block diagram and signal flow graph representation

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1. Block Diagram and Signal
Flow Representation
ELE3201 Control Engineering I
Lecture II & III
Compiled by Dr Mukhtar Fatihu Hamza

2. 2
Block Diagram Models
•Block diagrams are used as schematic representations
of mathematical models
•The various pieces correspond to mathematical entities
•Can be rearranged to help simplify the equations used
to model the system
•We will focus on one type of schematic diagram –
feedback control systems

3. 3
Processes
•Processes are represented by the blocks in block diagrams:
•Processes must have at least one input variable and at least one
output variable
•Reclassify processes without input or output:
Input
variable
Output
variable
Process
variable
variable

4. 4
Feedback Control Systems
•Many systems measure their output and use this measurement
to control system behavior
•This is known as feedback control –the output is “fed back”
into the system
•The summing junction is a special process that compares the
input and the feedback
•Inputs to summing junction must have same units!
process
sensor
input output

5. Automatic Control by Meiling CHEN 5
block
summer
pickoff point

6. 6)()()( sEsGsY )()()()( sYsHsRsE  )()()()()()]()()()[()( sYsHsGsRsGsYsHsRsGsY  )()(1
)(
)(
)(
)(
sHsG
sG
sR
sY
sT



7. 7

8. 8
original equivalentA B C BA CBA     A B C CA BCA     A B CBA    C A B C BA CBA    

9. 9
original equivalentA B BAG  
GAG A B BAG  
GG
B
A G
1 A B GBA)(  
GBA A B  
G
GGBA)(

10. 10
original equivalentA
G
GAG AG A
GAG AG A
GAG A A
GAG A G
1

11. 11
Example 1

12. 1212
Example 2

13. 13
Example 2 cont.

14. 14
Example 3

15. 15
Example 3 cont.

16. 16
Introduction to Signal Flow
•Alternativemethodtoblockdiagramrepresentation,
developedbySamuelJeffersonMason.
•Advantage:theavailabilityofaflowgraphgainformula,
alsocalledMason’sgainformula.
•Asignal-flowgraphconsistsofanetworkinwhichnodes
areconnectedbydirectedbranches.
•Itdepictstheflowofsignalsfromonepointofasystemto
anotherandgivestherelationshipsamongthesignals.

17. 17
Fundamentals of Signal Flow Graphs
•Considerasimpleequationbelowanddrawitssignalflowgraph:
•Thesignalflowgraphoftheequationisshownbelow;
•EveryvariableinasignalflowgraphisdesignedbyaNode.
•Everytransmissionfunctioninasignalflowgraphisdesignedbya
Branch.
•Branchesarealwaysunidirectional.
•Thearrowinthebranchdenotesthedirectionofthesignalflow.axy x y a

18. 18
Signal-Flow Graph ModelsY
1s()G
11s()R
1s() G
12s()R
2s()
Y
2s()G
21s()R
1s() G
22s()R
2s()

19. 19
Signal-Flow Graph Modelsa
11x
1a
12x
2 r
1 x
1
a
21x
1a
22x
2 r
2 x
2
r
1and r
2are inputs and x
1and x
2are outputs

20. 20
Signal-Flow Graph Models34
203
312
2101
hxx
gxfxx
exdxx
cxbxaxx




b
x
4x
3x
2
x
1
x
0
h
f
g
e
d
c
a
x
ois input and x
4is output

21. 21
Construct the signal flow graph for the following set of
simultaneous equations.
•Therearefourvariablesintheequations(i.e.,x
1,x
2,x
3,andx
4)
thereforefournodesarerequiredtoconstructthesignalflowgraph.
•Arrangethesefournodesfromlefttorightandconnectthemwith
theassociatedbranches.
•Another way to arrange
this graph is shown in
the figure.

22. 22
Terminologies
•Aninputnodeorsourcecontainonlytheoutgoingbranches.i.e.,X
1
•Anoutputnodeorsinkcontainonlytheincomingbranches.i.e.,X
4
•Apathisacontinuous,unidirectionalsuccessionofbranchesalongwhichnonode
ispassedmorethanones.i.e.,
•Aforwardpathisapathfromtheinputnodetotheoutputnode.i.e.,
X
1toX
2toX
3toX
4,andX
1toX
2toX
4,areforwardpaths.
•Afeedbackpathorfeedbackloopisapathwhichoriginatesandterminatesonthe
samenode.i.e.;X
2toX
3andbacktoX
2isafeedbackpath.
X
1to X
2to X
3to X
4
X
1to X
2to X
4
X
2to X
3to X
4

23. 23
Terminologies
•Aself-loopisafeedbackloopconsistingofasinglebranch.i.e.;A
33isaself
loop.
•Thegainofabranchisthetransmissionfunctionofthatbranch.
•Thepathgainistheproductofbranchgainsencounteredintraversingapath.
i.e.thegainofforwardspathX
1toX
2toX
3toX
4isA
21A
32A
43
•Theloopgainistheproductofthebranchgainsoftheloop.i.e.,theloopgain
ofthefeedbackloopfromX
2toX
3andbacktoX
2isA
32A
23.
•Twoloops,paths,orloopandapatharesaidtobenon-touchingiftheyhave
nonodesincommon.

24. 24
Consider the signal flow graph below and identify the
following
a)Input node.
b)Output node.
c)Forward paths.
d)Feedback paths (loops).
e)Determine the loop gains of the
feedback loops.
f)Determine the path gains of the forward
paths.
g)Non-touching loops

25. 25
Consider the signal flow graph below and identify the following
•There are two forward path
gains;

26. 26
Consider the signal flow graph below and identify the following
•There are four loops

27. 27
Consider the signal flow graph below and identify the following
•Nontouching loop
gains;

28. 28
Consider the signal flow graph below and identify the following
a)Input node.
b)Output node.
c)Forward paths.
d)Feedback paths.
e)Self loop.
f)Determine the loop gains of the
feedback loops.
g)Determine the path gains of the forward
paths.

29. 29
Input and output Nodes
a)Input node
b)Output node

30. 30
(c) Forward Paths

31. 31
(d) Feedback Paths or Loops

32. 32
(d) Feedback Paths or Loops

33. 33
(d) Feedback Paths or Loops

34. 34
(d) Feedback Paths or Loops

35. 35
(e) Self Loop(s)

36. 36
(f) Loop Gains of the Feedback Loops

37. 37
(g) Path Gains of the Forward Paths

38. 38
Mason’s Rule (Mason, 1953)
•Theblockdiagramreductiontechniquerequiressuccessive
applicationoffundamentalrelationshipsinordertoarriveatthe
systemtransferfunction.
•Ontheotherhand,Mason’sruleforreducingasignal-flowgraph
toasingletransferfunctionrequirestheapplicationofoneformula.
•TheformulawasderivedbyS.J.Masonwhenherelatedthe
signal-flowgraphtothesimultaneousequationsthatcanbewritten
fromthegraph.

39. 39
Mason’s Rule:
•Thetransferfunction,C(s)/R(s),ofasystemrepresentedbyasignal-flowgraph
is;
Where
n= number of forward paths.
P
i= the i
th
forward-path gain.
∆ = Determinant of the system
∆
i= Determinant of the i
th
forward path
•∆ is called the signal flow graph determinant or characteristic function. Since
∆=0 is the system characteristic equation.



n
i
iiP
sR
sC
1
)(
)(

40. 40
Mason’s Rule:
∆=1-(sumofallindividualloopgains)+(sumoftheproductsofthegains
ofallpossibletwoloopsthatdonottoucheachother)–(sumofthe
productsofthegainsofallpossiblethreeloopsthatdonottoucheach
other)+…andsoforthwithsumsofhighernumberofnon-touchingloop
gains
∆
i=valueofΔforthepartoftheblockdiagramthatdoesnottouchthei-th
forwardpath(Δ
i=1iftherearenonon-touchingloopstothei-thpath.)



n
i
iiP
sR
sC
1
)(
)(

41. 41
Systematic approach
1.CalculateforwardpathgainP
iforeachforward
pathi.
2.Calculatealllooptransferfunctions
3.Considernon-touchingloops2atatime
4.Considernon-touchingloops3atatime
5.etc
6.CalculateΔfromsteps2,3,4and5
7.CalculateΔ
iasportionofΔnottouchingforward
pathi
41

42. 42
Example#1: Apply Mason’s Rule to calculate the transfer function of
the system represented by following Signal Flow Graph


2211 PP
R
C
Therefore,24313242121411 HGGGLHGGGLHGGL  ,,
There are three feedback loops

43. 43
Example#1: Apply Mason’s Rule to calculate the transfer function of
the system represented by following Signal Flow Graph
∆ = 1-(sum of all individual loop gains)
There are no non-touching loops, therefore 
3211 LLL   
243124211411 HGGGHGGGHGG 

44. 44
Example#1: Apply Mason’s Rule to calculate the transfer function of
the system represented by following Signal Flow Graph
∆
1= 1-(sum of all individual loop gains)+...
Eliminate forward path-1
∆
1= 1
∆
2= 1-(sum of all individual loop gains)+...
Eliminate forward path-2
∆
2= 1

45. 45
Example#1: Continue

46. 46
Example#2: Apply Mason’s Rule to calculate the transfer function
of the system represented by following Signal Flow Graph
2. Calculate all loop gains.
3. Consider two non-touching loops.
L
1L
3L
1L
4
L
2L
4 L
2L
3
1. Calculate forward path gains for each forward path.
P
1
P
2

47. 47
4.Consider three non-touching loops.
None.
5.Calculate Δfrom steps 2,3,4.  
4232413143211 LLLLLLLLLLLL   
 
7733663377226622
776633221
HGGHHGGHHGHGHGHG
HGHGGHHG



Example#2: continue

48. 48
48
Example#2: continue
Eliminate forward path-1 
4311 LL  
2121 LL
Eliminate forward path-2 
776611 HGHG   
332221 HGHG 

49. 49


2211 PP
sR
sY
)(
)(
Example#2: continue     
  
773366337722662277663322
3322876577664321
1
11
HGGHHGGHHGHGHGHGHGHGGHHG
HGHGGGGGHGHGGGGG
sR
sY



)(
)(

50. 50
Example#3
•Findthetransferfunction,C(s)/R(s),forthesignal-flow
graphinfigurebelow.

51. 51
Example#3
•ThereisonlyoneforwardPath.)()()()()( sGsGsGsGsGP
543211

52. 52
Example#3
•Therearefourfeedbackloops.

53. 53
Example#3
•Non-touchingloopstakentwoatatime.

54. 54
Example#3
•Non-touchingloopstakenthreeatatime.

55. 55
Example#3
Eliminate forward path-1

56. 56
Example#4: Apply Mason’s Rule to calculate the transfer function
of the system represented by following Signal Flow Graph





 332211
3
1 PPP
P
sR
sC
i
ii
)(
)(
There are three forward paths, therefore n=3.

57. 57
Example#4: Forward Paths722AP 76655443321 AAAAAP 766554423 AAAAP

58. 58
Example#4: Loop Gains of the Feedback Loops23321 AAL 34432 AAL 45543 AAL 56654 AAL 67765 AAL 776AL 2334427 AAAL 6776658 AAAL 23344557729 AAAAAL 23344556677210 AAAAAAL

59. 59
Example#4: two non-touching loops31LL 41LL 51LL 61LL 81LL 42LL 52LL 62LL 82LL 53LL 63LL 64LL 74LL 75LL 87LL

60. 60
Example#4: Three non-touching loops31LL 41LL 51LL 61LL 81LL 42LL 52LL 62LL 82LL 53LL 63LL 64LL 74LL 75LL 87LL

61. 61
G
1 G
4G
3
From Block Diagram to Signal-Flow Graph Models
Example#5
－
－
－
C(s)R(s)
G
1 G
2
H
2
H
1
G
4G
3
H
3
E(s) X
1
X
2
X
3
R(s) C(s)
－H
2
－H
1
－H
3
X
1
X
2 X
3E(s)1 G
2

62. 621;
)(1
143211
14323234321


GGGGP
HGGHGGHGGGG 14323234321
4321
1)(
)(
HGGHGGHGGGG
GGGG
sR
sC
G


R(s)
－H
2
1G
4G
3
G
2
G
11 C(s)
－H
1
－H
3
X
1
X
2 X
3E(s)
From Block Diagram to Signal-Flow Graph Models
Example#5

63. 63
G
1
G
2
+
－
+
－
－
－
+C(s)R(s) E(s)
Y
2
Y
1
X
1
X
2
－
1
-1
1
-1
-1
-1
-1
1
1
G
1
G
2
1
R(s) E(s) C(s)
X
1
X
2
Y
2
Y
1
Example#6

64. 64
Example#6
1
-1
1
-1
-1
-1 -1
1
1
G
1
G
2
1
R(s) E(s) C(s)
X
1
X
2
Y
2
Y
1
7 loops:
3 ‘2 non-touching loops’ :

65. 65
Example#6
1
-1
1
-1
-1
-1 -1
1
1
G
1
G
2
1
R(s) E(s) C(s)
X
1
X
2
Y
2
Y
1212
G4G2G1Δ 
Then:
4 forward paths:211
G1 Δ1G1)(p 
1 1Δ1G1)(G1)(p
221

2 132
G1Δ1G1p 
3 1Δ1G1G1p
412

4

66. 66
Example#6
We have212
2112
421
2
GGG
GGGG
p
sR
sC
kk








)(
)(

67. 67
Example-7:Determine the transfer function C/R for the block diagram below
by signal flow graph techniques.
•The signal flow graph of the above block diagram is shown below.
•There are two forward paths. The path gains are
•The three feedback loop gains are
•No loops are non-touching, hence
•Since no loops touch the nodes of P2,
therefore
•Because the loops touch the nodes of P1,
hence
•Hence the control ratio T = C/R is

68. 68
Example-6: Find the control ratio C/R for the system given below.
•The two forward path gains are
•The signal flow graph is shown in the figure.
•The five feedback loop gains are
•Hence the control ratio T =
•There are no non-touching loops, hence
•All feedback loops touches the two forward
paths, hence

69. 69
Design Example#1RsIsI
Cs
sV )()()(
111
1
 RsIsV )()(
12 )()()( sIsCsVsCsV
121  )(sV
1 )(sI
1 )(sV
2 Cs R Cs

70. 70
Design Example#2)(
2111
2
1 XXkXsMF  221212
2
20 XkXXkXsM  )(

71. 71
Design Example#2

72. 72
Design Example#2