有限时间控制问题综述-丁世宏

262Vol.26No.2ControlandDecision20112Feb.2011:1001-0920(2011)02-0161-091,2(1.2120132.210096):.,,;,;,.:Lyapunov:TP273:AAsurveyforfinite-timecontrolproblemsDINGShi-hong1,LIShi-hua2(1.CollegeofElectricalandInformationEngineeringJiangsuUniversityZhenjiang212013China2.SchoolofAutomationSouthea[e-mail])Abstract:Finite-timecontroltheoryhasattractedmuchattentioninrecentyears,sincefinite-timestablesystemsusuallypossessbetterrobustnessanddisturbancerejectionproperties.Firstofall,theoriginforfinite-timecontrolmethodisdiscussed,andthefrequently-usedcriteriaforfinite-timecontrolsystemsislisted.Thenthepresentresearchdevelopmentforfinite-timecontrolsystemsissummarized.Finally,thefutureoutlookonfinite-timecontrolisdiscussed.Keywords:finite-timecontrolhomogeneoussystemLyapunovmethodstatefeedbackoutputfeedback1,.,,,.:Lipschitz.,.,.,,,,[1-3].,.,[4].[4]Äx=u;x(0)=x0;_x(0)=y0;u(x0;y0;t)=¡2t2f(3x0+2y0tf)+6t3f(2x0+y0tf)t;tf,J=wtf0u2(t)dt.,.,[5-8].[6],,,,.:[6];[7][6].[5,8].:2010-07-11:2010-09-21.:(61074013)(20090092110022)(BK2008295,BK2010200)(201004)(10JDG112):(1983¡),,,,(1975¡),,,,.DOI10.13195/j.cd.2011.02.4.dingshh.00816226s=_x+cxp=q=0:(1):0p=q1,pq.,.,,.:;,;,.,,.,2060,..[9],,,.,[10],,.,[11],,.,,.,Lipschitz,Lipschitz,Lyapunov..2090,[12-13]Lyapunov[1,14],.22.1,:_x=f(x);f(0)=0:(2):x2Rn;f:U!RnUnRn.Lipschitz,[15];Lipschitz,[16].,,,.,.1[14](2),x=0,.T(x):U0nf0g!(0;+1)8x02U0½U,(2)x(t;x0);t2[0;T(x0)),x(t;x0)2U0nf0glimt!T(x0)x(t;x0)=0;tT(x0),x(t;x0)=0.U=U0=Rn,.,,.2.2.[1,12-14],:Lyapunov..,.2[17]f(x):Rn!Rn.0,(r1;r2;¢¢¢;rn)2Rn,ri0(i=1;2;¢¢¢;n),f(x)fi(r1x1;¢¢¢;rnxn)=k+rifi(x)i=1;2;¢¢¢;n;k¡maxfri;i=1;2;¢¢¢;ng.f(x)(r1;r2;¢¢¢;rn)k.3V(x):Rn!R.0,¾0(r1;r2;¢¢¢;rn)2Rn,ri0(i=1;2;¢¢¢;n),V(r1x1;¢¢¢;rnxn)=¾V(x);8x2Rn:V(x)(r1;r2;¢¢¢;rn)¾.4f(x),_x=f(x)(x2D).[13,18-19].[12-13].1(2)k0,,.,[1,14]2:163Lyapunov.2V:U!R,:1)V.2)c0®2(0;1),U0½U,:_V(x)+cV®(x)60;x2U0nf0g;(2);U=U0=Rn,(2).33.1Lyapunov,,.:_x1=x2;_x2=u+d(t):(3):x=(x1;x2)T,u,d(t).(3)d(t)=0,[12]:u=¡k1sign(x1)jx1j®1¡k2sign(x2)jx2j®2:(4):k10;k20;0®1;®2=2®11+®2.Lyapunov,[1](3)..,.,,[2,14,20],.[14]Lyapunov,.,.[20],().,[14,20].,[2](3),:­k,d,­/(d=k)1=®(0®1).,kkd,®.,[21]:_x1=xm2;_x2=u;[22],3.[23],,,.,.,,.,.,.3.2.,.[13]:_x1=x2;¢¢¢;_xn¡1=xn;_xn=u:u=¡k1sig®1(x1)¡k2sig®2(x2)¡¢¢¢¡knsig®n(xn)::ki0(i=1;2;¢¢¢;n)sn+knsn¡1+¢¢¢+k2s+k1=0Hurwitz;®i=®i®i+12®i+1¡®i(i=1;2;¢¢¢;n),®n+1=1;®n=®;®2(1¡²;1);²2(0;1).,².,,.,().,,1,.,,,.[24][22].,[24]:_x1=xm12;_x2=xm23;¢¢¢;_xn=u;mi(i=1;2;¢¢¢;n¡1),,.[24],.16426,,[25][22],:_xi=xi+1+fi(x1;x2;¢¢¢;xi;u;t);i=1;2;¢¢¢;n.[26][24-25],P-,[24-25]Lyapunov,.[27],,,..,,[28].[29],(ISS),Lyapunov,.,.,.,:(PVTOL).:_xi=xi+1+fi(xi+1;¢¢¢;xn;u;t);i=1;2;¢¢¢;n¡1;_xn=u+fn(u;t):(5),Lyapunov.[30](5).[31][30]P-,,.,.,.,.,,,.[32],,..,[33],,.4,.,.Lipschitz,,..,:_x1=x2;_x2=u;y=x1;[3]:(_³1=³2¡k1j³1¡yj¾1sign(³1¡y);_³2=u¡k2j³1¡yj¾2sign(³1¡y):(6):0¾21,¾2=(1+¾2)=2,k10,k20.,.,.[34],_x1=xp12+Á1(x1);_x2=u+Á2(x1;x2);y=x1::p1;Á1(x1);Á2(x1;x2).,.Á1(x1);Á2(x1;x2),:_z=@L(x1)@x1((z+L(x1)p=(p+1¡h))+Á1(x1));u=u(x1;(z+L(x1))p=(p+1¡h))::h,u(x1;(z+L(x1))p=(p+1¡h)).,,.,[34-35].,;,,,.[36][35]P-.[37],[35-36],,.,[35],,2:165,.,,.,[39]:_x=Ax+f(y;u;_u;¢¢¢;u(r));y=Cx:(7):x;y;u;(A;C).Tube[40],[39],2664_^x1..._^xn3775=A2664^x1...^xn3775+f(y;u;_u;¢¢¢;u(r))¡2664k1sig®1(y1¡^x1)...knsig®n(y1¡^x1)3775::ki0;®i0.[35-38],,®i.,(7)f(¢),,.,[41-42](7),,[39],.5,,.:1,,,;2,..,:s=_x+cx=0(c0).,.,[43](1).,(1),,.,,:SISO[8],MIMO[44],[45].,_sx(q=p¡1),x(q=p¡1)..,[46].,,;,,..,[5]s=x+c_xp=q:(8):c0,1p=q2,pq.,,(8).,.(8),p;q1p=q2,p=q..,[47],:s=x+csign(_x)j_xj®;(9)1®2.,,[5,47].,.6,.:,,.1),,.,.[48][12],DuffingLorenz.,[49].,[50].,16626,.2).,,,.,,.,.,.[51-52],.,.,[53-54].[53].,,;,..[54],.,;,,,.3),,.,,.,,.,.,._xi=ui;i=1;2;¢¢¢;n;xi2Rui2Ri.T,tTxi=xj(i;j=1;2;¢¢¢;n):_xi=yi;_yi=ui;i=1;2;¢¢¢;n::xi2R,yi2Rui2Ri.T,tTxi=xj;yi=yj;i;j=1;2;¢¢¢;n:,[55]Lyapunov,:ui=¯sig³Xj2NiWij(xj¡xi)´®+°Xj2NiWij(xj¡xi)::0®11,¯0,°0,Wij0;Ni.[56],:ui=kdg=dxiXj2Nisig(Wij(#(xj¡xi)))®::0®11;Wij0;Ni;g(¢),#(¢).[57],,,..,.,[58],,:ui=nXi=1Wij[Ã1(sig(xj¡xi)®1)+Ã2(sig(yj¡yi)®2)]::Wij0;0®11;®2=2®11+®1;Ã1(¢),Ã2(¢).4),.,.,PI,.,.[59],.,,;,.PI,.[59],[60],2:167,,.[61],,.,.[62],,,.,.[63],,;,.,,[64][5,47][65],,.7,.Lips-chitz,.,,,,:1),.,.,,,.[13],,.,,.2)[31],fi(xi+1;¢¢¢;xn),xi+1.fi(xi+1;¢¢¢;xn)xi+1,..,.3),,.,,.,,,.,,.,,.,,.(References)[1]BhatSP,BernsteinDS.Continuousfinite-timestabilizationofthetranslationalandrotationaldoubleintegrators[J].IEEETransonAutomaticControl,1998,43(5):678-682.[2]DingSH,LiSH,LiQ.Stabilityanalysisforasecond-ordercontinuousfinite-timecontrolsystemsubjecttoadisturbance[J].JofControlTheoryandApplications,2009,7(3):271-176.[3]HongY,HuangJ,XuY.Onanoutputfeedbackfinite-timestabilizationproblem[J].IEEETransonAutomaticControl,2001,46(2):305-309.[4]AthansM,FalbPL.Optimalcontrol:Anintroductiontothetheoryanditsapplications[M].NewYorK:McGraw-Hill,1985.[5]FengY,YuXH,ManZ.Non-singularterminalslidingmodecontrolofrigidmanipulators[J].Automatica,2002,38(9):2159-2167.[6]RyanEP.Singularoptimalcontrolsforsecond-ordersaturatings