nise. Copyright © 2015. All rights reserved. 12
Derivation of Steady State Error -e() oress
Parabola input: R(s) = 1/s
3
For zero steady state error:
These lead to defining:
a)System type
b)Static error constants

13. Unity Feedback Systems
Steady State Errors
Control Systems Engineering, 7
th
Edition by Norman S. Nise. Copyright © 2015. All rights reserved. 13
Example: (7.2)
Find the steady state errors for step,
ramp and parabola signals of
magnitude 5:
•5u(t)
•5tu(t)
•5t
2
u(t)
Finite
Infinite
Infinite

14. Unity Feedback Systems
Steady State Errors
Control Systems Engineering, 7
th
Edition by Norman S. Nise. Copyright © 2015. All rights reserved. 14
Example: (7.3)
Find the steady state errors for step,
ramp and parabola signals of
magnitude 5:
•5u(t)
•5tu(t)
•5t
2
u(t)
Zero
Finite
Infinite
R(s) = 5u(t)
R(s) = 5tu(t)
R(s) = 5t
2
u(t)

15. Static Error Constants
Steady State Errors
Control Systems Engineering, 7
th
Edition by Norman S. Nise. Copyright © 2015. All rights reserved. 15
.. Define Static Error Constants

16. Static Error Constants
Example: 7.7
Skills Assessment 7.3
x
x
x
‣x
‣x
x
x
Steady State Errors
Control Systems Engineering, 7
th
Edition by Norman S. Nise. Copyright © 2015. All rights reserved. 16

17. System Type
System Type:
With reference to the figure below:
System Type is defined as the value
of nin the denominator
ie. the number of pure integrators in
the forward path
Steady State Errors
Control Systems Engineering, 7
th
Edition by Norman S. Nise. Copyright © 2015. All rights reserved. 17
Type 0
Type 1
Type 2

18. System Type and Error Constants
Steady State Errors
Control Systems Engineering, 7
th
Edition by Norman S. Nise. Copyright © 2015. All rights reserved. 18
Summary–Static Error Constants and System Type
Static Error Constants
Input Type 0 Type 1 Type 2
Step ??????
??????=lim
??????→0
??????(??????) ??????
??????= ??????
??????=
Ramp ??????
??????=0 ??????
??????=lim
??????→0
????????????(??????) ??????
??????=
Parabola ??????
??????=0 ??????
??????=0 ??????
??????=lim
??????→0
??????
2
??????(??????)

19. System Type and Steady State Errors
Steady State Errors
Control Systems Engineering, 7
th
Edition by Norman S. Nise. Copyright © 2015. All rights reserved. 19
Summary–Steady State Errors and System Type
Steady State Errors
Input Type 0 Type 1 Type 2
Step
1
1+??????
??????
0 0
Ramp 
1
??????
??????
0
Parabola  
1
??????
??????
Highlights the role of integrators (integral controller) in steady state error elimination

20. Summary
Steady State Errors
Control Systems Engineering, 7
th
Edition by Norman S. Nise. Copyright © 2015. All rights reserved. 20
System Type … Static Error Constant … Steady State Error

21. Error Specifications
Steady State Errors
Control Systems Engineering, 7
th
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** Very Important **
.

22. Error Specifications
Steady State Errors
Control Systems Engineering, 7
th
Edition by Norman S. Nise. Copyright © 2015. All rights reserved. 22
** Very Important **
.

23. Errors Due To Disturbances
Steady State Errors
Control Systems Engineering, 7
th
Edition by Norman S. Nise. Copyright © 2015. All rights reserved. 23
C(s) = E(s)G
1(s)G
2(s) + D(s)G
2(s)
also C(s) = R(s) -E(s)
hence
E(s){1 + G
1(s)G
2(s)} = R(s)+ D(s)G
2(s)

is the transfer function relating E(s) to R(s)
and is the transfer function relating E(s) to D(s)
**G(s) = G
1(s)G
2(s) **

24. Errors Due To Disturbances
So:
Apply the final value theorem to obtain the steady state error:

…
Steady State Errors
Control Systems Engineering, 7
th
Edition by Norman S. Nise. Copyright © 2015. All rights reserved. 24
e
R(s) = error due to R(s) e
D(s) = error due to D(s)
--

25. Errors Due To Disturbances
Example: 7.7
The result shows that
the steady-state error
e
D() produced by the
step disturbance, is
inversely proportional to
the dc gain of G
1(s).
The dc gain of G
2(s)is
infinite in this example.
Steady State Errors
Control Systems Engineering, 7
th
Edition by Norman S. Nise. Copyright © 2015. All rights reserved. 25
.

26. Non-Unity Feedback Systems
Steady State Errors
Control Systems Engineering, 7
th
Edition by Norman S. Nise. Copyright © 2015. All rights reserved. 26
Strategy:
Reduce the non-unity feedback
system to a unity feedback system
… then apply the unity feedback
system steady state error
solutions.
Step 1:
If there is an input transducer
G
1(s), then move it past the
summing point:
where:
G(s)= G
1(s)G
2(s)
H(s)= H
1(s)/G
1(s)

27. Non-Unity Feedback Systems
Steady State Errors
Control Systems Engineering, 7
th
Edition by Norman S. Nise. Copyright © 2015. All rights reserved. 27
Step 2: Expand H(s)
H(s)= H(s)–1 + 1
Step 3: Simplify -H(s)= (H(s)–1) + 1
… then to unity feedback system

28. Non-Unity Feedback Systems
Equivalent unity feedback system and steady state error:
where:
hence:
Summary:
Steady State Errors
Control Systems Engineering, 7
th
Edition by Norman S. Nise. Copyright © 2015. All rights reserved. 28
??????∞=lim
??????→�
????????????(??????)
�+??????
??????(??????)
Input Error Constants Error
Step ??????
??????=lim
??????→0
??????
??????(??????) ??????(∞)=1/(1+??????
??????)
Ramp ??????
??????=lim
??????→0
????????????
??????(??????) ??????(∞)=1/??????
??????
Parabola ??????
??????=lim
??????→0
??????
2
??????
??????(??????) ??????(∞)=1/??????
??????

29. Non-Unity Feedback Systems
Your thoughts …. Is the system Type 1?
Steady State Errors
Control Systems Engineering, 7
th
Edition by Norman S. Nise. Copyright © 2015. All rights reserved. 29

30. Non-Unity Feedback Systems
Example: 7.8
Skills Assessment 7.4
x
x
x
‣x
‣x
x
x
Steady State Errors
Control Systems Engineering, 7
th
Edition by Norman S. Nise. Copyright © 2015. All rights reserved. 30
.
.

31. Non-Unity Feedback Systems
Skills Assessment x
x
x
‣x
‣x
x
x
Steady State Errors
Control Systems Engineering, 7
th
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32. Next Topic
Steady State Errors
Control Systems Engineering, 7
th
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Frequency Response Techniques
Bode Plots
Nise-Chapter 10
10.1, 10.2 and 10.7 (In class -Assignment 2)

33. Next Topic
Steady State Errors
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th
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Bode Plots –Case Study