MATLAB-LMI工具箱使用教程算例及论文原文

LMI工具箱使用教程算例算例是论文：H∞statefeedbackcontrolforgeneralizedcontinuous/discretetime-delaysystem中第三部分里面的一个例子。论文原文附在后面。算例程序如下：A=[21;01];Ad=[0.20.1;00.1];B1=[0.10.1]';B2=[11]';Bd=[0.10.1]';C=[1,1];Cd=[0.1,0.1];D11=0.1;D12=1;Dd=0.1;gammar=1;%InitialaLMIsystemsetlmis([]);%DefineVariables%Qisasymmetricmatrix,hasablocksizeof2andthisblockissymmetricQ=lmivar(1,[21]);%S1asymmericmatrix,size2S1=lmivar(1,[21]);%S2is1by1matrixS2=lmivar(1,[10]);%Typeof2,size1by2M=lmivar(2,[12]);%Q,S1,S20QN=newlmi;lmiterm([-QN11Q],1,1);S1N=newlmi;lmiterm([-S1N11S1],1,1);S2N=newlmi;lmiterm([-S2N11S2],1,1);MAllN=newlmi;%posin(1,1)lmiterm([MAllN11Q],-1,1);lmiterm([MAllN11S2],Bd,Bd');lmiterm([MAllN11S1],Ad,Ad');%pos(1,2)lmiterm([MAllN12Q],A,1);lmiterm([MAllN12M],B2,1);%pos(1,3)lmiterm([MAllN130],B1);%pos(1,4)lmiterm([MAllN14S2],Bd,Dd');lmiterm([MAllN14S1],Ad,Cd');%pos(2,2)lmiterm([MAllN22Q],-1,1);%pos(2,4)lmiterm([MAllN24Q],1,C');lmiterm([MAllN24-M],1,D12');%pos(2,5)lmiterm([MAllN25-M],1,1);%pos(2,6)lmiterm([MAllN26Q],1,1);%pos(3,3)lmiterm([MAllN330],-(gammar^2));%pos(3,4)lmiterm([MAllN340],D11');%pos(4,4)lmiterm([MAllN440],-1);lmiterm([MAllN44S1],Cd,Cd');lmiterm([MAllN44S2],Dd,Dd');lmiterm([MAllN55S2],-1,1);lmiterm([MAllN66S1],-1,1);lmis=getlmis;[tmin,feas]=feasp(lmis)运行上述程序后，在命令窗口（1）输入：Q=dec2mat(lmis,feas,Q)得出：Q=1.9253-2.2338-2.23389.1054（2）输入：P=Q^-1P=0.72610.17810.17810.1535（3）输入：S1=dec2mat(lmis,feas,S1)得出：S1=16.0928-14.9665-14.966530.3214输入：R1=S1^-1得出：R1=0.11490.05670.05670.0610（４）输入：S2=dec2mat(lmis,feas,S2)得出：S2=16.0495输入：R2=S2^-1得出：R2=0.0623（5）输入：M=dec2mat(lmis,feas,M)得出：M=-0.2279-6.2240（6）输入：K=M*P得出：K=-1.2741-0.9961与论文中结果保持一致！Automatica35(1999)1443}1451TechnicalCommuniqueH=statefeedbackcontrolforgeneralizedcontinuous/discretetime-delaysystem1JongHaeKima,*,HongBaeParkbaSensorTechnologyResearchCenter,KyungpookNationalUniversity,Taegu,702-701,SouthKoreabSchoolofElectronicandElectricalEngineering,KyungpookNationalUniversity,Taegu,702-701,SouthKoreaReceived5February1998;revised20July1998;receivedinnalform3February1999AbstractInthispaper,weconsidertheproblemofdesigningH=statefeedbackcontrollerforthegeneralizedtime-delaysystemswithdelayedstatesandcontrolinputsincontinuousanddiscretetimecases,respectively.Thegeneralizedtime-delaysystemproblemsaresolvedonthebasisoflinearmatrixinequality(LMI)techniqueconsideringtimedelays.Thesu$cientconditionfortheexistenceofcontrollerandH=statefeedbackcontrollerdesignmethodsarepresented.Also,usingsomechangesofvariablesandSchurcomplements,theobtainedsu$cientconditioncanberewrittenasanLMIformintermsoftransformedvariables.TheproposedcontrollerdesignmethodcanbeextendedintotheproblemofrobustH=statefeedbackcontrollerdesignmethodeasily.(1999ElsevierScienceLtd.Allrightsreserved.Keywords:H=control;Statefeedback;Delayedsystem;Linearmatrixinequality1.IntroductionSincethetimedelayisfrequentlyasourceofinstabilityandencounteredinvariousengineeringsystemssuchaschemicalprocesses,longtransmissionlinesinpneumaticsystems,etc.,thestudyoftime-delaysystemshasreceivedconsiderableattentionoverthepastyears.BecausesomeworksofanalyticH=controllerdesignmethod(seee.g.Doyleetal.,1989;Gahinet,1996)andsoftwaretoolbox(Gahinetetal.,1995)havebeendeveloped,manystatefeedbackcontrollerdesignmethodsoftime-delaysys-temswerepresented(Shenetal.,1991;Leeetal.,1994;MahmoudandAl-Muthairi,1994;ChoiandChung,1995,1996;Kimetal.,1996;Geetal.,1996;Yuetal.,1996).Leeetal.(1994)presentedamemorylessH=con-trollerdesignwhichisadelay-independentstabilizerfor******Correspondingauthor.Tel.:#[phone];fax:#[phone];e-mail:[e-mail]orpublicationinrevisedformbyEditorPeterDorato.thestatedelayedsystem.AndtheworkproposedbyChoiandChung(1995)wasextendedtotheproblemofmemorylessH=controllerdesignforlinearsystemswithdelayedstateandcontrolusingtheRiccatiequationapproach.Butnotonlytheirworks(Leeetal.,1994;ChoiandChung,1995,1996)butalsootherresultswerecon-servativeinpre-determinationofsomestartingvaluesdeterminedwhetherthereexistsapositive-denitesolu-tion,andwerenotconsidereddelayedstateandcontrolinputinthecontrolledsignaloutput.AlsoNiculescu(1995)presentedH=memorylesscontrolwithana-stab-ilityconstraintfortime-delayssystemsusingthelinearmatrixinequality(LMI)approach.However,theworkdidnotconsidertime-varyingdelayinstatesandcontrolinputs.Foralinearsystemwithtime-varyingdelayinallstatesandcontrolinputs,itismorecomplicatedtoob-tainthecontroller.AlsoNiculescu(1995)didnotcon-siderthecontrollerdesignmethodfordiscretetime-delaysystem.Therefore,ourresultsdealwithcontrollerdesignmethodsofgeneralizedtime-delaysystemsincontinuoustimecaseanddiscretetimecase,respectively.Jeungetal.(1996)proposedrobustcontrollerdesignmethodforuncertainsystemswithtimedelaysusingtheLMIapproach.Buttheworkdidnottreatdisturbanceat-tenuationH=problem.Alsosomestartingvariables0005-1098/99/$-seefrontmatter(1999ElsevierScienceLtd.AllrightsreservedPII:S0005-1098(99)00038-2arepre-selectedinordertoobtainpositive-denitesolutions.Thisisrest