Topic #8 Signal Flow Graphs Reference textbook : Control Systems, - - PowerPoint PPT Presentation

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ME 779 Control Systems

Signal Flow Graphs

Topic #8

Reference textbook:

Control Systems, Dhanesh N. Manik, Cengage Publishing, 2012

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Control Systems: Signal Flow Graphs

Learning Objectives

• Definition
• Canonical feedback system
• Mason’s formula
• Examples

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A signal flow graph is a diagram consisting of nodes that are connected by several directed branches, each node representing a variable of the system.

( ) ( ) ( ) Y s G s X s 

Control Systems: Signal Flow Graphs

Definition

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A forward path is any path which goes from the input node to the output node along which no node is traversed more than once. A loop is any path which originates and terminates at the same node along which no node is traversed more than once

Control Systems: Signal Flow Graphs

Definition

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Canonical feedback system

( ) ( ) ( ) ( ) E s R s C s H s  

forward path is the one that goes from R(s)—E(s)—C(s)

G(s) is the product of all the gains in the forward path Loop: E(s)-C(s)-E(s)

• G(s)H(s) is the product of all the gains in the loop

Control Systems: Signal Flow Graphs

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Mason’s formula

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( ) ( )

n k k k

T C s R s



  



Tk is the gain of the kth forward path from the input node R(s) to the output node C(s)

∆=1-(sum of all individual loop gains)+(sum of gain products of all

combinations of two non-touching loops)-(sum of gain products of all combinations of three non-touching loops)+.. ∆k=determinant of graph in which all loops touching the kth forward path are removed

Control Systems: Signal Flow Graphs

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1

1 ( ) ( ) T G s G s   

Canonical feedback system

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1 1 ( ) ( ) L G s H s     

∆1 =1

1

( ) ( ) ( ) 1 ( ) ( )

n k k k

T C s G s R s G s H s



    



Control Systems: Signal Flow Graphs

Mason’s formula

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Control Systems: Signal Flow Graphs

Example

signals: R(s), E1(s), E2(s), C1(s), C(s)

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Loop1: E2(s)-C1(s)-E2(s) Loop2: E1(s)-E2(s)-C1(s)-C(s)-E1(s) Touching loops

 

1 1 2 3 4

( ) ( ) ( ) ( ) ( ) T s G s G s G s G s  

1 2

Control Systems: Signal Flow Graphs

Example

R(s) E1(s) E2(s) C1(s) C(s)

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 

 

1 2 1 1 2 3 4 2

1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) G s G s H s G s G s G s G s H s      

∆=1-(sum of all individual loop gains)+(sum of gain products of all

combinations of two non-touching loops)-(sum of gain products of all combinations of three non-touching loops)+..

∆1=1 after removing all the loops in the forward path

Control Systems: Signal Flow Graphs

Example

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   

1 2 3 4 1 2 1 1 2 3 4 2

( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) G s G s G s G s C s T s R s G s G s H s G s G s G s G s H s      

Control Systems: Signal Flow Graphs

Example

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( ) ( )

n k k k

T C s R s



  



 

1 1 2 3 4

( ) ( ) ( ) ( ) ( ) T s G s G s G s G s  

 

1 2 1 1 2 3 4 2

( ) ( ) ( ) 1 ( ) ( ) ( ) ( ) ( ) G s G s H s G s G s G s G s H s                

∆1=1

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Control Systems: Signal Flow Graphs

Example

signals: Q1(s), E1(s), H1(s), E2(s), Q2(s), E3(s), H2(s), Q3(s)

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Loop1: E1(s)-H1(s)-E2(s)-Q2(s)-E1(s) Loop2: E3(s)-H2(s)-Q3(s)-E3(s) Loop3:E2(s)-Q2(s)-E3(s)-H2(s)-E2(s) 1

2 3

Loops 1 and 2 are non-touching Loops 1 and 3 are touching Loops 2 and 3 are touching

Control Systems: Signal Flow Graphs

Example

Q1(s) E1(s) H1(s) E2(s) Q2(s) E3(s) H2(s) Q3(s)

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1 1 1 2 2 2 1 1 2 2 2 1 1 1 2 2 2 1 1 2 2

1 1 1 1 1 1 1 1 1 1 1 R C s R C s R C s R C s R C s R C s R C s R C s R C R C s                                

2 1 1 1 2 2 3 2 1 1 1 2 2 2 1 1 2 2 2 1 1 2 2 1 1 2 2 2 1

1 ( ) 1 1 1 1 ( ) 1 1 ( ) 1 Q s R C R C s Q s R C s R C s R C s R C R C s R C R C s s R C R C R C               Δ1=1

Control Systems: Signal Flow Graphs

Example

1 2 1 1 2 2

1 T R C R C s 

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( ) ( )

n k k k

T C s R s



  



∆=1-(sum of all individual loop gains)+

(sum of gain products of all combinations of two non-touching loops