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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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--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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--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)

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