现代控制理论课件

(Foundationof)

ModernControlTheory10.Introduction

0.1.What？-how？-why？Anexample（processcontrolshownbyblockdiagram）OutputY(controlledvariable)-process(controlledobject,plant)-actuator-poweramplifier(open-loopcontrol,Manual/automatic

)-disturbanceinput-controller-sensor(f1)-referenceinput(R)-Difference(R-f1)0.2.Typicallythinkofclosed-loopcontrol→sowewouldanalyzetheclosed-loopdynamics–Open-loopcontrolalsopossible(called“feedforward”)–morepronetomodelingerrorssinceinputsnotchangedasaresultofmeasurederror.Openloopcontrolsystem.AsysteminwhichtheoutputhasnoeffectupontheinputsignalClosed-loopcontrolsystem.Asysteminwhichtheoutputhasaneffectupontheinputquantityinsuchamannerastomaintainthedesiredoutputvalue.

0.3.Goal:

DesignacontrollerGc(s)sothatthesystemhassomedesiredcharacteristics.Typicalobjectives:-Stabilizethesystem(Stabilization)–Regulatethesystemaboutsomedesignpoint(Regulation)–Followagivenclassofcommandsignals(Tracking)-Reduceresponsetodisturbances.(DisturbanceRejection)0.4.Notethatatypicalcontrolsystemincludesthesensors,actuators,andthecontrollaw.–Thesensorsandactuatorsneednotalwaysbephysicaldevices(e.g.,economicsystems).–Agoodselectionofthesensorandactuatorcangreatlysimplifythecontroldesignprocess.–Courseconcentratesonthedesignofthecontrollawgiventherestofthesystem(althoughwewillneedtomodelthesystem).20.5.FeedbackControlApproach•0.5.1.Establishcontrolobjectives–Qualitative–don’tusetoomuchfuel–Quantitative–settlingtimeofstepresponse<3sec–Typicallyrequiresthatyouunderstandtheprocess(expectedcommandsanddisturbances)andtheoverallgoals(bandwidths).–Oftenrequiresthatyouhaveastrongunderstandingofthephysicaldynamicsofthesystemsothatyoudonot“fight”themininappropriate(i.e.,inefficient)ways.0.5.2.Selectsensors&actuators–Whataspectsofthesystemaretobesensedandcontrolled?–Considersensornoiseandlinearityaskeydiscriminators.–Cost,reliability,size,...0.5.3.Obtainmodel–Analyticorfrommeasureddata(systemID)–Evaluationmodel→reducesize/complexity→Designmodel–Accuracy?Errormodel?0.5.4.Designcontroller–Selecttechnique(SISO,MIMO),(classical,state-space)–Chooseparameters(optimization)•0.5.5.Analyzeclosed-loopperformance.Meetobjectives?–Analysis,simulation,experimentation,...–Yes⇒done,No⇒iterate...30.6.WhyUseControl?

0.6.1•Typicallyeasyquestiontoanswerforaerospacebecausemanyvehicles(spacecraft,aircraft,rockets)andaerospaceprocesses(propulsion)needtobecontrolledjusttofunction–Example:theF-117doesnotevenflywithoutcomputercontrol,andtheX-29isunstableOperationIraqifreedomAnF-117fromthe8thexpeditionaryFighterSquadronoutofHollomannAirForcebace,N.M.,fliesoverthePersianGolfonApril14,2003(U.S.AirforcephotobyStaffSgt.DerrickC.Goode)Fig1.1.F11740.6.2.Betterperformance•Buttherearealsomanystablesystemsthatsimplyrequirebetterperformanceinsomesense(e.g.,faster,lessoscillatory),andwecanusecontroltomodifythisbehavior.50.7.ApplicationoftheProportional-Integral-Derivative

(PID)Controller•Somefeaturesalreadypresent(inherently,i.e.built-in)insteamgovernors,butnotclearlyrecognized;•FirstpaperbyN.Minorsky(USNavy)in1922:“Directionalstabilityofautomaticallysteeredbodies”,J.Am.Soc.NavalEng.,34.•LandmarkpaperbyJ.G.Ziegler(TaylorInstruments)andN.B.Nichols(MIT)in1942:“Optimumsettingsforautomaticcontrollers”,Trans.ASME,64,759-768.•PIDcontrollersareusedinover90%ofindustrialcontrolloopsworldwide.•ControlsystemsengineeringeducationbyK.J.Aströmet.60.8.Conclusion•Controlofsystemsanddevicesisaveryoldhumanneedand,therefore,isanancientconcept;•Asadiscipline,ithadastartwiththeanalysisofthegovernor(late19thcentury);•IthadamajorboostwithWWII,ColdWarandthepaceRace;"Theartofinteractionindynamicnetworks."-RoyAscott70.9.CourseDescriptionThiscourseisintendedtointroducetheanalysisandsynthesismethodsoflinearsystemsbasedonstatespacedescriptions.Themaincontentsofthecourseareasfollows:statespacedescriptionofcontrolsystemsandthedifferenceandconnectionbetweenstatespacedescriptionandthetransferfunctiondescriptioninclassicalcontroltheory,definitionsandcriterionsofcontrollabilityandobservability,definitionsandcriterionsofLyapunovstability,methodsofconstantlinearsystemsynthesissuchaspoleassignment,stabilizationandobserverdesign.80.10.Myreasonsforthereviewofclassicaldesign:

–State-spacetechniquesarejustanotherwaytodesignacontroller,anditisessentialthatyouunderstandthebasicsofthecontroldesignprocess.Otherwisethesearejusta“bunchofnumericaltools”.–Totrulyunderstandtheoutputofthestate-spacecontroldesignprocess,Ithinkitisimportantthatyoubeabletoanalyzeitfromaclassicalperspective._Trytoanswer“whydiditdothat”?_Notalwayspossible,butalwaysagoodgoal.•Matlabwillberequiredextensively.Ifyouhavenotuseditbefore,thenstartpracticing.Data---information---understanding---knowledge---wisdom9

0.11.ControlTheoryHistory1769, JamesWatt’ssteamengineandgovernor-FlyingBall1875,1895,RouthandHurwittzdevelopedstabilityanalysiscriterion1932,Nyquistdevelopedamethodforsystemstabilityanalysis1948，NorbertWiener-《Cybernetics》1892,Lyapunovstability,interestinitstartedduringthe ColdWar(1953-1962)1960,Kalmancontrollabilityandobservability1960,ModerncontroltheorybasedonStatevariablemethodsClassicalControl-ModernControl10WATT’sFlyballGovernorFig1.2.WATT’sFlyballGovernor11CourserequirementPrerequisitesPrinciplesofAutomaticcontrolLinearAlgebraMatlabEvaluationAttendance (10%)Assignment (20%)Exam (70%)12作业1、什么是高性能的控制系统？什么因素影响控制系统性能？高性能控制系统的控制结构是什么？2、为了提高实际环节K/(Ts+1)的快速性，1、采用串联(Ts+1)/(0.01Ts+1)，2、采用在前向通道加入一个Kc构成反馈环节，再在输入加入一个Kc1，求Kc、Kc1使这两种方法的传递函数完全一样，问这两种方法的区别是什么？13Systemp=q=1:SISOOtherwise:MIMO0.12.1.Input-outputdescriptionofsystems0.12.Introduction140.12.2.Relaxedproperty

(y(t0)=f(t0)

→

memorylesssystemseg:onlyresistors)(y(t1)=f(t,u(t)),-∞<t<t1)→

memorysystems)H:u→y;y=Hu (1-1)(Hisanoperatororfunctionwhichcanbedecidedbysystemproperties)Eg:Forasystemrelaxedattimet0,itexistsY[t0,∞)=Hu[t0,∞)150.12.3.CausalProperty

Asystemiscalledacausalsystemifitscurrentoutputattimetdependsonlyontheinputbeforetratherthanthefutureinputaftertimet.

Foracausalrelaxedsystem：y(t)=Hu(-∞,t),∀t>-∞0.12.4.LinearityDefinition1-1:Supposethatasystemisrelaxedatt=t0

α1,α2arearbitraryrealconstantandu1(t),u2(t)aretwoarbitraryinputs(t≥t0),ify=H(α1u1+α2u2)=α1Hu1+α2Hu2=α1y1+α2y2 (1-3)hold,

Thenthesystemsatisfieswithlinearty,oritiscalledalinearsystem.(Arelaxedsystemiscalledalinearsystemifithasthesuperpositionproperty)y=H(α1u1)=α1Hu1homogeneity,y=H(u1+u2)=Hu1+Hu2=y1+y2additivityy(t)=u2(t)/u(t-1),u(t-1)≠0=0,u(t-1)=0,homogeneity,×additivity160.12.5.Time-invariantpropertyTime-invariantpropertyofasystemmeansthatthedynamicalcharacteristicofinput-outputdoesnotchangewithtime.TheeffectofshiftingoperatorQaisshowninfollowingFig.Definition1-2Arelaxedsystemiscalledatime-invariantsystemifandonlyifforanyinputuandanyrealnumberα,thefollowingequationhold.Otherwiseitistime-variant171StateSpaceDescriptionofDynamicsystems1.1.StateVariableandStateSpaceExpression1.2.StateVariableExpression

fromdifferentialoperatorrepresentation1.3.Transferfunctionmatrix1.4.Mathematicalmodelofadiscrete-timesystem1.5.Equivalentstateequations

1.6.Thestate-spacerepresentationofcompositesystem1.7.ModeltypeconversionsbasedonMATLAB181.1.StateVariableandStateSpaceExpressionEg.2-1Fig1.3.RLCcurcirtInitialvalues:19L作业：1、Couldweassume:x1=uc(t),x2=i(t),orx1=3x1,x2=x1+x2,…列写新的状态方程。2、一个机械系统具有弹性系数K的弹簧，串联质量为M质量块，并在上连接系数为f阻尼器，在质量块上输入外力F(t)，质量块的运动位移为y(t)，按上述方式建立系统的数学模型，并与上述电气系统对比，是否适当选取参数使这两个系统动态性能完全一样？20Assume，theresistanceRandcapacitorCarebothnonlinear(R=R0*i(t),C=2*C0*uc)ForNLTI211.1.1StateandStateSpaceDefinitionofstate:Thestateofadynamicsystemisamathematicalstructurecontainingasetofnvariablesx1(t),x2(t),…xn(t),calledstate,suchthattheinitialvaluesxi(t0),i=1,…n,ofthissetattimet0andthesysteminputsuj(t),j=1,2,…,p,t≥t0，aresufficienttodetermineuniquelythesystem’sfuturebehaviorofoft≥t0.Aminimumsetofstatevariablesisrequiredtorepresentthesystemcompletely.Definitionofstatevector:ThestatevectorofadynamicsystemisdefinedasthecolumnvectorX(t);thatis22StatespaceandstatetrajectoryDefinitionofstatespace:Statespaceisdefinedasthen-dimensionalspaceinwhichthecomponentsofthestatevectorrepresentitscoordinateaxesDefinitionofstatetrajectory:StatetrajectoryisdefinedasthepathproducedinstatespacebythestatevectorX(t)asitchangeswiththepassageoftime.Eg.2-dimensional,thephaseplane-SS,phasetrajectory-ST23Stateequationandoutputequationtogetherarecalledstatespaceexpression

1.1.2.Stateequation：îíìÛFirst-order

differenceequation

First-orderDifferentialEquation

uxOutputequation：yxuìÛîAlgebraicequation

Stateequation24Lineartimevaringsystem

time-invariantdiscretelinearsystem

A–StateMatrixB–InputMatrixC–OutputMatrixD–DirecttransmissionMatrixf,g-linearfunction

LinearsystemLineartime-invariantsystemT-samplingperiod25ClassicalcontroltheoryVSModerncontroltheory

SISO,linear,time-invariantsystemLaplacetransformTime,complexnumberFrequencydomainoutputMIMO,Nonlinear,time-varingsystemMatrix,vector,linearalgebraTimedomainstate26•Mostgeneralcontinuous-timelineardynamicalsystemhasformY(t)=C(t)x(t)+D(t)u(t)

where:–t∈Rdenotestime, –x(t)∈Rnisthestate(vector)–u(t)∈Rmistheinputorcontrol, –y(t)∈Rpistheoutput–A(t)∈Rn×nisthedynamicsmatrix, –B(t)∈Rn×mistheinputmatrix–C(t)∈Rp×nistheoutputorsensormatrix–D(t)∈Rp×misthefeedthro