现代控制理论课件

,(Foundationof)ModernControlTheory,0.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-loopcontrolsowewouldanalyzetheclosed-loopdynamicsOpen-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).,0.5.FeedbackControlApproach,0.5.1.EstablishcontrolobjectivesQualitativedontusetoomuchfuelQuantitativesettlingtimeofstepresponse3secTypicallyrequiresthatyouunderstandtheprocess(expectedcommandsanddisturbances)andtheoverallgoals(bandwidths).Oftenrequiresthatyouhaveastrongunderstandingofthephysicaldynamicsofthesystemsothatyoudonot“fight”themininappropriate(i.e.,inefficient)ways.0.5.2.SelectsensorsFirstpaperbyN.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.Astrmet.,0.8.Conclusion,Controlofsystemsanddevicesisaveryoldhumanneedand,therefore,isanancientconcept;Asadiscipline,ithadastartwiththeanalysisofthegovernor(late19thcentury);IthadamajorboostwithWWII,ColdWarandthepaceRace;Theartofinteractionindynamicnetworks.-RoyAscott,0.9.CourseDescription,Thiscourseisintendedtointroducetheanalysisandsynthesismethodsoflinearsystemsbasedonstatespacedescriptions.Themaincontentsofthecourseareasfollows:statespacedescriptionofcontrolsystemsandthedifferenceandconnectionbetweenstatespacedescriptionandthetransferfunctiondescriptioninclassicalcontroltheory,definitionsandcriterionsofcontrollabilityandobservability,definitionsandcriterionsofLyapunovstability,methodsofconstantlinearsystemsynthesissuchaspoleassignment,stabilizationandobserverdesign.,0.10.Myreasonsforthereviewofclassicaldesign:,State-spacetechniquesarejustanotherwaytodesignacontroller,anditisessentialthatyouunderstandthebasicsofthecontroldesignprocess.Otherwisethesearejusta“bunchofnumericaltools”.Totrulyunderstandtheoutputofthestate-spacecontroldesignprocess,Ithinkitisimportantthatyoubeabletoanalyzeitfromaclassicalperspective._Trytoanswer“whydiditdothat”?_Notalwayspossible,butalwaysagoodgoal.Matlabwillberequiredextensively.Ifyouhavenotuseditbefore,thenstartpracticing.Data-information-understanding-knowledge-wisdom,0.11.ControlTheoryHistory,1769,JamesWattssteamengineandgovernor-FlyingBall1875,1895,RouthandHurwittzdevelopedstabilityanalysiscriterion1932,Nyquistdevelopedamethodforsystemstabilityanalysis1948，NorbertWiener-Cybernetics1892,Lyapunovstability,interestinitstartedduringtheColdWar(1953-1962)1960,Kalmancontrollabilityandobservability1960,ModerncontroltheorybasedonStatevariablemethods,ClassicalControl-ModernControl,WATTsFlyballGovernor,Fig1.2.WATTsFlyballGovernor,Courserequirement,PrerequisitesPrinciplesofAutomaticcontrolLinearAlgebraMatlabEvaluationAttendance(10%)Assignment(20%)Exam(70%),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？,p=q=1:SISOOtherwise:MIMO,0.12.1.Input-outputdescriptionofsystems,0.12.Introduction,0.12.2.Relaxedproperty(y(t0)=f(t0)memorylesssystemseg:onlyresistors)(y(t1)=f(t,u(t),-t-0.12.4.LinearityDefinition1-1:Supposethatasystemisrelaxedatt=t01,2arearbitraryrealconstantandu1(t),u2(t)aretwoarbitraryinputs(tt0),ify=H(1u1+2u2)=1Hu1+2Hu2=1y1+2y2(1-3)hold,Thenthesystemsatisfieswithlinearty,oritiscalledalinearsystem.(Arelaxedsystemiscalledalinearsystemifithasthesuperpositionproperty)y=H(1u1)=1Hu1homogeneity,y=H(u1+u2)=Hu1+Hu2=y1+y2additivityy(t)=u2(t)/u(t-1),u(t-1)0=0,u(t-1)=0,homogeneity,additivity,0.12.5.Time-invariantpropertyTime-invariantpropertyofasystemmeansthatthedynamicalcharacteristicofinput-outputdoesnotchangewithtime.TheeffectofshiftingoperatorQaisshowninfollowingFig.,Definition1-2Arelaxedsystemiscalledatime-invariantsystemifandonlyifforanyinputuandanyrealnumber,thefollowingequation,hold.Otherwiseitistime-variant,1StateSpaceDescriptionofDynamicsystems,1.1.StateVariableandStateSpaceExpression,1.2.StateVariableExpressionfromdifferentialoperatorrepresentation,1.3.Transferfunctionmatrix,1.4.Mathematicalmodelofadiscrete-timesystem,1.5.Equivalentstateequations,1.6.Thestate-spacerepresentationofcompositesystem,1.7.ModeltypeconversionsbasedonMATLAB,1.1.StateVariableandStateSpaceExpression,Eg.2-1,Fig1.3.RLCcurcirt,Initialvalues:,L,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？,Assume，theresistanceRandcapacitorCarebothnonlinear(R=R0*i(t),C=2*C0*uc),ForNLTI,1.1.1StateandStateSpace,Definitionofstate:Thestateofadynamicsystemisamathematicalstructurecontainingasetofnvariablesx1(t),x2(t),xn(t),calledstate,suchthattheinitialvaluesxi(t0),i=1,n,ofthissetattimet0andthesysteminputsuj(t),j=1,2,p,tt0，aresufficienttodetermineuniquelythesystemsfuturebehaviorofoftt0.Aminimumsetofstatevariablesisrequiredtorepresentthesystemcompletely.,Definitionofstatevector:ThestatevectorofadynamicsystemisdefinedasthecolumnvectorX(t);thatis,Statespaceandstatetrajectory,Definitionofstatespace:Statespaceisdefinedasthen-dimensionalspaceinwhichthecomponentsofthestatevectorrepresentitscoordinateaxes,Definitionofstatetrajectory:StatetrajectoryisdefinedasthepathproducedinstatespacebythestatevectorX(t)asitchangeswiththepassageoftime.,Eg.2-dimensional,thephaseplane-SS,phasetrajectory-ST,Stateequationandoutputequationtogetherarecalledstatespaceexpression,1.1.2.Stateequation：,Outputequation：,y,Algebraicequation,Stateequation,Lineartimevaringsystem,time-invariantdiscretelinearsystem,AStateMatrixBInputMatrixCOutputMatrixDDirecttransmissionMatrix,Linearsystem,Lineartime-invariantsystem,T-samplingperiod,ClassicalcontroltheoryVSModerncontroltheory,SISO,linear,time-invariantsystemLaplacetransformTime,complexnumberFrequencydomainoutput,MIMO,Nonlinear,time-varingsystemMatrix,vector,linearalgebraTimedomainstate,Mostgeneralcontinuous-timelineardynamicalsystemhasform,Y(t)=C(t)x(t)+D(t)u(t)where:tRdenotestime,x(t)Rnisthestate(vector)u(t)Rmistheinputorcontrol,y(t)RpistheoutputA(t)Rnnisthedynamicsmatrix,B(t)RnmistheinputmatrixC(t)RpnistheoutputorsensormatrixD(t)RpmisthefeedthroughmatrixNotethattheplantdynamicscanbetime-varying.Alsonotethatthisisamulti-input/multi-output(MIMO)system.,Examplesoflinear,nonlinear,time-varying,periodically-varying,An