Masons-gain-formula

P.C.Chau©2002Mason’sGainFormulaToreduceablockdiagram,onemayusetheMason’sgainformula(alsocalledMason’srule).Masonfirstderivedtheideausingwhathecalledasignal-flowgraph,whichisadifferentgraphicalrepresentationofablockdiagram.Asignal-flowgraphisdrawnwithpaths(lines)andnodes.Thetransferfunctionsinablockdiagrambecomethepathsandthevariablesinbetweentheblocksbecomethenodes.Theinputandoutputvariablesofablockdiagramaredesignatedthesourceandsinknodes.Inthisbriefintroduction,weshallskipdrawingthesignal-flowgraphsbecausewecanexplainandapplyMason’sformulatoablockdiagramjustaswell.Thisisthesimpleideabehindtheformula.Weknowthatablockdiagramisagraphicalrepresentationofalgebraicrelations.Ifwewriteouttheequations,weshouldbeabletosolvethemwith,forexample,theCramer’srule.IfweanalyzeandcomparecarefullythedeterminanttermsresultingfromtheuseofCramer’srulewithablockdiagram,wemaymakesomemeaningfulassociationsbetweenthealgebraandthediagram,andthisiswhatMasondid.SowenowstatetheMason’sgainformulawithoutproof.1TherulestatesthatthetransferfunctionbetweentheinputandoutputvariablesofablockdiagramisG(s)=1∆Fi∆iΣi=1f,(1)where∆isthedeterminantofthesystem,Fiisthegainofthei-thforwardpath,and∆iisthedeterminantofthei-thforwardpath.Thesummationisoverallfforwardpaths;wearesuperimposingallthetermsinalinearsystem.Moreover,thedeterminant∆isthecharacteristicpolynomialofthesystem.Wenowneedtodefinesomemoretermsandshowhoweachofthesequantitiescanbecalculated:Systemdeterminant∆=1–(sumofallindividualloopgains)+(sumoftheproductsofthegainsofallpossibletwoloopsthatdonottoucheachother)–(sumoftheproductsofthegainsofallpossiblethreeloopsthatdonottoucheachother)+…andsoforthwithsumsofhighernumberofnon-touchingloopgainsForwardpathgainFi=productofallthetransferfunctionsalongthei-thforwardpathForwardpathdeterminant∆i=valueof∆forthepartoftheblockdiagramthatdoesnottouchthei-thforwardpath(∆i=1iftherearenonon-touchingloopstothei-thpath.)ForwardpathApaththatgoesfromtheinputtotheoutput,andinawaythatnovariables(nodes)areencounteredmorethanonce.LooppathApaththatleadsfromonevariableandbacktothesamevariable.PathgainTheforwardpathgainistheproductofallthetransferfunctionsalongthepath.Similarly,thelooppathgainistheproductofallthetransferfunctionsthatformtheloop.Non-touchingloopTwoloopsarenottouchingiftheydonotshareacommonvariable.1Hardlyanyintroductorytextprovidestheproof,butthetextbyPhillipsandHarbor(1996)hasaniceexampletoillustratetheassociationofthedeterminantswithCramer’srule.2Toseehowtoapplytherule,weneedtorevisitourexamplesinthetext.Beforewedothat,beforewarnedthatitisextremelyeasytomakeanerrorapplyingtheMason’sformula;wecaneasilyoverlookandomitoneoftheterms.Weneedtoapplytherulewithextremecare.ToapplyMason’sformula,wefirstidentifythevariablesintheblockdiagram.Theyaredenotedwithencirclednumbersintheblockdiagramsofthefollowingexamples.Generally,wehaveanewvariablewheninformationischanged,eitherafteratransferfunctionorafterasummingpoint.Wealsolabeltheinputandoutputvariables.Itisnotastrictrule,butweusuallyassignthenumbersalongthemostobviousforwardpathfirst.Example1.Findtheclosedlooptransferfunctionofasimplefeedbackloop(Fig.E.1).ThisproblemisessentiallytheblockdiagraminFig.2.11inthetextwiththeservotransferfunctionderivedinSection5.2.1.Itisagoodhabittomakeatableofthepathsandloopsinordertoavoiderrors.Forthisproblem,therearenonon-touchingloops.Wehaveonlyoneforwardpath,andoneloopthatbeginsandendsafterthesummingpointatvariablenumber2.Theloopgainisnegativebecausetheminussignisessentiallyagainof–1.ForwardpathPathgainDeterminant12345F1=GcGaGp∆1=1LoopLoopgain234562GcGaGpGmx–1Sowehave∆=1–(–GcGaGpGm),andsincethereisonlyoneforwardpath,wearriveatG(s)=GcGaGp1+GcGaGpGm.RCH++–+KGp654321RCG1–G2G3G4H1H2––+++871234569FigureE.2FigureE.3–+GcGmGpGaRC123456FigureE.13Example2.RepeatExample2.14inthetext.FigureE2.14isduplicatedinFig.E.2withthelocationsofthevariablesadded.Therearetwoforwardpathsandonelooppath,alltouchingeachother.Therearenonon-touchingparts.Sowehave:ForwardpathPathgainDeterminant123F1=Gp∆1=1145623F2=KHGp∆2=1LoopLoopgain23562Gpx–1xHThesystemdeterminantis∆=1–(–GpH),andforthetwoforwardpaths,Fi∆iΣi=12=Gp+KHGp.Finally,withEq.(1),G(s)=Gp(1+KH)1+GpH.Example3.RepeatExample2.15inthetext.FigureE2.15aisduplicatedinFig.E.3withthelocationsofthevariablesadded.(Strictly,weshouldassignavariablelabelimmediatelyaftertheblockG1,butwecheatandskipthatbecauseomittingthatlabelwillnotaffectourresultshere.)Thereisoneforwardpathandthreelooppaths.Twoofthelooppathsdonottoucheachother,butallthreelooppathstouchtheforwardpath.ForwardpathPathgainDeterminant1234567F1=G1G2G3G4∆1=1LoopLoopgain345693G2G3H1x–156785*G3G4H2x–12342*G1G2x–1*ThesetwoloopsdonottoucheachotherBecausetwoofthelooppathsdonottoucheachother,thesystemdeterminanthasanextraproducttermofthesetwonon-touchingloops:∆=1+(G2G3H1+G3G4H2+G1G2)+(G3G4H2xG1G2).4Theforwardpathtouchesallthreeloops,and∆1=1.Hence,thetransferfunctionofthissystemisG(s)=G1G2G3G41+G2G3H1+G3G4H2+G1G2+G1G2G3G4H2.Ifwefactorout(1+G1G2)inthedenominator,wecanarriveatexactlythesameformaspresentedinExampl