PDF-一类MIMO状态可测的非线性连续系统的激励辨识

3310Vol.33,No.10200710ACTAAUTOMATICASINICAOctober,2007MIMO121MIMO.,.Girsanov.NNR..,,NNR120.3020Identi¯cationofaClassofMIMONonlinearContinuous-timeSystemswithObservableStatesUsingDrivingSignalSUNXi-Ping1WANGYong-Ji2QIANXin-En1AbstractWeproposeanidenti¯cationapproachforatypeofMIMOnonlinearcontinuous-timesystemswithobservablestatesusingdrivingsignals.ThedrivingsignalisGaussionwhitenoise,thestateoutputsaresampledevenly.ThemaximumlikelihoodestimatesofthemodelparametersarederivedbyusingtheGir-sanovtheorem.Thenumericalsimulationsillustratethee±-ciencyoftheestimatesandtheNNRphenomenonofcouplingmulti-variableappearsinthenumericalsimulations.Astep-typeidenti¯cationalgorithmissuggestedattheend.KeywordsNonlinearcontinuous-timesystems,systemiden-ti¯cationusingdrivingsignals,NNRphenomenon1,[1],,,,,,,,NARMAX,[2],,,.,.,,,,,,,,,.2006-3-32007-2-6ReceivedMarch3,2006;inrevisedformFebruary6,2007(60674105),(D200723001)SupportedbyNationalNaturalScienceFoundationofChina(60674105),ScienceandTechnologyResearchProjectofEducationDepartmentofHubeiProvince(D200723001)1.4420022.4300741.DepartmentofElectricityEngineering,HubeiAutomobileIndus-triesInstitute,Shiyan4420022.DepartmentofControlScienceandEngineering,HuazhongUniversityofScienceandTechnology,Wuhan430074DOI:10.1360/aas-007-1105,,,.,.,.,,[3]SISO,Sagara[4],,Voros[5],[6],,[7,8],[9]ARMA,[10].,.[11],[12»14],,,[15],.[16].?_x=x1+®®0;x(0)=1(1)(1)x(t)=1=(1¡®t)1®,0t®,t!®,x(t)!1,,(1),,().,,,,..,,,.,.,_x=ax+bx3a=2;b=¡1;x(0)=1(2),(2),,,,,().,110633(,),(),,(),().,,(),,(1),(),(),,.,,,(),.(6),,()[17],.(),,.,,,,.,,.2,_x1(t)=a11+a12x1(t)+a13x2(t)+a14x1(t)x2(t)+e1(t)+u1_x2(t)=a21+a22x1(t)+a23x2(t)+a24x32(t)+e2(t)+u2(3)xxx=(x1(t);x2(t))T,0·t·T,uuu=(u1(t);u2(t))T,0·t·T,eee=(e1(t);e2(t))T,0·t·T,.(3),,A=Ãa11a12a13a14a21a22a23a24!(4),,,ei(t)+ui(t)=bidwi(t);i=1;2(5),(3)dx1(t)=(a11+a12x1(t)+a13x2(t)+a14x1(t)x2(t))dt+b1dw1(t)dx2(t)=(a21+a22x1(t)+a23x2(t)+a24x1(t)x2(t))dt+b2dw2(t)(6)(6),f­;=;f=tg;Pg,=(w1(t);w2(t))T,0·t·TP,BBB=(b1;b2)T,,,bi(i=1;2),,bi(i=1;2).bi=0(i=1;2),(6),,(5).3PYY=f(x1(t)=b1;x2(t)=b2);0·t·Tg,Girsanov[18],dPY=dP=H(A;BBB)=expf¡RT0ha11+a12x1(t)+a13x2(t)+a14x1(t)x2(t)b21dx1(t)+a21+a22x1(t)+a23x2(t)+a24x1(t)x2(t)b22dx2(t)]+12ZT0[(a11+a12x1(t)+a13x2(t)+a14x1(t)x2(t)b1)2+(a21+a22x1(t)+a23x2(t)+a24x1(t)x2(t)b2)2]dtg(7)A^a11;^a12;^a13;^a14@H@a11=exp(¸)¢f¡RT0dx1(t)+RT0(^a11+¢¢¢+^a14x1(t)x2(t))dtg=b21=0@H@a12=exp(¸)¢f¡RT0x1(t)dx1(t)+RT0(^a11+¢¢¢+^a14x1(t)x2(t))x1(t)dtg=b21=0@H@a13=exp(¸)¢f¡RT0x2(t)dx1(t)+RT0(^a11+¢¢¢+^a14x1(t)x2(t))x2(t)dtg=b21=0@H@a14=exp(¸)¢f¡RT0x1(t)x2(t)dx1(t)+RT0(^a11+¢¢¢+^a14x1(t)x2(t))x1(t)x2(t)dtg=b21=0(8)¸=¡RT0[a11+a12x1(t)+a13x2(t)+a14x1(t)x2(t)b21dx1(t)+a21+a22x1(t)+a23x2(t)+a24x32(t)b22dx2(t)]+12ZT0[((a11+a12x1(t)+a13x2(t)+a14x1(t)x2(t)b1)2+(a21+a22x1(t)+a23x2(t)+a24x32(t)b2)2]dt(9)10MIMO1107(8)0BBBB@RT0dtRT0x1dtRT0x2dtRT0x1x2dtRT0x1dtRT0x21dtRT0x1x2dtRT0x21x2dtRT0x2dtRT0x1x2dtRT0x22dtRT0x1x22dtRT0x1x2dtRT0x21x2dtRT0x1x22dtRT0x21x22dt1CCCCA¢0BBBB@^a11^a12^a13^a141CCCCA=0BBBB@RT0dx1(t)RT0x1(t)dx1(t)RT0x2(t)dx1(t)RT0x1(t)x2(t)dx1(t)1CCCCA(10)fxik;i=1;2;k=0;1;2;:::;mg,(x1(0);x2(0))=(x10;x20),¢t=¢tk:=tk+1¡tk,¢xik:=xik+1¡xik,(10):0BBBBBBBBBB@mm¡1Pk=0x1km¡1Pk=0x2km¡1Pk=0x1kx2km¡1Pk=0x1km¡1Pk=0x21km¡1Pk=0x1kx2km¡1Pk=0x21kx2km¡1Pk=0x2km¡1Pk=0x1kx2km¡1Pk=0x22km¡1Pk=0x1kx22km¡1Pk=0x1kx2km¡1Pk=0x21kx2km¡1Pk=0x1kx22km¡1Pk=0x21kx22k1CCCCCCCCCCA¢0BBBB@^a11^a12^a13^a141CCCCA=0BBBBBBBBB@(x1m¡x10)=¢tm¡1Pk=0x1k¢x1k=¢tm¡1Pk=0x2k¢x1k=¢tm¡1Pk=0x1kx2k¢x1k=¢t1CCCCCCCCCA(11)0BBBBBBBBBB@mm¡1Pk=0x1km¡1Pk=0x2km¡1Pk=0x32km¡1Pk=0x1km¡1Pk=0x21km¡1Pk=0x1kx2km¡1Pk=0x1kx32km¡1Pk=0x2km¡1Pk=0x1kx2km¡1Pk=0x22km¡1Pk=0x42km¡1Pk=0x32km¡1Pk=0x1kx32km¡1Pk=0x42km¡1Pk=0x62k1CCCCCCCCCCA¢0BBBB@^a21^a22^a23^a241CCCCA=0BBBBBBBBB@(x2m¡x20)=¢tm¡1Pk=0x1k¢x2k=¢tm¡1Pk=0x2k¢x2k=¢tm¡1Pk=0x32k¢x2k=¢t1CCCCCCCCCA(12)(11)(12)A,1),2),.4Runge-Kutta[19]dx1(t)=a1(x1(t);x2(t))dt+¾1(x1(t);x2(t))dw1(t)dx2(t)=a1(x1(t);x2(t))dt+¾1(x1(t);x2(t))dw2(t)(x1(t);x2(t))=(x10;x20)(13)xik+1=xik+ai(x1k;x2k)¢tk+¾i(x1k;x2k)»ik+12f1p¢t[¾i(x1k+¾1(x1k;x2k)p¢t)¡¾i(x1k;x2k)]g¢(»2ik¡¢t)»ik»N(0;¢t);i:i:d;k=0;1;:::;m¡1;i=1;2(x1(0);x2(0)=(x10;x20)(14)(6),A=Ãa11a12a13a14a21a22a23a24!;BBB=Ãb1b2!(15)BBB=(b1;b2)T,r:=b1=b2NNR(Noiseintensityratioofonestatetoanother),.[0;100],T=100,n=5000,dt=T=n=0:02,(x1(0);x2(0))=(1;¡1).1)A=á0:200:1¡0:10¡0:1¡0:2¡0:3!;BBB=Ã0:0010:001!;r=1(16)12A.2)A=á0:200:1¡0:10¡0:1¡0:2¡0:3!;BBB=Ã0:0010:1!;r=0:01(17)34A.1r=1AFig.1Identi¯cationofthe¯rstrowofAwithr=11108332r=1AFig.2Identi¯cationofthesecondrowofAwithr=13r=0:01AFig.3Identi¯cationofthe¯rstrowofAwithr=0:014r=0:01AFig.4Identi¯cationofthesecondrowofAwithr=0:013):A=á0:200:1¡0:10¡0:1¡0:2¡0:3!;BBB=Ã0:010:001!;r=10(18)56A.NNRSNR(),,(),,,NNR,.,..NNR[15],,,,,.5r=10AFig.5Identi¯cationofthe¯rstrowofAwithr=106r=10AFig.6Identi¯cationofthesecondrowofAwithr=105[15,20]LorenzRossler,()[15,20],,,,.7[20](8)Lorenzb,(¾;r;b)=(10;28;8=3),bt=608=3+5.9[15](10)Lorenz,(¾;r;b)=(10;28;8=3),t=5(¾;r;b)=(11;35;3).[15,20],.,.10MIMO11097(T=100;n=5000)Fig.7Themaximumlikelihoodestimatesofthispaper8[20](T=100;n=50