This document provides an overview of control systems and techniques for manipulating block diagrams that represent control systems. It discusses canonical forms of feedback control systems and characteristics equations. It then provides examples of transforming block diagrams into canonical form and using the canonical forms to determine open loop transfer functions, closed loop transfer functions, and characteristics equations. Examples also demonstrate the use of superposition to determine system outputs from multiple inputs.

1. University of Hormuud
Subject: control systems
Lecturer: Eng. Abdullahi Abubakar
Master of Engineering(Telecommunication)
University Of Malay (UM), Malaysia
Email: Abdullahiabukar99@gmail.com
Tel: 0615169452
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2. Block Diagram Manipulation
(reduction)

3. Canonical form of a feedback control
system
• R= reference input
• C=output
• E=actuating signal or Error
• H= feedback transfer function
• G= direct transfer function or forward transfer function
• H= feedback transfer function
• GH= loop transfer function or open loop transfer function
• C/R= G/1+GH
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4. Characteristics equations
• the control ratio is the closed loop transfer
function of the system
• The denominator of closed loop transfer
function determines the characteristics
equation of the system
• Which is always determined as:
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5. • We often represent control systems using
block diagrams. A block diagram consists of
blocks that
represent transfer functions of the different
variables of interest.
• If a block diagram has many blocks, not all of
which are in cascade, then it is useful to have
rules for rearranging the diagram such that
you end up with only one block.

6. For example, we would want to
transform the following diagram

7. Block Diagram Transformations
1.Combining blocks in cascade
2. combining blocks in parallel

8. 3. Eliminating a
feedback loop

9. Example 1
• Reduce the block diagram to canonical form
• Step 1: combine all cascaded blocks
• Step 2: combine all parallel blocks
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10. Example 1
• Step 3: Eliminate all minor feedback loops
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11. 4. Moving a summing
point behind a block

12. 5. Moving a pickoff point ahead
of a block

13. 6. Moving a pickoff
point behind a block

14. 7. Moving a summing
point ahead of a block

15. 8. replacing summing points

16. 9. Combining summing points

17. Example 2
• For the system represented by the following
block diagram determine:
1. Open loop transfer function
2. Feed forward transfer function
3. Closed loop transfer function
4. Characteristics equation
5. Closed loop poles and zeros if K=10
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18. Example 2
• First we will reduce the given block diagram to
canonical form
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19. Example 2
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20. Example 3
• Reduce the following diagram to canonical
form.
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21. Example 3
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22. Example 3
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23. Example 3
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24. Example 4
• Find the transfer of following diagram
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25. Example 4
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26. Example 5

28. Superposition of multiple inputs
Step 1: set all inputs except one equal to zero
Step 2: transform the block diagram to canonical
Step 3: calculate the response due to the chosen
input acting alone
Step 4: repeat step 1 to 3 for each of the remaining
inputs
Step 5: Algebraically add all of the responses
determined in step 1 to 4. this sum is the total
output of the system with all input acting
simultaneously
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29. Example 6: multiple input system
• Determine the output C due to inputs R and U
using the superposition method.
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30. Example 6
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31. Example 6
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32. Example 7
• Determine the output C due to inputs R, U1
and U2 using the superposition method.
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33. Example 7
• Where C1 is the response due to U1 acting alone.
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34. Example 7
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35. Example 8 multiple input multiple
output system
• Determine C1 and C2 due to the R1 and R2
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36. Example 8
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37. EXAMPLE 8
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