1. Feedback Control Systems (FCS)
Lecture-8-9
Block Diagram Representation of Control Systems
Dr. Imtiaz Hussain
email: imtiaz.hussain@faculty.muet.edu.pk
URL :http://imtiazhussainkalwar.weebly.com/

2. Introduction
• A Block Diagram is a shorthand pictorial representation of
the cause-and-effect relationship of a system.
• The interior of the rectangle representing the block usually
contains a description of or the name of the element, gain,
or the symbol for the mathematical operation to be
performed on the input to yield the output.
• The arrows represent the direction of information or signal
flow.
x
d
dt
y

3. Introduction
• The operations of addition and subtraction have a special
representation.
• The block becomes a small circle, called a summing point, with
the appropriate plus or minus sign associated with the arrows
entering the circle.
• The output is the algebraic sum of the inputs.
• Any number of inputs may enter a summing point.
• Some books put a cross in the circle.

4. Introduction
• In order to have the same signal or variable be an input
to more than one block or summing point, a takeoff (or
pickoff) point is used.
• This permits the signal to proceed unaltered along
several different paths to several destinations.

5. Example-1
• Consider the following equations in which 𝑥1 , 𝑥2 , 𝑥3 , are
variables, and 𝑎1 , 𝑎2 are general coefficients or mathematical
operators.
x3  a1 x1  a2 x2  5

6. Example-1
x3  a1 x1  a2 x2  5

7. Example-2
• Draw the Block Diagrams of the following equations.
(1)
( 2)
dx1 1
x2  a1
  x1dt
dt
b
x3  a1
d 2 x2
dt 2
dx1
3
 bx1
dt

8. Canonical Form of A Feedback Control System

9. Characteristic Equation
• The control ratio is the closed loop transfer function of the system.
C( s )
G( s )

R( s ) 1  G( s ) H ( s )
• The denominator of closed loop transfer function determines the
characteristic equation of the system.
• Which is usually determined as:
1  G( s )H ( s )  0

10. Example-3
1. Open loop transfer function
B( s )
 G( s ) H ( s )
E( s )
2. Feed Forward Transfer function
C( s )
G( s )

R( s ) 1  G( s ) H ( s )
3. control ratio
4. feedback ratio
5. error ratio
C (s)
 G (s)
E (s)
G(s )
B( s )
G( s ) H ( s )

R( s ) 1  G ( s ) H ( s )
E( s )
1

R( s ) 1  G( s ) H ( s )
6. closed loop transfer function
H (s )
C( s )
G( s )

R( s ) 1  G( s ) H ( s )
7. characteristic equation 1  G( s )H ( s )  0
8. Open loop poles and zeros if 9. closed loop poles and zeros if K=10.

11. Reduction techniques
1. Combining blocks in cascade
G2
G1
G1G2
2. Combining blocks in parallel
G1
G2
G1  G2

12. 3. Eliminating a feedback loop
G
G
1  GH
H
G
H 1
G
1 G

13. Example-4: Reduce the Block Diagram to Canonical Form.

14. Example-4: Continue.

15. Example-5
• For the system represented by the following block diagram
determine:
1.
2.
3.
4.
5.
6.
7.
8.
Open loop transfer function
Feed Forward Transfer function
control ratio
feedback ratio
error ratio
closed loop transfer function
characteristic equation
closed loop poles and zeros if K=10.

16. Example-5
– First we will reduce the given block diagram to canonical form
K
s 1

17. Example-5
K
s 1
K
G
 s 1
K
1  GH
1
s
s 1

18. Example-5 (see example-3)
1. Open loop transfer function
B( s )
 G( s ) H ( s )
E( s )
2. Feed Forward Transfer function
C( s )
 G( s )
E( s )
C( s )
G( s )
3. control ratio

R( s ) 1  G( s ) H ( s )
4. feedback ratio
5. error ratio
G(s )
B( s )
G( s ) H ( s )

R( s ) 1  G ( s ) H ( s )
E( s )
1

R( s ) 1  G( s ) H ( s )
6. closed loop transfer function
C( s )
G( s )

R( s ) 1  G( s ) H ( s )
7. characteristic equation 1  G( s )H ( s )  0
8. closed loop poles and zeros if K=10.
H (s )

19. Example-6
• For the system represented by the following block diagram
determine:
1.
2.
3.
4.
5.
6.
7.
8.
Open loop transfer function
Feed Forward Transfer function
control ratio
feedback ratio
error ratio
closed loop transfer function
characteristic equation
closed loop poles and zeros if K=100.

20. Reduction techniques
4. Moving a summing point behind a block
G
G
G
5. Moving a summing point ahead a block
G
G
1
G

21. 6. Moving a pickoff point behind a block
G
G
1
G
7. Moving a pickoff point ahead of a block
G
G
G

22. 8. Swap with two neighboring summing points
A
B
B
A

23. Example-7
• Reduce the following block diagram to canonical form.
H2
_
R
+_
+
+
G1
+
H1
C
G2
G3

24. Example-7
H2
G1
_
R
+_
+
+
+
C
G1
H1
G2
G3

25. Example-7
H2
G1
_
R
+_
+
+
C
+
G1G2
H1
G3

26. Example-7
H2
G1
_
R
+_
+
C
+
G1G2
+
H1
G3

27. Example-7
H2
G1
_
R
+_
+
G1G2
1  G1G2 H1
C
G3

28. Example-7
H2
G1
_
R
+_
+
G1G2G3
1  G1G2 H1
C

29. Example-7
R
+_
G1G2G3
1  G1G2 H 1  G2G3 H 2
C

30. Example 8
Find the transfer function of the following block diagram
G4
R (s )
Y (s )
G1
G2
G3
H2
H1

31. I
G4
R(s)
B
G1
G2
A
G3
H2
H1
G2
Solution:
1. Moving pickoff point A ahead of block
2. Eliminate loop I & simplify
B
G4  G2G3
G2
Y (s )

32. G4
R(s)
G1
G A G
G 4  G2G3
B
Y (s )
3
2
H2
H1G2
G4  G2G3
3. Moving pickoff point B behind block
II
R(s)
G1
B
G4  G2G3
H2
H1G2
1 /(G4  G2G3 )
C
Y (s )

33. 4. Eliminate loop III
R(s)
G1
G4 4 G2G3
G G2G3
H
1  H 2 (G4 2 G2G3 )
C
C
Y (s )
G2 H1
G4  G2G3
R(s)
G1 (G4  G2G3 )
1  G1G 2 H1  H 2 (G4  G2G3 )
G1 (G4  G2G3 )
Y (s)

R( s ) 1  G1G 2 H1  H 2 (G4  G2G3 )  G1 (G4  G2G3 )
Y (s )

34. Example 9
Find the transfer function of the following block diagrams
H4
R(s)
Y (s )
G1
G3
G2
H3
H2
H1
G4

35. Solution:
1. Moving pickoff point A behind block
G4
I
H4
R(s)
Y (s )
G1
G3
G2
H3
H2
H3
G4
H2
G4
H1
1
G4
1
G4
A
G4
B

36. 2. Eliminate loop I and Simplify
R(s)
G2G3G4
1  G3G4 H 4
G1
II
Y (s )
B
H3
G4
H2
G4
III
H1
II
feedback
G2G3G4
1  G3G4 H 4  G2G3 H 3
III
Not feedback
H 2  G4 H 1
G4

37. 3. Eliminate loop II & IIII
R(s)
G1G2G3G4
1  G3G4 H 4  G2G3 H 3
Y (s )
H 2  G4 H 1
G4
G1G2G3G4
Y (s)

R( s ) 1  G2G3 H 3  G3G4 H 4  G1G2G3 H 2  G1G2G3G4 H1

38. Example-10: Reduce the Block Diagram.

39. Example-10: Continue.

40. Example-11: Simplify the block diagram then obtain the closeloop transfer function C(S)/R(S). (from Ogata: Page-47)

41. Example-11: Continue.

42. Superposition of Multiple Inputs

43. Example-12: Multiple Input System. Determine the output C
due to inputs R and U using the Superposition Method.

44. Example-12: Continue.

45. Example-12: Continue.

46. Example-13: Multiple-Input System. Determine the output C
due to inputs R, U1 and U2 using the Superposition Method.

47. Example-13: Continue.

48. Example-13: Continue.

49. Example-14: Multi-Input Multi-Output System. Determine C1
and C2 due to R1 and R2.

50. Example-14: Continue.

51. Example-14: Continue.
When R1 = 0,
When R2 = 0,

52. Block Diagram of Armature Controlled D.C Motor
Ra
La
c
Va
ia
eb
T
J

La s  Ra I a(s)  K b(s)  Va(s)
Js  c (s)  K m I a(s)

53. Block Diagram of Armature Controlled D.C Motor
La s  Ra I a(s)  K b(s)  Va(s)

54. Block Diagram of Armature Controlled D.C Motor
Js  c (s)  K ma I a(s)

55. Block Diagram of Armature Controlled D.C Motor

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END OF LECTURES-8-9