第一节系统的稳定性

第一节系统的稳定性第一页，共四十四页，编辑于2023年，星期四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)第二页，共四十四页，编辑于2023年，星期四第一节系统的稳定性和

代数稳定判据

Section1Stabilityofthecontrolsystemandtheitsalgebraevaluationcriteria第三页，共四十四页，编辑于2023年，星期四1.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四页，共四十四页，编辑于2023年，星期四

稳定的基本概念(Basicconceptofstability)：Ifthesystemisinthestateofbalance,itwilldepartthestateundertheeffectofoutsideexciting.whentheoutsideexcitingisdisappeared,thesystemwillreturntotheoriginalstateasitrunsforsolongtime.Thesystemisstable,orthesystemisoffstability.Orelsethesystemisunstable,orthesystemisofinstability.

第五页，共四十四页，编辑于2023年，星期四Considerthefollowdifferentialequation：+polynomialrelativewithinitialvalue稳定的充要条件和属性DoLaplacetransform：where：x(t)—inputy(t)—output;isconstants.第六页，共四十四页，编辑于2023年，星期四Thefirstitemiszerostatesolution,andisrelatedwiththeresponseexcitedbytheinput.Theseconditemiszeroinputsolution,andisrelatedwiththeresponseexcitedbytheinitialvalue.第七页，共四十四页，编辑于2023年，星期四necessaryandsufficientconditionoflinearsystemstabilizingis:allthesystemcharacteristicrootsareofnegativerealpart(Eigenvalue),orallthesystemcharacteristicrootsarelainonthelefthalfplaneofscomplexplane.稳定的充要条件和属性第八页，共四十四页，编辑于2023年，星期四充要条件说明ifthereisapositiverealrootforasystem,itmeansthesystemresponseisdispersive.ifthereisapairofpositiverealpartcomplexrootforasystem,itmeansthesystemresponseisperioddispersiveoscillation.bothcasesareunstable.

ifthereisazerorootforasystem,itmeansthesystemresponseisrandombalancestate.ifthereisapairofimaginaryrootsforasystem,itmeansthesystemresponseisoscillatingstateinaconstantsize.StablezoneNonstablezoneCriticalStablezoneSplane第九页，共四十四页，编辑于2023年，星期四Fortheoneordersystem,ifandonlyifarepositive,thesystemisstable.Ifandonlyifarepositive,thesystemisstable.For3ormoreordersystem,itismoredifficulttogettherootsofaalgebraequation.Howcanwedo?充要条件说明Notice：Systemstabilityisaqualityoflinearsystem,andisjustrelatedwiththesystemstructureandparameter,butnotrelatedwiththeinputsignal.

notrelatedwiththe

initialcondition.

Systemstabilityisjustrelatedwiththepolar,notrelatedwiththezero.Forthe2ordersystem第十页，共四十四页，编辑于2023年，星期四2.Routh-Hurwitzcriterion

Consideringthecharacteristicequationofthelinearsystem劳斯判据1).Routhcriterion：Thenecessaryandsufficientconditionofthesystemisasfollow:b.AlltheelementswhicharelainonthefirstcolumnoftheRoutharray,andarecomposedofthecoefficientsofthecharacteristicequationshouldbepositive.a.Allthepolynomialcoefficientsofthecharacteristicequationshouldbepositive;第十一页，共四十四页，编辑于2023年，星期四ThefirsttwolineelementsoftheRoutharrayconsistofthecoefficientsofcharacteristicequation.Thefirstrowelementsarecomposedofthecoefficientsan,an-2,an-4,...;Thesecondrowelementsarecomposedofthecoefficients

an-1,an-3,an-5,….HowtoconstructtheRouthtable?第十二页，共四十四页，编辑于2023年，星期四劳斯判据Therulestocalculatethe3throwelementsisasfollow：第十三页，共四十四页，编辑于2023年，星期四劳斯判据Therulestocalculatethe4throwelementsisasfollow：第十四页，共四十四页，编辑于2023年，星期四Accordingtotheabovesimilarmethod,theremainelementscanalsobelead.Therulestocalculatethe5throwelementsisasfollow：第十五页，共四十四页，编辑于2023年，星期四劳斯判据例子[example]consideringthesystemwhichcharacteristicequationis:①TowritedowntheRoutharrayasrightposition②Accordingtothenecessaryandsufficientconditionofastablesystem,wecanget：andTrytodeterminethesystemstability.productof2innercoefficientsminusproductof2outcoefficientsispositive.第十六页，共四十四页，编辑于2023年，星期四2)DiscussionofthespecialconditionoftheRoutharrayandsomeconclusionb.

ThesystemisunstableifalltheelementsinthefirstcolumnofRoutharrayarenotzerobutnotallarepositive.劳斯判据特殊情况a.ItwillnotaffectthesystemstabilitytomultipleordivideallelementsinarowoftheRoutharraywithapositivenumber;c.Italsoindicatethattherearesomecharacteristicrootsintherighthalfplanofcomplexnumberplanes.d.ThenumberoftheunstablerootsisequaltothechangedsignnumberofelementsinthefirstrankofRoutharray.第十七页，共四十四页，编辑于2023年，星期四[example]Assumethatthesystemcharacteristicequationis:-130（2）100（）③Thereis2signchangesinthefirstcolumn.Trytofindthenumberofunstablerootofthesystem.Discuss:

①TolisttheRoutharray;②Thereisanegativenumberinthefirstcolumn.Thesystemisunstable.④

2unstablecharacteristicrootsareintherighthalfsplane.第十八页，共四十四页，编辑于2023年，星期四劳斯判据特殊情况

e.IfthefirstelementiszerobuttheothersinonelineoftheRouthtablearenotallzero,Anewmethodshouldbeconsidered.[Solution]：tosubstitutetheelement‘0’withaverysmallpositivenumber.Atlastcountingthesign-changednumber.Afterthentocalculatetheotherelementsonthelineorbelowtheline.第十九页，共四十四页，编辑于2023年，星期四[example]Consideringthecharacteristicequation:Let,then③

Thereis2signchangesthatmeans2unstablerootsintherighthalfplaneofcomplexnumberplanes.

②toanalyse：Trytofindthenumberofunstablerootofthesystem.Discuss:

①tolisttheRoutharray;Clearly,thereisanegativenumberinfirstcolumn.Thesystemisunstable.第二十页，共四十四页，编辑于2023年，星期四f.AllthenumbersinaRoutharrayrowarezero.Itmeansthatthereisapairofcharacteristicrootswhichareequalinsizeandoppositeinsign.劳斯判据特殊情况Thereare3casesas:apairofrealrootswhichareequalinsizeandoppositeinsign;orapairofconjugateimaginaryroots;or2pairofconjugatecomplexnumberrootswhicharesymmetrictotheimaginaryaxis.╳╳╳╳╳╳╳╳第二十一页，共四十四页，编辑于2023年，星期四[example][Solution]：toconstructanassistantalgebraequationofcomplexnumbervariablesaccordingtothecoefficientsofthelastrowinwhichthecoefficientsarenonzero.b.Todifferentiatetheassistantequationandgetthenewequationc.Tosubstitutethecoefficientsofthezerorowwiththecoefficientsofthenewequation.Notice:theassistantequationmustbeevenorder.第二十二页，共四十四页，编辑于2023年，星期四[example]todiscussthestabilityofthebelowsystem.168168130380劳斯判据特殊情况④tosimplifyit;Analyse:①tolisttheRoutharray;②tobuildtheassistantequation;③todifferentiatetheaboveequationtogetanewone;⑤tosubstitutethecoefficientswiththenewcoefficientsfromthesimplifiedequation.⑥tocontinuethelisttheRoutharray⑦todeterminetheunstableroots.第二十三页，共四十四页，编辑于2023年，星期四Itseemsthatthesystemisstablebecausetheelementsarebiggerthanorequaltozero.Clearlythesystemiscriticalstablethatmeansunstableinanengineeringmeaning.168168130380Tobuildanassistantequationasbelowandtosolveit,wemayget:第二十四页，共四十四页，编辑于2023年，星期四3).Hurwitzcriterion赫尔维茨判据ConsideringthecharacteristicequationofthelinearsystemThenecessaryandsufficientconditionofthesystemisasfollow:andWhere:ΔistheHurwitzdeterminant,andΔiisthehostsub-determinant.第二十五页，共四十四页，编辑于2023年，星期四①Eachofthehostdiagonalelementsisthecoefficientsofcharacteristicpolynomialfromthesecondtothelast.Problem1.HowtoconstructtheHurwitzarray?②Eachelementofeachrowbelowthehostdiagonalissomecoefficientsaccordingtosubscriptincreasing.③Eachelementofeachrowabovethehostdiagonalissomecoefficientsaccordingtosubscriptdecreasing.④Alltheelementis0whenthesubscriptisbiggerthannorsmallerthan0.subscriptdecreasingsubscriptincreasing第二十六页，共四十四页，编辑于2023年，星期四Problem2.Howtoconstructthehostsub-determinant?第二十七页，共四十四页，编辑于2023年，星期四赫尔维茨判据[example]：todiscussthestabilityof4ordersystemasbelow.Hurwitzdeterminantis：Thenecessaryandsufficientconditionis:第二十八页，共四十四页，编辑于2023年，星期四赫尔维茨判据的另一种形式ItisanotherformofHurwitzcriterion.where:isallthehostsub-determinantswithdifferentorder.4).Lienard-Chipard

criterionThenecessaryandsufficientconditionofthesystemis:Forthesystemwiththecharacteristicequationas:or第二十九页，共四十四页，编辑于2023年，星期四3.ApplicationofRouth–Hurwitzcriterion1)Todeterminethesystemstability[example]ifthesystemcharacteristicequationis：,trytodeterminethesystemstability.Analyse:①

TolisttheRoutharrayasbelow：2unstablerootsareintherighthalfplane.

Thesystemisunstable.②③

Thereis2signchanges第三十页，共四十四页，编辑于2023年，星期四[example]

ifthesystemcharacteristicequationis：trytodeterminethesystemstability.Analyse:systemcharacteristicequationcanberewriteas:TheHurwitzdeterminantis：Thehostsub-determinantcanbecalculatedasTheconclusionisthatthesystemisstable.第三十一页，共四十四页，编辑于2023年，星期四2)Toanalyzetheinfluenceofsystemparameterchanging[example]thesystemblockdiagramisgivenasbelow,trytodeterminethecriticalamplifyingcoefficient.[Solution]closedlooptransferfunctionis：Thecharacteristicequationis：AnimportanteffectoftheRouth-Hurwitzcriterionistoanalyzetheinfluenceofsomesystemparametersvaryingsuchastheopen-loopsystemamplifyingcoefficientK.Wecanusethecriteriontodeterminethemaximum–criticalamplifyingcoefficient.第三十二页，共四十四页，编辑于2023年，星期四TowriteouttheRoutharrayasbelow：Accordingtothenecessaryandsufficientcondition:①allthecoefficientsmustbebiggerthan0.②theelementslainonthefirstcolumnoftheRoutharrayshouldbepositive.Thenwemayget:Thecriticalamplifyingcoefficientis.Thecharacteristicequationis：第三十三页，共四十四页，编辑于2023年，星期四3)Todeterminetherelativesystemstability(stabilizationabundance)Asweknow,wecanusetheRouth-Hurwitzcriteriontodeterminewhetheracontrolsystemisstableorunstable.Itisaabsolutestability.ifwewanttoknowtherelativestabilityofacontrolsystem,orhowcanitbedetermined?Usually,thedistancebetweenthecharacteristicrootpofthemaximumrealpartandtheimaginaryaxisisusedvirtuallytoexpressthesystemstabilizationabundance.Clearly,ifpislainontheimaginaryaxis,,thatmeansthesystemstabilizationabundanceis0.Howdowedeterminethesystemstabilizationabundance?第三十四页，共四十四页，编辑于2023年，星期四Todrawaverticallineonthecomplexplaneswhichisparalleltotheimaginaryaxis,andifallthecharacteristicrootsareontheleftoftheline,thesystemiscalledofstabilizationabundance.Thebiggertheis,themorestablethesystemis.Problem:Howtofindtheinacontrolsystem?①Let,andsubstitutethecomplexvariableswithinthecharacteristicequation,thenleadanewcharacteristicequationwithanewcomplexvariablez.②UseRouth-Hurwitzcriteriontoanalyzethesystemstabilityaccordingtothenewcharacteristicequation.③Ifthenewsystemisstable,theoriginalsystemiscalledofstabilizationabundance.第三十五页，共四十四页，编辑于2023年，星期四[example]asystemcharacteristicequationis,Thereisapairofimaginaryroots.Thenewsystemiscriticalstable.Andtheoriginalsystemisof1abundancestability.Howabouttherelativesystemstability?[solution]clearly,AndThesystemisstable.Let,substitutetheswithz-1,thenewequationis:or第三十六页，共四十四页，编辑于2023年，星期四Usually,therealpartofthecogentcomplexrootrepresentstheattenuationspeedofsystemresponse,whereastheimaginarypartofthecogentcomplexrootrepresentstheoscillatingofsystemresponse.istheanglebetweenthepolarandthenegativerealaxis.Thesmallertheangleis,thebetterthesystemqualityis.Anotherformtodiscusstherelativestability,therelativestabilityisworst.第三十七页，共四十四页，编辑于2023年，星期四3.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第三十八页，共四十四页，编辑于2023年，星期四结构不稳定系统及其改进措施Closedlooptransferfunction：let：Characteristicequation：or：clearly：Routharray：Nomatterhowtochangeth