Chapter 3Chapter3

BlockDiagramsandSignalFlowGraphs

AutomaticControlSystems,9thEdition

Farid Golnaraghi Simon Fraser UniversityFaridGolnaraghi,SimonFraserUniversityBenjaminC.Kuo,UniversityofIllinois

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IntroductionIntroduction

Inthischapter,wediscussgraphicaltechniquesformodelingcontrolsystemsandtheirunderlyingmathematics.

WealsoutilizetheblockdiagramreductiontechniquesandtheMasonsgainformulatofindthetransferfunctionoftheoverallcontrolsystem.

LateroninChapters4and5,weusethematerialpresentedinthischapterandChapter2tofullyp p p ymodelandstudytheperformanceofvariouscontrolsystems.y

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Objectives of this ChapterObjectivesofthisChapter1. Tostudyblockdiagrams,theircomponents,andtheir

d l hunderlyingmathematics.

2. Toobtaintransferfunctionofsystemsthroughblockdiagrami l ti d d timanipulationandreduction.

3. Tointroducethesignalflowgraphs.

4 T t bli h ll l b t bl k di d i l4. Toestablishaparallelbetweenblockdiagramsandsignalflowgraphs.

5 To use Masons gain formula for finding transfer function of5. TouseMason sgainformulaforfindingtransferfunctionofsystems.

6 To introduce state diagrams6. Tointroducestatediagrams.

7. TodemonstratetheMATLABtoolsusingcasestudies.

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31BLOCKDIAGRAMS

Block diagrams provide a better understanding of the composition and interconnection of the components of a system. It can be used, together with transfer functions, to describe the cause-and-effect relationships throughout the system.

Figure31Asimplifiedblockdiagramrepresentationofaheatingsystem.

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311TypicalElementsofBlockDiagramsinControlSystems

The common elements in block diagrams of most control systems include:The common elements in block diagrams of most control systems include:

Comparators Blocks representing individual component transfer functions, including:oc s ep ese g d v du co po e s e u c o s, c ud g:

Reference sensor (or input sensor) Output sensor

Actuator Controller Plant (the component whose variables are to be controlled) Input or reference signals Output signals Disturbance signal Feedback loops

Figure33Blockdiagramrepresentationofageneralcontrolsystem.

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Figure34Blockdiagramelementsoftypicalsensingdevicesofcontrolsystems.(a)Subtraction.(b)Addition.(c)Additionandsubtraction.

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Figure35TimeandLaplacedomainblockdiagrams.

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EXAMPLE311

Figure36BlockdiagramsG1(s)andG2(s)connectedinseries.

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EXAMPLE312

Figure37BlockdiagramsG1(s)andG2(s)connectedinparallel.

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Basicblockdiagramofafeedbackcontrolsystem

Figure 38 Basic block diagram of a feedback control systemFigure38Basicblockdiagramofafeedbackcontrolsystem.

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Feedback Control System

R(s) : (reference input), (input), command( ) ( p ),( p ), Y(s) : (output, controlled variable), (response)B(s) : (feedback signal)E(s) :(error signal) actuating signalE(s) : (error signal) actuating signalG(s) : (forward-path transfer function)H(s) : (feedback transfer function, feedback gain)G(s)H(s) : (loop transfer function), (open-loop transfer function)M(s) = Y(s)/R(s) : (closed-loop transfer function, system transfer function)B( ) H( )Y( )B(s)=H(s)Y(s)

E(s)=R(s) B(s)

Y(s)=G(s)E(s)=G(s)R(s) G(s)B(s)

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( ) ( ) ( ) ( ) ( ) ( ) ( )

M(s) = Y(s) / R(s) = G(s) / (1 + G(s)H(s))

312RelationbetweenMathematicalEquationsandBlockDiagrams

Figure39GraphicalrepresentationofEq.(316)usingacomparator.

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Figure312(a)Factorizationof1/stermintheinternalfeedbackloopofFig.311.(b)FinalblockdiagramrepresentationofEq.(317)inLaplacedomain.

14Figure313BlockdiagramofEq.(317)inLaplacedomainwithV(s)representedas

theoutput.

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Figure314(a)Factorizationof.(b)AlternativediagramrepresentationofEq.(317)inLaplacedomain.

2n

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Figure315AblockdiagramrepresentationofEq.(319)inLaplacedomain.

313BlockDiagramReduction:Branchpointrelocation

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Figure316(a)BranchpointrelocationfrompointP to(b)pointQ.

313BlockDiagramReduction:Comparatorrelocation

18Figure317(a)ComparatorrelocationfromtherighthandsideofblockG2(s)to

(b)thelefthandsideofblockG2(s).

EXAMPLE315Findtheinputoutputtransferfunctionofthesystem

Figure318(a)Originalblockdiagram.(b)MovingthebranchpointatY1 totheleftofblockG2.(c)CombiningtheblocksG1,G2,andG3.(d)Eliminatingtheinnerfeedbackloop.

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20 Figure318(Continued)

314BlockDiagramofMultiInputSystemsSpecialCase:SystemswithaDisturbance

Figure 319 Block diagram of a system undergoing disturbance.

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Figure3 19Blockdiagramofasystemundergoingdisturbance.

Figure320BlockdiagramofthesysteminFig.319whenD(s)=0.

Figure321BlockdiagramofthesysteminFig.319whenR(s)=0.

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Figure322Blockdiagramrepresentations ofamultivariablesystem. Figure322Blockdiagramrepresentations ofamultivariablefeedbackcontrolsystem.

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32SIGNALFLOWGRAPHS(SFGs)

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SignalFlowGraphs(SFG,) causeandeffect(node)(branch), node(variable) branch(gain).

[xj=aijxi node branch]

output= gainxinput ,jthoutput= (gainfrom k toj)x(kthcause)Yj(s)= Gkj(s)Yk(s)

SFG Terms(Inputnode,Source)

branch node] x

SFGTerms

] x1(Outputnode,Sink)

branch node] x26

] x4

(Gain)branch) x1 x2 branch a21,

x2 =a21x1+().( :x2/x1 =a21)

(Path) branch,

.,,.,, node.) x1 x3 path.

(Forwardpath) node node path

) x1 x4 forwardpath 2.) 1 4 p

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(Feedbackpath) d h node path.

Loop,Selfloop(loop),

node ( branch loop)selfloop.)

Nontouchingloops

(Pathgain) h b h i

g pLoop node loop

path branchgain.) path: pathgain a21a42

( : pathgain a21a42 x4/x1=a21a42) Loopgain

loop branchgain (loop pathgain)

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)loop: loopgain a23a32

324SFGAlgebra

Figure329~31Signalflowgraph.

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327GainFormulaforSFG

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M:Thegainbetweeninputnodeyin andoutputnodeyout

GainFormulaforSFG(Mason'sgainrule)

g p yin p yout

M=yout/yin =Mkk / ,k=1, ,N

,N:TotalnumberofforwardpathMk :k forwardpath gain : signal flow graph determinant characteristic function :signalflowgraphdeterminant characteristicfunction =1 Li1+Lj2 Lk3+..L = r nontouching loops mth possible combination gain product ( 1 r L )Lmr=rnontouchingloops m possiblecombination gainproduct(1 r L) =1 ( loop)

+(2 loop)(3 loop) (3 loop)

+..L=loopsk:kth forwardpath nontouching partk=k graph k b h

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k branch

Figure332SignalflowgraphofthefeedbackcontrolsystemshowninFig.38.

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Figure 333 Signalflow graph for Example 323.

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Figure3 33Signal flowgraphforExample3 2 3.

Figure333SignalflowgraphforExample324.34

Ex.322M1 =G(s)

L = G(s)H(s)L11=G(s)H(s)

1=1 =1+G(s)H(s)

Closedlooptransferfunction

M=Y(s) /R(s)=M1 1 / =G(s)/(1+G(s)H(s))

y2/y1 =

/

Ex.324

y4/y1 =

* chosenoutput sameN i t d t t d i

yout/y2 =(yout/yin)/(y2/yin)=(Mkk from yintoyout/ )/(Mkk from yintoy2/ )

Noninputnode outputnode gain

=(Mkk from yintoyout)/(Mkk from yin toy2 )

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( k k yin y2) Ex.325&326

329ApplicationoftheGainFormulatoBlockDiagrams EXAMPLE326

Figure334(a)Blockdiagramofacontrolsystem.(b)Equivalentsignalflowgraph.36

3 2 10 Simplified Gain Formula3210SimplifiedGainFormula

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