reviewChapter1MathematicalmodelChapter2TimeresponseChapter3QualitativeanalysisChapter4StabilityofLinearsystemTheoutline4-1Introduction4-2TheStabilityandTheFirstMethod4-3TheSecondMethodLTIS

Controllabilityandobservability–twoimportantqualitativeproperties.Nowintroduceanotherqualitativeproperty—stability.4-1IntroductionTherootsdescribingthedynamiccharacteristicsofthedynamicsystemlocatesinleftpartoftherootplaneStabilityCriterionNyquiststabilitycriterion,Routhstabilitycriterion,etc.

time-varyingsystemsandnonlinearsystemLyapunovstabilityproposedbyRussianmathematicianLyapunovisanimportantmethodtoanalyzethestabilityofsystemLyapunovmethod

thefirstmethodthesecondmethodThefirstmethodThefirstmethodasapproximatemethodistosolvethedifferentialequationsofsystems,andtodeterminethestabilityofthesystemaccordingtothenatureofsolutions。對于非線性系統(tǒng)，在工作點(diǎn)附近的一定范圍內(nèi)，可以用線性化了的微方程式來近似地加以描述。如果線性化特征方程式的根全都是負(fù)實(shí)數(shù)根，或者是具有負(fù)實(shí)部的復(fù)根，則該系統(tǒng)在工作點(diǎn)附近是穩(wěn)定的，否則是不穩(wěn)定的。

ThesecondmethodThesecondmethodiscalleddirectmethod,andthebasicideaistoanalyazeenergychangesinthestabilityofthesystem.Iftheenergystoredinthemotionprocessgraduallyreduce,thesystemwillbestable;theotherhand,ifthesystemcontinuetoabsorbenergyfromtheoutsideworldthesystemwillnotbeabletostable.Stabilityreferstothemovementofthesysteminequilibriumdisturbed.4-2TheStabilityandtheFirstMethodHomogeneousstateequationThesystemisstableifIfexist

Thenthesystemisstable

--(Euclidian)norm

ThesystemisstableaccordingtoLyapunovmethod

。

Equilibriumstate，ifIfexist

ThensystemisAsymptotically

stable

Asymptoticallystable--(Euclidian)norm。

thestateequation

TheoremForLTIS,thenecessaryandsufficientconditionofstablilityisthattheeigenvaluesofAveIfthecoefficientmatrixAhasapositiverealparteigenvalues,thenthedynamicresponseofthecorrespondingcomponentmustincreasetoinfinity.Asresultthesystemisnotstable.So，arelinearcombinationIfitisJordanmatrix，include。Onlytherealpartsofeigenvaluesarenegative,isthesystemstableNonsingularmartrixP，makeisdiagonalmatrix，Thestateequation:areeigenvalue式中

Lyapunovfirstmethod李亞普諾夫第一方法指出：對于非線性特性不很嚴(yán)重的實(shí)際系統(tǒng)，可利用線性化的數(shù)學(xué)模型按線性系統(tǒng)的穩(wěn)定條件去分析它的穩(wěn)定性。1、若線性化系統(tǒng)的系數(shù)矩陣A的特征值全部具有負(fù)實(shí)部，則實(shí)際系統(tǒng)就是漸近穩(wěn)定的。線性化過程中被忽略的高階導(dǎo)數(shù)項(xiàng)對系統(tǒng)的穩(wěn)定性沒有影響（或影響甚微）。2、如果系數(shù)矩陣A中只要有一個實(shí)部為正的特征值，那么實(shí)際系統(tǒng)就是不穩(wěn)定的，與被忽略的高階導(dǎo)數(shù)項(xiàng)無關(guān)。3、如果系數(shù)矩陣A的特征值中，即使只有一個實(shí)部為零，其余的都具有負(fù)實(shí)部，那么實(shí)際系統(tǒng)就不能靠線性化的數(shù)學(xué)模型來判別其穩(wěn)定性。這時，系統(tǒng)的穩(wěn)定與否，與被忽略的高階導(dǎo)數(shù)項(xiàng)有關(guān)，必須分析原始的非線性數(shù)學(xué)模型才能決定它的穩(wěn)定性。

設(shè)系統(tǒng)狀態(tài)方程假定系統(tǒng)的平衡狀態(tài)，將系統(tǒng)的狀態(tài)方程在平衡的鄰域展成泰勒級數(shù),得4-3TheDirectMethodofLyapunov

Iftheenergystoredinthesystemreducesinworkingprocess,thesystemwillbestable;ontheotherhand,ifthesystemabsorbsenergyfromtheoutsideintheworkingprocesssothatmoreandmoreenergywillbestored,thesystemisnotbeabletobestable.IdeaHereusescalarfunctiontorepresenttheenergy，thenitshouldmeet

representstherateofenergychange

Showsthattheenergyreduceswithtime；when

ShowsthattheenergyincreaseswithtimeiscalledLyapunovfunction。

Lyapunovsecondmethoddeterminesthestabilityofthesystembasedon.Becauseitdirectlyusesenergyfunctionratherthansolvesthedifferentialequationstodeterminethestabilityofsystems.Therefore,itisalsoknownasLyapunovdirectmethod.St侍ab斧il現(xiàn)it白y混Cr乞it際er訪io柿nTheorem

selecttheinitialstateand1．If，systemisstable(ifSystemisasymptoticstabilityinanyinitialstate

)；2．If，systemisnotstable；3．If，butitdoesnotidenticallyequaltozeroforever,thesystemisasymptoticallystable;However,ifitidenticallyequaltozero,thenaccordingtothedefinitionofLyapunovstability,thesystemisstablebutnotasymptoticallystableHo乳w酷t(yī)o卸o愿bt滲ai馬nLy其ap檔un叔ovfu喇nc場ti淋onex抗am埋pl撞eso風(fēng)lu倆ti所on：Gi利ve嗽n拍th選e輔st墓at跨e鬼eq棋ua進(jìn)ti樹onSupposeaenergyfunction

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It瞇i咬s壩n耕ot堆d葛ef陡in涂it考e，so四w塘e陜ca吉nn殼ot睬k烘no社w轎th療e墻st斷ab廊il雪it鹿y奏.子Bu童t良we抵c壤an撒f默in兇d鵲an會ot脊he現(xiàn)r醫(yī)fu題nc摘ti摟on莊.，thesystemisstableWe波c悉ho掩os期e劍po還si伯ti家ve孟d崇ef令in伍it數(shù)e惰fu始nc采ti查onso

then

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Q＝I

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