The-Root-Locus

2019/11/[e-mail]3Unit132019/11/29rootlocus根轨迹unfactoreda.不可分解的highorder高阶gainn增益degreeofstability稳定程（裕）度complexvariable复变量complexfunction复变函数complexconjugates共轭复数zeron零点polen极点AngleCriteria相角条件MagnitudeCriteria幅值条件parameterofinterest增益参数argumentn幅角thecounterclockwisedirection逆时针方向polarform极坐标形式eigenvaluen.特征值asymptoten.渐近线breakawaypoint分离点centroidn.质心，心形曲线intersectv.相交arrivalanglen.入射角departureanglen.出射角dynamiccompensation动态补偿pole-zeropattern零、极点图activea.主动的，有源的passivea.被动的，无源的phase-leadn.相位超前phase-lagn..相位滞后的Proportion-Differential(PD)比例微分控制Proportion-integral(PI)比例积分控制dashedcurves虚线constraintn.约束条件NewWords2019/11/29TheRootLocusThethreebasicperformancecriteriaforacontrolsystemarestability,acceptablesteady-stateaccuracy,andanacceptabletransientresponse.Withthesystemtransferfunctionknown,theRouth-Hurwitzcriterionwilltelluswhetherornotasystemisstable.控制系统的三个基本性能准则为：稳定性，容许的稳态精度和满意的动态响应。由系统的传递函数知，劳斯-郝耳维兹判据用来判断系统的稳定性。2019/11/29Ifitisstable,thesteady-stateaccuracycanbedeterminedforvarioustypesofinputs.Todeterminethenatureofthetransientresponse,weneedtoknowthelocationinthesplaneoftherootsofthecharacteristicequation.Unfortunately,thecharacteristicequationisnormallyunfactoredandofhighorder.Note:Nature--Theessentialcharacteristicsandqualitiesofapersonorthing；n.性质,特性如系统稳定，不同类型输入下的稳态精度可以求得。为确定瞬态响应的特性，我们有必要知道系统特征方程根在s平面的位置。遗憾的是，特征方程通常是高阶不可分解的。2019/11/29Therootlocustechniqueisagraphicalmethodofdeterminingthelocationoftherootsofthecharacteristicequationasanysingleparameter,suchasagainortimeconstant,isvariedfromzerotoinfinity.Therootlocus,therefore,providesinformationnotonlyastotheabsolutestabilityofasystembutalsoastoitsdegreeofstability,whichisanotherwayofdescribingthenatureofthetransientresponse.Translation根轨迹是一种图解方法，是当系统的一个参数——如系统增益或时间常数从零变到无穷时，确定系统特征方程根的位置的方法。因此，由根轨迹不仅可以判断系统稳定，而且可知其稳定程度，根轨迹法是描述动态响应特性的另一种方法。2019/11/29Ifthesystemisunstableorhasanunacceptabletransientresponse,therootlocusindicatespossiblewaystoimprovetheresponseandisaconvenientmethodofdepictingqualitativelytheeffectsofanysuchchanges.Withouttransportlagthetransferfunctionofasystemcanbereducedtoaratioofpolynomialssuchthat.Translation如果系统不稳定或动态响应不满意，根轨迹方法给出改进动态响应的可能方法，也是定量地描述校正效果的一种简便方法。系统如无传输延时，其传递函数可以简写成：2019/11/29TherootlocustechniqueisdevelopedbyexpressingthecharacteristicfunctionD(s)asthesumoftheintegerunityandanewratioofpolynomialsins.thecharacteristicequationwillbewrittenasKisindependentofsandmustnotappearinthepolynomialsZ(s)andP(s).TheformofKZ(s)/P(s)isimportant;thesepolesandzerosmayberealorcomplexconjugates.NoteinEq.(2-3A-2)thatthecoefficientofsisalwayssetequaltounityforrootlocusoperations.Translation根轨迹方法可进一步将特征函数D(s)写为整数1与一个新的s多项式分式之和。特征方程可以写为：K与s无关，且不会出现在多项式Z(s)和P(s)中。KZ(s)/P(s)的形式很重要；零点和极点可以是实数或共轭复数。在根轨迹法中，注意公式(2-3A-2)中s前的系数总为1。2019/11/29AzeroisavalueofsthatmakesZ(s)equaltozeroandisgiventhesymbol.Donotautomaticallyassumethatthiszeroisalsoaclosed-loopzerothatmakesN(s)equaltozerointhesystem(open-loop)transferfunction;itmaybe,butisnotnecessarilyso.ApoleisavalueofsthatmakesP(s)equaltozeroandisgiventhesymbol.Thetermrepresentsnpoles,allequaltozeroandlocatedattheoriginofthesplane.ArootofthecharacteristicequationhaspreviouslybeendefinedasavalueofsthatmakesD(s)equaltozeroandisgiventhesymbol.Translation零点为使Z(s)的等于零的s的值，用符号表示。不能自然认定这一零点是闭环零点，即开环传递函数中使N(s)等于零的点，这只是一种可能，但不是必要条件。极点为使P(s)等于零的s的值，用符号表示。项表示n个位于s平面原点的零极点，特征方程的根定义为使D(s)等于零的s的值，用符号表示。nsns2019/11/29TheCharacteristicEquationSincesisacomplexvariableandthepolesandzerosmaybecomplex,KZ(s)/P(s)isacomplexfunctionandmay,therefore,behandledasavectorhavingamagnitudeandanassociatedangleorargument.Translation因为s是复变量，所以零、极点也可能为复数；KZ(s)/P(s)为复变函数而可作为向量处理，具有幅值及相应的相角或幅角。2019/11/29EachofthefactorsontherightsideofEq.(2-3A-2)canalsobetreatedasavectorwithanindividualmagnitudeandassociatedangle,asshowninFig.2-3A-1.Noticethattheangleismeasuredfromthehorizontalandispositiveinthecounterclockwisedirection.Translation公式(2-3A-2)右边的每个元素可以当成向量，如图2-3A-1有各自独立的幅值和相关的相角。注意相角由水平向右方向起始逆时针方向为正。2019/11/29Since-1canberepresentedbyavectorofunitymagnitudeandananglethatisanoddmultipleof.AccordingtoEq.(2-3A-3)andEq.(2-3A-4),wecanfindtwocriteriathatmakethecharacteristicfunctionD(S)equaltozero,i.e.,therearetwocriteriawhichcanfindsystem(close-loop)polesasKisincreasedfrom0to.Translation因为-1可以用单位幅值和的奇数倍的相角的向量来表示。根据公式(2-3A-3)和(2-3A-4)我们可以得出使特征方程D(S)等于零的两条准则。例如：当K由零变到时，闭环系统极点的两个准则为：2019/11/29RulesforRootLocusPlotting（根轨迹绘制法则）Applyingangleandmagnitudecriterion,therootlocicanobviouslybeplottedbyacomputer,however,we’llintroducerapidsketchingtechniques.Thefollowingguidesareprovidedtofacilitatetheplottingofrootloci:Translation根据幅值和相角条件，用计算机可以很容易绘制出根轨迹，但这里我们将介绍快速绘制技术。下列法则可以为绘制根轨迹提供方便：2019/11/29Thereareasmanyaslocusbranchesasthereareopen-looppoles.Abranchstarts,fork=0,ateachopen-looppole.Askisincreased,theclose-looppolepositionstraceoutloci,whichend,for,attheopen-loopzeros.Iftherearefeweropen-loopzerosthanpoles(ji),thosebranchesforwhi