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Automatic ControlAutomatic Control
Block DiagramsBlock Diagrams
Dr. Aly Mousaad AlyDepartment of Mechanical EngineeringDepartment of Mechanical Engineering
Faculty of Engineering, Alexandria University
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SystemSystem
• A system is an arrangement of physical components connected or related in such a pmanner as to form and/or act as an entire unitunit.
Slides 3: Block diagrams 2
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SystemSystem
1 1
1 0 1 01 1..... .....n n m m
n n m mn n m m
d y d y d x d xb b b y a a a xdt dt dt dt
− −
− −− −+ + + = + + +
• Transformed Eq. (I.C. = 0):
( ) ( ) ( )( ) ( ) ( )
11 0
1
.....n nn n
m m
b s y s b s y s b y s−−
−
+ + +
( ) ( ) ( )11 0.....m m
m ma s x s a s x s a x s−−= + + +
( ) 1( )( )
11 0
11 0
..........
m mm m
n nn n
y s a s a s a outputx s b s b s b input
−−
−
+ + += =+ + +
Slides 3: Block diagrams 3
( ) 1 0n n p−
Transfer Function, m<n
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Transfer FunctionTransfer Function
P tiProperties:• Transfer functions are defined only for a linear Time
Invariant(LTI) Systems. They are not defined for non-linear ( ) y ysystems.
• All initial conditions of the system are zero.Wh h d f h f f i ’ i• When the order of the transfer function’s numerator is equal to that of the denominator, the transfer function is called proper.p p
• If the order of the numerator is less than that of the denominator, the transfer function is called strictly proper.If th d f th t i t th th t f th• If the order of the numerator is greater than that of the denominator, the transfer function is called improper.
Slides 3: Block diagrams 4
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Transfer FunctionTransfer Function
• EX: M
• Basic Law: ( ) ( )F t Mx t=• Laplace: 2( ) ( )F s M s x s=
• T.F. 2
( ) 1( )( )x sG sF s M s
= =( )F s M s
Slides 3: Block diagrams 5
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Elements of Block DiagramsElements of Block Diagrams
• Summing point:
• Takeoff point:Takeoff point:
Slides 3: Block diagrams 6
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Elements of Block DiagramsElements of Block Diagrams
• Arrow:
• Block symbol:
Slides 3: Block diagrams 7
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Block DiagramsBlock Diagrams
• Blocks in series:
X 21
1
XGX
= 32
2
XGX
=1 2
3 321 2
X XXGGX X X
= =1 2 1X X X
Slides 3: Block diagrams 8
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Block DiagramsBlock Diagrams
• Blocks in parallel:
Slides 3: Block diagrams 9
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Block DiagramsBlock Diagrams
• Feedback loop:
Slides 3: Block diagrams 10
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Block DiagramsBlock Diagrams
Slides 3: Block diagrams 11
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Block DiagramsBlock Diagrams
• Moving a summing point behind a block
≡• Moving a summing point a head of a blockMoving a summing point a head of a block
≡≡Slides 3: Block diagrams 12
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Block DiagramsBlock Diagrams
• Moving a takeoff point a head of a block
≡• Moving a takeoff point behind a blockMoving a takeoff point behind a block
≡≡Slides 3: Block diagrams 13
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Example 1Example 1
• Reduce the block diagram shown in figure to a single transfer function.g
Slides 3: Block diagrams 14
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Slides 3: Block diagrams 15
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Example 2Example 2
• Reduce the following block diagram and find the overall transfer function.
Slides 3: Block diagrams 16
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Slides 3: Block diagrams 17
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HW 1HW 1
• Reduce the following block diagram to a single transfer function.
Slides 3: Block diagrams 18
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HW 2HW 2
Slides 3: Block diagrams 19