Control system block diagram

Introduction to Block diagram, rules for block diagram reduction, and examples

1. Automatic Control Engineering
Block Diagram Representation
Dr. Nilesh Bhaskarrao Bahadure
[email protected]
https://www.sites.google.com/site/nileshbbahadure/home
June 29, 2021
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 1 / 86

2. Overview
1
Introduction
2
Canonical form of feedback control system
3
Rules for block diagram reduction
Critical rule
4
Problems
5
Analysis of multiple input multiple output systems
Problems
6
Thank You
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 2 / 86

3. What is Block Diagram
A control system may consist of a number of components. In control
engineering to show the functions performed by each component, we
commonly use a diagram called the block diagram. A block diagram of a
system is a pictorial representation of the functions performed by each
component and or the ow of signals. It is a very simple way of
representing the given complicated practical system. Such diagram depicts
the interrelationships that exist among the various components. The
elements of a block diagram are
block,branch pointandsumming
point.
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 3 / 86

4. What is Block Diagram
A control system may consist of a number of components. In control
engineering to show the functions performed by each component, we
commonly use a diagram called the block diagram. A block diagram of a
system is a pictorial representation of the functions performed by each
component and or the ow of signals. It is a very simple way of
representing the given complicated practical system. Such diagram depicts
the interrelationships that exist among the various components. The
elements of a block diagram are
block,branch pointandsumming
point.
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 3 / 86

5. What is Block Diagram
A control system may consist of a number of components. In control
engineering to show the functions performed by each component, we
commonly use a diagram called the block diagram. A block diagram of a
system is a pictorial representation of the functions performed by each
component and or the ow of signals. It is a very simple way of
representing the given complicated practical system. Such diagram depicts
the interrelationships that exist among the various components. The
elements of a block diagram are
block,branch pointandsumming
point.
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 3 / 86

6. What is Block Diagram
A control system may consist of a number of components. In control
engineering to show the functions performed by each component, we
commonly use a diagram called the block diagram. A block diagram of a
system is a pictorial representation of the functions performed by each
component and or the ow of signals. It is a very simple way of
representing the given complicated practical system. Such diagram depicts
the interrelationships that exist among the various components. The
elements of a block diagram are
block,branch pointandsumming
point.
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 3 / 86

7. What is Block Diagram...
To draw the block diagram of a practical system, each element of practical
system is represented by a block. The block is called as functional block. It
means, block explains mathematical operation on the input by the element
to produce the corresponding output. The actual mathematical function is
indicated by inserting corresponding transfer function of the element inside
the block. Figure 1 shows the block diagram of functional block.
Figure :
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 4 / 86

8. What is Block Diagram...
To draw the block diagram of a practical system, each element of practical
system is represented by a block. The block is called as functional block. It
means, block explains mathematical operation on the input by the element
to produce the corresponding output. The actual mathematical function is
indicated by inserting corresponding transfer function of the element inside
the block. Figure 1 shows the block diagram of functional block.
Figure :
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 4 / 86

9. Constructing block diagram for control System
A control system can be represented diagrammatically by block diagram.
The dierential equations governing the system are used to construct the
block diagram. By taking Laplace transform the dierential equations are
converted to algebraic equations. The equations will have variables and
constants. From the working knowledge of the system the input and
output variables are identied and the block diagram for each equation can
be drawn. Each equation gives one section of block diagram. The output
of one section will be input for another section. The various sections are
interconnected to obtain the overall block diagram of the system.
In short any block diagram has the following ve basic elements associated
with it:
1
Blocks
2
Transfer functions of elements shown inside the blocks
3
Summing points
4
Take o points
5
Arrows
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 5 / 86

10. Constructing block diagram for control System
A control system can be represented diagrammatically by block diagram.
The dierential equations governing the system are used to construct the
block diagram. By taking Laplace transform the dierential equations are
converted to algebraic equations. The equations will have variables and
constants. From the working knowledge of the system the input and
output variables are identied and the block diagram for each equation can
be drawn. Each equation gives one section of block diagram. The output
of one section will be input for another section. The various sections are
interconnected to obtain the overall block diagram of the system.
In short any block diagram has the following ve basic elements associated
with it:
1
Blocks
2
Transfer functions of elements shown inside the blocks
3
Summing points
4
Take o points
5
Arrows
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 5 / 86

11. Constructing block diagram for control System
A control system can be represented diagrammatically by block diagram.
The dierential equations governing the system are used to construct the
block diagram. By taking Laplace transform the dierential equations are
converted to algebraic equations. The equations will have variables and
constants. From the working knowledge of the system the input and
output variables are identied and the block diagram for each equation can
be drawn. Each equation gives one section of block diagram. The output
of one section will be input for another section. The various sections are
interconnected to obtain the overall block diagram of the system.
In short any block diagram has the following ve basic elements associated
with it:
1
Blocks
2
Transfer functions of elements shown inside the blocks
3
Summing points
4
Take o points
5
Arrows
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 5 / 86

12. Constructing block diagram for control System
A control system can be represented diagrammatically by block diagram.
The dierential equations governing the system are used to construct the
block diagram. By taking Laplace transform the dierential equations are
converted to algebraic equations. The equations will have variables and
constants. From the working knowledge of the system the input and
output variables are identied and the block diagram for each equation can
be drawn. Each equation gives one section of block diagram. The output
of one section will be input for another section. The various sections are
interconnected to obtain the overall block diagram of the system.
In short any block diagram has the following ve basic elements associated
with it:
1
Blocks
2
Transfer functions of elements shown inside the blocks
3
Summing points
4
Take o points
5
Arrows
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 5 / 86

13. Constructing block diagram for control System
A control system can be represented diagrammatically by block diagram.
The dierential equations governing the system are used to construct the
block diagram. By taking Laplace transform the dierential equations are
converted to algebraic equations. The equations will have variables and
constants. From the working knowledge of the system the input and
output variables are identied and the block diagram for each equation can
be drawn. Each equation gives one section of block diagram. The output
of one section will be input for another section. The various sections are
interconnected to obtain the overall block diagram of the system.
In short any block diagram has the following ve basic elements associated
with it:
1
Blocks
2
Transfer functions of elements shown inside the blocks
3
Summing points
4
Take o points
5
Arrows
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 5 / 86

14. Constructing block diagram for control System
A control system can be represented diagrammatically by block diagram.
The dierential equations governing the system are used to construct the
block diagram. By taking Laplace transform the dierential equations are
converted to algebraic equations. The equations will have variables and
constants. From the working knowledge of the system the input and
output variables are identied and the block diagram for each equation can
be drawn. Each equation gives one section of block diagram. The output
of one section will be input for another section. The various sections are
interconnected to obtain the overall block diagram of the system.
In short any block diagram has the following ve basic elements associated
with it:
1
Blocks
2
Transfer functions of elements shown inside the blocks
3
Summing points
4
Take o points
5
Arrows
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 5 / 86

15. Constructing block diagram for control System
A control system can be represented diagrammatically by block diagram.
The dierential equations governing the system are used to construct the
block diagram. By taking Laplace transform the dierential equations are
converted to algebraic equations. The equations will have variables and
constants. From the working knowledge of the system the input and
output variables are identied and the block diagram for each equation can
be drawn. Each equation gives one section of block diagram. The output
of one section will be input for another section. The various sections are
interconnected to obtain the overall block diagram of the system.
In short any block diagram has the following ve basic elements associated
with it:
1
Blocks
2
Transfer functions of elements shown inside the blocks
3
Summing points
4
Take o points
5
Arrows
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 5 / 86

16. Constructing block diagram for control System..
Figure :
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17. Example
Example For example consider the liquid level system as shown in Figure 3
Figure :
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18. Example
Example For example consider the liquid level system as shown in Figure 3
Figure :
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 7 / 86

19. Example
Example For example consider the liquid level system as shown in Figure 3
Figure :
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 7 / 86

20. Solution
Solution So to represent this by block diagram, identify the elements which are
1
Controller
2
Pneumatic valve
3
Tank
4
Float
Hence indicating them by blocks, the block diagram can be developed as
in Figure 4.
Figure :
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 8 / 86

21. Solution
Solution So to represent this by block diagram, identify the elements which are
1
Controller
2
Pneumatic valve
3
Tank
4
Float
Hence indicating them by blocks, the block diagram can be developed as
in Figure 4.
Figure :
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 8 / 86

22. Solution
Solution So to represent this by block diagram, identify the elements which are
1
Controller
2
Pneumatic valve
3
Tank
4
Float
Hence indicating them by blocks, the block diagram can be developed as
in Figure 4.
Figure :
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 8 / 86

23. Solution
Solution So to represent this by block diagram, identify the elements which are
1
Controller
2
Pneumatic valve
3
Tank
4
Float
Hence indicating them by blocks, the block diagram can be developed as
in Figure 4.
Figure :
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 8 / 86

24. Example
Example
Figure :
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25. Example
Example
Figure :
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 9 / 86

26. Solution
Solution
Figure :
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 10 / 86

27. Solution
Solution
Figure :
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 10 / 86

28. Advantages of block diagram
Advantages of block diagram
1
Very simple to construct the block diagram for complicated systems
2
The function of individual elements can be visualized from block
diagram
3
Individual as well as overall performance of the system can be studied
by using transfer functions shown in the block diagram
4
Overall closed loop TF can be easily calculated by using block
diagram reduction rules
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 11 / 86

29. Advantages of block diagram
Advantages of block diagram
1
Very simple to construct the block diagram for complicated systems
2
The function of individual elements can be visualized from block
diagram
3
Individual as well as overall performance of the system can be studied
by using transfer functions shown in the block diagram
4
Overall closed loop TF can be easily calculated by using block
diagram reduction rules
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 11 / 86

30. Advantages of block diagram
Advantages of block diagram
1
Very simple to construct the block diagram for complicated systems
2
The function of individual elements can be visualized from block
diagram
3
Individual as well as overall performance of the system can be studied
by using transfer functions shown in the block diagram
4
Overall closed loop TF can be easily calculated by using block
diagram reduction rules
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 11 / 86

31. Advantages of block diagram
Advantages of block diagram
1
Very simple to construct the block diagram for complicated systems
2
The function of individual elements can be visualized from block
diagram
3
Individual as well as overall performance of the system can be studied
by using transfer functions shown in the block diagram
4
Overall closed loop TF can be easily calculated by using block
diagram reduction rules
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 11 / 86

32. Disadvantages of block diagram
Disadvantages of block diagram
1
Block diagram does not include any information about the physical
construction of the system
2
Source of energy is generally not shown in the block diagram. So
number of dierent block diagrams can be drawn depending upon the
point of view of analysis. So block diagram for given system is not
unique.
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 12 / 86

33. Disadvantages of block diagram
Disadvantages of block diagram
1
Block diagram does not include any information about the physical
construction of the system
2
Source of energy is generally not shown in the block diagram. So
number of dierent block diagrams can be drawn depending upon the
point of view of analysis. So block diagram for given system is not
unique.
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 12 / 86

34. Videos on LaTeX, CorelDraw and Many More
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 13 / 86

35. Canonical form of feedback/closed loop control system
Figure :A block diagram in which, forward path consists only one block, feedback
path contains only one block, one summing point and one take o point
represents simple or canonical form of a closed loop system. This can be
achieved by using block diagram reduction rules without disturbing output
of the system. This form is very useful as its closed loop transfer function
can be easily calculated by using the standard result.
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 14 / 86

36. Canonical form of feedback/closed loop control system
Figure :A block diagram in which, forward path consists only one block, feedback
path contains only one block, one summing point and one take o point
represents simple or canonical form of a closed loop system. This can be
achieved by using block diagram reduction rules without disturbing output
of the system. This form is very useful as its closed loop transfer function
can be easily calculated by using the standard result.
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 14 / 86

37. Canonical form of feedback/closed loop control system
Figure :A block diagram in which, forward path consists only one block, feedback
path contains only one block, one summing point and one take o point
represents simple or canonical form of a closed loop system. This can be
achieved by using block diagram reduction rules without disturbing output
of the system. This form is very useful as its closed loop transfer function
can be easily calculated by using the standard result.
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 14 / 86

38. Canonical form of feedback/closed loop control system
1
G= direct transfer function = forward transfer function
2
H= feedback transfer function
3
GH= loop transfer function = open loop transfer function
4C
R
= closed loop transfer function = control ration
5E
R
= actuating signal ratio = error ratio
6B
R
= primary feedback ratio
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 15 / 86

39. Canonical form of feedback/closed loop control system
1
G= direct transfer function = forward transfer function
2
H= feedback transfer function
3
GH= loop transfer function = open loop transfer function
4C
R
= closed loop transfer function = control ration
5E
R
= actuating signal ratio = error ratio
6B
R
= primary feedback ratio
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 15 / 86

40. Canonical form of feedback/closed loop control system
1
G= direct transfer function = forward transfer function
2
H= feedback transfer function
3
GH= loop transfer function = open loop transfer function
4C
R
= closed loop transfer function = control ration
5E
R
= actuating signal ratio = error ratio
6B
R
= primary feedback ratio
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 15 / 86

41. Canonical form of feedback/closed loop control system
1
G= direct transfer function = forward transfer function
2
H= feedback transfer function
3
GH= loop transfer function = open loop transfer function
4C
R
= closed loop transfer function = control ration
5E
R
= actuating signal ratio = error ratio
6B
R
= primary feedback ratio
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 15 / 86

42. Canonical form of feedback/closed loop control system
1
G= direct transfer function = forward transfer function
2
H= feedback transfer function
3
GH= loop transfer function = open loop transfer function
4C
R
= closed loop transfer function = control ration
5E
R
= actuating signal ratio = error ratio
6B
R
= primary feedback ratio
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 15 / 86

43. Canonical form of feedback/closed loop control system
1
G= direct transfer function = forward transfer function
2
H= feedback transfer function
3
GH= loop transfer function = open loop transfer function
4C
R
= closed loop transfer function = control ration
5E
R
= actuating signal ratio = error ratio
6B
R
= primary feedback ratio
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 15 / 86

44. Derivation of T.F. of closed loop system
Refereeing to Figure 7 we can write following equations,
E=RB (1)
B=CH (2)
C=EG (3)
by substitutingB=CHin equation 1
E=RCH (4)
by substituting Equation 4 in equation 3
C= (RCH)G (5)
C= (RGCGH) =C+CHG=RG (6)
C(1 +GH) =RG (7)
C
R
=
G
(1 +GH)
This is for negative feedback (8)
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 16 / 86

45. Derivation of T.F. of closed loop system
C
R
=
G
(1GH)
(9)
+ sign!This is for negative feedback
- sign!This is for positive feedback
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 17 / 86

46. Derivation of T.F. of closed loop system
C
R
=
G
(1GH)
(9)
+ sign!This is for negative feedback
- sign!This is for positive feedback
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 17 / 86

47. Derivation of T.F. of closed loop system
C
R
=
G
(1GH)
(9)
+ sign!This is for negative feedback
- sign!This is for positive feedback
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 17 / 86

48. Rules for block diagram reduction
Rule:01 - Combining block in cascade/series The transfer functions of the blocks which are connected in series or
cascade get multiplied with each other.Figure :
Solution
Figure :
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 18 / 86

49. Rules for block diagram reduction
Rule:01 - Combining block in cascade/series The transfer functions of the blocks which are connected in series or
cascade get multiplied with each other.Figure :
Solution
Figure :
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 18 / 86

50. Rules for block diagram reduction
Rule:01 - Combining block in cascade/series The transfer functions of the blocks which are connected in series or
cascade get multiplied with each other.Figure :
Solution
Figure :
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 18 / 86

51. Rules for block diagram reduction
Rule:01 - Combining block in cascade/series The transfer functions of the blocks which are connected in series or
cascade get multiplied with each other.Figure :
Solution
Figure :
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52. Rule:02 - Combining block in Parallel The transfer functions of the blocks which are connected in parallel get
added algebraically.Figure :
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 19 / 86

53. Rule:02 - Combining block in Parallel The transfer functions of the blocks which are connected in parallel get
added algebraically.Figure :
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 19 / 86

54. Rule:02 - Combining block in Parallel The transfer functions of the blocks which are connected in parallel get
added algebraically.Figure :
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 19 / 86

55. Solution
Figure :
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 20 / 86

56. Solution
Figure :
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 20 / 86

57. Rule:03 - Shifting summing point before of the block
Figure :
C(s) =RG+y (10)now we have to shift summing point ahead of the block, and should be
veried for the output and output should remain same
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 21 / 86

58. Rule:03 - Shifting summing point before of the block
Figure :
C(s) =RG+y (10)now we have to shift summing point ahead of the block, and should be
veried for the output and output should remain same
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 21 / 86

59. Rule:03 - Shifting summing point before of the block
Figure :
C(s) =RG+y (10)now we have to shift summing point ahead of the block, and should be
veried for the output and output should remain same
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 21 / 86

60. Solution
Figure :
(R+x)G=C(s) (11)(R+x)G=RG+y (12)y=xG (13)Therefore,x=
1
G
yso signalxshould be replaced with
1
G
to keep output
same
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 22 / 86

61. Solution
Figure :
(R+x)G=C(s) (11)(R+x)G=RG+y (12)y=xG (13)Therefore,x=
1
G
yso signalxshould be replaced with
1
G
to keep output
same
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 22 / 86

62. Solution
Figure :
(R+x)G=C(s) (11)(R+x)G=RG+y (12)y=xG (13)Therefore,x=
1
G
yso signalxshould be replaced with
1
G
to keep output
same
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 22 / 86

63. Solution
Figure :
(R+x)G=C(s) (11)(R+x)G=RG+y (12)y=xG (13)Therefore,x=
1
G
yso signalxshould be replaced with
1
G
to keep output
same
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 22 / 86

64. Solution
Figure :
(R+x)G=C(s) (11)(R+x)G=RG+y (12)y=xG (13)Therefore,x=
1
G
yso signalxshould be replaced with
1
G
to keep output
same
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 22 / 86

65. Solution
Figure :
Thus while shifting a summing point behind the block i.e. before the
block, add a block having transfer function as reciprocal of the transfer
function of the block before which summing point is to be shifted, in series
with all the signals at that summing point
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 23 / 86

66. Solution
Figure :
Thus while shifting a summing point behind the block i.e. before the
block, add a block having transfer function as reciprocal of the transfer
function of the block before which summing point is to be shifted, in series
with all the signals at that summing point
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 23 / 86

67. Solution
Figure :
Thus while shifting a summing point behind the block i.e. before the
block, add a block having transfer function as reciprocal of the transfer
function of the block before which summing point is to be shifted, in series
with all the signals at that summing point
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 23 / 86

68. Rule:04 - Shifting summing point ahead/beyond the block
Figure :
C(s) = (R+y)G (14)now we have to shift summing point after the block, and should be veried
for the output and output should remain same
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 24 / 86

69. Rule:04 - Shifting summing point ahead/beyond the block
Figure :
C(s) = (R+y)G (14)now we have to shift summing point after the block, and should be veried
for the output and output should remain same
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 24 / 86

70. Rule:04 - Shifting summing point ahead/beyond the block
Figure :
C(s) = (R+y)G (14)now we have to shift summing point after the block, and should be veried
for the output and output should remain same
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 24 / 86

71. Solution
Figure :
C(s) =RG+x (15)RG+x=RG+yG (16)x=yG (17)i.e. signalymust get multiplied with TF of block beyond which summing
point is to be shifted
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 25 / 86

72. Solution
Figure :
C(s) =RG+x (15)RG+x=RG+yG (16)x=yG (17)i.e. signalymust get multiplied with TF of block beyond which summing
point is to be shifted
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 25 / 86

73. Solution
Figure :
C(s) =RG+x (15)RG+x=RG+yG (16)x=yG (17)i.e. signalymust get multiplied with TF of block beyond which summing
point is to be shifted
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 25 / 86

74. Solution
Figure :
C(s) =RG+x (15)RG+x=RG+yG (16)x=yG (17)i.e. signalymust get multiplied with TF of block beyond which summing
point is to be shifted
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 25 / 86

75. Solution
Figure :
C(s) =RG+x (15)RG+x=RG+yG (16)x=yG (17)i.e. signalymust get multiplied with TF of block beyond which summing
point is to be shifted
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 25 / 86

76. Solution
Figure :
Thus while shifting a summing point after the block, add a block having
transfer function same as that of block after which summing point is to be
shifted, in series with all the signals at that summing point
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 26 / 86

77. Solution
Figure :
Thus while shifting a summing point after the block, add a block having
transfer function same as that of block after which summing point is to be
shifted, in series with all the signals at that summing point
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 26 / 86

78. Solution
Figure :
Thus while shifting a summing point after the block, add a block having
transfer function same as that of block after which summing point is to be
shifted, in series with all the signals at that summing point
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 26 / 86

79. Rule:05 - Shifting a take o/branch point before/behind the block
Figure :
C=RG;Y=RG (18)To shift take o point behind block, value of signal taking o must remain
same.Through shifting of take o point without any change does not aect
output directly, the value of feedback signal which is changed aects the
output indirectly which must be kept same. But without any change it is
just R as shown in Figure 19
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 27 / 86

80. Rule:05 - Shifting a take o/branch point before/behind the block
Figure :
C=RG;Y=RG (18)To shift take o point behind block, value of signal taking o must remain
same.Through shifting of take o point without any change does not aect
output directly, the value of feedback signal which is changed aects the
output indirectly which must be kept same. But without any change it is
just R as shown in Figure 19
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 27 / 86

81. Rule:05 - Shifting a take o/branch point before/behind the block
Figure :
C=RG;Y=RG (18)To shift take o point behind block, value of signal taking o must remain
same.Through shifting of take o point without any change does not aect
output directly, the value of feedback signal which is changed aects the
output indirectly which must be kept same. But without any change it is
just R as shown in Figure 19
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 27 / 86

82. Rule:05 - Shifting a take o/branch point before/behind the block
Figure :
C=RG;Y=RG (18)To shift take o point behind block, value of signal taking o must remain
same.Through shifting of take o point without any change does not aect
output directly, the value of feedback signal which is changed aects the
output indirectly which must be kept same. But without any change it is
just R as shown in Figure 19
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 27 / 86

83. Rule:05 - Shifting a take o/branch point before/behind the block
Figure :
C=RG;Y=RG (18)To shift take o point behind block, value of signal taking o must remain
same.Through shifting of take o point without any change does not aect
output directly, the value of feedback signal which is changed aects the
output indirectly which must be kept same. But without any change it is
just R as shown in Figure 19
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 27 / 86

84. Solution
Figure :
C=RG;Y=R (19)But it must be equal to RG. So a block with TFGmust be introduced i.e.
signal taking o after the block must be multiplied with TF of that block
while shifting behind the block.
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 28 / 86

85. Solution
Figure :
C=RG;Y=R (19)But it must be equal to RG. So a block with TFGmust be introduced i.e.
signal taking o after the block must be multiplied with TF of that block
while shifting behind the block.
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 28 / 86

86. Solution
Figure :
C=RG;Y=R (19)But it must be equal to RG. So a block with TFGmust be introduced i.e.
signal taking o after the block must be multiplied with TF of that block
while shifting behind the block.
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 28 / 86

87. Solution
Figure :
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 29 / 86

88. Solution
Figure :
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 29 / 86

89. Rule:06 - Shifting a take o/branch point beyond/ahead the block
Figure :
To shift take o point beyond the block, value ofYmust remain same.
To keep value ofYconstant it must be multiplied by
1
G
. While shifting a
take o point beyond the block, add a block in series with all the signals
which are taking o from that point, having TF as reciprocal of the TF of
the block beyond which take o point is to be shifted.
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 30 / 86

90. Rule:06 - Shifting a take o/branch point beyond/ahead the block
Figure :
To shift take o point beyond the block, value ofYmust remain same.
To keep value ofYconstant it must be multiplied by
1
G
. While shifting a
take o point beyond the block, add a block in series with all the signals
which are taking o from that point, having TF as reciprocal of the TF of
the block beyond which take o point is to be shifted.
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 30 / 86

91. Rule:06 - Shifting a take o/branch point beyond/ahead the block
Figure :
To shift take o point beyond the block, value ofYmust remain same.
To keep value ofYconstant it must be multiplied by
1
G
. While shifting a
take o point beyond the block, add a block in series with all the signals
which are taking o from that point, having TF as reciprocal of the TF of
the block beyond which take o point is to be shifted.
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 30 / 86

92. Rule:07 - Interchanging summing point
Figure :
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 31 / 86

93. Rule:07 - Interchanging summing point
Figure :
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 31 / 86

94. Rule:08 - Splitting summing point
Figure :
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 32 / 86

95. Rule:08 - Splitting summing point
Figure :
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 32 / 86

96. Rule:09 - Combining summing point
Figure :
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 33 / 86

97. Rule:09 - Combining summing point
Figure :
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 33 / 86

98. Rule:10 - Elimination of negative feedback loop
Figure :
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 34 / 86

99. Rule:10 - Elimination of negative feedback loop
Figure :
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 34 / 86

100. Rule:11 - Elimination of positive feedback loop
Figure :
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 35 / 86

101. Rule:11 - Elimination of positive feedback loop
Figure :
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 35 / 86

102. Rule:12 - For multiple input system use superposition theorem
Figure :
Consider only one input at a time treating all other as zero.
ConsiderR1,R2=R3=:Rn= 0 and nd outputC1,Then considerR2,R1=R3=:Rn= 0 and nd outputC2At the end when all inputs are covered take algebraic sum of all the outputs.Total outputC=C1+C2+CnSame logic can be extended to nd the outputs if system is multiple input multiple
output type. Separate ratio of each output with each input is to be calculated, assuming
all other input and outputs zero. Then such components of the outputs can be added to
get resultant outputs of the system.
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 36 / 86

103. Rule:12 - For multiple input system use superposition theorem
Figure :
Consider only one input at a time treating all other as zero.
ConsiderR1,R2=R3=:Rn= 0 and nd outputC1,Then considerR2,R1=R3=:Rn= 0 and nd outputC2At the end when all inputs are covered take algebraic sum of all the outputs.Total outputC=C1+C2+CnSame logic can be extended to nd the outputs if system is multiple input multiple
output type. Separate ratio of each output with each input is to be calculated, assuming
all other input and outputs zero. Then such components of the outputs can be added to
get resultant outputs of the system.
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 36 / 86

104. Rule:12 - For multiple input system use superposition theorem
Figure :
Consider only one input at a time treating all other as zero.
ConsiderR1,R2=R3=:Rn= 0 and nd outputC1,Then considerR2,R1=R3=:Rn= 0 and nd outputC2At the end when all inputs are covered take algebraic sum of all the outputs.Total outputC=C1+C2+CnSame logic can be extended to nd the outputs if system is multiple input multiple
output type. Separate ratio of each output with each input is to be calculated, assuming
all other input and outputs zero. Then such components of the outputs can be added to
get resultant outputs of the system.
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 36 / 86

105. Rule:12 - For multiple input system use superposition theorem
Figure :
Consider only one input at a time treating all other as zero.
ConsiderR1,R2=R3=:Rn= 0 and nd outputC1,Then considerR2,R1=R3=:Rn= 0 and nd outputC2At the end when all inputs are covered take algebraic sum of all the outputs.Total outputC=C1+C2+CnSame logic can be extended to nd the outputs if system is multiple input multiple
output type. Separate ratio of each output with each input is to be calculated, assuming
all other input and outputs zero. Then such components of the outputs can be added to
get resultant outputs of the system.
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 36 / 86

106. Rule:12 - For multiple input system use superposition theorem
Figure :
Consider only one input at a time treating all other as zero.
ConsiderR1,R2=R3=:Rn= 0 and nd outputC1,Then considerR2,R1=R3=:Rn= 0 and nd outputC2At the end when all inputs are covered take algebraic sum of all the outputs.Total outputC=C1+C2+CnSame logic can be extended to nd the outputs if system is multiple input multiple
output type. Separate ratio of each output with each input is to be calculated, assuming
all other input and outputs zero. Then such components of the outputs can be added to
get resultant outputs of the system.
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 36 / 86

107. Rule:12 - For multiple input system use superposition theorem
Figure :
Consider only one input at a time treating all other as zero.
ConsiderR1,R2=R3=:Rn= 0 and nd outputC1,Then considerR2,R1=R3=:Rn= 0 and nd outputC2At the end when all inputs are covered take algebraic sum of all the outputs.Total outputC=C1+C2+CnSame logic can be extended to nd the outputs if system is multiple input multiple
output type. Separate ratio of each output with each input is to be calculated, assuming
all other input and outputs zero. Then such components of the outputs can be added to
get resultant outputs of the system.
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 36 / 86

108. Rule:12 - For multiple input system use superposition theorem
Figure :
Consider only one input at a time treating all other as zero.
ConsiderR1,R2=R3=:Rn= 0 and nd outputC1,Then considerR2,R1=R3=:Rn= 0 and nd outputC2At the end when all inputs are covered take algebraic sum of all the outputs.Total outputC=C1+C2+CnSame logic can be extended to nd the outputs if system is multiple input multiple
output type. Separate ratio of each output with each input is to be calculated, assuming
all other input and outputs zero. Then such components of the outputs can be added to
get resultant outputs of the system.
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 36 / 86

109. Rule:12 - For multiple input system use superposition theorem
Figure :
Consider only one input at a time treating all other as zero.
ConsiderR1,R2=R3=:Rn= 0 and nd outputC1,Then considerR2,R1=R3=:Rn= 0 and nd outputC2At the end when all inputs are covered take algebraic sum of all the outputs.Total outputC=C1+C2+CnSame logic can be extended to nd the outputs if system is multiple input multiple
output type. Separate ratio of each output with each input is to be calculated, assuming
all other input and outputs zero. Then such components of the outputs can be added to
get resultant outputs of the system.
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 36 / 86

110. Rule:13 - Shifting take o point after summing point
Figure :
Now after shifting the take o point, let signal taking o be 'z' as shown
in Figure 28.
Nowz=R1yBut we want feedback signal asx=R1only.So signalymust be inverted and added toC1to keep feedback signal
value same. And to add the signal, summing point must be introduced in
series with takeo signal. So modied conguration becomes as shown in
Figure 29.
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 37 / 86

111. Rule:13 - Shifting take o point after summing point
Figure :
Now after shifting the take o point, let signal taking o be 'z' as shown
in Figure 28.
Nowz=R1yBut we want feedback signal asx=R1only.So signalymust be inverted and added toC1to keep feedback signal
value same. And to add the signal, summing point must be introduced in
series with takeo signal. So modied conguration becomes as shown in
Figure 29.
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 37 / 86

112. Rule:13 - Shifting take o point after summing point
Figure :
Now after shifting the take o point, let signal taking o be 'z' as shown
in Figure 28.
Nowz=R1yBut we want feedback signal asx=R1only.So signalymust be inverted and added toC1to keep feedback signal
value same. And to add the signal, summing point must be introduced in
series with takeo signal. So modied conguration becomes as shown in
Figure 29.
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 37 / 86

113. Rule:13 - Shifting take o point after summing point
Figure :
Now after shifting the take o point, let signal taking o be 'z' as shown
in Figure 28.
Nowz=R1yBut we want feedback signal asx=R1only.So signalymust be inverted and added toC1to keep feedback signal
value same. And to add the signal, summing point must be introduced in
series with takeo signal. So modied conguration becomes as shown in
Figure 29.
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 37 / 86

114. Rule:13 - Shifting take o point after summing point
Figure :
Now after shifting the take o point, let signal taking o be 'z' as shown
in Figure 28.
Nowz=R1yBut we want feedback signal asx=R1only.So signalymust be inverted and added toC1to keep feedback signal
value same. And to add the signal, summing point must be introduced in
series with takeo signal. So modied conguration becomes as shown in
Figure 29.
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 37 / 86

115. Rule:13 - Shifting take o point after summing point
Figure :
Now after shifting the take o point, let signal taking o be 'z' as shown
in Figure 28.
Nowz=R1yBut we want feedback signal asx=R1only.So signalymust be inverted and added toC1to keep feedback signal
value same. And to add the signal, summing point must be introduced in
series with takeo signal. So modied conguration becomes as shown in
Figure 29.
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 37 / 86

116. Rule:13 - Shifting take o point after summing point
Figure :
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 38 / 86

117. Rule:13 - Shifting take o point after summing point
Figure :
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118. Rule:14 - Shifting take o point before summing point
Figure :
Now after shifting the take o point, let signal taking o be 'z' as shown
in Figure 30.
Nowz=R1only because nothing is changed.
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 39 / 86

119. Rule:14 - Shifting take o point before summing point
Figure :
Now after shifting the take o point, let signal taking o be 'z' as shown
in Figure 30.
Nowz=R1only because nothing is changed.
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 39 / 86

120. Rule:14 - Shifting take o point before summing point
Figure :
Now after shifting the take o point, let signal taking o be 'z' as shown
in Figure 30.
Nowz=R1only because nothing is changed.
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 39 / 86

121. Rule:14 - Shifting take o point before summing point
Figure :
Now after shifting the take o point, let signal taking o be 'z' as shown
in Figure 30.
Nowz=R1only because nothing is changed.
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 39 / 86

122. Rule:14 - Shifting take o point after summing point
Figure :
But we want feedback signal x again which isR1y. hence toz, signaly
must be added with same sign as it is present at summing point which can
be achieved by using summing point in series with takeo signal as shown
in Figure 31.
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 40 / 86

123. Rule:14 - Shifting take o point after summing point
Figure :
But we want feedback signal x again which isR1y. hence toz, signaly
must be added with same sign as it is present at summing point which can
be achieved by using summing point in series with takeo signal as shown
in Figure 31.
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 40 / 86

124. Rule:14 - Shifting take o point after summing point
Figure :
But we want feedback signal x again which isR1y. hence toz, signaly
must be added with same sign as it is present at summing point which can
be achieved by using summing point in series with takeo signal as shown
in Figure 31.
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 40 / 86

125. Procedure to solve block diagram reduction problems
1
Step 1: Reduce the blocks connected in series
2
Step 2: Reduce the blocks connected in parallel
3
Step 3: Reduce the minor internal feedback loops
4
Step 4: As far as possible try to shift take o point towads right and
summing
5
points to the left. Unless and until it is the requirement of problem do
not use rule 13 and 14.
6
Step 5: Repeat step 1 to 4 till simple form is obtained
7
Step 6: Using standard TF of simple closed loop system obtain the
closed loop TF
C(S)
R(S)
of the overall system
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 41 / 86

126. Procedure to solve block diagram reduction problems
1
Step 1: Reduce the blocks connected in series
2
Step 2: Reduce the blocks connected in parallel
3
Step 3: Reduce the minor internal feedback loops
4
Step 4: As far as possible try to shift take o point towads right and
summing
5
points to the left. Unless and until it is the requirement of problem do
not use rule 13 and 14.
6
Step 5: Repeat step 1 to 4 till simple form is obtained
7
Step 6: Using standard TF of simple closed loop system obtain the
closed loop TF
C(S)
R(S)
of the overall system
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 41 / 86

127. Procedure to solve block diagram reduction problems
1
Step 1: Reduce the blocks connected in series
2
Step 2: Reduce the blocks connected in parallel
3
Step 3: Reduce the minor internal feedback loops
4
Step 4: As far as possible try to shift take o point towads right and
summing
5
points to the left. Unless and until it is the requirement of problem do
not use rule 13 and 14.
6
Step 5: Repeat step 1 to 4 till simple form is obtained
7
Step 6: Using standard TF of simple closed loop system obtain the
closed loop TF
C(S)
R(S)
of the overall system
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 41 / 86

128. Procedure to solve block diagram reduction problems
1
Step 1: Reduce the blocks connected in series
2
Step 2: Reduce the blocks connected in parallel
3
Step 3: Reduce the minor internal feedback loops
4
Step 4: As far as possible try to shift take o point towads right and
summing
5
points to the left. Unless and until it is the requirement of problem do
not use rule 13 and 14.
6
Step 5: Repeat step 1 to 4 till simple form is obtained
7
Step 6: Using standard TF of simple closed loop system obtain the
closed loop TF
C(S)
R(S)
of the overall system
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 41 / 86

129. Procedure to solve block diagram reduction problems
1
Step 1: Reduce the blocks connected in series
2
Step 2: Reduce the blocks connected in parallel
3
Step 3: Reduce the minor internal feedback loops
4
Step 4: As far as possible try to shift take o point towads right and
summing
5
points to the left. Unless and until it is the requirement of problem do
not use rule 13 and 14.
6
Step 5: Repeat step 1 to 4 till simple form is obtained
7
Step 6: Using standard TF of simple closed loop system obtain the
closed loop TF
C(S)
R(S)
of the overall system
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 41 / 86

130. Procedure to solve block diagram reduction problems
1
Step 1: Reduce the blocks connected in series
2
Step 2: Reduce the blocks connected in parallel
3
Step 3: Reduce the minor internal feedback loops
4
Step 4: As far as possible try to shift take o point towads right and
summing
5
points to the left. Unless and until it is the requirement of problem do
not use rule 13 and 14.
6
Step 5: Repeat step 1 to 4 till simple form is obtained
7
Step 6: Using standard TF of simple closed loop system obtain the
closed loop TF
C(S)
R(S)
of the overall system
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 41 / 86

131. Procedure to solve block diagram reduction problems
1
Step 1: Reduce the blocks connected in series
2
Step 2: Reduce the blocks connected in parallel
3
Step 3: Reduce the minor internal feedback loops
4
Step 4: As far as possible try to shift take o point towads right and
summing
5
points to the left. Unless and until it is the requirement of problem do
not use rule 13 and 14.
6
Step 5: Repeat step 1 to 4 till simple form is obtained
7
Step 6: Using standard TF of simple closed loop system obtain the
closed loop TF
C(S)
R(S)
of the overall system
Dr. Nilesh Bhaskarrao Bahadure (Ph.D.) Automatic Control Engineering June 29, 2021 41 / 86

132. Example
Reduce the given block diagram to its canonical (simple) form and hence
obtain the equivalent transfer function
C(s)
R(s)
Figure :
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133. Example
Reduce the given block diagram to its canonical (simple) form and hence
obtain the equivalent transfer function
C(s)
R(s)
Figure :
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134. Solution
Figure :C(s)
R(s)
=
G1G2(G3+G4)
1+G1G2H1G1G2(G3+G4)H2
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135. Solution
Figure :C(s)
R(s)
=
G1G2(G3+G4)
1+G1G2H1G1G2(G3+G4)H2
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136. Solution
Figure :C(s)
R(s)
=
G1G2(G3+G4)
1+G1G2H1G1G2(G3+G4)H2
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137. Example
Reduce the given block diagram to its canonical (simple) form and hence
obtain the equivalent transfer function
C(s)
R(s)
Figure :
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138. Example
Reduce the given block diagram to its canonical (simple) form and hence
obtain the equivalent transfer function
C(s)
R(s)
Figure :
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139. Solution
Figure :C(s)
R(s)
=
G1G2
1+G1G2+G2H2+G1H1+G1G2H1H2
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140. Solution
Figure :C(s)
R(s)
=
G1G2
1+G1G2+G2H2+G1H1+G1G2H1H2
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141. Solution
Figure :C(s)
R(s)
=
G1G2
1+G1G2+G2H2+G1H1+G1G2H1H2
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142. Example
Reduce the given block diagram to its canonical (simple) form and hence
obtain the equivalent transfer function
C(s)
R(s)
Figure :
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143. Example
Reduce the given block diagram to its canonical (simple) form and hence
obtain the equivalent transfer function
C(s)
R(s)
Figure :
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144. Solution
Figure :C(s)
R(s)
=
G1G2G3
1+G1H1+G1G2H2+G1G2G3H3
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145. Solution
Figure :C(s)
R(s)
=
G1G2G3
1+G1H1+G1G2H2+G1G2G3H3
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146. Solution
Figure :C(s)
R(s)
=
G1G2G3
1+G1H1+G1G2H2+G1G2G3H3
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147. Example
Reduce the given block diagram to its canonical (simple) form and hence
obtain the equivalent transfer function
C(s)
R(s)
Figure :
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148. Example
Reduce the given block diagram to its canonical (simple) form and hence
obtain the equivalent transfer function
C(s)
R(s)
Figure :
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149. Solution
Figure :C(s)
R(s)
=
G1G2G3G4
1+G3G4H3+G2G3H2+G1G2G3G4H1
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150. Solution
Figure :C(s)
R(s)
=
G1G2G3G4
1+G3G4H3+G2G3H2+G1G2G3G4H1
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151. Solution
Figure :C(s)
R(s)
=
G1G2G3G4
1+G3G4H3+G2G3H2+G1G2G3G4H1
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152. Example
Reduce the given block diagram to its canonical (simple) form and hence
obtain the equivalent transfer function
C(s)
R(s)
Figure :
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153. Example
Reduce the given block diagram to its canonical (simple) form and hence
obtain the equivalent transfer function
C(s)
R(s)
Figure :
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154. Solution
Figure :C(s)
R(s)
=
G1G2(G3G4+G5)
1+G2G3H1+G1G2(G3G4+G5)H2
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155. Solution
Figure :C(s)
R(s)
=
G1G2(G3G4+G5)
1+G2G3H1+G1G2(G3G4+G5)H2
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156. Solution
Figure :C(s)
R(s)
=
G1G2(G3G4+G5)
1+G2G3H1+G1G2(G3G4+G5)H2
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157. Example
Reduce the given block diagram to its canonical (simple) form and hence
obtain the equivalent transfer function
C(s)
R(s)
Figure :
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158. Example
Reduce the given block diagram to its canonical (simple) form and hence
obtain the equivalent transfer function
C(s)
R(s)
Figure :
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159. Solution
Figure :C(s)
R(s)
=
G1G4(G2+G3)
1+G1G2H2+G4H1+G1G2G4H1H2+G1G4(G2+G3)
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160. Solution
Figure :C(s)
R(s)
=
G1G4(G2+G3)
1+G1G2H2+G4H1+G1G2G4H1H2+G1G4(G2+G3)
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161. Solution
Figure :C(s)
R(s)
=
G1G4(G2+G3)
1+G1G2H2+G4H1+G1G2G4H1H2+G1G4(G2+G3)
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162. Problems
Figure :
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163. Problems
Figure :
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164. Problems
Figure :
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165. Problems
Figure :
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166. Problems
Figure :
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167. Problems
Figure :
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168. Problems
Figure :
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169. Problems
Figure :
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170. Problems
Figure :
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171. Problems
Figure :
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172. Problems
Figure :
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173. Problems
Figure :
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174. Problems
Figure :
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175. Problems
Figure :
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176. Problems
Figure :
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177. Problems
Figure :
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178. Problems
Figure :
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179. Problems
Figure :
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180. Problems
Figure :
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181. Problems
Figure :
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182. Problems
Figure :
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183. Problems
Figure :
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184. Problems
Figure :
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185. Problems
Figure :
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186. Analysis of multiple input multiple output systems
In these problems, the law of superposition is to be used, considering each
input separately. While assuming the other inputs as zero, most of the
times if only input is applied to the summing point, summing point is to
be removed if not necessary. While removing summing point if sign of the
signal present at that summing point which is to be removed is negative
must be carried forward in the further analysis.This can be achieved by
introducing a block of transfer function 1 in series with that signal.
This is the important step to be remembered while solving problems on
multiple input multiple output systems.
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187. Analysis of multiple input multiple output systems
In these problems, the law of superposition is to be used, considering each
input separately. While assuming the other inputs as zero, most of the
times if only input is applied to the summing point, summing point is to
be removed if not necessary. While removing summing point if sign of the
signal present at that summing point which is to be removed is negative
must be carried forward in the further analysis.This can be achieved by
introducing a block of transfer function 1 in series with that signal.
This is the important step to be remembered while solving problems on
multiple input multiple output systems.
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188. Analysis of multiple input multiple output systems
WhenR(s) is considered alone,Y(s) must be assumed zero and summing
point atY(s) can be removed as withY(s) = 0 there remains only a
single signal present at that point so system gets modied as shown in
gure (2nd)
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189. Analysis of multiple input multiple output systems
WhenR(s) is considered alone,Y(s) must be assumed zero and summing
point atY(s) can be removed as withY(s) = 0 there remains only a
single signal present at that point so system gets modied as shown in
gure (2nd)
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190. Analysis of multiple input multiple output systems
Figure :
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191. Analysis of multiple input multiple output systems
Figure :
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192. Analysis of multiple input multiple output systems
Now whenR(s) = 0 withY(s) active, the summing point atR(s) also
can be removed. But now sign of the signalxat that summing point is
negative which must be considered and carried forward for further analysis.
This is possible by adding a block of1 in series withxwithout altering
any other sign. This avoids the confusion and problem can be solved
without any error.
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193. Analysis of multiple input multiple output systems
Now whenR(s) = 0 withY(s) active, the summing point atR(s) also
can be removed. But now sign of the signalxat that summing point is
negative which must be considered and carried forward for further analysis.
This is possible by adding a block of1 in series withxwithout altering
any other sign. This avoids the confusion and problem can be solved
without any error.
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194. Problems
Obtain the resultant outputC(s) in terms of the inputsR(s) andY(s).
Figure :
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195. Problems
Figure :
C(s) =
G1G2R(s) +G2Y(s)
1 +G1G2H1
(20)
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196. Problems
Obtain the expression forC1andC2for the given multiple input multiple
output system.
Figure :
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197. Problems
Figure :
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198. Obtain the output in terms of input for the system shown in Figure 73.
Figure :
C(s) =
G1G2R(s) +G2U(s)
1 +G1G2H1H2
(21)
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199. Thank you
Please send your feedback at [email protected]
For download and more information
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200. Thank you
Please send your feedback at [email protected]
For download and more information
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