第4章-线性二次型最优控制

41LQLQRLinearQuadraticRegulator1234LQR4.11)()()()()(tutBtxtAtx+=&(4-1-1)()()(txtCty=(4-1-2lmnRtyRtuRtx∈∈∈)(,)(,)(A(t)B(t)C(t)nnnmln0lmnu(t)yr(t)lrRty∈)()()()(tytyter−=(4-1-3u*(t)∫++=fttffdttutRtutetQtetFeteuJ0)]()()()()()([21)()(21)((4-1-4LQR(4-1-4FllQ(t)llR(t)mmznnAn1x0AxxTAznnAn1x0≥AxxTA422(4-1-4∫++=fttueffdtLLtFeteuJ0][21)()(21)((4-1-5)()()(tetQteLe=e(t)∫fttdtte0)(212)()()(tutRtuLu=u(t)u2(t))()(fftFete3FQRFQRzzzQRQ(t)R(t)t=t0e(t)Q(t)t→tfe(t)Q(t)FQRFQR4(4-1-1~(4-1-3I.C(t)=Iyr(t)=0y(t)=x(t)=-e(t)II.yr(t)=0y(t)=-e(t)III.yr(t)≠0)()()(tytyter−=434.2Riccati1)()()()()(tutBtxtAtx+=&(4-2-1mnRtuRtx∈∈)(,)(A(t)B(t)nnnmx(t0)=x0u(t)u*(t)∫++=fttffdttutRtutxtQtxtFxtxuJ0)]()()()()()([21)()(21)((4-2-2(4-2-2FQ(t)R(t)2Hamilton)]()()()([)()()(21)()()(21tutBtxtAtutRtutxtQtxH+++=λ(4-2-3u(t)0)(=∂∂tuH0)()()()()(=+=∂∂ttBtutRtuHλ(4-2-4)()()()(1ttBtRtuλ−−=(4-2-5R(t)t[t0tf]R-1(t)t[t0tf]0)()(22=∂∂tRtuH)()()()(1ttBtRtuλ−−=Hu(t)x(t)(t)x(t))()()()()()(ttAtxtQtxHtλλ−−=∂∂−=&(4-2-6)()()()()()()()()()()()(1ttBtRtBtxtAtutBtxtAtHtxλλ−−=+=∂∂=&(4-2-744)()()()(1tBtRtBtS−=(4-2-8S(t)nn(4-2-6(4-2-7S(t)−−−=)()()()()()()()(ttxtAtQtStAttxλλ&&(4-2-92nx(t0)=x0nn(t)(tf))()]()(21[)()(ffffftFxtFxtxtxt=∂∂=λ(4-2-10tt0(4-2-92n2n(t0)(4-2-9Ω=)()(),()()(000ttxttttxλλ(4-2-11Ω=)()(),()()(ttxttttxfffλλ(4-2-12tft4nnΩΩΩΩ=Ω),(),(),(),(),(22211211ttttttttttfffff(4-2-13(4-2-12)(),()(),()(1211ttttxtttxfffλΩ+Ω=(4-2-14)(),()(),()(2221ttttxtttfffλλΩ+Ω=(4-2-15(4-2-10)()],(),([)],(),([)(211111222txttttFttFtttffffΩ−ΩΩ−Ω=−λ(4-2-16(t)x(t))()()(txtPt=λ(4-2-17tftf=I2n2n11tftf=22tftf=Inn12tftf=21tftf=0Ptf=F(4-2-18(4-2-17)()()()()(txtPtxtPt&&&+=λ(4-2-1945)()()()()()()()()()()()()()(txtPtStxtAttStxtAtutBtxtAtx−=−=+=λ&(4-2-20(4-2-19)()]()()()()()([)(txtPtStPtAtPtPt−+=&&λ(4-2-21(4-2-6(4-2-17)()]()()([)(txtPtAtQt−−=λ&(4-2-22x(t)0)()()()()()()()()(=++−+tQtPtAtPtStPtAtPtP&(4-2-23)()()()(1tBtRtBtS−=)()()()()()()()()()()(1tQtPtAtPtBtRtBtPtAtPtP−−+−=−&(4-2-24(4-2-24RiccatiRiccati(4-2-5(4-2-17)()()()()()()(1txtKtxtPtBtRtu−=−=−(4-2-25K(t)(4-2-25)()()(txtPtxt)()()()()()()()()()]()()([txtPtxtxtPtxtxtPtxtxtPtxdtd&&&++=(4-2-26(4-2-1(4-2-24)]()()()()()[()]()()()()([)]()()()()()([)]()()([txtPtBtRtutPtxtPtBtRtututRtutxtQtxtxtPtxdtd++++−=(4-2-27t0tf∫∫+++=++ffttttffdttxtPtBtRtutPtxtPtBtRtutxtPtxtutRtutxtQtxtFxtx00)]}()()()()()[()]()()()()({[)()()()]()()()()()([)()(000(4-2-281/2(4-2-246∫+++=fttdttxtPtBtRtutPtxtPtBtRtutxtPtxuJ0)]}()()()()()[()]()()()()({[21)()()(21)(000(4-2-29)()()()()()()(1txtKtxtPtBtRtu−=−=−J(u))()()(21)(*000txtPtxuJ=(4-2-30x(t))()()(21],[)(*txtPtxtxVuJ==(4-2-31(4-2-1(4-2-5(4-2-174.14.2)()()()(1tPtBtRtK−=3I.(4-2-24(4-2-18)()(1tBtR−−)()()()()(tutBtxtAtx+=&P(t)4.1u(t)λ(t)x(t)B(t)A(t)-K(t)x(t)u(t))(tx&++4.247II.III.P(t)nnQ(t)R(t)n(n+1)/2IV.P(t)x(t)V.ABQRP(t)tf4.1.)()()()(221tutxtxtx==&&u*(t)∫+++++=3022122212221)](21)()(2)(4)(2[21)]3(2)3([21dttutxtxtxtxxxJ3,21,4112,2001,10,0010======ftRQFBAn=2P(t)22=)()()()()(22211211tPtPtPtPtP)()(1221tPtP=u*(t))]()()()([2)()()()()()(]10[2)()()()(*22212121222112111txtPtxtPtxtxtPtPtPtPttBtRtu+−=−=−=−λ[]−⋅⋅+−−=4112)()()()(10210)()()()()()()()(01000010)()()()()()()()(2212121122121211221212112212121122121211tPtPtPtPtPtPtPtPtPtPtPtPtPtPtPtPtPtPtPtP&&&&=2001)3()3()3()3(22121211PPPP484)(2)(2)(1)()(2)()(2)(2)(2212222212111221211−+−=−+−=−=tPtPtPtPtPtPtPtPtP&&&2)3(0)3(1)3(221211===PPP4.3tf4.1P(t)100)()()()(xtxtButAxtx=+=&(4-3-1mnRtuRtx∈∈)(,)(ABnnnmu(t)∫∞+=0)]()()()([21tdttRututQxtxJ(4-3-2QR(4-3-2fffftttttJdttRututQxtxJ∞→∞→=+=∫lim)]()()()([21lim0(4-3-3(4-3-3ftJF=0ftJ)()()(*1txtPBRtu−−=(4-3-4P(t)QtPAtPBBRtPAtPtP−−+−=−)()()()()(1&(4-3-50)(=ftP(4-3-6(4-3-1P49ftttPtPf≤=∞→0,)(lim(4-3-7t0)(=tP&0)(=tP&4.3∞→ft01=++−−QPAPBBRPAP(4-3-8P)()(*1txPBRtu−−=(4-3-9x*(t)001)(),()(][)(*xtxtxAtxPBBRAtxc==−=−&(4-3-102I(4-3-1∞→ftII(4-3-10AcIIIF=0∞→ft4.2.)()()()(221tutxtxtx==&&u*(t)∫∞+++=02222121)]()()()(2)([21dttutaxtxtbxtxJa-b201,1,10,0010====RabbQBAt0PtP=)(tf04.350a-b20Q[]20110==ABBRank01=++−−QPAPBBRPAP[]01010010110001022211211222112112221121122211211=++⋅⋅−abbPPPPPPPPPPPPPPPP1221PP=0200122212221211212=−+−=−+−=−aPPbPPPP122222121121221PaPbPPPP+±=−=±=P11P02122211PPP−022P012222PaP+=112−=P222−=aPa202221211−−−=−=babPPP02−−ab112−=PbaP−−−=211222−=aP2122211PPP−012)2(−−−−abaaaaaab−+=−−212)1(22a-b20112+=P222+=aPbaP−+=211[])(2)()()()()(101)()(*2122211221221212111txatxtxPtxPtxtxPPPPtxPBRtu+−−=−−=⋅−=−=−4.24.1514.41)()()(tButAxtx+=&(4-4-1)()(tCxty=(4-4-2∫++=fttffdttRututQytytFytyJ0)]()()()([21)()(21(4-4-3lmnRtyRtuRtx∈∈∈)(,)(,)(ABCnnnmlnu(t)FQRu*(t)J)()(tCxty=J∫++=fttffdttRututQCxCtxtFCxCtxJ0)]()()()([21)()(21(4-4-4(4-4-1(4-4-2FQCTFCCTQC2tfFQR)()()(*1txtPBRtu−−=(4-4-5nnP(t)QCCtPAtPBBRtPAtPtP−−+−=−)()()()()(1&(4-4-6FCCtPf=)((4-4-7x*(t))()]([)(1txtPBBRAtx−−=&4.4BR1−−)()()(tButAxtx+=&P(t)4.4u(t)x(t)Cy(t)52u*(t)x(t)y(t)x(t)y(t)y(t