Transcript Feedback Control Systems (FCS) - Dr. Imtiaz Hussain
Feedback Control Systems (FCS)
Lecture-14-15 Block Diagram Representation of Control Systems Dr. Imtiaz Hussain email: [email protected]
URL : http://imtiazhussainkalwar.weebly.com/
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, 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
•
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.
Introduction
• In order to have the same signal or variable be an input to more than one block or summing point, a takeoff point is used.
• This permits the signal to proceed unaltered along several different paths to several destinations.
Example-1
• Consider the following equations in which x 1 , x 2 , x 3 , are variables, and a
1 , a 2
are general coefficients or mathematical operators.
x
3
a
1
x
1
a
2
x
2 5
Example-1
• Consider the following equations in which x 1 , x 2 , x 3 , are variables, and a
1 , a 2
are general coefficients or mathematical operators.
x
3
a
1
x
1
a
2
x
2 5
Example-2
• Consider the following equations in which x 1 , x 2 ,. . . , x n , are variables, and a
1 , a 2 ,. . . , a n
, are general coefficients or mathematical operators.
x n
a
1
x
1
a
2
x
2
a n
1
x n
1
Example-3
• Draw the Block Diagrams of the following equations.
( 1 )
x
2 ( 2 )
x
3
a
1
a
1
dx
1
dt d
2
dt x
2 2 1
b
x
1
dt
3
dx dt
1
bx
1
Canonical Form
of A Feedback Control System
Characteristic Equation
• The control ratio is the closed loop transfer function of the system.
C
(
s
)
R
(
s
) 1
G
(
s
)
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
1. Open loop transfer function 2. Feed Forward Transfer function
Example-4
B
(
s
)
G
(
s
)
H
(
s
)
E
(
s
)
C
(
s
)
E
(
s
)
G
(
s
) 3. control ratio
C
(
s
)
R
(
s
) 1
G
(
s
)
G
(
s
)
H
(
s
) 4. feedback ratio
B
(
s
)
R
(
s
)
G
(
s
)
H
(
s
) 1
G
(
s
)
H
(
s
) 5. error ratio
E
(
s
)
R
(
s
) 1 1
G
(
s
)
H
(
s
) 6. closed loop transfer function
C
(
s
)
R
(
s
) 1
G
(
s
)
G
(
s
)
H
(
s
) 7. characteristic equation 1
G
(
s
)
H
(
s
) 0 8. closed loop poles and zeros if K=10.
G
(
s
)
H
(
s
)
Reduction techniques 1. Combining blocks in cascade
G
1
G
2 2. Combining blocks in parallel
G
1
G
2
G
1
G
2
G
1
G
2
Example-5: Reduce the Block Diagram to Canonical Form .
Example-5: Continue.
However in this example step-4 does not apply.
However in this example step-6 does not apply.
•
Example-6
4.
5.
6.
7.
8.
For the system represented by the following block diagram determine: 1.
2.
3.
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.
Example-6
– First we will reduce the given block diagram to canonical form
s K
1
1
G GH
1
s K
1
s K
1
s
Example-6
s K
1
1. Open loop transfer function 2. Feed Forward Transfer function
Example-6
B
(
s
)
G
(
s
)
H
(
s
)
E
(
s
)
C
(
s
)
E
(
s
)
G
(
s
) 3. control ratio
C
(
s
)
R
(
s
) 1
G
(
s
)
G
(
s
)
H
(
s
) 4. feedback ratio
B
(
s
)
R
(
s
)
G
(
s
)
H
(
s
) 1
G
(
s
)
H
(
s
) 5. error ratio
E
(
s
)
R
(
s
) 1 1
G
(
s
)
H
(
s
) 6. closed loop transfer function
C
(
s
)
R
(
s
) 1
G
(
s
)
G
(
s
)
H
(
s
) 7. characteristic equation 1
G
(
s
)
H
(
s
) 0 8. closed loop poles and zeros if K=10.
G
(
s
)
H
(
s
)
•
Example-7
4.
5.
6.
7.
8.
For the system represented by the following block diagram determine: 1.
2.
3.
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.
Reduction techniques 3. Moving a summing point behind a block
G G G
3. Moving a summing point ahead of a block
G
4. Moving a pickoff point behind a block
G
5. Moving a pickoff point ahead of a block
G G G
1
G G G
1
G
6. Eliminating a feedback loop
G H G
1
GH G H
1 7. Swap with two neighboring summing points
A B B G
1
G A
R
+ _ + +
Example-8
H
2
G
1 + _
H
1
G
2
G
3
C
R
+ _ + +
Example-8
+ _
G
1
H
1
G
2
H
2
G
1
G
3
C
R
+ _ + +
Example-8
+ _
H
1
G
1
G
2
H
2
G
1
G
3
C
R
+ _ + _
Example-8
+ +
H
1
G
1
G
2
H
2
G
1
G
3
C
block diagram: reduction example
H
2
G
1
R
+ _ + _ 1
G
1
G
2
G
1
G
2
H
1
G
3
C
block diagram: reduction example
H
2
G
1
R
+ _ + _ 1
G
1
G
2
G
3
G
1
G
2
H
1
C
block diagram: reduction example
R
+ _ 1
G
1
G G
1
G
2
G
3 2
H
1
G
2
G
3
H
2
C
R
Example-8
1
G
1
G
2
H
1
G
1
G
2
G
3
G
2
G
3
H
2
G
1
G
2
G
3
C
Example 9 Find the transfer function of the following block diagrams
R
(s )
G
1
H
1
G
2
G
4
G
3
H
2
Y
(
s
)
R
(
s
)
G
1
B H
1
G
2 Solution: 1. Moving pickoff point A ahead of block
G
2 2. Eliminate loop I & simplify
B G
4
G
2
G
3
G
2
G
4
A G
3
H
2
I Y
(
s
)
R
(
s
)
G
1
H
1
G
2
B G G
2 4
G
4
G H
2 2
G G
3 3
Y
(
s
) 3. Moving pickoff point B behind block
R
(
s
)
G
1
H
1
G
2
B G
4
G
2
G
3
G
4
G
2
G
3
H
2 1 /(
G
4
G
2
G
3 )
II C Y
(
s
)
4. Eliminate loop III
R
(
s
)
R
(
s
)
G
1 1
H G G
4
G
2 (
G H
4 2
G
2
G
3
G
2
G
3 )
C G
4
G
2
H
1
G
2
G
3
Using rule 6
1
G
1
G G
1 2
H
(
G
4 1
H
2
G
2
G
3 (
G
4 )
G
2
G
3 )
C Y
(
s
)
T
(
s
)
Y
(
s
)
R
(
s
) 1
G
1
G
2
H
1
H
2
G
( 1 (
G
4
G
4
G G
2 2
G G
3 ) 3 )
G
1 (
G
4
G
2
G
3 )
Y
(
s
)
Example 10 Find the transfer function of the following block diagrams
R
(
s
)
G
1
H
1
G
2
H
2
H
3
Y
(
s
)
Solution: 1. Eliminate loop I
R
(
s
)
G
1
H
1
A
1
G G
2 2
G H
2 2
H
2
I B Y
(
s
) 2. Moving pickoff point A behind block
R
(
s
)
G H
1 1
A H
3 1
G
2
H
2
G
2
H
3 1
G
2
G
2
H
2 1
G
2
G
2
H
2
B Y
(
s
)
II H
3
H
1 ( 1
G G
2 2
H
2 ) Not a feedback loop
3. Eliminate loop II
R
(
s
) 1
G
1
G
2
G
2
H
2
Y
(
s
)
H
3
H
1 ( 1
G
2
H
2 )
G
2
Using rule 6 T
(
s
)
Y
(
s
)
R
(
s
) 1
G
2
H
2
G
1
G
2
H G
1
G
2 3
G
1
H
1
G
1
G
2
H
1
H
2
Example 11 Find the transfer function of the following block diagrams
H
4
R
(
s
)
G
1
G
2
G
3
G
4
H
3
H
2
H
1
Y
(
s
)
Solution: 1. Moving pickoff point A behind block
R
(
s
)
G
1
G
4
G
2
G
3
H
3
H
2
H
3
G
4
H
2
G
4
H
1 1
G
4 1
G
4
H
4
A I G
4
B Y
(
s
)
2. Eliminate loop I and Simplify
R
(
s
)
G
1
II
feedback 1
G
3
G G
2 4
H G
3
G
4 4
G
2
G
3
H
3 1
G
2
G
3
G
4
G
3
G
4
H
4
H
3
G
4
H
2
G
4
H
1
III II B Y
(
s
) Not feedback
H
2
G
4
H
1
G
4
III
3. Eliminate loop II & IIII
R
(
s
) 1
G
3
G
1
G
2
G
4
H
4
G
3
G
4
G
2
G
3
H
3
H
2
G
4
H
1
G
4
Using rule 6 Y
(
s
)
T
(
s
)
Y
(
s
)
R
(
s
) 1
G
2
G
3
H
3
G
3
G
4
H G
1
G
2
G
3
G
4 4
G
1
G
2
G
3
H
2
G
1
G
2
G
3
G
4
H
1
Example 12 Find the transfer function of the following block diagrams
R
(
s
)
G
1
H
2
G
2
A G
3
B H
1
G
4
Y
(
s
)
Solution: 1. Moving pickoff point A behind block
R
(
s
)
G
1
G
3
H
2
H
1
G
4
G
2
H
1
A
1
G
3
G
3
B
1
G
3
I Y
(
s
)
2. Eliminate loop I & Simplify
H
2
R
(
s
)
G
2
H
1
G
1 1
G
3
G
3
B II
1
G
2
H G
2 1
G
3
G
2
G
3
H
2
G
4
H
1
G
3
G
2
G
3
H
1
H
2
G
3
Y
(
s
)
B
3. Eliminate loop II
R
(
s
) 1
G
2
H
1
G
1
G
2
G
3
G
2
G
3
H
2
G
1
G
2
H
1
G
4
Y
(
s
)
T
(
s
)
Y
(
s
)
R
(
s
)
G
4 1
G
2
H
1
G
1
G
2
G
3
G
2
G
3
H
2
G
1
G
2
H
1
Example-13: Simplify the Block Diagram.
Example-13: Continue.
Example-14: Reduce the Block Diagram.
Example-14: Continue.
Example-15: Reduce the Block Diagram. (from Nise: page-242)
Example-15: Continue.
Example-16: Reduce the system to a single transfer function. (from Nise:page-243).
Example-17: Simplify the block diagram then obtain the close loop transfer function C(S)/R(S). (from Ogata: Page-47)
Example-18: Multiple Input System. Determine the output C due to inputs R and U using the Superposition Method.
Example-18: Continue.
Example-18: Continue.
Example-19: Multiple-Input System . Determine the output C due to inputs R, U1 and U2 using the Superposition Method.
Example-19: Continue.
Example-19: Continue.
Example-20: Multi-Input Multi-Output System . Determine C1 and C2 due to R1 and R2.
Example-20: Continue.
When R1 = 0 , When R2 = 0 ,
Example-20: Continue.
Block Diagram of Armature Controlled D.C Motor
V a R
a
L
a
i
a
e
b
T c
J
L a s
R a
I a (s)
K b
(s)
V a (s)
Js
c
(s)
K ma I a (s)
Block Diagram of Armature Controlled D.C Motor
L a s
R a
I a (s)
K b
(s)
E a (s)
Block Diagram of Armature Controlled D.C Motor
Js
c
(s)
K ma I a (s)
Block Diagram of Armature Controlled D.C Motor
Block Diagram of liquid level system
q
1
h
1
h
2
R
1
q
2
h
2
R
2
C
1
dh
1
dt
q
q
1
C
2
dh
2
dt
q
1
q
2
Block Diagram of liquid level system
L
q
1
h
1
h
2
R
1
Q
1 (
s
)
H
1 (
s
)
H
2 (
s
)
R
1
C
1
dh
1
dt
q
q
1
C
1
sH
1 (
s
)
Q
(
s
)
Q
1 (
s
) L L
q
2
h
2
R
2
Q
2 (
s
)
H
2 (
s
)
R
2
C
2
dh
2
dt
q
1
q
2 L
C
2
sH
2 (
s
)
Q
1 (
s
)
Q
2 (
s
)
Block Diagram of liquid level system
Q
1 (
s
)
H
1 (
s
)
H
2 (
s
)
R
1
C
1
sH
1 (
s
)
Q
(
s
)
Q
1 (
s
)
Q
2 (
s
)
H
2 (
s
)
R
2
C
2
sH
2 (
s
)
Q
1 (
s
)
Q
2 (
s
)
Block Diagram of liquid level system
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