Signal flow diagrams

8/22/2019 Signal flow diagrams

1/20

Lecture -5: Signal Flow Graphs 1 hour

By Mr S WijewardanaPhD student QMUL 20-04-2013

Learning Objectives:

1. Elements of signal flow graphs.

2. Block diagrams vs signal flow graphs.3. Maisons formula.


2/20

Signal Flow Graphs(SFG)

The other way of representing the control systemsdynamics(Pl. See Lecture notes on Lecture-5) is with theuse of Signal Flow Graphs. Even though the Block diagramrepresentation and the signal flow graph methods are

quite similar, there are some notable differences betweenthem.

Signal flow graph method is easier to construct.

System equation is readily available than the Blockdiagram method.

SFG method gives a visual representation of the systemequation and hence logical reduction is much easier.


3/20

Elements of a SFG

A SFG is a network of directed branches which

connect at Nodes.

The Nodes are connected by line segments

and these line segments are called branches.

Node: is defined as a junction point to

represent variables in a SFG.

The branches have associated branch gains

and directions


4/20

Consider a linear system described by a set of N algebraic

equations as shown below:

N

kkkjj NjyaY 1 ,....,2,1

These N equations can be written in the form of causeand

effect relationship as shown below:

inputgainoutput

causekjtokfromgaineffectjN

k

thth

1

)()(

Equation-1

Equation-1 can be written in Laplace domain as shownbelow:

N

k

kkjj NjsYsGsY1

,...,2,1)()()(


5/20

Input Node:

An input node is a node that has only out going branches.

Output Node:

Output node is a node that has only incoming branches.The small circles represent nodes(e.g: y1,y2,x4, y3,y4) The arrows represent the

flow of data from one node to the other. The arrow head indicates the gain of the

signal.

In signal flow graphs we cannot normally find the condition which satisfies

according to the definition of an output node. Because, in SFG as shown in fig

below does not have the output node which satisfies all the above condition.

Therefore we draw a similar node which is equivalent to the output node with a

UNITY GAIN.

x4

y1a12 y2 a23 y3 a34

y4

a42


6/20

a12 y2 a23 y3

To derive the equation from graph is easy:

y2= a12y1 + a24y4

x4

y1a12 y2 a23 y3 a34 y4

a42

Y3

B1

Unity gain

SFG with a unity gain: y3=y3(this is not

possible for an input node)

y4

y1

a24


7/20

G(s) C(s)

R(s)

x2

C(s)

a

x3x1 x1

R(s)

G2(s)

C(s)

G1(s)

+

+

G1(s)

G2(s)

G(s)

C(s)R(s) R(s)

+

b

+

a

b

x2

x3


8/20

x1

x

h

Y = gx + hz ( addition)

x2

x1

b

a

c

x2

x3

a

x3

-

c

+

b

y

z

g

x

h

y

z

-g

Y = hzgx (subtraction)


9/20

Y= gx (Multiplication)

Y= -x (negative unit

transmittance)

Y= x/g (division)

Z=x (Unit

transmittance)

x

1/g

y

x

g

x

y

z

1

xy

-1


10/20

R(s)

G(s)

E(s)

E(s)

C(s)

-

H(s)

+R(s)

1 G(s) C(s) 1 C(s)

x1

b

-H(s)

y

x2

x3a

Y = ax1 + bx2cx3 +dy

-c

d


11/20

Construct the signal flow diagram for two simultaneous equations given below :

t11x + t12y + t13z = 0 -----------------eq1

t21x+ t22y + t23z = 0 -----------------eq2

From eq1 and 2 we can write:

xt

tzt

ty

yt

tz

t

tx

22

21

22

23

11

12

11

13

x

z

y

11

13

t

t

11

12

t

t

x

z

y

22

23

t

t

22

21

t

t

Example-1:


12/20

Combining both SFGs:

z

y

x

11

13

t

t

11

12

t

t

22

23

t

t

22

21

t

t


13/20

Gain Formula for Signal Flow Graphs

Maisons gain formula gives us an easy

method to find the input-output relationship.

Input-output relationship of any control

system is important for stability analysis.

Transmittance:

Is defined as the overall gain between an input

node an output node.


14/20

Maisons gain formula:

kkPP1

Pk = Path gain or Transmittance of kth

forward path.

= determinant of graph

= 1(sum of all individual loop gains)

+ ( sum of gain products of all possible combinations of two non-

touching loops)

- (sum of gains of all combinations of three non touching loops)+ (sum of ........) - .......

cb fed

fedca

a

a LLLLLL, ,,

...1


15/20

a

aL

cb

caLL,

= Sum of gain products of all possible combinations of two

non-touching loops

fed

fed LLL,,

= Sum of gain products of all possible combinations of

three-non-touching loops.

= Sum of all individual loop gains

k = Cofactor of kth forward path is determined from the graph with the loopstouching the kth forward path removed. That is, the cofactor k is obtained from by removing the loops that touch the path Pk .

= or we can say -loop gains touching the kth forward path.


16/20

R(s)

G(s)

E(s)

E(s)

C(s)

-

H(s)

+R(s)

1 G(s) C(s) 1 C(s)

Example-2:

Determine the TF of the control system by using (Maisons Gain Formula) MGF

-H(s)

(i) From the SFG shown above there is only one forward path

between R(s) and C(s) and the forward path gain is

P1= G(s)

(ii) There is only one loop in the SFG.

E(s)-G(s) through H(s) and E(s)

Hence: L1 = -G(s).H(s)

Solution:


17/20

(iii) There are no non-touching loops since there is only one loop. The

forward path R(s)-E(s)-C(s)-C(s) is in touch with the only loop.

Therefore 1 = 1

= 1- L1= 1 + G(s).H(s)

(iv) Using gain formula now we can write the C.L .T.F.:

)()(1

)(

)(

)( 11

sHsG

sGP

sR

sC

R(s) +

Example-3: Convert the block diagram into SFG

and use MGF to find the TF or C/R gain ratio. G4

G2G1

H1

+

C(s)

G3

H2

+-

+

SFG f th b bl k di i h b l


18/20

R 1 a1 G1 a2 G2 a3 G3 a4 1 a5 1 C

SFG for the above block diagram is shown below:

R 1 a1

G1

a2

G2

a3

G3

a4

1 a5

1 C

R 1 a1 G1 a2 G2 a3 a4 1 a5 1 C

Path 1:

Path 2:

R(s) +

-H1

C(s)

H2

G4

There are two forward paths that

can be drawn as shown below:

G4

Gain for two forward paths:

P1 = G1G2G3

P2 = G1G2G4


19/20

(ii) There are three non-touching feedback loops in the SFG as shown below:

G4

a1 G1 a2 G2 a3

H2

a1 G1 a2 G2 a3 G3 a4 1 a5

-H1

-H1

a1 G1 a2 G3 a3 a4 1 a5

Three non-touching feedback loops.


20/20

Loop Gain:

L1 = G2 G2 H2

L2 = -G1G2G3H1

L3= -G1G2G4H1

(III) There are no two non-touching loops in the SFG.Therefore

= 1-(G1G2H2-G1G2G3H1-G1G2G4H1)

(iv)

As P 1 and P2 are both touching the feedback loops

1 = 1, 2=1Hence the required gain is:

)(1

)(

1

1.1.

4113221

4321

13211321221

421321

2211

GHHGHGG

GGGG

R

C

HGGGHGGGHGG

GGGGGG

PP

R

C

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