bnj30703-02

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14/09/20

BNJ 30703

Contents

Block Diagram

Transfer Function

Signal Flow Graph

Transfer Function

Transfer function G(s) of a linear system is definedas the ratio of the Laplace transform of the outputvariable to the Laplace transform of the inputvariable, with all initial conditions assumed to bezero

)s(Input

)s(Output)s(G =

Definition Transfer FunctionExercise:Find the transfer function of the following

differential equation

)()(4)(5)( t ut yt yt y =++ &&&

4s5s

1

)s(U

)s(Y2

++=

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5

Laplace Transform Operations

( ) f t ( )F s

'( ) f t ( ) (0)sF s f − O-1

0( )

t

f t dt ∫ ( )F s

s

O-2

( )t e f t

α − ( )F s α + O-3

( ) ( ) f t T u t T − − ( )sT e F s− O-4

(0) f lim ( )s

sF s→∞

O-5

lim ( )t

f t →∞

*

0lim ( )s

sF s→

O-6

*Poles of ( )sF s must have negative real parts.

6

Significant Operations forSolving Differential Equations

[ '( )] ( ) (0) L f t sF s f = −

2[ "( )] ( ) (0) '(0) L f t s F s sf f = − −

0

( )( )

t F s L f t dt

s

=

∫

7

Procedure for Solving DEs2

2 1 02 ( )

d y dyb b b y f t

dt dt + + =

[ ]2

2 1 02 ( )

d y dy L b b b y L f t

dt dt

+ + =

[ ]

2

2

1 0

( ) (0) '(0)

( ) (0) ( ) ( )

b s Y s sy y

b sY s y b Y s F s

− −

+ − + =

2 2 1

2 2

2 1 0 2 1 0

(0) '(0) (0)( )

( )

sb y b y b yF s

Y s b s b s b b s b s b

+ +

= ++ + + +

8

Example. Solve DE shownbelow.

2 12dy

ydt

+ = (0) 10 y =

[ ] [ ]2 12dy

L L y Ldt

+ =

12( ) 10 2 ( )sY s Y s

s− + =

( ) 12

2 ( ) 10s Y ss

+ = +

10 12( )2 ( 2)

Y ss s s

= ++ +

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Transfer Function Given the following transfer function G(s)

)s(B

)s(A)s(G =

The roots of the numerator A(s) are called zeros. The roots of the denominator B(s) are called poles. The denominator of a transfer function is called

characteristic equation.

Block Diagram A block diagram is a convenient tool to visualize the

systems as a collection of interrelated subsystemsthat emphasize the relationships among the systemvariables.

)(sGinput output

)()()( s RsGsC =

Block Diagram Three Elementary Block Diagrams

Series connection

Parallel connection

Negative Feedback connection

Block Diagram Block diagram algebra

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Block DiagramExercise:Find the transfer function of the following block

diagram

4s2s

4s2

)s(R

)s(Y2

++

+=

Transfer function?

Signal Flow Graph

For a system with complex interrelationships, the blockdiagram reduction is cumbersome and often quitedifficult to complete.

Signal-flow graph is an alternative method fordetermining relationship between system variables.

Signal Flow Graph

A signal flow-graph consists of branches, whichrepresents system/subsystem transfer function,and nodes which represent signal (input/outputpoint)

Input node Output node

System

)()()( sV sGs f =θ

Basic Concept

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Signal Flow GraphBlock Diagram to Signal Flow Graph




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Signal Flow Graph Signal Flow GraphExercise:

Signal Flow Grapherminologies

Signal flow graph terminologies which are used todetermine the transfer function

Forward-path gain

Loop gain

Non-touching loop gain

Summar!

Signal Flow GraphForward "ath Gain

Forward-path gain: The product of gains found bytraversing a path from the input node to the outputnode of the signal-flow graph.

Two forward#path gains: ?

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Signal Flow Graph

Loop gain: The product of branch gains bytraversing a path that starts at a node and ends atsame node without passing trough any other nodemore than once and following the direction of thesignal flow.

Four loop gains: $

%oop GainSignal Flow Graph

&on#touching %oop Gain

Nontouching loop: Loops that do not have anynode in common.

&ontouching loops $

Signal Flow Graph&on#touching %oop Gain

Nontouching loop gain: The product of loop gainsfrom nontouching loops taken two, three, four, etc.

&ontouching loop gains$

Signal Flow Graph

%oop'(G)*+'%oop)(G,*+)%oop-(G,*G.*+-%oop,(G,*G/*+-

%oop' does not touch other loops0%oops )1- 2 , ha3e node 4- in common0

Forward"ath'(G'*G)*G-*G,*G.*G5Forward"ath)(G'*G)*G-*G,*G/*G5

&ontouching %oop Gains:

%oop'*%oop)%oop'*%oop-%oop'*%oop,

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Signal Flow Graph

The transfer function T between output variableY(s) and input variable R(s) is

∆

∆==

∑k k k P

s R

sY sT

)(

)()(

Where:

K :number of forward path

Pk : the kth forward-path gain

6ason7s Gain FormulaSignal Flow Graph

Where:

∆ = System determinant

∆ 1 -Σ (all individual loop gain)

+ Σ (non-touching loop gain of two loops)

- Σ (non-touching loop gain of three loops)

+ …..

∆k =kth forward path determinant

∆

∆==

∑k k k P

s R

sY sT

)(

)()(

6ason7s Gain Formula

Signal Flow Graph

∆k = ∆ -Σ loop gain terms in ∆ that touch the kthforward path.

In other words, ∆k is formed by eliminating from ∆

those loop gain touch the kth forward path.

6ason7s Gain FormulaSignal Flow Graph

ExampleExample

?)s(R

)s(C=

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Signal Flow GraphExample

?)s(R

)s(Y=

Summary Transfer Function is the ratio of the Laplace transform

of the output variable to the Laplace transform of theinput variable, with all initial conditions assumed to bezero.

A block diagram is a convenient tool to visualize thesystems as a collection of interrelated subsystems thatemphasize the relationships among the systemvariables.

Signal flow graph and Mason7s gain formula are usedto determine the transfer function of the complex blockdiagram.

Further Reading

Franklin, et. al., Chapter 3

Section 3.2

Dorf, Modern Control System

Chapter 2, Signal Flow Graph

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