EXERCISE SIGNAL FLOW GRAPH

Block Diagram Reduction & Signal Flow GraphBlock DiagramsBasic components of a block diagram for a LTIV system

Cascade or series subsystems,

Parallel Subsystems,

Feedback Form

a. Feedback control system;b. simplified model;c. equivalent transfer functionMoving blocks to create familiar forms,

Example 1

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

Solution,

Example 2

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

Solution,

1.1 Control signal

R(s) +

E(s)

Y(s)

_

B(s)

E(s) error signal

B(s) feedback signal

R(s) reference signal

Y(s) output signal

Feed forward transfer function

Feedback transfer function

Open-loop transfer function Closed-loop transfer function

E(s)

Y(s)

B(s)

E(s)

B(s)

Open-loop

Assume, gives .

LISTNUM

LISTNUM

LISTNUM

Variable difference

LISTNUM Characteristic equation

LISTNUM Signal Flow Graphs

SFG may be viewed as a simplified form of block diagram. SFG consists of arrows (represent systems) and nodes (represent signals).

Signal-flow graph components:a. system;b. signal;c. interconnection of systems and signals

Converting common block diagrams to SFG

Converting a block diagram to SFG

Signal-flow graph development:a. signal nodes;b. signal-flow graph;c. simplified signal-flow graph

Mason Gain Formula

The transfer function of a given system represented by a SFG is:

where

k = no. of paths

= the kth forward-path gain

= 1 - loop gains + non-touching loop gains 2 at a time -

non-touching loop gains 3 at a time + non-touching

loop gains 4 at a time -

k= - ( loop gain terms in that do not touch the k-th forwad-path. In

other words, k is formed by eliminating from those loop gains that do

touch the k-th forward path.SIGNAL FLOW GRAPHDefinitions:

i)Loop Gains:

G2(s)H1(s), G4(s)H2(s), etc

ii)Forward-path gains:

G1(s)G2(s)G3(s)G4(s)G5(s)G7(s), G1(s)G2(s)G3(s)G4(s)G6(s)G7(s)

iii)Non-touching Loopsiv)Non-touching Loop Gains

[G2(s)H1(s)][G4(s)H2(s)]

[G2(s)H1(s)][G4(s)G5(s)H3(s)]

[G2(s)H1(s)][G4(s)G6(s)H3(s)]

Example

Find the transfer function, C(s)/R(s), for the signal-flow graph below:

Solution:i)Firstly, identify the forward-path gains.G1(s)G2(s)G3(s)G4(s)G5(s)ii)Secondly, identify the loop gains.

G2(s)H1(s), G4(s)H2(s), G7(s)H4(s), G2(s)G3(s)G4(s)G5(s)G6(s)G7(s)G8(s)

iii)Thirdly, identify the non-touching loops taken two at a time.

Loop 1 & Loop 2

:G2(s)H1(s)G4(s)H2(s)

Loop 1 and Loop 3:G2(s)H1(s)G7(s)H4(s)

Loop 2 and Loop 3:G4(s)H2(s)G7(s)H4(s)

iv)Finally, identify the non-touching loops taken three at a time.

Loops 1, 2 and 3

:G2(s)H1(s)G4(s)H2(s)G7(s)H4(s)

Hence,

=1 [G2(s)H1(s) + G4(s)H2(s) + G7(s)H4(s) +

G2(s)G3(s)G4(s)G5(s)G6(s)G7(s)G8(s)] + [G2(s)H1(s)G4(s)H2(s) +

G2(s)H1(s)G7(s)H4(s) +G4(s)H2(s)G7(s)H4(s)]

[G2(s)H1(s)G4(s)H2(s)G7(s)H4(s)]

Then form k by eliminating from the loop gains that do not touch the kth forward-path :

1=1 - G7(s)H4(s)

Hence:

EXERCISE

1.

Find the transfer function, G(s) = C(s)/R(s) of the figure above using:1) Block diagram reduction technique2) Masons ruleSOLUTION:1) Block diagram reduction technique:Combine the parallel blocks in the forward path. Then, push 1/s to the left past the pickoff point.

Combine the parallel feedback paths and get 2s. Then, apply the feedback formula, simplify and get

2) Signal flow graph technique:

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H(s)G(s)

G(s)

H(s)s

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S

1/S

S2+1/S

+

-

-

R(s)

C(s)

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