第一节系统的稳定性

第一节系统的稳定性第1页，课件共44页，创作于2023年2月稳定性和代数稳定判据(Stabilityofthesystemandthealgebracriteria)典型输入作用和时域性能指标(Typicalinputandtimeperformanceindex)一阶系统的瞬态响应(Transientresponseofoneorderdynamicalsystem)二阶系统的瞬态响应(Transientresponseoftwoorderdynamicalsystem)稳态误差分析(Steadyerroranalyse)主要内容(Mainissues)第2页，课件共44页，创作于2023年2月第一节系统的稳定性和

代数稳定判据

Section1Stabilityofthecontrolsystemandtheitsalgebraevaluationcriteria第3页，课件共44页，创作于2023年2月1.稳定的基本概念和线性系统稳定的充要条件(Basicconceptofsystemstabilityanditsthesufficient,necessaryconditionofthelinearcontrolsystem)1)Stabilizingofcontrolsystemisthemostimportantconditionforsystemtorunproperly.2)Infact,therealsystemisalwaysaffectedbytheoutsideorinsidedisturbances,suchasloadvarying,energywave,systemparameterchanging,environmentchangingetc.3)Ifthesystemisunstable,systemwilldeparturetheinitialbalancestateunderanysmallwave,andwilldispersewiththetimegoing.4)Toanalyzethesystemstabilityandtomakeouttheplantoensurethesystemstableisthebasicgoalofcontroltheory.稳定的充要条件和属性第4页，课件共44页，创作于2023年2月

稳定的基本概念(Basicconceptofstability)：Ifthesystemisinthestateofbalance,itwilldepartthestateundertheeffectofoutsideexciting.whentheoutsideexcitingisdisappeared,thesystemwillreturntotheoriginalstateasitrunsforsolongtime.Thesystemisstable,orthesystemisoffstability.Orelsethesystemisunstable,orthesystemisofinstability.第5页，课件共44页，创作于2023年2月Considerthefollowdifferentialequation：+polynomialrelativewithinitialvalue稳定的充要条件和属性DoLaplacetransform：where：x(t)—inputy(t)—output;isconstants.第6页，课件共44页，创作于2023年2月Thefirstitemiszerostatesolution,andisrelatedwiththeresponseexcitedbytheinput.Theseconditemiszeroinputsolution,andisrelatedwiththeresponseexcitedbytheinitialvalue.第7页，课件共44页，创作于2023年2月necessaryandsufficientconditionoflinearsystemstabilizingis:allthesystemcharacteristicrootsareofnegativerealpart(Eigenvalue),orallthesystemcharacteristicrootsarelainonthelefthalfplaneofscomplexplane.稳定的充要条件和属性第8页，课件共44页，创作于2023年2月充要条件说明ifthereisapositiverealrootforasystem,itmeansthesystemresponseisdispersive.ifthereisapairofpositiverealpartcomplexrootforasystem,itmeansthesystemresponseisperioddispersiveoscillation.bothcasesareunstable.

ifthereisazerorootforasystem,itmeansthesystemresponseisrandombalancestate.ifthereisapairofimaginaryrootsforasystem,itmeansthesystemresponseisoscillatingstateinaconstantsize.StablezoneNonstablezoneCriticalStablezoneSplane第9页，课件共44页，创作于2023年2月Fortheoneordersystem,ifandonlyifarepositive,thesystemisstable.Ifandonlyifarepositive,thesystemisstable.For3ormoreordersystem,itismoredifficulttogettherootsofaalgebraequation.Howcanwedo?充要条件说明Notice：Systemstabilityisaqualityoflinearsystem,andisjustrelatedwiththesystemstructureandparameter,butnotrelatedwiththeinputsignal.

notrelatedwiththe

initialcondition.

Systemstabilityisjustrelatedwiththepolar,notrelatedwiththezero.Forthe2ordersystem第10页，课件共44页，创作于2023年2月2.Routh-Hurwitzcriterion

Consideringthecharacteristicequationofthelinearsystem劳斯判据1).Routhcriterion：Thenecessaryandsufficientconditionofthesystemisasfollow:b.AlltheelementswhicharelainonthefirstcolumnoftheRoutharray,andarecomposedofthecoefficientsofthecharacteristicequationshouldbepositive.a.Allthepolynomialcoefficientsofthecharacteristicequationshouldbepositive;第11页，课件共44页，创作于2023年2月ThefirsttwolineelementsoftheRoutharrayconsistofthecoefficientsofcharacteristicequation.Thefirstrowelementsarecomposedofthecoefficientsan,an-2,an-4,...;Thesecondrowelementsarecomposedofthecoefficients

an-1,an-3,an-5,….HowtoconstructtheRouthtable?第12页，课件共44页，创作于2023年2月劳斯判据Therulestocalculatethe3throwelementsisasfollow：第13页，课件共44页，创作于2023年2月劳斯判据Therulestocalculatethe4throwelementsisasfollow：第14页，课件共44页，创作于2023年2月Accordingtotheabovesimilarmethod,theremainelementscanalsobelead.Therulestocalculatethe5throwelementsisasfollow：第15页，课件共44页，创作于2023年2月劳斯判据例子[example]consideringthesystemwhichcharacteristicequationis:①TowritedowntheRoutharrayasrightposition②Accordingtothenecessaryandsufficientconditionofastablesystem,wecanget：andTrytodeterminethesystemstability.productof2innercoefficientsminusproductof2outcoefficientsispositive.第16页，课件共44页，创作于2023年2月2)DiscussionofthespecialconditionoftheRoutharrayandsomeconclusionb.

ThesystemisunstableifalltheelementsinthefirstcolumnofRoutharrayarenotzerobutnotallarepositive.劳斯判据特殊情况a.ItwillnotaffectthesystemstabilitytomultipleordivideallelementsinarowoftheRoutharraywithapositivenumber;c.Italsoindicatethattherearesomecharacteristicrootsintherighthalfplanofcomplexnumberplanes.d.ThenumberoftheunstablerootsisequaltothechangedsignnumberofelementsinthefirstrankofRoutharray.第17页，课件共44页，创作于2023年2月[example]Assumethatthesystemcharacteristicequationis:-130（2）100（）③Thereis2signchangesinthefirstcolumn.Trytofindthenumberofunstablerootofthesystem.Discuss:①TolisttheRoutharray;②Thereisanegativenumberinthefirstcolumn.Thesystemisunstable.④

2unstablecharacteristicrootsareintherighthalfsplane.第18页，课件共44页，创作于2023年2月劳斯判据特殊情况

e.IfthefirstelementiszerobuttheothersinonelineoftheRouthtablearenotallzero,Anewmethodshouldbeconsidered.[Solution]：tosubstitutetheelement‘0’withaverysmallpositivenumber.Atlastcountingthesign-changednumber.Afterthentocalculatetheotherelementsonthelineorbelowtheline.第19页，课件共44页，创作于2023年2月[example]Consideringthecharacteristicequation:Let,then③Thereis2signchangesthatmeans2unstablerootsintherighthalfplaneofcomplexnumberplanes.

②toanalyse：Trytofindthenumberofunstablerootofthesystem.Discuss:

①tolisttheRoutharray;Clearly,thereisanegativenumberinfirstcolumn.Thesystemisunstable.第20页，课件共44页，创作于2023年2月f.AllthenumbersinaRoutharrayrowarezero.Itmeansthatthereisapairofcharacteristicrootswhichareequalinsizeandoppositeinsign.劳斯判据特殊情况Thereare3casesas:apairofrealrootswhichareequalinsizeandoppositeinsign;orapairofconjugateimaginaryroots;or2pairofconjugatecomplexnumber

rootswhicharesymmetrictotheimaginaryaxis.╳╳╳╳╳╳╳╳第21页，课件共44页，创作于2023年2月[example][Solution]：toconstructanassistantalgebraequationofcomplexnumbervariablesaccordingtothecoefficientsofthelastrowinwhichthecoefficientsarenonzero.b.Todifferentiatetheassistantequationandgetthenewequationc.Tosubstitutethecoefficientsofthezerorowwiththecoefficientsofthenewequation.Notice:theassistantequationmustbeevenorder.第22页，课件共44页，创作于2023年2月[example]todiscussthestabilityofthebelowsystem.168168130380劳斯判据特殊情况④tosimplifyit;Analyse:①tolisttheRoutharray;②tobuildtheassistantequation;③todifferentiatetheaboveequationtogetanewone;⑤tosubstitutethecoefficientswiththenewcoefficientsfromthesimplifiedequation.⑥tocontinuethelisttheRoutharray⑦todeterminetheunstableroots.第23页，课件共44页，创作于2023年2月Itseemsthatthesystemisstablebecausetheelementsarebiggerthanorequaltozero.Clearlythesystemiscriticalstablethatmeansunstableinanengineeringmeaning.168168130380Tobuildanassistantequationasbelowandtosolveit,wemayget:第24页，课件共44页，创作于2023年2月3).Hurwitzcriterion赫尔维茨判据ConsideringthecharacteristicequationofthelinearsystemThenecessaryandsufficientconditionofthesystemisasfollow:andWhere:ΔistheHurwitzdeterminant,andΔiisthehostsub-determinant.第25页，课件共44页，创作于2023年2月①Eachofthehostdiagonalelementsisthecoefficientsofcharacteristicpolynomialfromthesecondtothelast.Problem1.HowtoconstructtheHurwitzarray?②Eachelementofeachrowbelowthehostdiagonalissomecoefficientsaccordingtosubscriptincreasing.③Eachelementofeachrowabovethehostdiagonalissomecoefficientsaccordingtosubscriptdecreasing.④Alltheelementis0whenthesubscriptisbiggerthannorsmallerthan0.subscriptdecreasingsubscriptincreasing第26页，课件共44页，创作于2023年2月Problem2.Howtoconstructthehostsub-determinant?第27页，课件共44页，创作于2023年2月赫尔维茨判据[example]：todiscussthestabilityof4ordersystemasbelow.Hurwitzdeterminantis：Thenecessaryandsufficientconditionis:第28页，课件共44页，创作于2023年2月赫尔维茨判据的另一种形式ItisanotherformofHurwitzcriterion.where:isallthehostsub-determinantswithdifferentorder.4).Lienard-Chipard

criterionThenecessaryandsufficientconditionofthesystemis:Forthesystemwiththecharacteristicequationas:or第29页，课件共44页，创作于2023年2月3.ApplicationofRouth–Hurwitzcriterion1)Todeterminethesystemstability[example]ifthesystemcharacteristicequationis：,trytodeterminethesystemstability.Analyse:①

TolisttheRoutharrayasbelow：2unstablerootsareintherighthalfplane.

Thesystemisunstable.②③

Thereis2signchanges第30页，课件共44页，创作于2023年2月[example]

ifthesystemcharacteristicequationis：trytodeterminethesystemstability.Analyse:systemcharacteristicequationcanberewriteas:TheHurwitzdeterminantis：Thehostsub-determinantcanbecalculatedasTheconclusionisthatthesystemisstable.第31页，课件共44页，创作于2023年2月2)Toanalyzetheinfluenceofsystemparameterchanging[example]thesystemblockdiagramisgivenasbelow,trytodeterminethecriticalamplifyingcoefficient.[Solution]closedlooptransferfunctionis：Thecharacteristicequationis：AnimportanteffectoftheRouth-Hurwitzcriterionistoanalyzetheinfluenceofsomesystemparametersvaryingsuchastheopen-loopsystemamplifyingcoefficientK.Wecanusethecriteriontodeterminethemaximum–criticalamplifyingcoefficient.第32页，课件共44页，创作于2023年2月TowriteouttheRoutharrayasbelow：Accordingtothenecessaryandsufficientcondition:①allthecoefficientsmustbebiggerthan0.②theelementslainonthefirstcolumnoftheRoutharrayshouldbepositive.Thenwemayget:Thecriticalamplifyingcoefficientis.Thecharacteristicequationis：第33页，课件共44页，创作于2023年2月3)Todeterminetherelativesystemstability(stabilizationabundance)Asweknow,wecanusetheRouth-Hurwitzcriteriontodeterminewhetheracontrolsystemisstableorunstable.Itisaabsolutestability.ifwewanttoknowtherelativestabilityofacontrolsystem,orhowcanitbedetermined?Usually,thedistancebetweenthecharacteristicrootpofthemaximumrealpartandtheimaginaryaxisisusedvirtuallytoexpressthesystemstabilizationabundance.Clearly,ifpislainontheimaginaryaxis,,thatmeansthesystemstabilizationabundanceis0.Howdowedeterminethesystemstabilizationabundance?第34页，课件共44页，创作于2023年2月Todrawaverticallineonthecomplexplaneswhichisparalleltotheimaginaryaxis,andifallthecharacteristicrootsareontheleftoftheline,thesystemiscalledofstabilizationabundance.Thebiggertheis,themorestablethesystemis.Problem:Howtofindtheinacontrolsystem?①Let,andsubstitutethecomplexvariableswithinthecharacteristicequation,thenleadanewcharacteristicequationwithanewcomplexvariablez.②UseRouth-Hurwitzcriteriontoanalyzethesystemstabilityaccordingtothenewcharacteristicequation.③Ifthenewsystemisstable,theoriginalsystemiscalledofstabilizationabundance.第35页，课件共44页，创作于2023年2月[example]asystemcharacteristicequationis,Thereisapairofimaginaryroots.Thenewsystemiscriticalstable.Andtheoriginalsystemisof1abundancestability.Howabouttherelativesystemstability?[solution]clearly,AndThesystemisstable.Let,substitutetheswithz-1,thenewequationis:or第36页，课件共44页，创作于2023年2月Usually,therealpartofthecogentcomplexrootrepresentstheattenuationspeedofsystemresponse,whereastheimaginarypartofthecogentcomplexrootrepresentstheoscillatingofsystemresponse.istheanglebetweenthepolarandthenegativerealaxis.Thesmallertheangleis,thebetterthesystemqualityis.Anotherformtodiscusstherelativestability,therelativestabilityisworst.第37页，课件共44页，创作于2023年2月3.Essentialunstablesystemandtheplantoimprove1)Whatistheessentialunstablesystem?Itisthesystemwhoseperformancecannotbeimprovedjustbyadjustingthesystemparameters.结构不稳定系统及其改进措施-杠杆和放大器的传递函数执行电机的传递函数进水阀门的传递函数控制对象水箱的传递函数2)Example:liquidheightcontrolsystem第38页，课件共44页，创作于2023年2月结构不稳定系统及其改进措施Closedlooptransferfunction：let：Characteristicequation：or：clearly：Routharray：Nomatterhowtochan