1、Vol.32,No.5ACTAAUTOMATICASINICASeptember,2006ConstructionofControlLyapunovFunctionsforaClassofNonlinearSystemsCAIXiu-Shan11(College2(DepartmentHANZheng-Zhi2WANGXiao-Dong1ofInformationScienceandEngineering,ZhejiangNormalUniversity,Jinhua321004)ofAutomation,ShanghaiJiaotongUniversity,Shanghai200030)(E

2、-mail:xiushan)Keywordsics.Nonlinearsystems,controlLyapunovfunctions,semiglobalstabilization,zerodynam-IntroductionWithanarbitrarychoiceofV(x)thisattemptmayfail,butifV(x)isaCLF,therearemanycontrol(x,u(x)negativedenite,oneofwhichisgivenbyaformuladuetoSontag3.ThelawsthatrenderVconstructionofaCLFisahard

3、problem,whichhasbeensolvedforspecialclassesofsystems.Forexample,whenthesystemisinthestrictfeedbackform,CLFscanbeconstructedbybackstepping4.Foralinearsystem,wehaveobtainedauniversalformulatoconstructCLFs5.Inthispaper,theconstructionofcontrolLyapunovfunctionsforaclassofnonlinearsystemsisconsidered.For

4、thefeedbacklinearizablepart,CLFscanbeconstructedbythemethodpresentedin5.BasedonaCLFofthefeedbacklinearizablepartandaLyapunovfunctionofthezerodynamics,wepresentamethodtoobtainaCLFfortheoverallnonlinearsystem.2SystemdescriptionandpreliminariesConsideranonlinearsystemdescribedby=Q(z,x)z=Ax+BF(z,x)+G(z,

5、x)uxy=Cx(1a)(1b)(1c)1wherexRr,zRnrarethestates,uRmistheinput,yRlistheoutput.Q(z,x):RnRnrissmooth.fi,gij:RnR,areassumedtobesmoothwithfi(0,0)=0,i=1,2,···l.F(z,x)=f1(z,x)f2(z,x)···fl(z,x)T,G(z,x)=(gij(z,x)l×mandrank(G(z,x)=l.r1,r2,···rlisavectorrelative

6、degreeofsystem(1),andr1+r2+···+rl=r<n.(1b)hasthefollowingcannonicalform:01···0.A=blockdiagA1,···,Al,Ai=,B=blockdiagB1,···,Bl00···100···0Bi=0···01T1×ri,C=blockdiagC1,···,Cl,

7、=f(x)+g(x)uxy=h(x)Ci=10···01×ri.FromIsidori6,ifananenonlinearsystem(2)No.5CAIXiu-Shanetal.:ConstructionofControlLyapunovFunctionsfor···797798ACTAAUTOMATICASINICAcV(x)h+1U(z)+2c2+h2+1+c,U(z)󰀁(h+1)c+h2+1c(c+1)c+1󰀁c(c+1)hc2+h2+1+h)2(h+1U)2󰀁(UV(

8、h+1U(z)2(c+1V(x)2V(c+1V(x)2xB=0,wehavexT(PA+ATP)x=KXTrlFXrlSincerank(G(z,x)=l,by(17)wehaveLgW(z,x)=0VUT(h+1U(z)2(c+1V(x)2x(PA+ATP)x=h(h+1)Q(z,x)Kc(c+1)zVol.32(13)(18)No.5CAIXiu-Shanetal.:ConstructionofControlLyapunovFunctionsfor···799c+1h(h+1)XTrlFXrl+(c2+h2+1+h)2z(Q(z,x)Q(z,0)|2(h+1U

9、(z)21(z),(x)=|KcU(z)h(h+1)2(c+1)2(c+1)XTrlFXrl+XTrlFXrlh(h+1)h(h+1)1(z).Inconclusion,LfW(z,x)<0,forLgW(z,x)=0,(z,x)=(h+1h+10.ThusW(z,x)isaCLFforsystem(1)onS.U(z)24ConclusionReferences2SontagED.ALyapunov-likecharacterizationofasymptoticcontrollability.SIAMJournalControlandOptimization,1983,21(3):4624713SontagD.AUniversalconstructionofArtsteinstheoremonnonlinearstabilization.SystemsandControlLetters,1989,13(2):1171234KokotovicPV,ArcakM.Constructivenonlinearcontrol:Ahistoricalperspective.Automatica,200