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©1994-2009ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.(,210096):,,,:PD,,,:;;;;:TM351:A:1007-449X(2009)03-0424-07Finite2timecontrolofPMSMpositionservosystemLIUHui2xian,DINGShi2hong,LIShi2hua,CHENXi2song(SchoolofAutomation,SoutheastUniversity,Nanjing210096,China)Abstract:Basedonthefinite2timecontroltechnique,thepositioncontrolproblemofpermanentmagnetsynchronousmotor(PMSM)servosystemisstudied.Usingfinite2timecontroltheoryandbackingsteppingmethod,acontrolschemebasedonfeedbacklinearizationandfinite2timecontroltechniquesisproposedforpositionloop.Rigorousmathematicalanalysisisgivenforthecloseloopsystemperformanceinthepresenceofdisturbances.Theresultsshowthat,comparedwiththeconventionalcontrolschemebasedonPDandfeedbacklinearization,thismethodnotonlymakesthepositiontrackingerroroftheclosedloopsystemwithafasterconvergencerate,butalsomakestheboundaryofsteady2stateerrorsmallerbyregula2tingthecontrollerparameters,whichmeanstheclosedloopsystemhasstrongerdisturbancerejectionproperty.Thesimulationresultsvalidatetheefficiencyofthismethod.Keywords:permanentmagnetsynchronousmotors;positioncontrol;finite2timecontrol;backsteppingmethod;feedbacklinearization:2008-03-21:(60504007);(BK2008295);(KXJ07125):(1984-),,,;(1983-),,,;(1975-),,,;(1970-),,,1,[1]PMSM,[2],,©1994-2009ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.[3][4][5,6][7],,(),,(),,,,bang2bang,[8-14],,[9],,,,,,,PD,2PMSM,,,d2q,[2]id=-Rsid/Ld+npiq+ud/Ld,iq=-npid-Rsiq/Lq-npf/Lq+uq/Lq,Û=npfiq/J-B/J-Tl/J+d1(t),=(1):uduqd2q;idiqd2q;LdLq;Rs;f;np;J;B,Tl;;;d1(t),d2qidiq,,,,id=0,PI,,d2qidiqi3di3q(iqi3q)i3d=0,PMSM,i3d1,,,PI1Fig.1SchematicdiagramofPMSMspeed2regulationsys2tembasedonvectorcontrol3:,,3,e=3-,(1)e=3-npfi3q/J+B/J+Tl/J+d(t),(2):d(t)=npf(i3q-iq)/J-d1(t),1(2)d(t),5243©1994-2009ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.|d(t)|l(3),Ûx=f(x),f(0)=0,xRn,(4)f(x)1[10]x,yR,0p=p1/p21,p1,p2,xq-yq21-qx-yq2[11]x,yR,c,dR+,(x,y),:xcydc(x,y)|x|c+d/(c+d)+c-cd(x,y)yc+d/(c+d)3[12]xiR,i=1,,n,0p1,(x1++xn)px1p++xnp1(2)1,(PMSM),i3q=Jnpf3+BJ+TlJ+2-1q21-1/qk1+q1k2(Ûeq+kq1e)2/q-1,(5)ee(k3/k1)q/(q+1)2l/2-1q21-1/qkq+11k3q/(2-q)(6):1q=q1/q22,q1q2;ld(t),k10,k221-1/q1+q23-1/qqk1(1+q)q+1k1+qq+14(1+q)k11/q,k3=k2-21-1/q1+q23-1/qqk1(1+q)q-1k1-q1+q4(1+q)k11/qx1=e,x2=Ûe,,(2)Ûx1=x2,Ûx2=u+d(t),(7)u=3-pfi3q/J+B/J+Tl/JLyapunovV1(x1)=x21/2,V1(x1)=x1x2x1(x2-x32)+x1x32,(8),k10,1q=q1/q22,q1q2x32=-k1x1/q1,V1(x1)x1(x2-x32)-k1x1+1/q1(9)LyapunovV2(x1,x2)=V1(x1)+12-1q21-1/qk1+q1x2x32(sq-x3q2)2-1qds,V2-k1x1+1/q1+x1(x2-x32)+121-1qk1+q1(5x3q2)5x1x2x2x32(sq-x3q2)1-1qds+12-1q21-1/qk1+q12-1q2u+12-1q21-1/qk1+q12-1q2d(t)(10)2=xq2-x3q25x3q2/5x1=-kq1,|d(t)|l,(9)V2-k1x1+1/q1+x1(x2-x32)+121-1qk1x2x2x32(sq-x3q2)1-1qds+12-1q21-1/qkq+112-1q2u+12-1q21-1/qk1+q12-1q2l(11)V2-k1x1+1/q1+x1(x2-x32)+121-1qk1|x2||x2-x32|1-1/q2+12-1q21-1/qkq+112-1q2u+12-1q21-1/qk1+q12-1q2l(12)1|x2-x32|21-1q|2|1q,(13)V2-k1x1+1/q1+21-1qx121/q+1k1|x2||2|+12-1q21-1/qkq+112-1q2u+12-1q21-1/qk1+q12-1q2l(14)2x1(x2-x32)21-1/q|x1|1/q2k1x1+1/q1/4+21-1/q11+q23-1/qqk1(1+q)q1+1/q2(15)x2=(2+x3q2)1q,362413©1994-2009ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.=21q+k1x11/q2x22/k11+1/q2/k1+x11/q21+1/q2/k1+k14x1+1/q1+11+q4k1(1+q)1/q1+1/q2(16)(14)(15)(16)V2-k1x1+1/q1+k14x11+1/q+21-1q11+q23-1/qqk1(1+q)q1+1/q2+1k11+1/q2+k14x1+1/q1+qq+14(1+q)k11/q1+1/q2+12-1q21-1/qkq+112-1q2(u+l)(17)uu=-(2-1/q)21-1/qk1+q1k2(xq2+kq1x1)2/q-1,(18),k10,k221-1/q1+q23-1/qqk1(1+q)q+1k1+qq+14(1+q)k11/q(17)V2-k12x1+1/q1-k31+1/q2+l2-1q21-1/qkq+112-1q2,(19):k3=k2-21-1/q11+q23-1/qqk1(1+q)q-1k1-q1+q41+q1/q1=(x1,x2):|2|2l2-1q21-1/qkq+11k3q2-q(x1,x2)1,V2-(k1x1+1/q1+k31+1/q2)/2(20)(x1(0),x2(0))1,ÛV20,t10(x1(t1),x2(t1))bd1,bd11;1,(x1,x2)1(x1,x2)1,(x1,x2)1,t10(x1(t1),x2(t1))bd1(x1(t),x2(t))1t[t1,)m=min(x1,x2)bd1k31+1/q2/2,m=k322l2-1q21-1/qkq+11k31+q2-q(x1(t),x2(t))bd1V2-m0V2(x1,x2),s10(x1(t),x2(t))1t[t1,t1+s1)h[t1,+),(x1(h),x2(h))1,(t1,h),(x1(),x2())bd1V2(x1(),x2())-m0V2,s20,V2(x1(t),x2(t))[-s2,)m=k31+1/q2()/2k31+1/q2(-s2)/2m,t[t1,+),(x1(t),x2(t))1|2|2l/2-1q21-1/qkq+11k3q/(2-q),tt1V2-k12x1+1/q1-k31+1/q2+l2-1q21-1/qkq+112l2-1q21-1/qkq+11k32q-12-q,tt1(21)k12x1+1/q1l(2-1q)21-1/qkq+112l2-1q21-1/qkq+11k32q-12-q,V2-k31+1/q2,x1x1(k3/k1)q/(1+q)(l/((2-1q)21-1/qkq+11k3))q/(2-q)(22)(18)(7)i3q=Jnpf3+BJ+TlJ+2-1q21-1/qk1+q1k2(Ûeq+kq1e)2/q-1(18)q=1,PD,u=-k21k2(x2+k1x1)(23),i3q=J[3+B/J+Tl/J+k21k2(Ûe+k1e)]/(npf),(24),k10,k23/k1,1,7243©1994-2009ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.|x1|(k3/k1)12(2l/k21k3),(25)k3=k2-3/k1(22)(25),,,,,k1k3(22)(2-1/q)21-1/qk1+q1k32lk1k3,,,k1k3,(25),q/(2-q),q2,,4Matlab:P=0175kW,d2qL=0101H,nN=2500r/min,np=4,J=712410-4kgm2,Rs=119,B=0102Nmsrad-1,f=01353Wb,TN=2167NmPI,(5)PD(24),,kp=30,kI=600,:1;;,k1=12,q=21/17,k2=118PD:k1=6108,k2=7104=20(rad),sin(10t)+cos(10t),115s,Tl=10Nm2(a)(b)2(a),,(3),,,2(b),,,PD,2Fig.2Stepresponseofposition3(a