第一節(jié)系統(tǒng)的穩(wěn)定性

第一節(jié)系統(tǒng)的穩(wěn)定性第一頁，共四十四頁，編輯于2023年，星期四穩(wěn)定性和代數(shù)穩(wěn)定判據(jù)(Stabilityofthesystemandthealgebracriteria)典型輸入作用和時(shí)域性能指標(biāo)(Typicalinputandtimeperformanceindex)一階系統(tǒng)的瞬態(tài)響應(yīng)(Transientresponseofoneorderdynamicalsystem)二階系統(tǒng)的瞬態(tài)響應(yīng)(Transientresponseoftwoorderdynamicalsystem)穩(wěn)態(tài)誤差分析(Steadyerroranalyse)主要內(nèi)容(Mainissues)第二頁，共四十四頁，編輯于2023年，星期四第一節(jié)系統(tǒng)的穩(wěn)定性和

代數(shù)穩(wěn)定判據(jù)

Section1Stabilityofthecontrolsystemandtheitsalgebraevaluationcriteria第三頁，共四十四頁，編輯于2023年，星期四1.穩(wěn)定的基本概念和線性系統(tǒng)穩(wěn)定的充要條件(Basicconceptofsystemstabilityanditsthesufficient,necessaryconditionofthelinearcontrolsystem)1)Stabilizingofcontrolsystemisthemostimportantconditionforsystemtorunproperly.2)Infact,therealsystemisalwaysaffectedbytheoutsideorinsidedisturbances,suchasloadvarying,energywave,systemparameterchanging,environmentchangingetc.3)Ifthesystemisunstable,systemwilldeparturetheinitialbalancestateunderanysmallwave,andwilldispersewiththetimegoing.4)Toanalyzethesystemstabilityandtomakeouttheplantoensurethesystemstableisthebasicgoalofcontroltheory.穩(wěn)定的充要條件和屬性第四頁，共四十四頁，編輯于2023年，星期四

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

第五頁，共四十四頁，編輯于2023年，星期四Considerthefollowdifferentialequation：+polynomialrelativewithinitialvalue穩(wěn)定的充要條件和屬性DoLaplacetransform：where：x(t)—inputy(t)—output;isconstants.第六頁，共四十四頁，編輯于2023年，星期四Thefirstitemiszerostatesolution,andisrelatedwiththeresponseexcitedbytheinput.Theseconditemiszeroinputsolution,andisrelatedwiththeresponseexcitedbytheinitialvalue.第七頁，共四十四頁，編輯于2023年，星期四necessaryandsufficientconditionoflinearsystemstabilizingis:allthesystemcharacteristicrootsareofnegativerealpart(Eigenvalue),orallthesystemcharacteristicrootsarelainonthelefthalfplaneofscomplexplane.穩(wěn)定的充要條件和屬性第八頁，共四十四頁，編輯于2023年，星期四充要條件說明ifthereisapositiverealrootforasystem,itmeansthesystemresponseisdispersive.ifthereisapairofpositiverealpartcomplexrootforasystem,itmeansthesystemresponseisperioddispersiveoscillation.bothcasesareunstable.

ifthereisazerorootforasystem,itmeansthesystemresponseisrandombalancestate.ifthereisapairofimaginaryrootsforasystem,itmeansthesystemresponseisoscillatingstateinaconstantsize.StablezoneNonstablezoneCriticalStablezoneSplane第九頁，共四十四頁，編輯于2023年，星期四Fortheoneordersystem,ifandonlyifarepositive,thesystemisstable.Ifandonlyifarepositive,thesystemisstable.For3ormoreordersystem,itismoredifficulttogettherootsofaalgebraequation.Howcanwedo?充要條件說明Notice：Systemstabilityisaqualityoflinearsystem,andisjustrelatedwiththesystemstructureandparameter,butnotrelatedwiththeinputsignal.

notrelatedwiththe

initialcondition.

Systemstabilityisjustrelatedwiththepolar,notrelatedwiththezero.Forthe2ordersystem第十頁，共四十四頁，編輯于2023年，星期四2.Routh-Hurwitzcriterion

Consideringthecharacteristicequationofthelinearsystem勞斯判據(jù)1).Routhcriterion：Thenecessaryandsufficientconditionofthesystemisasfollow:b.AlltheelementswhicharelainonthefirstcolumnoftheRoutharray,andarecomposedofthecoefficientsofthecharacteristicequationshouldbepositive.a.Allthepolynomialcoefficientsofthecharacteristicequationshouldbepositive;第十一頁，共四十四頁，編輯于2023年，星期四ThefirsttwolineelementsoftheRoutharrayconsistofthecoefficientsofcharacteristicequation.Thefirstrowelementsarecomposedofthecoefficientsan,an-2,an-4,...;Thesecondrowelementsarecomposedofthecoefficients

an-1,an-3,an-5,….HowtoconstructtheRouthtable?第十二頁，共四十四頁，編輯于2023年，星期四勞斯判據(jù)Therulestocalculatethe3throwelementsisasfollow：第十三頁，共四十四頁，編輯于2023年，星期四勞斯判據(jù)Therulestocalculatethe4throwelementsisasfollow：第十四頁，共四十四頁，編輯于2023年，星期四Accordingtotheabovesimilarmethod,theremainelementscanalsobelead.Therulestocalculatethe5throwelementsisasfollow：第十五頁，共四十四頁，編輯于2023年，星期四勞斯判據(jù)例子[example]consideringthesystemwhichcharacteristicequationis:①TowritedowntheRoutharrayasrightposition②Accordingtothenecessaryandsufficientconditionofastablesystem,wecanget：andTrytodeterminethesystemstability.productof2innercoefficientsminusproductof2outcoefficientsispositive.第十六頁，共四十四頁，編輯于2023年，星期四2)DiscussionofthespecialconditionoftheRoutharrayandsomeconclusionb.

ThesystemisunstableifalltheelementsinthefirstcolumnofRoutharrayarenotzerobutnotallarepositive.勞斯判據(jù)特殊情況a.ItwillnotaffectthesystemstabilitytomultipleordivideallelementsinarowoftheRoutharraywithapositivenumber;c.Italsoindicatethattherearesomecharacteristicrootsintherighthalfplanofcomplexnumberplanes.d.ThenumberoftheunstablerootsisequaltothechangedsignnumberofelementsinthefirstrankofRoutharray.第十七頁，共四十四頁，編輯于2023年，星期四[example]Assumethatthesystemcharacteristicequationis:-130（2）100（）③Thereis2signchangesinthefirstcolumn.Trytofindthenumberofunstablerootofthesystem.Discuss:

①TolisttheRoutharray;②Thereisanegativenumberinthefirstcolumn.Thesystemisunstable.④

2unstablecharacteristicrootsareintherighthalfsplane.第十八頁，共四十四頁，編輯于2023年，星期四勞斯判據(jù)特殊情況

e.IfthefirstelementiszerobuttheothersinonelineoftheRouthtablearenotallzero,Anewmethodshouldbeconsidered.[Solution]：tosubstitutetheelement‘0’withaverysmallpositivenumber.Atlastcountingthesign-changednumber.Afterthentocalculatetheotherelementsonthelineorbelowtheline.第十九頁，共四十四頁，編輯于2023年，星期四[example]Consideringthecharacteristicequation:Let,then③

Thereis2signchangesthatmeans2unstablerootsintherighthalfplaneofcomplexnumberplanes.

②toanalyse：Trytofindthenumberofunstablerootofthesystem.Discuss:

①tolisttheRoutharray;Clearly,thereisanegativenumberinfirstcolumn.Thesystemisunstable.第二十頁，共四十四頁，編輯于2023年，星期四f.AllthenumbersinaRoutharrayrowarezero.Itmeansthatthereisapairofcharacteristicrootswhichareequalinsizeandoppositeinsign.勞斯判據(jù)特殊情況Thereare3casesas:apairofrealrootswhichareequalinsizeandoppositeinsign;orapairofconjugateimaginaryroots;or2pairofconjugatecomplexnumberrootswhicharesymmetrictotheimaginaryaxis.╳╳╳╳╳╳╳╳第二十一頁，共四十四頁，編輯于2023年，星期四[example][Solution]：toconstructanassistantalgebraequationofcomplexnumbervariablesaccordingtothecoefficientsofthelastrowinwhichthecoefficientsarenonzero.b.Todifferentiatetheassistantequationandgetthenewequationc.Tosubstitutethecoefficientsofthezerorowwiththecoefficientsofthenewequation.Notice:theassistantequationmustbeevenorder.第二十二頁，共四十四頁，編輯于2023年，星期四[example]todiscussthestabilityofthebelowsystem.168168130380勞斯判據(jù)特殊情況④tosimplifyit;Analyse:①tolisttheRoutharray;②tobuildtheassistantequation;③todifferentiatetheaboveequationtogetanewone;⑤tosubstitutethecoefficientswiththenewcoefficientsfromthesimplifiedequation.⑥tocontinuethelisttheRoutharray⑦todeterminetheunstableroots.第二十三頁，共四十四頁，編輯于2023年，星期四Itseemsthatthesystemisstablebecausetheelementsarebiggerthanorequaltozero.Clearlythesystemiscriticalstablethatmeansunstableinanengineeringmeaning.168168130380Tobuildanassistantequationasbelowandtosolveit,wemayget:第二十四頁，共四十四頁，編輯于2023年，星期四3).Hurwitzcriterion赫爾維茨判據(jù)ConsideringthecharacteristicequationofthelinearsystemThenecessaryandsufficientconditionofthesystemisasfollow:andWhere:ΔistheHurwitzdeterminant,andΔiisthehostsub-determinant.第二十五頁，共四十四頁，編輯于2023年，星期四①Eachofthehostdiagonalelementsisthecoefficientsofcharacteristicpolynomialfromthesecondtothelast.Problem1.HowtoconstructtheHurwitzarray?②Eachelementofeachrowbelowthehostdiagonalissomecoefficientsaccordingtosubscriptincreasing.③Eachelementofeachrowabovethehostdiagonalissomecoefficientsaccordingtosubscriptdecreasing.④Alltheelementis0whenthesubscriptisbiggerthannorsmallerthan0.subscriptdecreasingsubscriptincreasing第二十六頁，共四十四頁，編輯于2023年，星期四Problem2.Howtoconstructthehostsub-determinant?第二十七頁，共四十四頁，編輯于2023年，星期四赫爾維茨判據(jù)[example]：todiscussthestabilityof4ordersystemasbelow.Hurwitzdeterminantis：Thenecessaryandsufficientconditionis:第二十八頁，共四十四頁，編輯于2023年，星期四赫爾維茨判據(jù)的另一種形式ItisanotherformofHurwitzcriterion.where:isallthehostsub-determinantswithdifferentorder.4).Lienard-Chipard

criterionThenecessaryandsufficientconditionofthesystemis:Forthesystemwiththecharacteristicequationas:or第二十九頁，共四十四頁，編輯于2023年，星期四3.ApplicationofRouth–Hurwitzcriterion1)Todeterminethesystemstability[example]ifthesystemcharacteristicequationis：,trytodeterminethesystemstability.Analyse:①

TolisttheRoutharrayasbelow：2unstablerootsareintherighthalfplane.

Thesystemisunstable.②③

Thereis2signchanges第三十頁，共四十四頁，編輯于2023年，星期四[example]

ifthesystemcharacteristicequationis：trytodeterminethesystemstability.Analyse:systemcharacteristicequationcanberewriteas:TheHurwitzdeterminantis：Thehostsub-determinantcanbecalculatedasTheconclusionisthatthesystemisstable.第三十一頁，共四十四頁，編輯于2023年，星期四2)Toanalyzetheinfluenceofsystemparameterchanging[example]thesystemblockdiagramisgivenasbelow,trytodeterminethecriticalamplifyingcoefficient.[Solution]closedlooptransferfunctionis：Thecharacteristicequationis：AnimportanteffectoftheRouth-Hurwitzcriterionistoanalyzetheinfluenceofsomesystemparametersvaryingsuchastheopen-loopsystemamplifyingcoefficientK.Wecanusethecriteriontodeterminethemaximum–criticalamplifyingcoefficient.第三十二頁，共四十四頁，編輯于2023年，星期四TowriteouttheRoutharrayasbelow：Accordingtothenecessaryandsufficientcondition:①allthecoefficientsmustbebiggerthan0.②theelementslainonthefirstcolumnoftheRoutharrayshouldbepositive.Thenwemayget:Thecriticalamplifyingcoefficientis.Thecharacteristicequationis：第三十三頁，共四十四頁，編輯于2023年，星期四3)Todeterminetherelativesystemstability(stabilizationabundance)Asweknow,wecanusetheRouth-Hurwitzcriteriontodeterminewhetheracontrolsystemisstableorunstable.Itisaabsolutestability.ifwewanttoknowtherelativestabilityofacontrolsystem,orhowcanitbedetermined?Usually,thedistancebetweenthecharacteristicrootpofthemaximumrealpartandtheimaginaryaxisisusedvirtuallytoexpressthesystemstabilizationabundance.Clearly,ifpislainontheimaginaryaxis,,thatmeansthesystemstabilizationabundanceis0.Howdowedeterminethesystemstabilizationabundance?第三十四頁，共四十四頁，編輯于2023年，星期四Todrawaverticallineonthecomplexplaneswhichisparalleltotheimaginaryaxis,andifallthecharacteristicrootsareontheleftoftheline,thesystemiscalledofstabilizationabundance.Thebiggertheis,themorestablethesystemis.Problem:Howtofindtheinacontrolsystem?①Let,andsubstitutethecomplexvariableswithinthecharacteristicequation,thenleadanewcharacteristicequationwithanewcomplexvariablez.②UseRouth-Hurwitzcriteriontoanalyzethesystemstabilityaccordingtothenewcharacteristicequation.③Ifthenewsystemisstable,theoriginalsystemiscalledofstabilizationabundance.第三十五頁，共四十四頁，編輯于2023年，星期四[example]asystemcharacteristicequationis,Thereisapairofimaginaryroots.Thenewsystemiscriticalstable.Andtheoriginalsystemisof1abundancestability.Howabouttherelativesystemstability?[solution]clearly,AndThesystemisstable.Let,substitutetheswithz-1,thenewequationis:or第三十六頁，共四十四頁，編輯于2023年，星期四Usually,therealpartofthecogentcomplexrootrepresentstheattenuationspeedofsystemresponse,whereastheimaginarypartofthecogentcomplexrootrepresentstheoscillatingofsystemresponse.istheanglebetweenthepolarandthenegativerealaxis.Thesmallertheangleis,thebetterthesystemqualityis.Anotherformtodiscusstherelativestability,therelativestabilityisworst.第三十七頁，共四十四頁，編輯于2023年，星期四3.Essentialunstablesystemandtheplantoimprove1)Whatistheessentialunstablesystem?Itisthesystemwhoseperformancecannotbeimprovedjustbyadjustingthesystemparameters.結(jié)構(gòu)不穩(wěn)定系統(tǒng)及其改進(jìn)措施-杠桿和放大器的傳遞函數(shù)執(zhí)行電機(jī)的傳遞函數(shù)進(jìn)水閥門的傳遞函數(shù)控制對(duì)象水箱的傳遞函數(shù)2)Example:liquidheightcontrolsystem第三十八頁，共四十四頁，編輯于2023年，星期四結(jié)構(gòu)不穩(wěn)定系統(tǒng)及其改進(jìn)措施Closedlooptransferfunction：let：Characteristicequation：or：clearly：Routharray：Nomatterhowtochangeth