現(xiàn)代控制理論ChaerSabiliy of Linear sysem

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)實數(shù)根，或者是具有負(fù)實部的復(fù)根，則該系統(tǒng)在工作點附近是穩(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

Th栗eo幸re居mFo刪r襪LT壯IS欠,筑th旺e污ne閑ce耗ss州ar許y當(dāng)an嚴(yán)d孟su倡ff裂ic錫ie樂nt喇c稿on盛di蠟ti滑on擇o濱fst軟ab善li隔li吊tyis稼t雅ha所t善th社eei教ge詳nv椒al繡ue零sof毛A護(hù)a帳ll賞h午av掉e迷ne痕ga瘦ti魂ve折r黃ea賴l買pa歌rt限s.pr掏ov登eIf虛t澇he假c許oe佩ff宅ic躲ie患nt毯m持at縫ri懼x腫A呀ha秤s壞a踏po絨si煩ti辜ve隙r昨ea濃l陵pa沿rtei嚴(yán)ge圣nv風(fēng)al棋ue足s,卻th球en司t良he箏d閘yn曉am芒ic逮r敘es卻po頓ns筑e章of弊t劫he塑c導(dǎo)or砌re蘿sp臘on精di便ng約c堪om平po纏ne勉nt計m喚us破t磨in絹cr獄ea技se遍t深o河in始fi慮ni翁ty贏.良As庫r夫es蹲u(yù)l歸t喂th墊e有sy定st母em砌i衰s絮no沈t燈st塞ab前l(fā)e添.So，arelinearcombinationIfitisJordanmatrix，include。On送ly蔑t須he摸r油ea梢l殼pa道rt戚s肚ofei乏ge觀nv匠al展ue撒sar迷e陡ne撥ga跟ti盛ve武,鵝is法t覆he拌s領(lǐng)ys獎te鉤m團(tuán)st岡ab之leNonsingularmartrixP，makeisdiagonalmatrix，Thestateequation:areeigenvalue式中Ly蠶ap項un忽ovfi吸rs嶄t刷me痕th幕od李亞鴨普諾直夫第變一方嫂法指俘出：顛對于非線鄭性特脅性不熊很嚴(yán)馳重的襲實際醋系統(tǒng)睛，可羨利用雷線性香化的挨數(shù)學(xué)仇模型嗓按線性遞系統(tǒng)歪的穩(wěn)暴定條貓件去斬分析英它的池穩(wěn)定源性。1、若線性揚(yáng)化系統(tǒng)忘的系數(shù)浪矩陣A的特鍋征值站全部極具有庫負(fù)實糟部，厘則實蘆際系末統(tǒng)就舞是漸奧近穩(wěn)弊定的渡。線叫性化生過程州中被歐忽略瓣的高畫階導(dǎo)對數(shù)項鵝對系瓣統(tǒng)的剖穩(wěn)定莖性沒賴有影速響（封或影塑響甚腦微）消。2、如霸果系數(shù)況矩陣A中只安要有困一個迫實部護(hù)為正啟的特栗征值炕，那催么實洋際系叮統(tǒng)就扔是不交穩(wěn)定用的，敏與被眨忽略會的高程階導(dǎo)箭數(shù)項賊無關(guān)各。3、如果系數(shù)錘矩陣A的特痰征值角中，微即使咽只有賓一個腿實部她為零站，其珍余的堵都具衡有負(fù)拆實部拘，那忌么實站際系造統(tǒng)就敗不能排靠線猜性化賀的數(shù)泰學(xué)模滿型來荷判別堪其穩(wěn)件定性警。這碗時，守系統(tǒng)盜的穩(wěn)織定與團(tuán)否，慶與被尚忽略搏的高聚階導(dǎo)桂數(shù)項胃有關(guān)芬，必排須分哲析原和始的稼非線利性數(shù)買學(xué)?？恍筒排つ軟Q抄定它費(fèi)的穩(wěn)臨定性根。設(shè)系貧統(tǒng)狀鼠態(tài)方啦程假定系統(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-賊3Th貝e遷Di紙re副ct塊M秒et榜ho寇d肥ofLy時ap冤un賭ovIf續(xù)t數(shù)he原e軍ne唱rg蒜y維st替or噸ed微i緒n蹦th城e封sy查st本em匹re誕du剛ce眨s欣in士w秩or醒ki較ng懲p迎ro宋ce央ss園,糕th斜e搬sy瞧st渴em蛙w燭il螞l印be炭s堆ta犧bl塵e;扒o竄n上th你e宜ot冊he摧r友ha討nd北,挺if嗓t頸he灶s倒ys斃te傻m桃ab啞so排rb土s縫en手er成gy故f哨ro浸m建th費(fèi)e垃ou袖ts硬id搭e幕in說t有he辱w驕or攤ki迅ng雜p攻ro暗ce煩ss怪s訓(xùn)o卡th亮at比m鍵or弄e滿an壞d管mo至re伍e賠ne碗rg銹y筑wi撓ll香b思e辜st泳or海ed邊,濾th鹿e鐘sy口st取em領(lǐng)i冷s貿(mào)no惠t日be脅a得bl轟e逆to至b魚e被st動ab搭le恥.Id喊eaHereusescalarfunctiontorepresenttheenergy，thenitshouldmeet

representstherateofenergychange

Sh頭ow結(jié)s寨th志at促t大he開e斑ne壯rg載y拉re玩du籮ce宮s謹(jǐn)wi獎th鼠t炕im管e；wh悶enSh商ow走s輝th液at開t號he協(xié)e百ne裹rg塊y抓in曉cr臂ea屬se公s從wi耗th蹄t歡im棚eiscalledLyapunovfunction。

Lyapunovsecondmethoddeterminesthestabilityofthesystembasedon.Becauseitdirectlyusesenergyfunctionratherthansolvesthedifferentialequationstodeterminethestabilityofsystems.Therefore,itisalsoknownasLyapunovdirectmethod.St爬ab惡il瓣it階y殺Cr退it延er胡io漫nTheorem

selecttheinitialstateand1．If，systemisstable(ifSystemisasymptoticstabilityinanyinitialstate

)；2．If，systemisnotstable；3．If，butitdoesnotidenticallyequaltozeroforever,thesystemisasymptoticallystable;However,ifitidenticallyequaltozero,thenaccordingtothedefinitionofLyapunovstability,thesystemisstablebutnotasymptoticallystableHo浩w脹to飽o鬼bt儀ai殊nLy已a(bǔ)p謝un痕ovfu弓nc椒ti姥onex喇a(chǎn)m惑pl漠eso腐lu榆ti款on：Gi插ve被n評th豈e沖st變at粒e朽eq辛ua席ti情onSupposeaenergyfunction

，thesystemisstableex媽am熄pl歪eso懇lu嘩ti襪on：Th翻e鬼se鍛co強(qiáng)nd淋m暈et臨ho丙d狀is松a粉n趣ec夸es筒sa材ry刑c倒on冒di少ti這on妥b食ut團(tuán)n胡ot壇s繩uf困fi青ci緞en沫t徐co仙nd普it明io屯nGiventhesystemst農(nóng)ab題le握?Theenergyfunctionispositivedefinite

It承i流s溜n匹ot探d四ef昂in氏it秒e，so棵w尸e麻ca賭nn脂ot喉k樣no樣w憂th穩(wěn)e禾st釘ab想il降it舅y陰.斗Bu氧t揉we唐c劣an消f銀in腿d歐an略ot哀he他r疼fu郵nc泰ti挖on州.，thesystemisstableWe修c貸ho秘os鄭e棋po埋si突ti料ve儲d部ef唱in辯it嗽e盛fu有nc亦ti超onso

then

Onlyif，even，isnotidenticallyequaltozero。Then,basedonstabilitycriterion(3),wecandeterminethesystemisasymptoticallystable.We鄭c涉ho宵os同eLy秒ap昏un抱ovse話co起nd鏟m另et許ho軟d不is棋a喘n澤ef窩fe慢ct類iv搭e歪wa針y銳to腰a脅na胞ly子ze勵t惱he嫁s皂ta述bi喚li役ty活o斷f選li裕ne損ar陰s收ys蘇te談ms悲n忠ot漏o篩nl趟y鋒fo劉r燭li甚ne遇ar斯t凝im園e-掩in封va且ri即an榜t閉sy互st賠em循s,攝b漸ut績a吩ls雙o酸fo蠢r乖ti其me趕-v村ar檔yi女ng測l挪in菊ea古r始sy巖st伸em筒s覺an裝d沙di宋sc予re斗te款-t系im這e子sy防st概em慰s.逃T陣hi憶s副se婆ct體io濤n叮wi豎ll麥f梁oc宵us讀o譽(yù)n臟th圓e嫂li寄ne誼ar祝t山im環(huán)e-榴in趴va囑ri士an礙t楊sy叢st勿em姨s日an做d乏li區(qū)ne崗ar疼t侍im捧e-寧in鐵va廊ri丙an倦t津di胸sc廟re回te達(dá)s諒ys細(xì)te伴ms總a拐na躺ly訪si應(yīng)s.P襲isre坐al斜s及ym霞me酷t(yī)r損ic纖p嚴(yán)os貞it蜻iv貍e兄de唇fi負(fù)ni雷te土m藏at虧ri牛xGi妙ve糞n窮th堤e饞sy都st都emAs媽su叮mi莫ng濾t搶he曾s河el擁ec略te技dLy尿ap準(zhǔn)un顏ovfu串nc陶ti昂on高a膨s縫fl偉ow第in沸g(shù),懸t撥he更re燭a箱rest喜ab猾il笑it植y你an野al譜ys顛is炕o擔(dān)f捧li齒ne完ar夢t劃im過e-童in哥va椒ri屆an挎t城sy舉st桿em創(chuàng)sP轎isre粉al錦s乒ym植me偉tr叔ic液p犬os發(fā)it重iv粉e詢de杜fi染ni謹(jǐn)te抱m遭at謠ri客xIf旦y職ou想c朝an賊f肢in翠d膀th字e皇p撐os甜it盞iv咳e信de相fi劍ni蠢te葵m夾at肝ri神ce尤s判P往an抱d稼Q,孔t黨he拉n肆th養(yǎng)e觀sy駐st廈em壘i晚s裝as傻ym雖pt喪ot比ic跑al欣ly乎s饞ta霉bl務(wù)e杰.。--Ly棚ap創(chuàng)un擇ovEq需ua聞ti勵onTheorem

Giventhesystemthenecessaryandsufficientconditionofasymptoticstability：GivenapositivedefinitesymmetricmatrixQ,thereisapositivedefinitesymmetricmatrixPtomeettheLyapunovequation.Andexistex捉am菜pl段eso果lu核ti尼on：In乏ve疤st漲ig躺at押e科th鏡e耕st氏ab激il袖it醉y惑of重t喊he碗g踢iv撥en京s繭ys供te棗mMake

Q＝I

P鉗is禿p眼os祖it撿iv尋e落de保fi椅ni堤te止a跨nd擊,降th陳er班ef蘇or燙e,妖t序he棵g逗iv飾en德s爛ys艘te志m逢is姿a道sy斯mp究to倍ti圍ca愿ll克y聲st藝ab席le欄.ex溜am制pl哈eso栽lu鍛ti事on：MakeQ＝I，In略ve墓st階ig顫at忠e意th覺e念st毅ab刃il益it驗y咽of追t幣he斯g卡iv即en花s插ys京te掏mGi往ve運(yùn)t扎he促s圍ta袍te獸e數(shù)qu摟at胞io直nP有is自n蓬eg啦at銹iv量e壞de峽fi驗ni哄te凈a核nd倚,斧th傍er麗ef降or魄e,膏t翻h(huán)e駕g攪iv鑰en掉s摩ys演te勾m步is聾n典ot疫a餃sy杯mp必to她ti托ca紐奉ll毀y浮st毀ab圾le統(tǒng).st棕ab光il肉it葛y猛an涌al燭ys哲is貞o泳fli的ne或ar告t雁im狗e-卸in慎va偷ri失an鞭t投di信sc盞re回te鳳s孔ys小te過msLy煉ap瓦un線ovEq厭ua鐵ti堆on窗:Th陶e伙ne柳ce川ss泰ar正y恥an哀d葬su徐ff概ic嚼ie全nt淚c屑o(jì)n傷di房誠ti滴on嗓o截f奴as別ym榮pt發(fā)ot押ic洲s究ta勤bi難li父ty我i客n雕ba語la登nc瓣e殊:刺gi頑ve竊n寫an披y盾po屆si跨ti晨ve寨d錦ef嚼in陣it潛e仆he萌rm順it刊m屈at唱ri奇x著Q,友t擱he偶re愉i麻s侍a擁po馬si健ti怠ve預(yù)d頑ef業(yè)in側(cè)it構(gòu)e鋤ma棉tr婆ix屯P否,秘sopr延ov閑eAc貓co偽rd醬in危g碰toLy姜ap芬un認(rèn)ovst沙ab臺il炎it圍y筋th嚇eo至re倚m,立r徒eq憑ui鎖reWe極c喊ho響os征eLy嚴(yán)ap沈un腥ovfu并nc洲ti種onQis套p統(tǒng)os典it溪iv驢e闖de塞fi明ni泊te。We走c奶an探g皮etth旱e怖n偏ec輝es糧sa淹ry貝a蒜nd緒s糕uf諸fi鈴ci揚(yáng)en牽t地co刃nd閑it杠io群n榜of痛a當(dāng)sy訴mp精to由ti古c闖st測ab原il幻玉it辭y如果沿任一解的序列不恒等于零，則矩陣Q也可取為正半定。Pl露ea枯se噸g據(jù)iv乘eth碑e圖n訴ec補(bǔ)es漏sa升ry捏a趣nd務(wù)s脖uf槽fi慣ci中en礎(chǔ)t騙co筒nd勤it嘩io靜nof勢a彩sy西mp備to律ti挨c趟st薪ab淡il砍it脹yex戲am串pl孤eso竿lu診ti向on：Q＝I

Th駁e走co扶nd章it亦io興n拋of替p劃os期it稱iv論e琴P禾issoAsLy矮ap何un莖ovfu癥nc別ti子on。st辰ab抖il歌it設(shè)y粗an嗽al警ys砍is澤o能f容Li俊ne林ar片t菠im考e-抹va鍋ry歡in潛g鐵sy握st閘em言s協(xié)ba窩se這d研onLy洞ap新un愧ovTh堂e當(dāng)st倘at鑼e蟻eq奏ua并ti振on漁i滑sSe林le踐ct寒s雙ca隔la隆r驗fu木nc決ti隸onTh奪e戶ne漸ce妄ss典ar怖y牛an惜d康su滔ff案ic榜ie襪nt晉a糕sy周mp寨to破ti邁c藝st懸ab父il