Automatic ControlAutomatic Control - School of …aly/Courses/ME221/Slides 3.pdf · Automatic ControlAutomatic Control Block DiagramsBlock Diagrams Dr. Aly Mousaad Aly ... systems.

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

Slides 3: Block diagrams 11

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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Slides 3: Block diagrams 17

HW 1HW 1

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

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HW 2HW 2

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