Automatic ControlAutomatic Control - School of …aly/Courses/ME221/Slides 3.pdf · Automatic...

Automatic ControlAutomatic Control

Block DiagramsBlock Diagrams

Dr. Aly Mousaad AlyDepartment of Mechanical EngineeringDepartment of Mechanical Engineering

Faculty of Engineering, Alexandria University

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

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

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

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

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Elements of Block DiagramsElements of Block Diagrams

• Summing point:

• Takeoff point:Takeoff point:

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Elements of Block DiagramsElements of Block Diagrams

• Arrow:

• Block symbol:

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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

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Block DiagramsBlock Diagrams

• Blocks in parallel:

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Block DiagramsBlock Diagrams

• Feedback loop:

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Block DiagramsBlock Diagrams

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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

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

Example 1Example 1

• Reduce the block diagram shown in figure to a single transfer function.g

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Slides 3: Block diagrams 15

Example 2Example 2

• Reduce the following block diagram and find the overall transfer function.

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HW 1HW 1

• Reduce the following block diagram to a single transfer function.

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HW 2HW 2

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