线性系统课后答案第5章

5.1IsthenetworkshowninFing5.2BIBOstable?Ifnot,findaboundedinputthatwillexciteanunboundoutput图5.2的网络BIBO是否稳定?如果不是.请举出一般激励无界输出的有界输入.FromFig5.2.wecanobtainthatxyyuxAssumingzeroinitialandapplyingLaplacetransformthenwehave1)(ˆ)(ˆ)(ˆ2sssusysytssLsgLtgcos]1[)]([)(211because002002322121cos)1(cos|cos||)(|xkkkktdttdtdttdttg=1+021231)sin()2[sin()1(kkk=1+021231]sin[sin)1()1(kkk=1+012122)1()1(kkk）（=1+201k=Whichisnotbounded.thusthenetworkshowninFin5.2isnotBIBOstable,Ifu(t)=sint,thenwehavey(t)=tttdt00)sin(cossin)cos(d=ttttd0sin21sin21wecanseeu(t)isbounded,andtheoutputexcitedbythisinputisnotbounded5.2considerasystemwithanirrationalfunction)(ˆsy.showthatanecessaryconditionforthesystemtobeBIBOstableisthat|g(s)|isfiniteforallRes0一个系统的传递函数是S的非有理式。证明该系统BIBO稳定的一个必要条件是对所有Res0.|g(s)|有界。Prooflets=,jthen0)()(ˆ)(ˆdteetgjgsgtjt00sin)(cos)(tdtetgjtdtetgttIfthesystemisBIBOstable,wehave0|)(|Mdttg.forsomeconstantMwhichimpliesshat0)(dttgisconvergent.ForallRes=,0Wehave|g(t)|costet|)(|tg|g(t0|)(||sintgtetso0cos)(tdtetgtand0sin)(tdtetgtarealsoconvergent,thatis.0cos)(Ntetgt0sin)(LtdtetgtwhereNandLarefiniteconstant.Then|21222122)())((|)(ˆLNLNsgisfinite,forallRes0.AnotherProof,IfthesystemisBIBOstable,wehave.|)(|0MdttgforsomeconstantM.And.ForallRes0.Weobtain||)(ˆsg00)(Re|0(|)(|Mdttgdtetgts.Thatis.||)(ˆsgisfiniteforallRes05.3Isasystemwithimpulseg(t)=11tBIBOstable?howaboutg(t)=ttefort?0脉冲响应为个g(t)=11t的系统是否BIBO稳定?0,)(ttetgt的系统又如何?Because)1ln(11|11|00ttdtdtt0001|)(||ttttetedttedttethesystemwithimpulseresponseg(t)=11tisnotBIBOstable,whilethesystemwithimpulseresponseg(t)=ttefor0tisBIBOstable.5．4Isasystemwithtransferfunction)1()(ˆ2sesgsBIBOstable一个系统的传递函数为)1()(ˆ2sesgs，问该系统是否BIBO稳定。Laplacetransformhastheproperty.If).()(atftgthen)(ˆ)(ˆsfesgsWeknow0,]11[1tesLt.Thus2,]11[)2(1teseLtsSotheimpulseresponseofthesystemis)2()(tetg.for2t2)2(01|)(|dtedttgtthesystemisBIBOstable5.5Showthatnegative-feedbackshowninFig.2.5(b)isBIBOstableifandonlyifandthegainahasamagnitudelessthan1.Fora=1,findaboundedinput)(trthatwillexciteanunboundedoutput.证明图2.5(b)中的负反馈系统BIBO稳定当且增到a的模小于1。对于a=1情况，找出一有界输入r(t),她所产生的输出是无界的。Poof,Ifr(t)=)(t.thentheoutputistheimpulseresponseofthesystemandequal)4()3()1()(43taratatg=11)()1(tiiitatheimpulseisdefindedasthelimitofthepulseinFig.3.2andcanbeconsideredtobepositive.thuswehave1)(|||)(|tiitatgand0101|)|1(||||)(|||)(|ttaaiiadtitadttg1||0||aifaifwhichimpliesthatthenegative-feedbacksystemshowninFig.2.5(b)isBIBOstableifandonlyifthegainahasmagitudelessthan1.For1a.ifwechoose).sin()(ttrclearlyitisboundedtheoutputexcitedby)sin()(ttris10111111111)sin()sin()1()1()sin()1()sin()1()sin()1()()()(itiiiiiiiiittitititdrtgtyAndy(t)isnotbounded5.6considerasystemwithtransferfunction)1()2()(ˆsssg。whatarethesteady-stateresponsesbyu(t)=3.for0t.andbyu(t)=sin2t.for?0t一个系统的传递函数是)1()2()(ˆsssg，分别由0.3)(ttu和0.2sin)(tttu所产生的稳态响应是什么？Theimpulseresponseofthesystemis0.3)(]13[`]12[)(11tetsLssLtgt。Andwehave000004313)()3||)((||3)(||)(|dtedttdtetdtetdttgtttsothesystemisBIBOstableusingTheorem5.2,wecanreadilyobtainthesteady-stateresponsesifu(t)=3for0t.,thenas)(,tytapproaches6323)0(ˆgifu(t)=sin2tfor,0tthen,as)(.tytapproaches)25.1sin(26.1)3arctansin(5102)(ˆsin(|)(ˆ|ttjgtjg）5.7considerxux0210101y=[-23]ux2isitBIBOstable?问上述状态方程描述的系统是否BIBO稳定？Thetransferfunctionofthesystemis202110)1)(1(1011322021010132)(ˆ1ssssssg=122214sssThepoleofis–1.whichliesinsidetheleft-halfs-plane,sothesystemisBIBOstable5.8consideradiscrete-timesystemwithimpulseresponsekkkg)8.0(][for,0kisthesystemBIBOstable?一个离散时间系统的脉冲响应序列是kkkg)8.0(][，,0k。问该系统是否BIBO稳定？Because000)8.0()8.01(2.01)8.0(|][|kkkkkkkkg=0001208.018.02.018.02.01])8.0()8.0([2.01kkkkkkkkG[k]isabsolutelysumablein,0[].Thediscrete-timesystemisBIBOstable.5.9Isthestateequationinproblem5.7marginallystable?Asymptoticallystable?题5.7中的状态方程描述的系统是否稳定？是否渐进稳定？Thecharacteristicpolynomial)1)(1(10101det)det()(AItheeigenralueIֹhaspositiverealpart,thustheequationisnotmarginallystableneitherisAsymptoticallystable.5.10Isthehomogeneousstableequationxx000000101.marginallystable?Asymptoticallystable?齐次状态方程xx000000101，是否限界稳定？Thecharacteristicpolynomialis)1(0000101det)det()(2AIAndtheminimalpolynomialis)1()(.Thematrixhaseigenvalues0,0.and–1.theeigenvalue0isasimplerootoftheminimal.thustheequationismarginallystableThesystemisnotasymptoticallystablebecausethematrixhaszeroeigenvalues5。11Isthehomogeneousstableequationxx000000101marginallystable?asymptoticallystable?齐次状态方程xx