67附錄IntroductiontoModernControlTheoryIntroductiontoModernControlTheorySeveralfactorsprovidedthestimulusforthedevelopmentofmoderncontroltheory:a.Thenecessaryofdealingwithmorerealisticmodelsofsystem.b.Theshiftinemphasistowardsoptimalcontrolandoptimalsystemdesign.c.Thecontinuingdevelopmentsindigitalcomputertechnology.d.Theshortcomingofpreviousapproaches.e.Recognitionoftheapplicabilityofwell-knownmethodsinotherfieldsofknowledge.Thetransitionfromsimpleapproximatemodels,whichareeasytoworkwith,tomorerealisticmodels,producestwoeffects.First,alargenumberofvariablesmustbeincludedinthemodels.Second,amorerealisticmodelismorelikelytocontainnonlinearitiesandtime-varyingparameters.Previouslyignoredaspectsofthesystem,suchasinteractionswithfeedbackthroughtheenvironment,aremorelikelytobeincluded.Withanadvancingtechnologicalsociety,thereisatrendtowardsmoreambitiousgoals.Thisalsomeansdealingwithcomplexsystemwithalargenumberofinteractingcomponents.Theneedforgreateraccuracyandefficiencyhaschangertheemphasisoncontrolsystemperformance.Theclassicalspecificationsintermsofpercentovershoot,settingtime,bandwidth,etc.haveinmanycasesgivenwaytooptimalcriteriasuchasminimumenergy,minimumcost,andminimumtimeoperation.Optimizationofthesecriteriamakesitevenmoredifficulttoavoiddealingwithunpleasantnonlinearities.Optimalcontroltheoryoftendictatesthatnonlineartime-varyingcontrollawsareused,evenifthebasicsystemislinearandtime-invariant.68Thecontinuingadvancesincomputertechnologyhavehadthreeprincipaleffectsonthecontrolsfield.Oneoftheserelatestothegiganticsupercomputers.Thesizeandtheclassoftheproblemsthatcannowbemodeled,analyzed,andcontrolledareconsiderablylargethantheywerewhenthefirsteditionofthisbookwaswritten.ThesecondimpactofthecomputertechnologyhastosowiththeproliferationandwideavailabilityofthemicrocomputersinhomesandItheworkplace,classicalcontroltheorywasdominatedbygraphicalmethodsbecauseatthetimethatwastheonlywaytosolvecertainproblems,Noweverycontroldesignerhaseasyaccesstopowerfulcomputerpackagesforsystemsanalysisanddesign.Theoldgraphicalmethodshavenotyetdisappeared,buthavebeenautomated.Theysurvivebecauseoftheinsightandintuitionthattheycanprovide,somedifferenttechniquesareoftenbettersuitedtoacomputer.Althoughacomputercanbeusedtocarryouttheclassicaltransform-inversetransformmethods,itisusedusuallymoreefficientforacomputertointegratedifferentialequationsdirectly.Thethirdmajorimpactofthecomputersisthattheyarenowsocommonlyusedasjustanothercomponentinthecontrolsystems.Thismeansthatthediscrete-timeanddigitalsystemcontrolnowdeservesmuchmoreattentionthanModerncontroltheoryiswellsuitedtotheabovetrendsbecauseitstime-domaintechniquesanditsmathematicallanguage(matrices,linearvectorspaces,etc.)areidealwhendealingwithacomputer.Computersareamajorreasonfortheexistenceofstatevariablemethods.Mostclassicalcontroltechniquesweredevelopedforlinearconstantcoefficientsystemswithoneinputandoneoutput(perhapsafewinputsandoutputs).ThelanguageofclassicaltechniquesistheLaplaceorZ-transformandtransferfunctions.Whennonlinearitiesadtimevariationsarepresent,theverybasisfortheseclassicaltechniquesisremoved.Somesuccessfultechniquessuchasphase-planemethods,describingfunctions,andotheradhocmethods,havebeendevelopedtoalleviantthisshortcoming.However,thegreatestsuccesshasbeenlimitedtolow-ordersystems.Thestatevariableapproachofmoderncontroltheoryprovidesauniformandpowerfulmethodofrepresentingsystemsofarbitraryorder,linearornonlinear,withtime-varying69orconstantcoefficient.Itprovidesanidealformulationforcomputerimplementationandisresponsibleformuchoftheprogressinoptimizationtheory.Moderncontroltheoryisarecentdevelopmentinthefieldofcontrol.Therefore,thenameisjustifiedatleastasadescriptivetitle.However,thefoundationsofmoderncontroltheoryaretobefoundinotherwell-establishedfields.RepresentingasystemintermsofstatevariablesisequivalenttotheapproachofHamiltonianmechanics,usinggeneralizedcoordinatesandgeneralizedmoment.Theadvantagesofthisapproachhavebeenwell-knownIclassicalphysicsformanyyears.Theadvantagesofusingmatriceswhendealingwithsimultaneousequationsofvariouskindshavelongbeenappreciatedinappliedmathematics.Thefieldoflinearalgebraalsocontributesheavilytomoderncontroltheory.Thisisduetotheconcisenotation,thegeneralityoftheresults,andtheeconomyofthoughtthatlinearalgebraprovides.MechanismofSurfaceFinishProductionTherearebasicallyfivemechanismswhichcontributetotheproductionofasurfacewhichhavebeenmachined.Thereare:(1)Thebasicgeometryofthecuttingprocess.In,forexample,singlepointturningthetoolwilladvanceaconstantdistanceaxiallyperrevolutionoftheworkpieceandtheresultantsurfacewillhaveonit,whenviewedperpendicularlytothedirectionoftoolfeedmotion,aseriesofcuspswhichwillhaveabasicformwhichreplicatestheshapeofthetoolincut.(2)Theefficiencyofthecuttingoperation.Ithasalreadybeenmentionedthatcuttingwithunstablebuilt-up-edgeswillproduceasurfacewhichcontainshardbuilt-up-edgefragmentswhichwillresultinadegradationofthesurfacefinish.Itcanalsobedemonstratedthatcuttingunderadverseconditionssuchasapplywhenusinglargefeedssmallrakeanglesandlowcuttingspeeds,besidesproducingconditionswhichcontinuousshearoccurringintheshearzone,tearingtakesplace,discontinuouschipsofuneventhicknessareproduced,andtheresultantsurfaceispoor.Thissituationisparticularlynoticeablewhenmachiningveryductilematerialssuchascopperandaluminum.70(3)Thestabilityofthemachinetool.Undersomecombinationsofcuttingconditions:workpiecesize,methodofclamping,andcuttingtoolrigidityrelativetothemachinetoolstructure,instabilitycanbesetupinthetoolwhichcausesittovibrate.Undersomeconditionsthevibrationwillbuiltupandunlesscuttingisstoppedconsiderabledamagetoboththecuttingtoolandworkpiecemayoccur.Thisphenomenonisknownaschatterandinaxialturningischaracterizedbylongpitchhelicalbandsontheworkpiecesurfaceandshortpitchundulationsonthetransientmachinedsurface.(4)Theeffectivenessofremovingsward.Indiscontinuouschipproductionmachining,suchasmillingorturningofbrittlematerials,itisexpectedthatthechip(sward)willleavethecuttingzoneeitherundergravityorwiththeassistanceofajetofcuttingfluidandthattheywillnotinfluencethecutsurfaceinanyway.However,whencontinuouschipproductionisevident,unlessstepsatetakentocontroltheswarfitislikelythatitwillimpingeonthecutsurfaceandmarkit.Inevitably,thismarkingbesidealookingunattractive,oftenresultsinapoorersurfacefinishing,(5)Theeffectiveclearanceangleonthecuttingtool.Forcertaingeometriesofminorcuttingedgereliefandclearanceanglesitispossibletocutonthemajorcuttingedgeandburnishontheminorcuttingedge.Thiscanproduceagoodsurfacefinishbut,ofcourse,itisstrictlyacombinationofmetalcuttingandmeta