A COMPILATION OF LECTURE NOTES IN CONTROL SYSTEMCHAPTER 7 CATADMAN, LIZETTE IVY G. CHAPTER 7 SIGNAL FLOW GRAPHS 7.1DEFINITIONS ASignalFlowGraphisapictorialrepresentationofthecause-and-effectrelationshipbetweentheinputandtheoutputofaphysicalsystem, justlikeBlockDiagram.Itgraphicallydisplaysthetransmissionofsignals through the system but is easier to draw and easier to manipulate than block diagram.The simplest form of a Signal Flow Graph would consist of a single branch, with an input, transmittance, and output Abranchisterminatedatbothendsbynodes,representedby thick dots, and the line whose arrow, situated in the middle, represents the direction of the signal.Branches are always unidirectional. 7.2SIGNAL FLOW GRAPH ALGEBRA 7.2.1THE ADDITION RULE Thevalueofthevariabledesignatedbyanodeisequalto the sum of all signals entering the node. ==n1 JJ IJ IR T Y 7.2.2THE TRANSMISSION RULE Thevalueofthevariabledesignatedbyanodeis transmitted on every branch leaving that node. K K 1 1R T Y =K K 2 2R T Y =K K 3 3R T Y =K K 4 4R T Y =K K 5 5R T Y = 7.2.3THE MULTIPLICATION RULE Acascaded(series)connectionofn-1branches(wherenisthenumberofnodes) withtheirrespectiveindividualtransmissionfunctionscanbereplacedbyasinglebranch with a new transmission function equal to the product of the original ones.

43 32 21 newT T T T = | |1 43 32 21 1 new 4X T T T X T X = =A COMPILATION OF LECTURE NOTES IN CONTROL SYSTEMCHAPTER 7 CATADMAN, LIZETTE IVY G. 7.3ADDITIONAL DEFINITIONS A Path is a continuous, unidirectional succession of branches along which no node is passed more than once.For example, 1Xto 2Xto 3Xto 4Xto 5X ; 1Xto 2Xto 4Xto 5X ; and 3Xto 2X . AnInputNodeorSourceisanodewithonlyoutgoingbranches.Forexample, 1X isan input node. AnOutputNodeorSinkisanodewithonlyincomingbranches.Forexample, 5X isan output node. A Forward Path is a path from the input node to the output node. For example, 1Xto 2Xto 3Xto 4Xto 5Xand 1Xto 2Xto 4Xto 5X . A Feedback Path or Feedback Loop is a path which originates and terminates on the same node. For example, 2Xto 3Xback to 2X . A Self-Loop is a feedback loop consisting of a single branch. For example, 3Xback to 3X . 7.4MASONS GAIN RULE MasonsGainRuleistheformulausedforfindingtheTransferFunctionofaSingle-Input, Single-Output System of a Signal Flow Graph. TheGainofaBranchisthetransmissionfunctionofthatbranchwhenthetransmission function is a multiplicative operator. The Path Gain is the product of the branch gains encountered in traversing a path. The Loop Gain is the product of the branch gains encountered in traversing a loop. TheDeterminantof theSignalFlowGraphorCharacteristic Function,=1 (Sumofall theLoopGains)+(Sumofproductsofthegainsofallcombinationsof2non-touchingloops) (Sum of products of gains of all combinations of 3 nontouching loops) + The Co-Factor of the Path is the determinant of the Signal Flow Graph formed by deleting all loops touching the path. TheCo-FactorofthePath,k=1(SumofalltheLoopGainsnottouchingthePath)+ (Sum of products of the gains of all combinations of 2 non-touching loops not touching the Path) (Sum of products of gains of all combinations of 3 nontouching loops not touching the Path) + A COMPILATION OF LECTURE NOTES IN CONTROL SYSTEMCHAPTER 7 CATADMAN, LIZETTE IVY G. Twoloopsaresaidtobetouchingiftheyhaveanynode/sorbranch/esincommon; otherwise, they are non-touching. 7.5TRANSFERFUNCTIONSFORSINGLE-INPUT,SINGLE-OUTPUTSIGNALFLOW GRAPHS USING MASONS GAIN RULE ... P P P) s ( T3 3 2 2 1 1+ + += Example: Solve for the Transfer Function of the Signal Flow Graph using Masons Gain Rule. Solution: Identify the Paths and Loops.Using the identified Paths and Loops, obtain the Path Gains and Loop Gains. For Paths: ( ) ( ) ( ) ( )3 ss 61 33 ss2 1 1 P+=|.|

\|+= ( ) ( ) ( ) ( ) 18 1 3 6 1 2 P = = For Loops: ( )3 ss 443 ss1 L+= |.|

\|+= ( )1 ss 31 ss3 2 L+= |.|

\|+= For Co-Factors: 1 1 = 1 2 = A COMPILATION OF LECTURE NOTES IN CONTROL SYSTEMCHAPTER 7 CATADMAN, LIZETTE IVY G. The Determinant: ( )( ) ( ) ( ) ( )( ) ( ) 1 s 3 s3 s s 3 1 s s 4 1 s 3 s1 ss 33 ss 41 2 L 1 L 1+ ++ + + + + +=)`((

++((

+ = + = The Transfer Function: ( ) ( )( )( ) ( ) ( ) ( )( ) ( )( )( ) ( ) ( ) ( )( ) ( )( ) ( ) | |( ) ( ) ( ) ( ) 3 s s 3 1 s s 4 1 s 3 s3 s 18 s 6 1 s1 s 3 s3 s s 3 1 s s 4 1 s 3 s3 s3 s 18 s 6) s ( R) s ( Y) s ( T1 s 3 s3 s s 3 1 s s 4 1 s 3 s1 18 13 ss 62 2 P 1 1 P) s ( R) s ( Y) s ( T+ + + + + ++ +=+ ++ + + + + +++ = =+ ++ + + + + + +|.|

\|+=+= = Example: Solve for the Transfer Function of the Signal Flow Graph using Masons Gain Rule. Solution: Identify the Paths and Loops.Using the identified Paths and Loops, obtain the Path Gains and Loop Gains. For Paths: ( ) ( )( )( ) 4 s 8 s s81 P1s1 s1 8 s8 4 s11 1 P2 22+ +=|.|

\||.|

\||.|

\|+|.|

\|+=

( ) ( ) ( )( ) ( )( ) ( )2 3 2 221 s s10s s s 1 s102 P1s1 s110s s1 1 s11 2 P+=+ +=|.|

\||.|

\||.|

\|+|.|

\|+= For Loops: A COMPILATION OF LECTURE NOTES IN CONTROL SYSTEMCHAPTER 7 CATADMAN, LIZETTE IVY G. ( )( ) 1 s s4s s14 1 L2+= |.|

\|+ = s 2 L = ( )( ) ( ) 2 s 3 s s302 ss 3 s3 s1 s110 3 L+ += |.|

\|+|.|

\|+|.|

\||.|

\|=( )8 s568 s87 4 L+= |.|

\|+ = ( )s6s16 5 L= |.|

\| = For Co-Factors: ( ) ( )( )( )( ) | |( ) ( )( ) 1 s ss 4 1 s s 4 1 s ss1 s s4s1 s s41 2 L 1 L 2 L 1 L 1 12++ + + + +=)`((

++((

++ = + + = ( ) ( ) 1 s s 1 s 1 2 L 1 2 + = + = = = The Determinant: ( ) ( ) ( ) 5 L 2 L 1 L 4 L 2 L 1 L 5 L 2 L 4 L 2 L 5 L 1 L 4 L 1 L 2 L 1 L 5 L 4 L 3 L 2 L 1 L 1 + + + + + + + + + + = ( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ))`((

((

++)`((

+((

+)`((

+)`((

+ +)`((

((

++)`((

+((

++)`((

++)`((

+((

++((

+ ++ +((

+ =s6s1 s s48 s56s1 s s4s6s8 s56ss61 s s48 s561 s s4s1 s s4s68 s562 s 3 s s30s1 s s41 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ))`++ +)`++++++ ++++)`+ + + ++ + + + + + + ++ + + ++ + + + + + + + + + =1 s s248 s 1 s22468 ss 561 s s248 s 1 s s2241 s48 s 3 s 2 s 1 s s8 s 3 s 2 s 1 s 6 3 s 2 s 1 s s 568 s 3 s 2 s 1 s s8 s 1 s 30 8 s 3 s 2 s 1 s s 8 s 3 s 2 s 4 -122 The Determinant: A COMPILATION OF LECTURE NOTES IN CONTROL SYSTEMCHAPTER 7 CATADMAN, LIZETTE IVY G. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )( ) ( )( )( ) ( ) )`+ ++)`+ ++ + + + + + + + + ++)`+ + + ++ + + + + + + + + + ++ + + + + + + + + + =8 s 1 s s8 s 24 - 224s -8 s 1 s s8 s 1 s s 6 8 s 1 s s 56 8 s 24 s 224 8 s s 48 s 3 s 2 s 1 s s8 s 3 s 2 s 1 s 6 3 s 2 s 1 s s 568 s 3 s 2 s 1 s s8 s 1 s 30 8 s 3 s 2 s 1 s s 8 s 3 s 2 s 4 -122 3 22 ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) 8 s 3 s 2 s 1 s s8 s 3 s 2 s s 24 3 s 2 s s 224 8 s 3 s 2 s 1 s s 68 s 3 s 2 s 1 s s8 s 3 s 2 s 1 s s 56 8 s 3 s 2 s 24 3 s 2 s s 224 8 s 3 s 2 s s 48 s 3 s 2 s 1 s s8 s 3 s 2 s 1 s s 6 3 s 2 s 1 s s 56 8 s 1 s s 308 s 3 s 2 s 1 s s8 s 3 s 2 s 1 s s 8 s 3 s 2 s s 48 s 3 s 2 s 1 s s8 s 3 s 2 s 1 s s22 223 222 223 222+ + + ++ + + + + + + + + + ++++ + + ++ + + + + + + + + + + + + + +++ + + ++ + + + + + + + + + + ++ + + ++ + + + + + + +++ + + ++ + + += The Transfer Function: ( )( )( ) ( )( ) ( )( )( ) ( ) | |( )( )( ) ( )( ) ( ) | | ( )( )( )( )( ) 4 s 8 s 1 s s4 s 8 s 108 s 4 1 s s 4 1 s s 81 s s104 s 8 s 1 s ss 4 1 s s 4 1 s s 8) s ( T1 s1 s s101 s ss 4 1 s s 4 1 s s 4 s 8 s s82 2 P 1 1 P) s ( R) s ( Y) s ( T2 32 23 2 322 322 2+ + ++ + + + + + + +=+++ + ++ + + + +=+((

++((

++ + + + +((

+ +=+= = 7.6TRANSFERFUNCTIONSFORMULTIPLE-INPUT,MULTIPLE-OUTPUTSIGNALFLOW GRAPHS USING SUPERPOSITION METHOD AND MASONS GAIN RULE TheSuperpositionMethodsolvesfortheTransferFunctionsbytransformingtheMultiple-Input,Multiple-OutputSystemintoasimplifiedSingle-Input,Single-OutputSystemwhichwould represent the relationship between a certain input with a particular output. Step 1: Ignore all outputs except the one being considered. Step 2: Set all inputs, except the one being considered, to be equal to zero. Step 3:Draw the resulting Single-Input, Single-Output Signal Flow Graph. Step 4:Solve the Transfer Function of the resulting Single-Input, Single-Output Signal Flow Graph using Masons Gain Rule. Step 5:Repeat Steps #2 to #4 for each of the remaining inputs. Step 6: When all inputs have been considered, take the next output and repeat Steps #1 to #5. Do this process for each of the remaining outputs. ... P P P) s ( R) s ( Y) s ( T3 3 2 2 1 1JIIJ+ + += = A COMPILATION OF LECTURE NOTES IN CONTROL SYSTEMCHAPTER 7 CATADMAN, LIZETTE IVY G. Example: Solve for the Transfer Functions of the Signal Flow Graph using Masons Gain Rule. Solution: It is advisable to obtain the Loop Gains first and the Determinant.The loops are usually the same regardless whichever input and output combination is being considered. For Loops:

( )2 ss 42 ss4 1 L+= |.|

\|+ = ( ) ( )( ) 3 s s40103 s1 s14 2 L+= |.|

\|+|.|

\| = The Determinant: ( )( )( ) ( ) ( ) ( )( ) ( ) 2 s 3 s s2 s 40 3 s s 4 2 s 3 s s3 s s402 ss 41 2 L 1 L 12+ ++ + + + + +=)`((

++((

+ = + = Ignore) s ( 2 Y , Set0 ) s ( 2 R = For Paths:

( ) ( )2 ss12 ss1 1 P+= |.|

\|+= ( ) ( ) ( )( ) 3 s s101 103 s1 s11 2 P+= |.|

\|+|.|

\|= For Co-Factors: 1 1 = 1 2 = A COMPILATION OF LECTURE NOTES IN CONTROL SYSTEMCHAPTER 7 CATADMAN, LIZETTE IVY G. The Transfer Function: ( )( ) ( )( ) ( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( ) 2 s 40 3 s s 4 2 s 3 s s2 s 10 3 s s2 s 3 s s2 s 40 3 s s 4 2 s 3 s s2 s 3 s s2 s 10 3 s s) s ( 1 R) s ( 1 Y) s ( T2 s 3 s s2 s 40 3 s s 4 2 s 3 s s13 s s1012 ss2 2 P 1 1 P) s ( 1 R) s ( 1 Y) s ( T222211211+ + + + + ++ + +=+ ++ + + + + ++ ++ + += =+ ++ + + + + +((

++ |.|

\|+=+= = Ignore) s ( 2 Y , Set0 ) s ( 1 R = For Path: For Co-Factor: 1 1 = The Transfer Function: ( )( ) ( ) ( ) ( )( ) ( )( )( ) ( ) ( ) ( ) 2 s 40 3 s s 4 2 s 3 s s2 s s 102 s 3 s s2 s 40 3 s s 4 2 s 3 s s13 s101 1 P) s ( 2 R) s ( 1 Y) s ( T2 2 12+ + + + + ++=+ ++ + + + + +|.|

\|+= = = Ignore) s ( 1 Y , Set0 ) s ( 2 R = ( ) ( )3 s10103 s11 1 P+=|.|

\|+=A COMPILATION OF LECTURE NOTES IN CONTROL SYSTEMCHAPTER 7 CATADMAN, LIZETTE IVY G. For Path: For Co-Factor: 1 1 = The Transfer Function: ( ) ( )( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( ) 2 s 40 3 s s 4 2 s 3 s s2 s2 s 3 s s2 s 40 3 s s 4 2 s 3 s s13 s s11 1 P) s ( 1 R) s ( 2 Y) s ( T2 2 21+ + + + + ++=+ ++ + + + + +((

+= = = Ignore) s ( 1 Y , Set0 ) s ( 1 R = For Path: For Co-Factor: ( )2 s2 s 52 ss 4 2 s2 ss 41 1 L 1 1++=+ + += |.|

\|+ = = ( ) ( )( ) 3 s s113 s1 s11 1 P+=|.|

\|+|.|

\|=( ) ( )3 s113 s11 1 P+= |.|

\|+=A COMPILATION OF LECTURE NOTES IN CONTROL SYSTEMCHAPTER 7 CATADMAN, LIZETTE IVY G. The Transfer Function: ( ) ( ) ( ) ( )( ) ( )( )( ) ( ) ( ) ( ) 2 s 40 3 s s 4 2 s 3 s s2 s 5 s2 s 3 s s2 s 40 3 s s 4 2 s 3 s s2 s2 s 5 3 s11 1 P) s ( 2 R) s ( 2 Y) s ( T2 2 22+ + + + + ++=+ ++ + + + + +|.|

\|++|.|

\|+= = = 7.7CONVERSION BETWEEN A SIGNAL FLOW GRAPH AND A BLOCK DIAGRAM BLOCK DIAGRAMSIGNAL FLOW GRAPH When given a Block Diagram to be converted to a Signal Flow Graph, assign node numbers to all inputs and outputs of the Block Diagram.For summing points and takeoff points, using the conversionabove,determinewhetheritwilltranslateintoasinglenodeortwoseparatenodes; then, assign their respective node numbers. Also,recalltheconceptofsummingpointswhereincascaded(series)summingpointscan becombinedintoasinglesummingpoint;andcascaded(series)takeoffpointscanalsobe combined into a single takeoff point. Thus, a single node will represent such summing points and takeoff points, respectively. Example: Convert the Block Diagram to its equivalent Signal Flow Graph and solve for the Transfer Function Using Masons Gain Rule. Solution: Convert the block diagram to its equivalent signal flow graph.Then, identify the paths and loops and obtain their gains. A COMPILATION OF LECTURE NOTES IN CONTROL SYSTEMCHAPTER 7 CATADMAN, LIZETTE IVY G. For Path: For Loops:

For Co-Factor: 1 1 = The Determinant: ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) ( ) 4 s 1 s3 4 s 2 s4 s 1 s4 s 3 4 s 1 s1 s14 s 1 s31 2 L 1 L 1+ ++ + +=+ ++ + + + +=((

|.|

\|+++ + = + = The Transfer Function: ( ) ( ) ( )( ) ( )( ) ( )( ) ( ) 3 4 s 2 s34 s 1 s3 4 s 2 s14 s 1 s31 1 P) s ( R) s ( Y) s ( T+ + +=+ ++ + +((

+ += = = Example: Convert the Block Diagram to its equivalent Signal Flow Graph and solve for the Transfer Functions Using Masons Gain Rule. Solution: ( ) ( )( ) ( ) 4 s 1 s314 s3 1 s11 1 P+ +=|.|

\|+|.|

\|+=( )1 s111 s12 L+= |.|

\|+=( )( ) ( ) 4 s 1 s314 s3 1 s11 L+ + = |.|

\|+|.|

\|+=A COMPILATION OF LECTURE NOTES IN CONTROL SYSTEMCHAPTER 7 CATADMAN, LIZETTE IVY G. For Loops: ( ) ( ) 1 H 2 G 1 G 1 H 1 2 G 1 G 1 L = = ( ) 2 H 2 G 2 H 2 G 2 L = = The Determinant: ( ) ( ) ( ) | | 2 H 2 G 1 H 2 G 1 G 1 2 H 2 G 1 H 2 G 1 G 1 2 L 1 L 1 + + = + = + = Set0 ) s ( 4 R ) s ( 3 R ) s ( 2 R = = = For Path: For Co-Factor:

1 1 = The Transfer Function: 2 H 2 G 1 H 2 G 1 G 12 G 1 G 1 1 P) s ( R) s ( Y) s ( T111+ += = = Set0 ) s ( 4 R ) s ( 3 R ) s ( 1 R = = = ( ) ( )( ) ( ) 2 G 1 G 1 2 G 1 G 1 1 P = =A COMPILATION OF LECTURE NOTES IN CONTROL SYSTEMCHAPTER 7 CATADMAN, LIZETTE IVY G. For Path: For Co-Factor:

1 1 = The Transfer Function: 2 H 2 G 1 H 2 G 1 G 12 G 1 1 P) s ( R) s ( Y) s ( T212+ += = = Set0 ) s ( 4 R ) s ( 2 R ) s ( 1 R = = = For Path: For Co-Factor:

1 1 = ( ) ( ) ( ) 2 G 1 2 G 1 1 P = =( ) ( ) ( ) 2 G 1 2 G 1 1 P = =A COMPILATION OF LECTURE NOTES IN CONTROL SYSTEMCHAPTER 7 CATADMAN, LIZETTE IVY G. The Transfer Function: 2 H 2 G 1 H 2 G 1 G 12 G 1 1 P) s ( R) s ( Y) s ( T313+ += = = Set0 ) s ( 3 R ) s ( 2 R ) s ( 1 R = = = For Path: For Co-Factor:

1 1 = The Transfer Function: 2 H 2 G 1 H 2 G 1 G 11 H 2 G 1 G 1 1 P) s ( R) s ( Y) s ( T414+ += = = 7.8PROBLEM SETS 7.8.1SINGLE-INPUT, SINGLE-OUTPUT SIGNAL FLOW GRAPHS A. ( ) ( ) ( ) ( ) ( ) 1 H 2 G 1 G 1 2 G 1 G 1 H 1 1 P = =A COMPILATION OF LECTURE NOTES IN CONTROL SYSTEMCHAPTER 7 CATADMAN, LIZETTE IVY G. B. C. D. 7.8.2MULTIPLE-INPUT, MULTPILE-OUTPUT SIGNAL FLOW GRAPHS A. A COMPILATION OF LECTURE NOTES IN CONTROL SYSTEMCHAPTER 7 CATADMAN, LIZETTE IVY G. B. C. 7.8.3TRANSFORMATIONOFBLOCKDIAGRAMSTOITSEQUIVALENTSIGNAL FLOW GRAPHS Instructions: Transform the BlockDiagrams to its equivalent Signal Flow Graphs; then, solve the Transfer Function using Masons Gain Rule. 1.Single-Input, Single-Output A. B. C. A COMPILATION OF LECTURE NOTES IN CONTROL SYSTEMCHAPTER 7 CATADMAN, LIZETTE IVY G. 2.Multiple-Input, Multiple-Output A. B. SOURCES/ REFERENCES Distefano, Joseph III J., Allen R. Stubberub, and Ivan J. Williams.Schaums Outlines: Feedback and Control Systems, Second Edition. USA: McGraw-Hill Companies, Inc., 1995. Hostetter, Gene H., Clement J. Savant Jr., and Raymond T. Stefani. Design of Feedback Control Systems, 2nd Edition. USA: Saunders College Publishing, 1989.