自动控制原理(双语)chapter2

ThePrincipleofAutomaticControl

自动控制原理Lecturers：Prof.JiangBinDr.LuNingyunCollegeofAutomationEngineeringNUAA，2010.Autumn1NUAA-ThePrincipleofAutomaticControl

Chapter2Modelingofcontrolsystems控制系统建模2ReviewofChap1ImportantconceptsOpen-loopandclosed-loopcontrolsystems?AdvantagesanddisadvantagesofOLandCLcontrolsystems?Basicrequirementsofacontrolsystem?3HowtoanalyzeanddesignacontrolsystemThefirstthingistoestablishsystemmodel(mathematicalmodel)RefinputcontrollerManipulatedvar.plantdisturbanceOutputerrorsensor4Chap2Modelingofcontrolsystems2-1Introduction2-2Establishmentofdifferentialequationandlinearization2-3Transferfunction2-4Structurediagram2-5Signal-flowgraphs52-1Introductiononmathematicalmodels6SystemmodelDefinition：Mathematicalexpressionofdynamicrelationshipbetweeninputandoutputofacontrolsystem.

MathematicalmodelisfoundationtoanalyzeanddesignautomaticcontrolsystemsNomathematicalmodelofaphysicalsystemisexact.Wegenerallystrivetodevelopamodelthatisadequatefortheproblemathandbutwithoutmakingthemodeloverlycomplex.7ThreemodelsDifferentialequation （微分方程）Transferfunction （传递函数）Frequencycharacteristic （频率特性）LaplacetransformFouriertransformTransferfunctionDifferentialequationFrequencycharacteristicLinearsystemStudy

time-domainresponsestudyfrequency-domainresponse8ModelingmethodsAnalyticmethod

解析方法/机理方法AccordingtoNewton’sLawofMotionLawofKirchhoffSystemstructureandparametersthemathematicalexpressionofsysteminputandoutputcanbederived.Thus,webuildthemathematicalmodel(suitableforsimplesystems).9ModelingmethodsSystemidentificationmethod系统辨识方法

Buildingthesystemmodelbasedonthesysteminput—outputsignalThismethodisusuallyappliedwhentherearelittleinformationavailableforthesystem.BlackboxInputOutputBlackbox:thesystemistotallyunknown.Greybox:thesystemispartiallyknown.NeuralNetworks,FuzzySystems10WhyFocusonLTISystem(线性时不变系统)Whatislinearsystem?Asystemiscalledlineariftheprincipleofsuperposition(叠加原理）appliessystemk1*u1k2*u2Y=k1*y1+k2*y2Isy(t)=u(t)+2alinearsystem？11Whatistime-invariantsystem?Asystemiscalledtime-invariantiftheparametersarestationarywithrespecttotimeduringsystemoperation时变系统的例子？12ForaLTIsystemTheoverallresponseofalinearsystemcanbeobtainedby--decomposingtheinputintoasumofelementarysignals--figuringouteachresponsetotherespectiveelementarysignal--addingall

theseresponsestogether.Thus,wecanusetypicalelementarysignal(e.g.unitstep,unitimpulse,unitramp)toanalyzesystemforthesakeofsimplicity.132.2Establishmentofdifferentialequationandlinearization14DifferentialequationLinearordinarydifferentialequations(线性常微分方程）AwiderangeofsystemsinengineeringaremodeledmathematicallybydifferentialequationsIngeneral,thedifferentialequationofann-thordersystemiswritten时域数学模型15HowtoestablishODEofacontrolsystem首先要根据各个元件的物理规律，列写各个元件的微分方程，得到一个微分方程组，然后消去中间变量，即得控制系统总的输入和输出的微分方程。---listdifferentialequationsaccordingtothephysicalrulesofeachcomponent;---obtainthedifferentialequationsetsbyeliminatingintermediatevariables;---gettheoverallinput-outputdifferentialequationofcontrolsystem.16Examples-1RLCcircuitRLCu(t)uc(t)i(t)Input（输入）u(t)systemOutput（输出）uc(t)17AccordingtoLawofKirchhoffinelectricityRLCu(t)uc(t)i(t)Itisre-writtenasinstandardform18关于例1的一个思考题能否直观地判断出一个电路系统是几阶微分方程？19Examples-2themass-spring-frictionsystem质量块－弹簧－阻尼系统mkF(t)Displacementx(t)ffrictionSpringWeareinterestedintherelationshipbetweenexternalforceF(t)andmassdisplacementx(t)Define:input—F(t);

output---x(t)Gravityisneglected.20Generally，wesettheoutputontheleftsideoftheequationtheinputontherightside

theinputisarrangedfromthehighestordertothelowestorder21Examples-3Nonlinearsystem现实中，绝大多数系统本质上都是非线性系统(nonlinearsystem)，例如钟摆系统(pendulumsystem)，需要用非线性微分方程描述。Itisdifficulttoanalyzenonlinearsystems,however,wecanlinearize(线性化）thenonlinearsystemnearitsequilibriumpoint(平衡点)undercertainconditions.22LinearizationofnonlineardifferentialequationsSeveraltypicalnonlinearcharacteristicsincontrolsystem23控制系统会有意引入非线性特性以便取得更好的控制效果。Forexample,inmanymissileorspacecraftcontrolsystems,toachieveminimum-timecontrol,anon-off(bang-bangorrelay)typecontrollerisused0xyML24Methodsoflinearization（1）Weaknonlinearityneglected（2）Smallperturbation/errormethodAssumption:Inthesystemcontrolprocess,therearejustsmallchangesaroundtheequilibriumpointintheinputandoutputofeachcomponent.Ifthenonlinearityofthecomponentisnotwithinitslinearworkingregion,itseffectonthesystemisweakandcanbeneglected.Thisassumptionisreasonableinmanypracticalcontrolsystem:inclosed-loopcontrolsystem,oncethedeviationoccurs,thecontrolmechanismwillreduceoreliminateit.Consequently,allthecomponentscanworkaroundtheequilibriumpoint.25A(x0,y0)isequilibriumpoint（平衡点）.Expandingthenonlinearfunctiony=f(x)intoaTaylorseriesaboutA(x0,y0)（泰勒展开）yields输入输出只是在平衡点附近作微小变化Thisislinearizedmodelofthenonlinearcomponent.

例26Note：thismethodisonlysuitableforsystemswithweaknonlinearity.

Forsystemswithstrongnonlinearity,wecannotusesuchlinearizationmethod.relaysaturation27Examples-4ModelinganonlinearsystemFlashtoiletProblem:Derivethedifferentialequationofwatertank(thecross-sectionalareaofthewatertankisC).Q1:inflowperunittimeQ2:outflowperunittimeInitialwaterlevel:H0Q10=Q20=0Define:Input—Q1,Output—H28Solution:

Outfloworinflowwithindttimeshouldbeequaltothetotalamountofwaterchange（Q1-Q2）dt,thatis:

Accordingto‘TorricelliTheorem’,thewateryieldisindirectproportiontothesquarerootoftheheightofwaterlevel,thus:

Itisobviousthatthisformulaisnonlinear,OnthebasisoftheTaylorSeriesexpansion

offunctionsaroundoperationpoints(Q10,H0)，Wehave292-3Transferfunction

(传递函数）30SolvingDifferentialEquations例Solvinglineardifferentialequationswithconstantcoefficients,•Tofindthegeneralhomogeneoussolution（奇次通解）(involvingsolvingthecharacteristicequation特征方程)•Tofindaparticularsolutionofthecompletenonhomogeneousequation（非奇次特解）(involvingconstructingthefamilyofafunction)•Tosolvetheinitialvalueproblem（初值问题）31WHYneedLAPLACEtransform?AlgebraproblemsSolutionsofalgebraproblemsInitialvalueproblemsODEsorPDEsSolutionsofInitialvalueproblemsLT拉氏变换InverseLT反拉氏变换32LaplaceTransformLaplace,Pierre-Simon1749-1827TheLaplacetransformofafunctionf(t)isdefinedas33Examples阶跃信号f(t)=A脉冲信号f(t)=σ(t)指数信号f(t)=34Laplacetransformtable35PropertiesofLaplaceTransform(1)Linearity(2)Differentiationf(0)istheinitialvalueoff(t).证明（用分部积分）36(3)Integration

whereistheinitialvalueofintegration

证明（用分部积分）37（4）Final-valueTheoremifthelimitexists.(5)Initial-valueTheoremifthelimitexists.Thefinal-valuetheoremrelatesthesteady-statebehavioroff(t)tothebehaviorofsF(s)intheneighborhoodofs=038(6)ShiftingTheorem：a.shiftintime(realdomain)b.shiftincomplexdomain(7)Realconvolution(Complexmultiplication)Theorem39InverseLaplacetransformDefinition：InverseLaplacetransform,denotedbyisgivenbywhereCisarealconstant。Note:TheinverseLaplacetransformoperationinvolvingrationalfunctionscanbecarriedoutusingLaplacetransformtableandpartial-fractionexpansion.40Partial-FractionExpansionmethod(因式分解）forfindinginverseLaplacetransformsIfF(s)isbrokenupintocomponentsIftheinverseLaplacetransformsofcomponentsarereadilyavailable,then41Case1:F(s)hassimplerealpoles(单实根）42Example143Case2:F(s)hassimplecomplex-conjugate(共轭复极点）polesExample244Case3:F(s)hasmultiple-orderpoles.(多重根）45Example346WithaidofMATLAB1.LaplaceTransformL=laplace(f)Note:f是t的符号表达式2.

InverseLaplaceTransform F=ilaplace(L)>>symst>>L=laplace(t)L=1/s^2>>L=laplace(sin(t))L=1/(s^2+1)>>F=ilaplace(L)F=sin(t)47Transferfunction（传递函数）SYSInputr(t)Outputc(t)ODEmodel:LaplaceTransform(withzeroinitialconditions)Transferfunctionmodel:时域复域48定义：传递函数是系统输出量拉氏变换与输入量拉氏变换之比。上式为多项式形式的传函，其中M(s)andN(s)arenumerator(分子）anddenominator（分母）polynomials（多项式）ofG(s).传函的分母多项式N(s)定义为系统的特征多项式。满足特征多项式的根定义为特征根，特征根的位置和系统的响应特性密切相关。49PolesandzerosofTFNote:Sincethereexistssomehowinertiainsystemcomponent,n>=m.Ifm>n,thesystemiscalledunachievable.增益零极点（gain-zero-pole)传函50关于传函的几点说明:1.传递函数是复变量s的有理真分式函数，具有复变函数的所有性质。2.传函与微分方程具有相通性，零初始条件下可通过n阶导数与n阶s的置换得到传函。3.传递函数表征了系统本身的动态特性。（传递函数只取决于系统本身的结构参数，而与输入和初始条件等外部因素无关，可见传递函数有效地描述了系统的固有特性。）4.只能描述线性定常系统与单输入单输出系统，且内部许多中间变量的变化情况无法反映，是系统一种外部描述。5.如果存在零极点对消情况，传递函数就不能正确反映系统的动态特性。517.传函是在零初始条件下定义的,只能反映零初始条件下输入信号引起的输出，不能反映非零初始条件引起的输出。在给定输入量时系统的零初始条件响应为：6.传函G(s)的反拉氏变换为系统的脉冲响应g(t)，即传函可定义为52传函的不同形式及相关概念PolynomialTF多项式传函ZPKTF零极点增益传函时间常数传函，多用于频域分析令G(s)中s＝0,可得开环增益53TransferfunctionmodelsinMATLABSupposealinearSISOsystemwithinputu(t),outputy(t),thetransferfunctionofthesystemisDescendingpowerofs多项式形式传函>>Sys=tf（num，den）>>[num,den]=tfdata(sys)54零极点形式传函>>sys=zpk（z，p,k）>>[z,p,k]=tfdata(sys)多项式传函变换为零极点传函>>[z,p,k]=tf2zp(num,den)55Howpolesandzerosrelatetosystemresponse

传递函数零极点与系统输出运动模态的关系Whywestrivetoobtainmathematicalmodels?WhycontrolengineersprefertouseTFmodel?HowtouseTFmodeltoanalyzeanddesigncontrolsystems?westartfromtherelationshipbetweenthelocationsofzerosandpolesofTFandtheoutputresponsesofasystem56-aji0Q:Whereisthezero?0PositionofpolesandzerosTransferfunctionTime-domainimpulseresponse57-ajib00PositionofpolesandzerosTransferfunctionTime-domainimpulseresponse58jib00PositionofpolesandzerosTransferfunctionTime-domainimpulseresponse59-aji0Transferfunction

Time-domainimpulseresponsePositionofpolesandzeros60TF：-ajib00Time-domainimpulseresponsePositionofpolesandzeros61Asummaryofpolepositionandsystemdynamicresponse62Transferfunctionoftypicalcomponents－163例：RCcircuitTakingLaplacetransformonbothsides,64Transferfunctionoftypicalcomponents－265例：位置随动系统--结构图如何求传递函数？如何分析设计系统？66Followedby…对于复杂的控制系统，很难直接得到整个系统的传递函数，因此下面学习系统结构图和信号流图，可简化求取系统传递函数的过程672-4Structurediagram

结构图68Concepts将方框图中各时间域中的变量用其拉氏变换代替，各方框中元件的名称换成各元件的传递函数，这时方框图就变成了结构图。结构图以及信号流图是控制理论中描述复杂系统的一种简便方法。69Composition（组成）(1)Functionblock：有输入信号，输出信号，传递线，方框内的函数为输入与输出的传递函数，一条传递线上的信号处处相同。

(2)Junctionpoint(汇合点/比较点）：

(3)Deducingpoint（引出信号点）：

G(s)X(s)Y(s)－70DrawblockdiagramProcedures:根据每个元件的拉氏变换方程，绘出其单元结构图。置系统的输入于最左端，输出于最右端，按照信号的流向，把各单元结构图中相同的信号连接起来，就得到整个系统的结构图.71Example：CascadeoftwoRCfilteringnetworksSTEP1：各信号与元件替换为s域描述（拉氏变换）72STEP2：从左到右写出信号之间的S域方程（lawofKirchhoff）73－－－74Equivalenttransform（等效变换）ofstructurediagram1cascadeconnection(级连)G(s)X(s)Y(s)X1(s)G1(s)G2(s)X(s)Y(s)752.ParallelconnectionG(s)X(s)Y(s)X(s)G2(s)G1(s)Y1(s)Y2(s)Y(S)763.FeedbackR(s)C(s)C(s)G(s)H(s)E(s)R(s)774.比较点和引出点的移动RCQRCQRCQRCCRCRRCQRCCRCR78－－－Example：CascadeoftwoRCfilteringnetworks－－－79－－－－－－80－－－－－81－－－－82－－－83说明结构图的等效变换和简化过程实际上对应于由元部件运动方程中消去中间变量求取系统传递函数的过程。但简化变换过程仍然比较复杂冗长，相比而言，信号流图符合简单，更便于绘制和应用。842-5Signal-flow

graph(信号流图)85TerminologySFGwasintroducedbyS.J.Masonforthecause-and-effectrepresentationoflinearsystems.Node（节点）结构图中所有的引出点，比较点称节点,代表方程式中变量，用小圆圈表示。支路(path)，连接两个节点的有向线段，支路增益表示两个变量之间的因果关系。RIU欧姆定律：U=R*I863.源节点（输入节点）：只有信号输出支路的节点4、阱节点（输出节点）：只有信号输入支路的节点5.混合节点：既有信号输入又有信号输出的节点6.Forwardpath（前馈通路）：从输入到输出，并与任何一个节点相交不多于一次的通路，叫前向通路，前向通路中各传递函数的乘积，叫前向通路增益P。7.Loop（回路）：起点和终点在同一节点，且与其他节点相交不多于一次的闭合通路叫单独回路，回路中所有传递函数的乘积叫回路增益L。8.Non-touchingloops（不接触回路）：相互间没有公共节点的回路称为不接触回路。1abcd1gef87DrawtheSFG由系统微分方程绘制SFG

通过拉氏变换将微分方程变为s的代数方程;为每个变量指定一个节点,按照系统中变量间关系由左至右排列节点;用标明支路增益的支路,按照微分方程式中关系连接节点.(Referto中文教材p53)881-1-111-1Example：CascadeoftwoRCfilteringnetworks892.由结构图绘制SFG 结构图的信号线上用小圆圈标记出传递的信号,得到节点;用标有传递函数的线段代替方框,得到支路.－－－－1－1－190MasonGainFormula梅森增益公式GivenaSFG,thegainbetweenainputnodeandaoutputnodeis目的：求出系统的传递函数P:TheoverallgainorthetransferfunctionofthesystemPk:Pathgainofkthforwardpathn:TotalnumberofforwardpathsΔ–Determinantfgraph(流图特征式)

91Procedures1.Findallloops,twonontouchingloops,threenontouchingloops…tocalculateΔ2.Findallforwardpath,andtheirgainPk3.Removethekthforwardpath(allnodesandpaths),calculatethecofactorofthekthforwardpathdeterminantofthegraphΔk4.CalculatetheoverallgainbytheMasonformula92Example1：求下图所示系统的传递函数

G4(s)

H1(s)H3(s)

G1(s)

G2(s)

G3(s)

R(s)

C(s)93R(s)C(s)L1=–G1H1L2=–G3H3L3=–G1G2G3H3H1L4=–G4G3L5=–G1G2G3L1L2=(–G1H1)(–G3H3)=G1G3H1H3L1L4=(–G1H1)(–G4G3)=G1G3G4H1

G4(s)

H1(s)H3(s)

G1(s)

G2(s)

G3(s)

G4(s)

H1(s)H3(s)

G1(s)

G2(s)

G3(s)

G4(s)

H1(s)H3(s)

G1(s)

G2(s)

G3(s)

G4(s)

H1(s)H3(s)

G1(s)

G2(s)

G3(s)

G4(s)

H1(s)H3(s)

G1(s)

G2(s)

G3(s)

G4(s)

H1(s)H3(s)

G1(s)

G2(s)

G3(s)

G4(s)

H1(s)H3(s)