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Chapter 3Chapter3
BlockDiagramsandSignalFlowGraphs
AutomaticControlSystems,9thEdition
Farid Golnaraghi Simon Fraser UniversityFaridGolnaraghi,SimonFraserUniversityBenjaminC.Kuo,UniversityofIllinois
1
IntroductionIntroduction
Inthischapter,wediscussgraphicaltechniquesformodelingcontrolsystemsandtheirunderlyingmathematics.
WealsoutilizetheblockdiagramreductiontechniquesandtheMasonsgainformulatofindthetransferfunctionoftheoverallcontrolsystem.
LateroninChapters4and5,weusethematerialpresentedinthischapterandChapter2tofullyp p p ymodelandstudytheperformanceofvariouscontrolsystems.y
2
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.
3
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.
4
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.
5
Figure34Blockdiagramelementsoftypicalsensingdevicesofcontrolsystems.(a)Subtraction.(b)Addition.(c)Additionandsubtraction.
6
Figure35TimeandLaplacedomainblockdiagrams.
7
EXAMPLE311
Figure36BlockdiagramsG1(s)andG2(s)connectedinseries.
8
EXAMPLE312
Figure37BlockdiagramsG1(s)andG2(s)connectedinparallel.
9
Basicblockdiagramofafeedbackcontrolsystem
Figure 38 Basic block diagram of a feedback control systemFigure38Basicblockdiagramofafeedbackcontrolsystem.
10
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)
11
( ) ( ) ( ) ( ) ( ) ( ) ( )
M(s) = Y(s) / R(s) = G(s) / (1 + G(s)H(s))
312RelationbetweenMathematicalEquationsandBlockDiagrams
Figure39GraphicalrepresentationofEq.(316)usingacomparator.
12
13
Figure312(a)Factorizationof1/stermintheinternalfeedbackloopofFig.311.(b)FinalblockdiagramrepresentationofEq.(317)inLaplacedomain.
14Figure313BlockdiagramofEq.(317)inLaplacedomainwithV(s)representedas
theoutput.
15
Figure314(a)Factorizationof.(b)AlternativediagramrepresentationofEq.(317)inLaplacedomain.
2n
16
Figure315AblockdiagramrepresentationofEq.(319)inLaplacedomain.
313BlockDiagramReduction:Branchpointrelocation
17
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.
19
20 Figure318(Continued)
314BlockDiagramofMultiInputSystemsSpecialCase:SystemswithaDisturbance
Figure 319 Block diagram of a system undergoing disturbance.
21
Figure3 19Blockdiagramofasystemundergoingdisturbance.
Figure320BlockdiagramofthesysteminFig.319whenD(s)=0.
Figure321BlockdiagramofthesysteminFig.319whenR(s)=0.
22
Figure322Blockdiagramrepresentations ofamultivariablesystem. Figure322Blockdiagramrepresentations ofamultivariablefeedbackcontrolsystem.
23
24
32SIGNALFLOWGRAPHS(SFGs)
25
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
27
(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)
28
)loop: loopgain a23a32
324SFGAlgebra
Figure329~31Signalflowgraph.
29
327GainFormulaforSFG
30
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
31
k branch
Figure332SignalflowgraphofthefeedbackcontrolsystemshowninFig.38.
32
Figure 333 Signalflow graph for Example 323.
33
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 )
35
( k k yin y2) Ex.325&326
329ApplicationoftheGainFormulatoBlockDiagrams EXAMPLE326
Figure334(a)Blockdiagramofacontrolsystem.(b)Equivalentsignalflowgraph.36
3 2 10 Simplified Gain Formula3210SimplifiedGainFormula
37