control systems (cs) · •the block diagram reduction technique requires successive application of...
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Control Systems (CS)
Dr. Imtiaz HussainAssociate Professor
Mehran University of Engineering & Technology Jamshoro, Pakistan
email: [email protected] :http://imtiazhussainkalwar.weebly.com/
Lecture-10-11Signal Flow Graphs
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Outline
• Introduction to Signal Flow Graphs
– Definitions
– Terminologies
– Examples
• Mason’s Gain Formula
– Examples
• Signal Flow Graph from Block Diagrams
• Design Examples
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Introduction
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• Alternative method to block diagram representation,developed by Samuel Jefferson Mason.
• Advantage: the availability of a flow graph gain formula,also called Mason’s gain formula.
• A signal-flow graph consists of a network in which nodesare connected by directed branches.
• It depicts the flow of signals from one point of a systemto another and gives the relationships among the signals.
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Fundamentals of Signal Flow Graphs
• Consider a simple equation below and draw its signal flow graph:
• The signal flow graph of the equation is shown below;
• Every variable in a signal flow graph is designed by a Node.• Every transmission function in a signal flow graph is designed by a
Branch.• Branches are always unidirectional.• The arrow in the branch denotes the direction of the signal flow.
axy
x ya
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Signal-Flow Graph Models
Y1 s( ) G11 s( ) R1 s( ) G12 s( ) R2 s( )
Y2 s( ) G21 s( ) R1 s( ) G22 s( ) R2 s( )
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Signal-Flow Graph Models
a11 x1 a12 x2 r1 x1
a21 x1 a22 x2 r2 x2
r1 and r2 are inputs and x1 and x2 are outputs
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Signal-Flow Graph Models
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203
312
2101
hxx
gxfxx
exdxx
cxbxaxx
b
x4x3x2x1
x0 h
f
g
e
d
c
a
xo is input and x4 is output
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Construct the signal flow graph for the following set of simultaneous equations.
• There are four variables in the equations (i.e., x1,x2,x3,and x4) therefore four nodes arerequired to construct the signal flow graph.
• Arrange these four nodes from left to right and connect them with the associatedbranches.
• Another way to arrange this graph is shown in the figure.
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Terminologies• An input node or source contain only the outgoing branches. i.e., X1
• An output node or sink contain only the incoming branches. i.e., X4
• A path is a continuous, unidirectional succession of branches along which no
node is passed more than ones. i.e.,
• A forward path is a path from the input node to the output node. i.e.,
X1 to X2 to X3 to X4 , and X1 to X2 to X4 , are forward paths.
• A feedback path or feedback loop is a path which originates and terminates on
the same node. i.e.; X2 to X3 and back to X2 is a feedback path.
X1 to X2 to X3 to X4 X1 to X2 to X4 X2 to X3 to X4
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Terminologies
• A self-loop is a feedback loop consisting of a single branch. i.e.; A33 is a self
loop.
• The gain of a branch is the transmission function of that branch.
• The path gain is the product of branch gains encountered in traversing a path.
i.e. the gain of forwards path X1 to X2 to X3 to X4 is A21A32A43
• The loop gain is the product of the branch gains of the loop. i.e., the loop gain
of the feedback loop from X2 to X3 and back to X2 is A32A23.
• Two loops, paths, or loop and a path are said to be non-touching if they haveno nodes in common.
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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
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Consider the signal flow graph below and identify the following
• There are two forward path gains;
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Consider the signal flow graph below and identify the following
• There are four loops
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Consider the signal flow graph below and identify the following
• Nontouching loop gains;
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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.
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Input and output Nodes
a) Input node
b) Output node
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(c) Forward Paths
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(d) Feedback Paths or Loops
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(d) Feedback Paths or Loops
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(d) Feedback Paths or Loops
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(d) Feedback Paths or Loops
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(e) Self Loop(s)
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(f) Loop Gains of the Feedback Loops
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(g) Path Gains of the Forward Paths
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Mason’s Rule (Mason, 1953)
• The block diagram reduction technique requires successive
application of fundamental relationships in order to arrive at the
system transfer function.
• On the other hand, Mason’s rule for reducing a signal-flow graph
to a single transfer function requires the application of one
formula.
• The formula was derived by S. J. Mason when he related the
signal-flow graph to the simultaneous equations that can be
written from the graph.
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Mason’s Rule:
• The transfer function, C(s)/R(s), of a system represented by a signal-flow graphis;
Where
n = number of forward paths.Pi = the i th forward-path gain.∆ = Determinant of the system∆i = Determinant of the ith forward path
• ∆ is called the signal flow graph determinant or characteristic function. Since ∆=0 is the system characteristic equation.
n
iiiP
sR
sC 1
)(
)(
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Mason’s Rule:
∆ = 1- (sum of all individual loop gains) + (sum of the products of the gainsof all possible two loops that do not touch each other) – (sum of theproducts of the gains of all possible three loops that do not touch eachother) + … and so forth with sums of higher number of non-touching loopgains
∆i = value of Δ for the part of the block diagram that does not touch the i-th forward path (Δi = 1 if there are no non-touching loops to the i-th path.)
n
iiiP
sR
sC 1
)(
)(
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Systematic approach
1. Calculate forward path gain Pi for each forwardpath i.
2. Calculate all loop transfer functions
3. Consider non-touching loops 2 at a time
4. Consider non-touching loops 3 at a time
5. etc
6. Calculate Δ from steps 2,3,4 and 5
7. Calculate Δi as portion of Δ not touching forwardpath i
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Example#1: Apply Mason’s Rule to calculate the transfer function of the system represented by following Signal Flow Graph
2211 PP
R
CTherefore,
24313242121411 HGGGLHGGGLHGGL ,,
There are three feedback loops
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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
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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
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Example#1: Continue
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Example#2: Apply Mason’s Rule to calculate the transfer function of the system represented by following Signal Flow Graph
774663332221 HGLHGLGHLHGL ,,,
2. Calculate all loop gains.
3. Consider two non-touching loops.L1L3 L1L4
L2L4 L2L3
2) (pathand1) (path 8765243211 GGGGPGGGGP
1. Calculate forward path gains for each forward path.
P1
P2
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4. Consider three non-touching loops.
None.
5. Calculate Δ from steps 2,3,4.
4232413143211 LLLLLLLLLLLL
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7733663377226622
776633221
HGGHHGGHHGHGHGHG
HGHGGHHG
Example#2: continue
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Example#2: continue
Eliminate forward path-1
431 1 LL
212 1 LL
Eliminate forward path-2
77661 1 HGHG
33222 1 HGHG
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2211 PP
sR
sY
)(
)(
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Example#2: continue
773366337722662277663322
3322876577664321
1
11
HGGHHGGHHGHGHGHGHGHGGHHG
HGHGGGGGHGHGGGGG
sR
sY
)(
)(
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Example#3
• Find the transfer function, C(s)/R(s), for the signal-flowgraph in figure below.
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Example#3
• There is only one forward Path.
)()()()()( sGsGsGsGsGP 543211
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Example#3
• There are four feedback loops.
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Example#3
• Non-touching loops taken two at a time.
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Example#3
• Non-touching loops taken three at a time.
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Example#3
Eliminate forward path-1
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Example#4: Apply Mason’s Rule to calculate the transfer function of the system represented by following Signal Flow Graph
332211
3
1 PPPP
sR
sC iii
)(
)(
There are three forward paths, therefore n=3.
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Example#4: Forward Paths
722 AP 76655443321 AAAAAP
766554423 AAAAP
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Example#4: Loop Gains of the Feedback Loops
23321 AAL
34432 AAL
45543 AAL
56654 AAL
67765 AAL
776 AL
2334427 AAAL
6776658 AAAL
23344557729 AAAAAL
23344556677210 AAAAAAL
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Example#4: two non-touching loops
31LL
41LL
51LL
61LL
81LL
42LL
52LL
62LL
82LL
53LL
63LL
64LL
74LL
75LL 87LL
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Example#4: Three non-touching loops
31LL
41LL
51LL
61LL
81LL
42LL
52LL
62LL
82LL
53LL
63LL
64LL
74LL
75LL 87LL
47
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G1 G4G3
From Block Diagram to Signal-Flow Graph ModelsExample#5
-
-
-
C(s)R(s)G1 G2
H2
H1
G4G3
H3
E(s) X1
X2
X3
R(s) C(s)
-H2
-H1
-H3
X1 X2 X3E(s)1 G2
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1;
)(1
143211
14323234321
GGGGP
HGGHGGHGGGG
14323234321
4321
1)(
)(
HGGHGGHGGGG
GGGG
sR
sCG
R(s)
-H2
1G4G3G2G11 C(s)
-H1
-H3
X1 X2 X3E(s)
From Block Diagram to Signal-Flow Graph ModelsExample#5
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G1
G2
+-
+
-
-
-
+ C(s)R(s) E(s)
Y2
Y1X1
X2
-
1
-1
1
-1
-1
-1-1
1
1
G1
G2
1
R(s) E(s) C(s)
X1
X2Y2
Y1
Example#6
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Example#6
1-1
1
-1
-1
-1 -1
1
1
G1
G2
1
R(s) E(s) C(s)
X1
X2Y2
Y1
7 loops:
3 ‘2 non-touching loops’ :
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Example#6
1-1
1
-1
-1
-1 -1
1
1
G1
G2
1
R(s) E(s) C(s)
X1
X2Y2
Y1
212 G4G2G1Δ Then:
4 forward paths:211 G1 Δ1G1)(p 1
1Δ1G1)(G1)(p 221 2
132 G1Δ1G1p 3
1Δ1G1G1p 412 452
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Example#6
We have
212
2112
421
2
GGG
GGGG
p
sR
sC kk
)(
)(
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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
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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
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Design Example#1
RsIsICs
sV )()()( 111
1
RsIsV )()( 12 )()()( sIsCsVsCsV 121
)(sV1 )(sI1)(sV2
Cs R
Cs
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Design Example#2
)( 21112
1 XXkXsMF 2212122
20 XkXXkXsM )(
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Design Example#2
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Design Example#2
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END OF LECTURES-10-11
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