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28320113ControlTheory&ApplicationsVol.28No.3Mar.2011:1000¡8152(2011)03¡0421¡06,(,410073):,,,..LyapunovLaSalle.,PD+I.,.:;;;:TP273:ASlidingmodecontrolapproachwithnonlinearintegratorLIPeng,ZHENGZhi-qiang(CollegeofMechatronicsEngineeringandAutomation,NationalUniversityofDefenseTechnology,ChangshaHunan410073,China)Abstract:Anonlinearintegralslidingmodecontrolapproachisproposedforaclassofuncertainnonlinearsystems.Topromotetheperformanceofthetraditionalintegralslidingmodecontrol,thisapproachincorporatesanewnonlinearsaturationfunctionwhichenhancessmallerrorsandwillbesaturatedwithlargeerrorsinshapingthetrackingerrors.Whilemaintainingthetrackingaccuracyofthetraditionalintegralslidingmodecontrol,thisapproachprovidesbettertransientperformances.UsingLyapunovstabilitytheoryandLaSalleinvarianceprinciple,weprovethattheproposedapproachensuresthezerosteady-stateerrorinthepresenceofaconstantdisturbanceoranasymptoticallyconstantdisturbance.Whenthecontrolinputisconstrained,thesaturatedcontrolleroperateslikeaPDcontrollerwithanonlinearIterm.Simulationexampleisgiventodemonstratetheeffectivenessandrobustnessoftheproposedapproach.Keywords:nonlinearsystem;nonlinearintegrator;slidingmodecontrol;controlconstraint1(Introduction),,[1],Slotine,,[2].,,.Chern[3].Baik[4].,,,,,Windup,,.,[5»8].Cho,[5].Lee[6].,,,,;,PD+I()..2(Problemstatement),2SISO,n.SISO::2009¡12¡02;:2010¡06¡21.:(60374006).422288:_x1=x2;_x2=f(xxx;t)+u(t)+d(t);y=x1:(1):xxx=[x1;x2]T2R2,u(t)2R,y2R.(1):1f(xxx;t),^f(xxx;t),¢f(xxx;t)=f(xxx;t)¡^f(xxx;t),j¢f(xxx;t)j6F(xxx;t):(2)2d(t),jd(t)j6D(t):(3)e=y¡yr:(4)yr,y.3(Nonlinearintegralslidingsurfacedesign)(1),,[2]S=_e+kPe;(5)kP2R+.,(5)wt0ed¿[2]S=_e+kPe+kIwt0ed¿;(6)kI2R+.,,Slotine[2,6],,S(0)=0,.:S=_e+kPe+kIwt0ed¿¡_e(0)¡kPe(0):(7)(6)(7),,,,Windup,.,,:½S=_e+kPe+kI¾;_¾=g(e):(8)g(e),,g(e),:G(e)=8:2¯2¼(1¡cos¼e2¯);jej¯;¯e¡¼¡2¼¯2;e¯;¡¯e¡¼¡2¼¯2;e6¡¯;(9)¯2R+.(9)e,:g(e)=8:¯sin¼e2¯;jej¯;¯;e¯;¡¯;e6¡¯:(10)(9)(10),.1G(e)g(e):1)e6=0,G(e)0;e=0,G(e)=0g(e)=0;2)G(e),jej¯,g(e),jej¯,g(e)..1¯=2G(e)g(e).1G(e)g(e)(¯=2)Fig.1PlotsofthequasipotentialfunctionG(e)anditsfirstderivativeg(e)(¯=2)1h=g(e)h=e,g(e),,(jej6¯),jg(e)jjej;(jej¯),jg(e)jjej§¯.¯.,¯.4(Slidingmodecontrollawdesign),:1);2).4.1(Caseofcontrolinputwithoutconstraints)S_S6¡´jSj,´0,sat(¢),1.3:4231(1),(8),u(t)=^u¡c(xxx;t)sat(S/¹);(11):^u=¡^f(xxx;t)¡kP_e¡kIg(e)+Äyr,c(xxx;t)=F(xxx;t)+D(t)+´,´0,¹,S.S,(11)u(t)=^u¡c(xxx;t)sgnS:(12)sgn(¢).St_S(t)=Äe+kP_e+kIg(e):(13)LyapunovV1=12S2,_V1=S(Äe+kP_e+kI_¾)=S(f(xxx;t)+u+d(t)¡Äyr+kP_e+kIg(e))=S(f(xxx;t)+^u¡c(xxx;t)sgn(S/¹)+d(t)¡Äyr+kP_e+kIg(e))6jSj(j¢f(xxx;t)+d(t)j¡(F(xxx;t)+D(t)+´))6¡´jSj;,Strtr6jS(0)j¡¹´:.1tr.,!(xxx;t)=¢f(xxx;t)+d(t),°=c(xxx;t)/¹.2:,.2(1),(8),(11),!(xxx;t),limt!1!(xxx;t)=l(l),limt!1e(t)=0.(1)jSj6¹_S(t)=Äe+kP_e+kIg(e)=!(t)¡°S(t):(14)S(s)=1s+°¢!(s):(15)sLaplace.limt!1S(t)=lims!0(ss+°¢!(s))=lims!01s+°¢lims!0s!(s)=lims!01s+°¢limt!1!(t)=l°:(16)_S(t),Barbalat,t!1,_S(t)!0,(14),t!1,Äe+kP_e+kIg(e)=0:(17)LyapunovV2=12_e2+kIG(e):(18)G(e),1G(0)=0,(18).(18)(17),_V2=_eÄe+kIg(e)_e=_e(¡kP_e¡kIg(e))+kIg(e)_e=¡kp_e2¡kIg(e)_e+kIg(e)_e=¡kp_e260:(19)_V2´0_e´0.LaSalle(e=0;_e=0)(17),limt!1e(t)=limt!1_e(t)=0..1limt!1!(xxx;t)=l,,_S(t)=Äe+kP_e=!(t)¡°S(t),[9]:limt!1e(t)=lkP°;limt!1e(t)=0.2:,.4.2(Caseofcontrolinputcon-straints)(11),^u,.,[10,11].(1),juj6umax;umax0:(20)umax,u=¡umaxsat(S=¹):(21)(6),(21):u=¡umaxsat((_e+kPe+kIwt0ed¿)umax¹umax)=¡umaxsat(umax¹_e+umax¹kPe+umax¹kIwt0ed¿umax)=umax¢sat(umax¹(¡_e)+umax¹kP(¡e)+umax¹kIwt0(¡e)d¿umax):(22)42428,(21)2PID[12].2(6)(21)Fig.2Theconfigurationofthesaturatedcontroller(21)usingintegralslidingsurface(6):KP=kPumax=¹,KI=kIumax=¹,KD=umax=¹.(8),(8)u=¡umaxsat((_e+kPe+kIwt0g(e)d¿)umax¹umax)=umaxsat((umax¹(¡_e)+umax¹kP(¡e)+umax¹kIwt0g(¡e)d¿)=(umax));(23)(21)PD()+I(),3.3(8)(21)Fig.3Theconfigurationofthesaturatedcontrollerusingnonlinearintegralslidingsurface(8):KP,KI,KD2.,umax,g(e)¯e,Windup[13,14].5(Simulationexamples),,,.:8:_x1=x2;_x2=5:2x1+(1+0:3sin(2t))x22+10+u;y=x1:(24):f(xxx)=5:2x1+(1+0:3sin2t)x22;^f(xxx)=5x1+x22;¢f(xxx)=0:2x1+0:3x22sin2t;j¢f(xxx)j=j0:2x1+0:3x22sin2tj60:2jx1j+0:3x22=F(xxx);d=10;jdj6D=10:5:x1(0)=¡0:5,x2(0)=¡5,yr=0:5.4:a:,S=_e+4e;b:,S=_e+4e+4wt0e(¿)d¿;c:,S=_e+4e+4wt0ed¿¡_e(0)¡4e(0).d:,S=_e+4e+4wt0g(e)d¿;g(e)¯=0:05.4+,¹=0:2,K=F+D+1.4.4x1Fig.4Theresponsecurvesofstatex1,a,0:0435,1limet!1=lkP°=10:14£0:1+10+10:2=0:0435;;b,c,d,21.db,c,,:,_S(t)=!(t)¡°S(t),t!1,S(t)=l°,e=_e=0.1)S=_e+4e+4wt0ed¿;t!1,3:425wt0ed¿=l4°;2)S=_e+4e+4wt0ed¿¡_e(0)¡4e(0);t!1,wt0ed¿=l4°+_e(0)+4e(0)4;3)S=_e+4e+4wt0g(e)d¿;t!1,wt0g(e)d¿=l4°:b5.5bFig.5Plotsoftrackingerror(Controllerb)SASBAB,w10ed¿=SB¡SA=l4°;SB=l4°+SA;¹,°=c(xxx;t)/¹,SB¼SA,e(0),SA,SB..c6.6cFig.6Plotsoftrackingerror(Controllerc),w10ed¿=SB¡SA=l4°+_e(0)+4e(0)4¼¡94::SB=0;SA¼¡_e(0)+4e(0)4:,,,_e(0)+4e(0)=0,;_e(0)+4e(0)0,,.c7.7dFig.7Plotsoftrackingerror(Controllerd)7:SB¡SA=w10g(e)d¿=l4°;g(e),SA,SB=SA+l4°,,.6(Conclusion),,:1),(+),:2),()Anti-WindupPD+I..(References):[1],.[J].,2007,24(3):407–418.(LIUJinkun,SUNFuchun.Researchanddevelopmentonthetheoryandalgorithmsofslidingmodecontro