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Signal Flow GraphsSignal Flow Graphs446446--99
Prof. Neil A.Prof. Neil A. DuffieDuffie
University of WisconsinUniversity of Wisconsin--MadisonMadison
Neil A. Neil A. DuffieDuffie, 1996, 1996
All rights reservedAll rights reserved
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Incorrect Block Diagram ManipulationsIncorrect Block Diagram Manipulations
++ ++
--
++Kp
s(ps + 1)
sKp
Kc
ProcessProcess
R(sR(s))
ControlControl
C(s)C(s)
++ ++
--
++
Kp
s(ps + 1)
s
Kp
Kc
ControlControl
ProcessProcessR(s)R(s)C(s)C(s)
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Correct Block Diagram ManipulationsCorrect Block Diagram Manipulations
++ ++
--
++Kp
s(ps + 1)
s
KcKp
Kc
ProcessProcess
R(s)R(s)
ControlControl
C(s)C(s)
++++
--
++ Kp
s(ps + 1)
s
KcKp
Kc
ProcessProcess
R(s)R(s)
ControlControlC(s)C(s)
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Block Diagram ReductionBlock Diagram Reduction
KcKp
s(
ps+
1)
1+ KcKps(
ps
+1)
1+ sKcKpR(s)R(s)
C(s)C(s)
1+ sK
cK
p
KcKps(
ps + 1) + K
cK
p
R(s)R(s) C(s)C(s)
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Block Diagram with Disturbance InputBlock Diagram with Disturbance Input
GG11
(s)(s)++ ++
--
++R(s)R(s) C(s)C(s)
GG22
(s)(s)
D(s)D(s)CommandCommand DisturbanceDisturbance
H(s)H(s)
A disturbance input is an unwanted orA disturbance input is an unwanted or
unavoidable input signal that affects aunavoidable input signal that affects asystems output. Examples:systems output. Examples:
-- load torque in motor controlload torque in motor control
-- open door in room climate controlopen door in room climate control
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Superposition: C(s) =Superposition: C(s) = CCrr(s) + C(s) + Cdd(s)(s)
GG11(s)(s)--
++
C(s)C(s)
GG22(s)(s)
D(s)D(s)
H(s)H(s)
GG11(s)(s)++
--
R(s)R(s)GG22(s)(s)
H(s)H(s)
++
++
CCdd(s)(s)
CCrr(s)(s)
Note signNote signchange!change!
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Disturbance Portion RedrawnDisturbance Portion Redrawn
C(s)C(s)
GG22(s)(s)
H(s)H(s)
GG11(s)(s)++
--
R(s)R(s)GG22(s)(s)
H(s)H(s)
++
++
CCdd(s)(s)
CCrr(s)(s)
++
--
D(s)D(s)
GG11(s)(s)
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Reduced Block DiagramReduced Block Diagram
C(s)C(s)
R(s)R(s)
++
++
CCdd(s)(s)
CCrr(s)(s)
G2 (s)
1+ G1(s)G2 (s)H(s)
G1(s)G2 (s)
1+ G1(s)G2 (s)H(s)
D(s)D(s)
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Individual Transfer FunctionsIndividual Transfer Functions
With R(s) = 0:With R(s) = 0:C(s)
D(s)
= G2 (s)1+
G1(s)G2 (s)H(s)
With D(s) = 0:With D(s) = 0:
C(s)
R(s)= G1(s)G2 (s)
1+ G1(s)G2 (s)H(s)Transfer equation:Transfer equation:
C(s) = G1(s)G2 (s)R(s) + G2 (s)D(s)1+ G1(s)G2 (s)H(s)
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Utility of Signal Flow GraphsUtility of Signal Flow Graphs
Alternative to block diagram approachAlternative to block diagram approach-- may be better for complex systemsmay be better for complex systems
-- good for highly interwoven systemsgood for highly interwoven systems
-- system variables represented as nodessystem variables represented as nodes
-- branches (lines) between nodes showbranches (lines) between nodes show
relationships between system variablesrelationships between system variables
The flow graph gain formula (Mason)The flow graph gain formula (Mason)allows the system transfer function to beallows the system transfer function to be
directly computed without manipulationdirectly computed without manipulationor reduction of the diagram.or reduction of the diagram.
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Basic Signal Flow GraphBasic Signal Flow Graph
G(s)G(s)
H(s)H(s)
++
--
R(s)R(s) E(s)E(s) C(s)C(s)
R(s)R(s) E(s)E(s)11 C(s)C(s)G(s)G(s)
--H(s)H(s)
InputInput
nodenode
OutputOutput
nodenode
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Signal Flow Graph ExampleSignal Flow Graph Example
GG11(s)(s)++
++
--
++R(s)R(s) C(s)C(s)GG22(s)(s)
D(s)D(s)
H(s)H(s)
++--
E(s)E(s) F(s)F(s) Q(s)Q(s)
R(s)R(s) E(s)E(s)11
F(s)F(s)GG11(s)(s)
--H(s)H(s)
11 GG22(s)(s)C(s)C(s)
Q(s)Q(s)
--11
D(s)D(s)11
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Terms for Masons Gain FormulaTerms for Masons Gain Formula
Path: A branch or sequence of branchesPath: A branch or sequence of branchesthat can be traversed from one node tothat can be traversed from one node to
another.another.
Loop: A closed path, along which noLoop: A closed path, along which no
node is met twice, that originates andnode is met twice, that originates and
terminates in the same node.terminates in the same node.
Nontouching: Two loops areNontouching: Two loops are
nontouching if they do not share anontouching if they do not share acommon node.common node.
Gain: Refers, in this case, to the productGain: Refers, in this case, to the productof transfer functions.of transfer functions.
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Masons Gain Formula:Masons Gain Formula:O(s)
I(s)= Pk k
k
PPkk = the gain of the k= the gain of the k
thth forward pathforward path
between I(s) and O(s).between I(s) and O(s). = 1 = 1 -- (sum of all individual loop gains)(sum of all individual loop gains)
+ (sum of gain products of all+ (sum of gain products of all
combinations of 2 nontouching loops)combinations of 2 nontouching loops)
-- (sum of gain products of all(sum of gain products of all
combinations of 3 nontouching loops)combinations of 3 nontouching loops)+ +
kk = value of for that part of graph= value of for that part of graphnontouching the knontouching the kthth forward path.forward path.
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Example of Gain Formula UseExample of Gain Formula Use
R(s)R(s) E(s)E(s)11 F(s)F(s) GG11(s)(s)
--H(s)H(s)
11 GG22(s)(s) C(s)C(s)
Q(s)Q(s)
--11
D(s)D(s)11
Assume R(s) = 0, desire to find theAssume R(s) = 0, desire to find the
transfer function C(s)/D(s).transfer function C(s)/D(s). There is only one forward path betweenThere is only one forward path between
D(s) and C(s), therefore k = 1.D(s) and C(s), therefore k = 1. There are two loops. They are touching.There are two loops. They are touching.
f G
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Example of Gain Formula UseExample of Gain Formula Use
R(s)R(s) E(s)E(s)11 F(s)F(s) GG11(s)(s)
--H(s)H(s)
11 GG22(s)(s) C(s)C(s)
Q(s)Q(s)
--11
D(s)D(s)11
PP
11 = G= G
22(s)(s)
= 1= 1 -- [[--GG11(s)G(s)G22(s)H(s)(s)H(s) -- GG11(s)G(s)G22(s)](s)]
11 = 1 (Both loops touch the k= 1 (Both loops touch the kthth path)path)
E l f G i F l U
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Example of Gain Formula UseExample of Gain Formula Use
R(s)R(s) E(s)E(s)11 F(s)F(s) GG11(s)(s)
--H(s)H(s)
11 GG22(s)(s) C(s)C(s)
Q(s)Q(s)
--11
D(s)D(s)11
C(s)
D(s) =Pk kk
C(s)
D(s)= G2 (s)
1+ G1(s)G2 (s)H(s) + G1(s)G2 (s)