现代控制工程-第六章

Chapter3TheStabilityofLinearFeedbackSystemsintheTimeDomain 3 1TheConceptofStability3 2TheRouth HurwitzStabilityCriterion3 3TheApplicationofRouth HurwitzCriterion 3 1TheConceptofStability Example3 1ThefirstbridgeacrosstheTacomaNarrows wasopenedtotrafficonJuly1 1940 Thebridgewasfoundedtooscillatewheneverthewindblew 1 Theconcept fourmonthslater awindproducedanoscillationthatgrewinamplitudeuntil Acatastrophicfailure Similarly donotmarchonbridges Example3 2Microphoneandaudioamplifier louder nearer squealing Anexampleofpositivefeedbacksystem a K 5 k 0 11 101 5 152 20 b K 5 k 0 21 c K 10 k 0 11 G s K 1 K k pickupechopositivefeedback StableSystem input outputstability Adynamicsystemwithaboundedresponsetoaboundedinput Thisisanabsolutestablesystem Givenanabsolutestablesystem wewilldiscussfurthertherelativestabilityofthesystem Commercialairplanesaremorestablethanfighteraircrafts Illustration Absolutestable Unstable Neutralstable where thenonzeropolesofthesystemare and 2 NecessaryandsufficientconditionforstabilityThegeneralformofatransferfunctionofaclosed loopsystemis where areconstantsthatdependonthesystemparameters whenandarepositive y t willbeboundedforallboundedinputs Inthiscase allpolesareintheleft hands plane WhenN 0 thegeneralformoftheimpulseresponseofthesystemwillbe forexample iftherootsontheimaginaryaxisrepeated2times thecorrespondingpartialfractiontermswillbe andthecorrespondingpartintheimpulseresponsewillbe Whenthesystemhasatleastonerootintherighthalfofthes plane someorisnegative orhasrepeatedrootsontheimaginaryaxis theoutputwillbeunboundedforanyinput inthiscase theresponsewillbesustainedoscillation orstep likeoutput foraboundedinput unlesstheinputisasinusoidwhosefrequencyisrightthemagnitudeoftheroots orstepinput explaintheseashomework Whenthesystemhassimplerootsontheimaginaryaxis includingN 1 withallotherrootsinthelefthalfs plane thesystemwillbecalledmarginallystable So marginallystablesystemwillhaveunboundedoutputonlyforcertainboundedinputs Allthepolesofthesystemtransferfunctionhavenegativerealparts ortheyarealllocatedinthelefthalfofthes plane assummary thenecessaryandsufficientconditionsforabsolutestabilityofLTICsystemsare Marginallystablewillbetreatedasunstable sincetherealwaysexistsapproximationinsystemmodeling andtheparametersarealwayschanging 3 2TheRouth HurwitzCriterionToascertainthestabilityofacontrolsystem thebasicapproachistosolvethecharacteristicequationtogetthepoles andthendeterminewhetheranypolesliesintherighthalfs plane Thiscanbeeasilydonebycomputer butdifficultbyhand TheRouthcriterioncanascertainthestabilitybydeterminingthesignoftherootswithoutsolvingthecharacteristicequation 1 ThecriterionThenumberoftherootswithpositiverealpartsisequaltothenumberofchangesinsignofthefirstcolumnoftheRoutharray Thisisnecessaryandsufficient 2 Subsidiarystatements 1 Forastablesystem thecoefficientsofthecharacteristicpolynomialmustbepositive havethesamesign andnonzero Thisrequirementisnecessarybutnotsufficient immediatelyknowitisunstable Itisstillunstable 2 Subsidiarystatements 2 ItissufficienttoassurethestabilityofthecontrolsystemthatalltheentriesinthefirstcolumnoftheRoutharrayarepositive havethesamesign Butnotnecessary Theentriesinthefirstcolumnmaybezero 2 Subsidiarystatements 3 ItissufficienttodeterminethatthecontrolsystemisunstableiftherearepositiveandnegativeentriesinthefirstcolumnoftheRoutharray Alsonotnecessary Theentriesinthefirstcolumnmaybezero Complexcaseswithzeroentriesinthefirstcolumnwillbediscussedlater TheRoutharrayisscheduledas 3 HowtoscheduletheRoutharrayconsiderthecharacteristicequationas Example3 3 Thesystemisunstable andhas2polesintherighthalfs plane Firstorder Ifa1 a0havethesamesign thesystemisstable Secondorder Ifa1 a2 a0havethesamesign thesystemisstable Thirdorder Ifa0 a1 a2 a3arepositive anda1a2 a3a0 thesystemisstable Fourdistinctcases Case1 NoentryinthefirstcolumniszeroCase2 Thereisazerointhefirstcolumn andtherowhasatleastonenonzeroentryCase3 Thereisazerointhefirstcolumn andentriesinthecorrespondingrowareallzeroCase4 Asincase3andwithrepeatedrootsontheimaginaryaxis Case1 Example3 4 withparameter thecharacteristicequationis s3 s2 s K 0 Kisnot1or0 Thus ThesystemisunstablewhenKislessthan0andwhenKisgreaterthan1 Else thesystemisstable Adjustingtheparametermaychangethestability Example3 5Thecharacteristicequationis s6 2s5 3s4 4s3 5s2 6s 7 0 Array 6 4 2 1 1 10 6 2 2 2 7 1 0 6 14 1 8 8 Case2 2 firstcolumnentriesunchangedinthe2ndorderdeterminant andadd 1 2ordersasstep add0whennecessary 3 denominatoristhefirstentryintheaboverow 4 Replacethezeroelementwithasmallpositive 2 8 7 8 7 2 8 7 5 let approachesto0 togetthearray Necessaryforstable Sufficientforunstable positiveandnegative orzerocoefficients Necessary sufficientforstable Nosignchangeinthefirstcolumn Numberofthechangesinsignisthenumberofrootsintherighthalfs plane Coefficientsareallpositive thesamesign unstable 2righthalfplaneroots Case2 Example3 6Thecharacteristicequationis s4 s3 s2 s 0 Let approachesto0 thereistheproductof andminusinfiniteinthefirstcolumn Thesystemisalwaysunstable Case3Zerorow Example3 7Thecharacteristicequationis s4 5s3 7s2 5s 6 0 Array 5 1 7 5 6 6 6 0 2 Buildtheauxiliarypolynomialusingtheaboverow itisafactorofthecharacteristicpolynomial Theorderwillbealwayseven s2 1 0 Sufficientforunstable Ifthereisazerorow thesystemisunstable s1 2 j Case4 Repeatedrootsontheimaginaryaxis Example3 8 Thecharacteristicequationis s5 s4 2s3 2s2 s 1 0 or Whentherearerepeatedrootsontheimaginaryaxis therewillbetermsliketsin t intheresponse whichmakethesystemunstable Inthearray let approachesto0 theyarestill0entriesinthefirstcolumn S 1 S j S j S j S j 0 3 3TheapplicationofRouth HurwitzCriterionExample3 9Analyzethestabilityofthegivensystem whereK 0 Thecharacteristicequationis Stable Unstable Example3 10weldingcontrol p256 Thecharacteristicequationis DeterminetherangeofKanda tomakethesystemstable RouthArray Letk 40 thenweneeda 0 639 Example3 11Thecharacteristicequationis determinetherangeofktomakethesystemstable Ifwerequirefurthertherootsareallinthele