现代控制理论Chapter4-Stability-of-Linear-system

ModernControlTheory沈阳航空航天大学自动化学院reviewChapter1MathematicalmodelChapter2TimeresponseChapter3QualitativeanalysisModernControlTheory沈阳航空航天大学自动化学院Chapter4StabilityofLinearsystemModernControlTheory沈阳航空航天大学自动化学院Theoutline4-1Introduction4-2TheStabilityandTheFirstMethod4-3TheSecondMethodModernControlTheory沈阳航空航天大学自动化学院LTISControllabilityandobservability–twoimportantqualitativeproperties.Nowintroduceanotherqualitativeproperty—stability.4-1IntroductionTherootsdescribingthedynamiccharacteristicsofthedynamicsystemlocatesinleftpartoftherootplaneStabilityCriterionNyquiststabilitycriterion,Routhstabilitycriterion,etc.ModernControlTheory沈阳航空航天大学自动化学院time-varyingsystemsandnonlinearsystemLyapunovstabilityproposedbyRussianmathematicianLyapunovisanimportantmethodtoanalyzethestabilityofsystemLyapunovmethodthefirstmethodthesecondmethodModernControlTheory沈阳航空航天大学自动化学院ThefirstmethodThefirstmethodasapproximatemethodistosolvethedifferentialequationsofsystems,andtodeterminethestabilityofthesystemaccordingtothenatureofsolutions。对于非线性系统，在工作点附近的一定范围内，可以用线性化了的微方程式来近似地加以描述。如果线性化特征方程式的根全都是负实数根，或者是具有负实部的复根，则该系统在工作点附近是稳定的，否则是不稳定的。ModernControlTheory沈阳航空航天大学自动化学院ThesecondmethodThesecondmethodiscalleddirectmethod,andthebasicideaistoanalyazeenergychangesinthestabilityofthesystem.Iftheenergystoredinthemotionprocessgraduallyreduce,thesystemwillbestable;theotherhand,ifthesystemcontinuetoabsorbenergyfromtheoutsideworldthesystemwillnotbeabletostable.ModernControlTheory沈阳航空航天大学自动化学院Stabilityreferstothemovementofthesysteminequilibriumdisturbed.4-2TheStabilityandtheFirstMethodHomogeneousstateequation()(),xtfxttThesystemisstableif0()exxS或IfexistThenthesystemisstable()()extxS或1222201122[()()()]eeennexxxxxxxx--(Euclidian)normModernControlTheory沈阳航空航天大学自动化学院0ex0()xt0()()Atxtext0lim()lim()0AtttxtextThesystemisstableaccordingtoLyapunovmethod。()(),xtfxttEquilibriumstate，ifex0()exxS或IfexistThensystemisAsymptoticallystable1222201122[()()()]eeennexxxxxxxxAsymptoticallystable--(Euclidian)norm。thestateequation()()xtAxt0)(limtxtModernControlTheory沈阳航空航天大学自动化学院TheoremForLTIS,thenecessaryandsufficientconditionofstablilityisthattheeigenvaluesofAallhavenegativerealparts.proveˆ10()()AtxtPePxtlim()0txtIfthecoefficientmatrixAhasapositiverealparteigenvalues,thenthedynamicresponseofthecorrespondingcomponentmustincreasetoinfinity.Asresultthesystemisnotstable.So，arelinearcombination12(),(),()nxtxtxt12,,...nttteeeˆA21,,2kkkttteteteIfitisJordanmatrix，include。Onlytherealpartsofeigenvaluesarenegative,isthesystemstableˆANonsingularmartrixP，make1ˆˆˆˆ()()()xtPAPxtAxtisdiagonalmatrix，12,,nxAxˆ()()xtPxtThestateequation:areeigenvalueModernControlTheory沈阳航空航天大学自动化学院式中Lyapunovfirstmethod李亚普诺夫第一方法指出：对于非线性特性不很严重的实际系统，可利用线性化的数学模型按线性系统的稳定条件去分析它的稳定性。1、若线性化系统的系数矩阵A的特征值全部具有负实部，则实际系统就是渐近稳定的。线性化过程中被忽略的高阶导数项对系统的稳定性没有影响（或影响甚微）。2、如果系数矩阵A中只要有一个实部为正的特征值，那么实际系统就是不稳定的，与被忽略的高阶导数项无关。3、如果系数矩阵A的特征值中，即使只有一个实部为零，其余的都具有负实部，那么实际系统就不能靠线性化的数学模型来判别其稳定性。这时，系统的稳定与否，与被忽略的高阶导数项有关，必须分析原始的非线性数学模型才能决定它的稳定性。设系统状态方程()(),xtfxtt假定系统的平衡状态，将系统的状态方程在平衡的邻域展成泰勒级数,得0ex0ex()xAxBxModernControlTheory沈阳航空航天大学自动化学院111122221212(,)nnTnnnnfffxxxffffxtxxxAdxfffxxxModernControlTheory沈阳航空航天大学自动化学院4-3TheDirectMethodofLyapunovIftheenergystoredinthesystemreducesinworkingprocess,thesystemwillbestable;ontheotherhand,ifthesystemabsorbsenergyfromtheoutsideintheworkingprocesssothatmoreandmoreenergywillbestored,thesystemisnotbeabletobestable.IdeaModernControlTheory沈阳航空航天大学自动化学院Hereusescalarfunctiontorepresenttheenergy，thenitshouldmeet()Vx()0Vxrepresentstherateofenergychange)(xV0)(xVShowsthattheenergyreduceswithtime；when0)(xVShowsthattheenergyincreaseswithtimeModernControlTheory沈阳航空航天大学自动化学院iscalledLyapunovfunction。()VxLyapunovsecondmethoddeterminesthestabilityofthesystembasedon.Becauseitdirectlyusesenergyfunctionratherthansolvesthedifferentialequationstodeterminethestabilityofsystems.Therefore,itisalsoknownasLyapunovdirectmethod.()VxModernControlTheory沈阳航空航天大学自动化学院StabilityCriterionTheoremselecttheinitialstateand1．If，systemisstable(ifSystemisasymptoticstabilityinanyinitialstate)；2．If，systemisnotstable；3．If，butitdoesnotidenticallyequaltozeroforever,thesystemisasymptoticallystable;However,ifitidenticallyequaltozero,thenaccordingtothedefinitionofLyapunovstability,thesystemisstablebutnotasymptoticallystable0x()0Vx()0Vxx()Vx()0Vx()0VxHowtoobtainLyapunovfunctionModernControlTheory沈阳航空航天大学自动化学院examplesolution：Giventhestateequation22121122221212()()xxxxxxxxxxSupposeaenergyfunction2212()Vxxx1122222212112212122212()22222Vxxxxxxxxxxxxxxxxx，thesystemisstable()0VxModernControlTheory沈阳航空航天大学自动化学院examplesolution：ThesecondmethodisanecessaryconditionbutnotsufficientconditionGiventhesystem0111xxstable?Theenergyfunctionispositivedefinite2212()2Vxxx2122212()222VxxxxxxxItisnotdefinite，sowecannotknowthestability.Butwecanfindanotherfunction.ModernControlTheory沈阳航空航天大学自动化学院，thesystemisst