非线性控制理论

1212331.1.∑=+=miiiuxgxfx1)()(&()Tnxxx,,1L=ffgg11……ggmmhh11……hhmmpixhyii≤≤=1),(RRnnU(1(1--1)1)⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛=),,(),,(),,()(11211nnnnxxfxxfxxfxfLLL⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛=),,(),,(),,()(11211nnininiixxgxxgxxgxgLLL),,()(1niixxhxhL=4A.A.UURRnnff::UURRffxx=(=(xx11……xxnn))ffCCffCCffCCxx00UU,,xx00AAxxAAffxx00TaylorTaylorff((xx))ffffP3P3ffP3P3ff5B.B.nnnnnnnn6P3P3ffgg11……ggmmUUxxRRnnff((xx))gg11((xx))……ggmm((xx))UUC.C.nnUUxx((RRnn))**7vvww**()[]⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡==∗∗∗∗211)(vvwwvwvwnLL⎟⎟⎠⎞⎜⎜⎝⎛∂∂∂∂∂∂=nxxxxdλλλλL21)(RRnnλλddλλTx∂∂=λ8D.D.D.1D.1λλff⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⎥⎦⎤⎢⎣⎡∂∂∂∂=211)(ffxxxLnfLLλλλ∑=∂∂=niiixf1λfxT∂∂=λfd,λ=λλλλff9D.1D.1λλ)(xLkfλfxLTkf∂∂=−)(1λ)()(0xxLfλλ=λλffgg=)(xLLfgλgxLTf∂∂)(λ10221221sinxyuxxxax=+−==&&⎟⎟⎠⎞⎜⎜⎝⎛−=212sin)(xxaxf⎟⎟⎠⎞⎜⎜⎝⎛=10)(xg2)(xxh=20)(xxhLf=[]21212sin10)()(xxxaxfxhxhLTf−=⎥⎦⎤⎢⎣⎡−=∂∂=[]2121212sin2sin02)())(()(xaxxxaxxfxxhLxhLTff−=⎥⎦⎤⎢⎣⎡−−=∂∂=)(xhLLfg)())((xgxxhLTf∂∂=[]010021=⎥⎦⎤⎢⎣⎡−=x11D.2D.2ffggRRnnUU[[ffgg]]UUxx[]gxffxgxgfxgadTTf∂∂−∂∂==)(,)(⎥⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎢⎣⎡∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂=∂∂nnnnnnTxgxgxgxgxgxgxgxgxgxgLMLL212221212111⎥⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎢⎣⎡∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂=∂∂nnnnnnTxfxfxfxfxfxfxfxfxfxfLMLL21222121211112ggff[])(,)(1xgadfxgadkfkf−=)()(0xgxgadf=1:(1:())[][][]12211112211,,,gfrgfrgfrfr+=+[][][]21211122111,,,gfrgfrgrgrf+=+D.2D.2132:(2:())[][]fggf,,−=3:(3:())[][][][][][]0,,,,,,=++gfhfhghgfD.3D.3ffωωUUxxTTTTTfxfxxxfxL∂∂+∂∂=)())(()(ωωωωωff14221221sinxyuxxxax=+−==&&⎟⎟⎠⎞⎜⎜⎝⎛−=212sin)(xxaxf⎟⎟⎠⎞⎜⎜⎝⎛=10)(xg2)(xxh=)()(0xgxgadf=gxffxgxgadTTf∂∂−∂∂=)()())(()(22xgadxffxxgadxgadfTTf∂∂−∂∂=⎥⎦⎤⎢⎣⎡⎥⎦⎤⎢⎣⎡−−0cos02cos0212xaxxa⎥⎦⎤⎢⎣⎡−⎥⎦⎤⎢⎣⎡=212sin0000xxa⎥⎦⎤⎢⎣⎡⎥⎦⎤⎢⎣⎡−−1002cos012xxa⎥⎦⎤⎢⎣⎡=0cos2xa⎥⎦⎤⎢⎣⎡−⎥⎦⎤⎢⎣⎡−=2122sin00sin0xxaxa21221cos2sinxaxxax+=15D.4D.4(1)(1)ααλλff)())(()(xxLxLffαλλα=(2)(2)ααββffgg[][])()())(()(,)()()(,xgxxLxgfxxxgffαββαβα+=)()())((xfxxLgβα−(3)(3)λλffgg[])()()(,xLLxLLxLfggfgfλλλ−=16(4)(4)ααββffωω)()(),()())()(()()(xdxfxxxLxxxLffαωβωβαβωα+=(5)(5)λλff)()(xdLxdLffλλ=(6)(6)ffggωω)](,[),()(),()(,xgfxxgxLxgLffωωω+=)()()(xxxLfωαβ+D.4D.41733(3)(3)λλffgg[])()()(,xLLxLLxLfggfgfλλλ−=[]],[)(,gfxxLTgf∂∂=λλTx∂∂=λ)(gxffxgTT∂∂−∂∂gxfxgxxffxgxfxxgTTTTTTTT∂∂∂∂−∂∂∂−∂∂∂∂+∂∂∂=λλλλ22gfxxfgxxTTTT)()(∂∂∂∂−∂∂∂∂=λλ)()(xLLxLLfggfλλ−=18..BuAxx+=&Txz=TBuzTATz+=−1&uBzAz~~+=&⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛=⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛==),,(),,(),,()()()()(1121121nnnnnxxΦxxΦxxΦxΦxΦxΦxΦzLLLLL19RRnn(1)(1)ΦΦ((xx))ΦΦ--11((zz))RRnnxx,,)())((11zΦxΦΦx−−==(2)(2)ΦΦ((xx))ΦΦ--11((zz))20F.F.F.1F.1RRnnUUffUUxxnnff((xx))ddff11……ffddUUUUxxff11((xx))……ffdd((xx))∆∆∆∆((xx)=span{)=span{ff11((xx))……ffdd((xx)})}∆∆=span{=span{ff11……ffdd}}21∆∆=span{=span{ff11……ffdd}}spanspan““””wxxuxxxxxxxx⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡+⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡++⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡+=0)1(11112232121&⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡+⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡+=0,)1(,11span)(112232121xxxxxxxxxx∆22F.2F.2RRnn((∆∆11∆∆22)()(xx)=)=∆∆11((xx))∆∆22((xx))((∆∆11∆∆22)()(xx)=)=∆∆11((xx))∆∆22((xx))(())⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=00,00span)(231xxx∆⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=31120,00span)(xxxx∆23⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=00,00span)(231xxx∆⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=31120,00span)(xxxx∆⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=31123210,00,00,00span))((xxxxxx∆∆U⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=00,00,00span123xxx⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=32100span))((xx∆∆I24F.3F.3xx∆∆((xx))∆∆xxdim(dim(∆∆((xx))))xxUU,,dim(dim(∆∆((xx))=))=ddUU∆∆∆∆=span{=span{ff11……ffdd}}UUxxUU00,,UU00∆∆xx∆∆25F.4F.4∆∆CCCC∆∆11P23P2322⎭⎬⎫⎩⎨⎧⎥⎦⎤⎢⎣⎡=11span)(1x∆⎭⎬⎫⎩⎨⎧⎥⎦⎤⎢⎣⎡+=11span)(12xx∆⎩⎨⎧=≠=0)(00))((11121xx∆xx∆∆IRR2226F.5F.5∆∆∆∆ffiiffjj[[ffiiffjj]]∆∆∆fi∈∆fj∈∆ffji∈],[RR33{}21,spanff∆=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=012)(21xxf⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=2201)(xxfRR3322⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛=⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛=10001000000020012010000000)](,[2221xxxff27⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=012)(21xxf⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=2201)(xxf⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛=⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛=10001000000020012010000000)](,[2221xxxff[]⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛=10001012)(],[222121xxxffff3328⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−=012)(31xxf⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−=32122)(xxxxf{}{}2123213,span0ff∆xxRxU=≠+∈=RR3322⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−−⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−−=3213212000000200012100020001)](,[xxxxxff⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−=0243x22f−=∆∈2229F.6F.6ddff11……ffddUUUUxxff11((xx))……ffdd((xx))∆∆ωω11……ωωddUUUUxxωω11((xx))……ωωdd((xx))ΩΩΩΩ((xx)=span{)=span{ωω11((xx))……ωωdd((xx)})}30F.7F.7∆∆UUxx∆∆((xx))∆∆((xx)){},0,:)()(=∈=∗∗∗⊥vwRwx∆n)(x∆v∈∀⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−=012)(31xxf⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−=32122)(xxxxf{}{}2123213,span0ff∆xxRxU=≠+∈=[]32123342)(xxxxxx+=ω31FrobeniusFrobeniusxx(0)=(0)=xx00xx((tt))ff((xx)))(xfx=&)(0xΦft)(xΦftttxxff((xx))G.G.FrobeniusFrobenius32G.1.G.1.)(xfx=&uxgxfx)()(+=&ff11……ffdd33G.1.1G.1.1(())∆∆=span{=span{ff11……ffdd}}∆∆RRnnUUddUUxx00UU00ddxx00UU00nn--dd(()){})(,),(span)(1xxx∆dn−⊥=ωωL340)(),(=xfxijωdi,,1L=dnj−=,,1L0)()(=xFxMjωdnj−=,,1Lωωjjλλjj((xx))xjjj∂∂==λλωgrad0)()(=xFxMjω[]0)()()(1=∂∂=∂∂xfxfxxFxdTjMTjLλλdnj−=,,1L(1(1--2)2)35[]0)()()(1=∂∂=∂∂xfxfxxFxdTjMTjLλλdnj−=,,1Lλλjj((xx))jj=1=1……nn--ddλλjj((xx))jj=1=1……nn--ddnn--ddλλjj((xx))jj=1=1……nn--dd36G.1.2G.1.2(1(1--2)2)jjλλjj((xx))[]0)()()(1=∂∂=∂∂xfxfxxFxdTjMTjLλλ(1(1--2)2)ttxx(t(t))λλjjλλjj((xx))λλjjλλjjxx000)()(0=−xxjjλλ37[]0)()()(1=∂∂=∂∂xfxfxxFxdTjMTjLλλ(1(1--2)2)jj=1=1……nn--ddλλjj((xx))nn--ddnn--dd(1(1--2)2)nn--dd(())uxxxuxg