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1、优化方法与最优控制例题1•Findthecurvex(t)thatminimizesthefunctionalAndpassesthroughthepoints^(0)=1andx(l)=3.4"g(x,x,t)=yx2(0+5x(t)x(t)+xSuchthat:f7e[x,x,t]dt=CWherewewillassumethattfisfreebutx(tf)isfixed.⑺+5x(0,可求得gv=5i⑺+2%⑺+5^=x(r)+5x(z)^=^(z)+5i(z)dt若J在x(z)处取极值,则有=即atX⑺一2x
2、⑺一5=0解微分方程沿7)-2x(z)=0,可得通解%(z)=cxe~<21+c2Z2/。设对)-2冰)-5=0的通解为%(0=<^,得力)=」。故原微分方程的解为2x(r)=c1e'^+c2e^+
3、又已知x(0)=l,x⑴=3,带入上式可得e^+32(^Oneimportantcalculusofvariationsproblemthatwedidnotdiscussinclasshasthesamebasicform,butwithconstraintsthataregivenbyanintegral-called-
4、1)e^+3C,2=2(eisoperimetricconstraints:minJ=[g[x,x,tlt山0-l)所以x⑺=c,e'r2t+c2^+-o2dSa_d(dSadxdtdx=0e[x,x,t]dt=CWherega=gvTe•(b)Usetheresultsofpart(a)toclearlystatethedifferentialequationsandcorrespondingboundaryconditionsthatmustbesolvedtofindthecurvey(x)ofaspecifi
5、edlengthLwithendpointsonthex-axis(i.e.,atx=0andx=xf)thatenclosesthemaximumarea,sothat7=£7ydxand£+y2dx=LWheret,free.(a)引入拉格朗口矢量因子V,另'g[x,x9t]dt+vT^fe(x,i,t)dt-CTcl求变分有ds,TSJa=7(§'—&+g^Sx)dt+g{tf)dtf+vT^J^{exSx+e{Sc)dt+e(tf)dtf+1edt(gx-::^-)Sxdt+++〔{edt-CSv+v*l
6、7{
7、ex-+e..(tf)4-e(tf)dtf有&,=&(◊)+々(,,)々,,并令么=g+vT《,带入上式,整理可得a=£[U.r+)—(音+^)]&cdt+7edt-Cj<5v+[A(,,)+vTG(〜)]&,+(U(z,)+’冲,)]-[心(z,)+vT¥(r)]地,))々,kdxdtSxdt+〔cedt-CI4--(rz)&cf+ga(tf)-^-dxdx因为zf自由,对rp固定,所以要使<5/=0,则需满足条件:dSa_d(^Sadxdtdxe[x,x,t]dt=C(b)令=>’++y2,则有dy2iodIty自
8、由。按题意耍取得极小的必耍条件为rf7+vW7=03,(^)+v(r/)Jl+y(r/)-+少丸y(tf)=otxf^1+y2dx=Landperformanceindex3.Considertheunstablesecondordersystem(a)ForRxx/Rl(u=Ishowanalytically(i.e.NotusingMatlab)thatthesteady-stateLQRgainsare:K=[73+1]Andthattheclosed-looppolesareat5=—(±J)/2.(b)Usi
9、ngthesteady-stateregulatorgainsfromMatlabineachcase,plot(ononegraph)theclosed-looplocationsforarangeofpossiblevaluesofR、JRl(lt.Showthepolelocationsfortheexpensivecontrolproblem.Doyouseeanytrendshere?(a)由己知条件"0f_0"rrvv0",B=,/?.=011X_00Rltu=r,其中r〉0,则不失一般性,不妨令r=l,满足
10、题R要求。由LQR理论可知,系统的最优反馈律为ut)=一dB]Px(t)=-Kx{t)AlPl2PiP22,满足Riccati方程A1P+PAV+-PBR-]BVP=OzViVifPnPiP11+A2-A2P221T-l0Al+/Al-P2lP22A24-2厂22+厂21一厂222,为使矩阵P为半正定,取Hoo